Golestan University
Faculty of Sciences
Department of Mathematics
The Extended Abstract of
The 6th National Group Theory
Conference
12–13 March 2014
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Invariants of a finite group acted on by a Frobenius-like group . . . . . . . . . . . . . . . . . . . 2
Ercan, G.
Representations of a Finite Group with an Extraspecial Normal Subgroup . . . . . . . 3
Güloğlu, İ. Ş.
The Classification of Groups via Capability; A Reality to Dream . . . . . . . . . . . . . . . . . 9
Kayvanfar, S.
Direct Limits of Finitary Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Kuzucuoğlu, M.
Algebraically closed groups and embedding theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Shahryari, M.
On the cover-avoiding properties in finite groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
Shum, K. P.
Some open problems in non-commuting graphs of groups . . . . . . . . . . . . . . . . . . . . . . . 28
Abdollahi, A.
The relative n−th nilpotency degree of two subgroups of a finite group . . . . . . . . . 31
Abdul Hamid*, M., Mohd Ali, N. M., Sarmin, N. H. and Erfanian, A.
Generalize commutator on polygroups and hypergroups. . . . . . . . . . . . . . . . . . . . . . . . .35
Aghabozorgi*, G. H., Jafarpour, M. and Davvaz, B.
Some solved and unsolved problems in loop theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Ahmadidelir, K.
Inequality for the number of generators of the c−nilpotent multiplier . . . . . . . . . . . 46
Alizadeh Sanati, M. and Mahdipour*, Z.
Characterization of 2 Dn (2) by the set of orders of maximal abelian subgroups . . 50
ii
Asadian*, B. and Ahanjideh, N.
A characterization of Sz(8) by nse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54
Asgary*, S. and Ahanjideh, N.
Symmetry classes of polynomials with respect to product of groups . . . . . . . . . . . . . 58
Babaei*, E. and Zamani, Y.
(Strongly) Gorenstein homological dimension of groups . . . . . . . . . . . . . . . . . . . . . . . . . 63
Bahlekeh, A.
OD-Characterization of the simple group G2 (p), where p < 100 . . . . . . . . . . . . . . . . . 67
Bibak*, M., Sajjadi*, M. and Rezaeezadeh, G.
Combinatorial conditions on groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Faramarzi Salles*, A. and Khosravi, H.
On the number of elements of a given order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Farrokhi D. G.*, M. and Saeedi, F.
On the lower autocentral series of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Gholamian*, A. and Nasrabadi, M. M.
On a conjecture about automorphisms of finite p-groups . . . . . . . . . . . . . . . . . . . . . . . . 80
Ghoraishi, S. M.
Embeddings of borel subgroup of the Ree groups of type 2 F4 (q 2 ) . . . . . . . . . . . . . . . 84
Ghorbany, M.
The commutativity degree of a polygroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Hokmabadi, A., Mohammadzadeh, F. and Mohammadzadeh*, E.
Finite p-groups whose order of their Schur multiplier is given(t=6). . . . . . . . . . . . . .92
Jafari, S. H.
Investigating equality of edge and vertex connectivity number in prime graph of
alternative groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Jahandideh*, M., Kazemi Esfeh, H. and Farhami, N.
iii
A note on the tensor and exterior center of a pair of Lie algebras . . . . . . . . . . . . . . 101
Johari*, F., Niroomand, P. and Parvizi, M.
Capability of finite nilpotent groups of class 2 with cyclic Frattini subgroups. . .105
Kaheni*, A., Hatamian, R. and Kayvanfar, S.
Support enumerators for some permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Kahkeshani, R. and Yazdany Moghaddam*, M.
On semigroups generated by vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Khoddami, A. R.
Engel degree and Isoclinism classes of finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Khosravi*, H. and Araskhan, M.
Burnside condition on some intersection subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Mirebrahimi*, H. and Ghanei, F.
Some properties of centralizer and autocommutator subgroup in auto-Engel groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Moghaddam, M. R. R. and Badrkhani Asl*, M.
Counting centralizers in non-abelian n-dimensional Lie algebras . . . . . . . . . . . . . . . 127
Moghaddam, M. R. R., Hoseini Ravesh, M. and Saffarnia*, S.
Some properties of 2-Engel transitive groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Moghaddam, M. R. R. and Rostamyari*, A.
Embedding a special subgroup in n-autocentral subgroups of a group . . . . . . . . . . 136
Moghaddam, M. R. R. and Sadeghifard*, M. J.
Triangle-free commuting conjugacy classes graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Mohammadian*, A., Erfanian, A. and Farrokhi D. G., M.
Isologism crossed modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Mohammadzadeh, H.
The structure of non-solvable CTI-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Mousavi, H., Rastgoo*, T. and Zenkov, V.
iv
P -semisimple BCI-algebras and adjoint groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Najafi*, A. and Rasouli, H.
The finite π-solvable groups with three conjugacy class sizes of primary and biprimary π-elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Najafi*, M. and Ahanjideh, N.
On the absolute center of some groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Nasrabadi*, M. M. and Gholamian, A.
On countability of homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Nasri*, T., Mashayekhy, B. and Mirebrahimi, H.
The schur multiplier of pairs for some finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Nawi*, A. A., Mohd Ali, N. M., Sarmin, N. H., Rashid, S. and Zainal, R.
Separation properties of topological fundamental groups . . . . . . . . . . . . . . . . . . . . . . . 174
Pakdaman*, A., Mashayekhi, B. and Torabi, H.
On the characterizations of finite groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .178
Parvizi Mosaed*, H., Iranmanesh, A. and Foroudi Ghasemabadi, M.
An approach to c-imperfect groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Pourmirzaei, A. and Hassanzadeh*, M.
The structure of Permutation Groups with t = 1/3(6m − 2) . . . . . . . . . . . . . . . . . . . 185
Razzaghmaneshi, B.
Permutation Groups with Three Constant Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Razzaghmaneshi, B.
On minimal non PST-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Rezaeezadeh, G. and Aghajari*, Z.
The structure of SS-semipermutable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Rezaeezadeh, G. and Mirdamadi*, S. E.
Movement of permutation groups with two orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
v
Rezaei, M.
Irreducible characters and conjugacy classes in finite groups . . . . . . . . . . . . . . . . . . . 203
Robati, S. M.
The classification of some nilpotent Leibniz 4-algebras . . . . . . . . . . . . . . . . . . . . . . . . . 207
Saeedi, F. and Akbarossadat*, S. N.
Finite groups with a given number of relative centralizers . . . . . . . . . . . . . . . . . . . . . . 212
Saeedi*, F. and Farrokhi D. G., M.
Cellular Automata and its application in group theory. . . . . . . . . . . . . . . . . . . . . . . . .217
Safa, H.
On t-extensions of abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Sahleh, H. and Alijani*, A. A.
OD-characterization of almost simple groups related to L2 (p2 ). . . . . . . . . . . . . . . . .224
Sajjadi*, M., Bibak, M. and Rezaeezadeh, G.
A new characterization of Ap where p and p − 2 are twin primes . . . . . . . . . . . . . . . 227
Salehi Amiri, S. S.
Commuting graphs on conjugacy classes of finite groups . . . . . . . . . . . . . . . . . . . . . . . 230
Shadab, M. and Saeidi*, A.
A Quotient OF Topological Fundamental Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Torabi*, H., Pakdaman, A. and Mashayekhy, B.
On 11−decomposable finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
Yousefi*, M. and Ashrafi, A. R.
The nonabelian tensor square of some finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Zainal*, R., Mohd Ali, N. M., Sarmin, N. H., Rashid, S. and Nawi, A. A.
Epicenter of Lie rings and the Lazard correspondence. . . . . . . . . . . . . . . . . . . . . . . . . .244
Zandi, S.
A generalization of Mohres’s Theorem on groups with all subnormal sub-
vi
groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
Zarrin, M.
* Speaker
vii
Preface
The 6th National Conference on Group Theory was held on the Faculty of Sciences
of Golestan University in Gorgan during 12-13 March 2014. The conference provides
a forum for mathematicians and scholar students to present their latest results about
all aspects of group theory and a means to discuss their recent researches with each
other.
The organizing committee of the conference warmly welcomes the participants
to Gorgan, hoping that their stay here will be happy and fruitful one.
The secretary office of the conference has received more than 115 papers from
which 62 papers have been accepted by the scientific committee.
Al in all, we have made every effort to make the conference as worthwhile as
possible. It is our pleasure to express our thanks to all whose help has made this
gathering possible, particularly the referees of the papers for spending many hours
reviewing papers and providing valuable feedback to the authors; the authors of all
submitted papers for their contributions and the administration of Golestan University.
Chair of Conference: Dr. M. Alizadeh Sanati
Scientific Chair: Dr. S. M. Taheri
Executive Chair: Dr. A. Pakdaman
viii
General Talks
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Invariants of a finite group acted on by a Frobenius-like
group
Gülİn Ercan
Middle East Technical University, Ankara, Turkey
[email protected]
Abstract
Let F be a finite group acted on by a finite group H via automorphisms. This action is said to be Frobenius if CF (h) = 1
for all nonidentity elements h ∈ H. Accordingly the semidirect product F H is called a Frobenius group with kernel F
and complement H whenever F and H are nontrivial. It
is well known that Frobenius actions are coprime actions
and the kernel F is nilpotent. A slight generalization of
the Frobenius group will be the object of this talk. More
precisely, we consider nontrivial finite groups F and H so
that H acts on F via automorphisms, F is nilpotent and
[F, h] = F for all nonidentity elements h ∈ H, and call the
semidirect product F H a “Frobenius-like group”. It should
be noted that the group F H is Frobenius-like if and only if
F is a nontrivial nilpotent group and the group F H/F ′ is
Frobenius with kernel F/F ′ and complement isomorphic to
H.
There have been a lot of research about the structure of
finite solvable groups admitting a Frobenius group F H of
automorphisms. In this talk the action of a Frobenius-like
group F H on a finite group G will be discussed and conclusions about some invariants of G and F under additional
hypothesis will be drawn.
2
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Representations of a Finite Group with an Extraspecial
Normal Subgroup
İsmail Ş.Güloğlu
Department of Mathematics, Doğuş University, İstanbul,Türkiye
[email protected]
Abstract
In this talk I will present a technical result which seems
to be of some independent interest and which was obtained
during our joint research with G.Ercan on the influence of
fixed-point free action of a Frobenius-like group on a solvable group which will appear in Jour. of Algebra with the
title “Action of a Frobenius-like Group”. We shall prove a
theorem about irreducible and faithful , complex representations of a finite group G=PH which has a normal subgroup
P isomorphic to an extraspecial group and a complement H
in which each Sylow subgroup is cyclic and H/F(H) is not a
nontrivial 2-group.
1
Introduction
Many questions about certain invariants of a finite group like nilpotent length, derived length, p-length are answered by reducing the relevant group structure to some
relatively simple configuration and then invoking some representation theoretic arguments. And most of these representation theoretic arguments are about linear
groups and say that a certain group has regular orbits on the natural module on
which this linear group acts. Regular modules force the existence of large dimension
and fixed points and hence there is a wide range of possible aplications,especially in
the study of groups admitting a fixed-point-free group of automorphisms.
The result I want to present in my talk in this conference is a generalization of
a very well known theorem which was proven by Dade, namely the following
Theorem Let H be a group in which each Sylow subgroup is cyclic. Assume
that H/F (H) is not a nontrivial 2-group. Let P be an extraspecial group of order
3
p2m+1 for some prime p not dividing |H|. Suppose that H acts on P in such a way
that H centralizes Z(P ), and [P, h] = P for any nonidentity element h ∈ H. Let k be
an algebraically closed field of characteristic not dividing the order of G = P H and
let V be a kG-module on which Z(P ) acts nontrivially and P acts irreducibly. Let χ
m −δ
be the character of G afforded by V. Then |H| divides pm − δ and χH = p |H|
ρ + δµ
where ρ is the regular character of H, µ is a linear character of H and δ ∈ {−1, 1}.
In particular, VH contains the regular kH-module as a direct summand if G is of
odd order.
2
Proof of the Theorem
In this section we present a proof of the theorem.
Lemma 2.1. Let F H be a group with F ▹ F H, F ′ ̸= F and [F, h] = F for all
nonidentity elements h ∈ H. Assume that all Sylow subgroups of H are cyclic.
Then
(i) the groups H ′ and H/H ′ are cyclic of coprime orders,
(ii) H = H ′ ⟨y⟩ = H0 ⟨y⟩ with H ′ ∩ ⟨y⟩ = 1 for some y ∈ H where H0 denotes the
Fitting subgroup of H, and H0 = H ′ × C⟨y⟩ (H ′ ) is cyclic,
(iii) π(H0 ) = π(H).
Proof. The group F H/F ′ is Frobenius with Frobenius complement isomorphic to
H. Then (i) follows by [3, Theorem 5.16]. In particular, H = H ′ ⟨y⟩ for some y ∈ H
with H ′ ∩ ⟨y⟩ = 1. On the other hand the group H has a unique subgroup of order p
for each prime p dividing its order by the argument applied in the proof of Theorem
6.19 in [3] which relies on [3, Theorem 6.9]. Hence π(H0 ) = π(H) as claimed in
(iii). Let now H0 denote the Fitting subgroup of H. Then H0 = H ′ (H0 ∩ ⟨y⟩) and
[H0 ∩ ⟨y⟩, H ′ ] = 1, that is, H0 ∩ ⟨y⟩ ⊆ C⟨y⟩ (H ′ ) ⊆ H0 . This establishes the claim
(ii).
Proof. of The Theorem Since all Sylow subgroups of H are cyclic and G/Z(P )
is a Frobenius group with a complement isomorphic to H, we see that H has the
properties described in Lemma 2.1. By [Huppert, V.17.13] we can assume that H is
not nilpotent and recall that H/F (H) is not a 2-group by hypothesis.
Note that dimV = pm as χP is a faithful irreducible character of P . Let D be the
representation of G afforded by the module V and let M be the k-space of square
matrices of size pm over k. We define a left kH-module structure on M by letting
h · X := D(h)XD(h−1 ), for any X ∈ M and for any h ∈ H.
4
It is known that H acts on Hom k (V, V ) via the multiplication (h · T )(v) = hT (h−1 v)
for any h ∈ H, T ∈ Hom k (V, V ), and v ∈ V. Then clearly M is isomorphic to the
k[H]-module Hom k (V, V ). Furthermore Hom k (V, V ) and V ∗ ⊗ V are∑isomorphic as
k[H]-modules. So by letting Irr(H) = {ψ1 , ψ2 , . . . ∑
, ψs } and χH = si=1 ni ψi with
nonnegative integers ni , i = 1, . . . s, we have Ψ = k,l=1,...,s nk nl ψk ψl where Ψ is
the character of H afforded by M .
Choose a transversal T for Z(P ) in P. Then the set {D(x)|x ∈ T } forms a basis
for M by a result of Burnside [Huppert, V.5.14] and the fact that D(zx) = λ(z)D(x)
for any x ∈ T and z ∈ Z(P ). Notice that P/Z(P ) is the union of one H-orbit of
2m
length 1 and d = p |H|−1 orbits of length |H|. Thus we have M = ⟨I⟩ ⊕ M1 ⊕ · · · ⊕ Md
with Mi ∼
= k[H] as H-module for any i = 1, 2, . . . , d. So we get
s
s
∑
∑
p2m −1
Ψ = 1H +
nk nl ψk ψl .
|H| ψi (1) ψi =
i=1
k,l=1
Thus the multiplicity of the principal character 1H in Ψ is
s
∑
p2m −1
n2k
[1H , Ψ]H = 1 + |H| =
k=1
and the multiplicity of any nonprincipal α ∈ Irr (H) in Ψ is
s
∑
p2m −1
[α, Ψ]H = |H| α(1) =
nk nl (ψl , ψk α).
k,l=1
In particular for any nonprincipal linear character γ of H we have
∑
p2m −1
α∈Irr(H) nα nαγ .
|H| =
This∑
gives
∑
∑
1 = α∈Irr (H) n2α − α∈Irr(H) nα nαγ , and hence 2 = α∈Irr (H) (nα − nαγ )2
for any nonprincipal linear character γ of H.
\′ of characters of the abelian group H/H ′ is isomorphic to H/H ′ .
The group H/H
\′ . It acts on Irr (H ) by
In particular it is cyclic. Let ϑ be a generator of H/H
multiplication. Let Φ
∑i , i =∑1, . . . , b be the orbits of ϑ on Irr (H) and let mi = |Φi | .
Then we have 2 = bi=1 α∈Φi (nα − nαϑ )2 . So there are exactly two elements β
and γ in Irr (H) such that |nβ −nβϑ | = 1 = |nγ −nγϑ |, and we have nα = nαϑ for any
α ∈ Irr (H) − {β, γ}. If β ∈ Φi and γ ∈
/ Φi , then nβ ̸= nβϑ = nβϑ2 = · · · = nβϑmi −1 =
nβ , which is not possible. So if necessary by reindexing the orbits, we can assume
that β and γ are both elements of Φb , and nα = nαϑ for any i = 1, 2, . . . , b − 1 and
any α ∈ Φi .
Suppose that γ = βϑu for some u ∈ {1, 2, . . . , mb − 1}. We have
nβ ̸= nβϑ = · · · = nβϑu ̸= nβϑu+1 = · · · = nβϑmb −1 = nβ .
Since each Φi is either a ϑ2 -orbit or the union of two ϑ2 -orbits of the same size
5
we get
2=
b ∑
∑
(nα − nαϑ2 )2 =
i=1 α∈Φi
∑
(nα − nαϑ2 )2 .
α∈Φb
So the differences nβ − nβϑ2 , nβϑmb −1 − nβϑ , nγ − nγϑ2 are all nonzero if
u ∈ {2, . . . , mb − 2}, which is a contradiction. If necessary by replacing ϑ by ϑ−1 we
can assume that nβ ̸= nβϑ ̸= nβϑ2 = · · · = nβϑmb −1 = nβ. We let nβϑ = nβ + δ, with
some δ ∈ {−1, 1}. Choose an element αi from Φi , i = 1, 2, . . . , b − 1, and let αb = β.
Then
b
∑
χH =
nαi (αi + αi ϑ + · · · αi ϑmi −1 ) + δµ, where µ = αb ϑ.
i=1
By [Huppert, V.17.13] we have
∑
pm − δ ′ ′
ρ + δ ′ µ′ = (
nαi mi αi )H ′ + δαb H ′
′
|H |
b
χH ′ =
i=1
for some δ ′ ∈ {−1, 1} and µ′ ∈ Irr (H ′ ) where ρ′ is the regular character of H ′ .
It follows by that if i ̸= j then the sets of irreducible constituents of the restrictions of αi and αj are disjoint. By Clifford’s theorem we have
ti
ti
∑
∪
αiH ′ = ei
λi,j where IH (λi,1 ) = Ti , ti = [H : Ti ], H =
Ti xi,j , and λi,j =
x
j=1
j=1
λi,1i,j , j = 1, 2, . . . , ti ; i = 1, 2, . . . , b. Now {λi,j |j = 1, 2, . . . , ti ; i = 1, 2, . . . , b} =
Irr (H ′ ).
It is known that there is a unique ξi ∈ Irr (Ti ) such that ξi H = αi and ξiH ′ =
ei λi,1 . On the other hand as Ti /H ′ is cyclic, λi,1 has an extension, say φ, to Ti . But
then φH must belong to the ϑ-orbit containing αi which implies αiH ′ = (φH )H ′ .
Therefore we have
ei = [αiH ′ , λi,1 ] = [(φH )H ′ , λi,1 ] = [φH ′ , λi,1 ] = 1 for any i = 1, 2, . . . , b.
m
′
c′ we have
and µ′ = λi0 ,j0 . Then for any υ ∈ H
Let now e = p |H−δ
′|
{
[χH ′ , υ]
H′
=
e
if υ ̸= µ′
.
e + δ ′ if υ = µ′
Set H0 = F (H). Applying [Huppert, V.17.13] to the action of P H0 on V we see
in particular that |H0 | divides pm − δ ∗ for some δ ∗ ∈ {−1, 1}. Then |H ′ | divides
δâe2 − δ ∗ = (pm − δ ∗ ) − (pm − δâe2 ) and so we have either δâe2 = δ ∗ or |H ′ | = 2.
If the latter holds then H ′ ≤ Z(H) and hence H is abelian, which is not the case.
Thus |H0 /H ′ | divides e. In particular e > 1 and so e + δ ′ > 0 which shows that
[χH , αi0 ]H ̸= 0.
6
If ti0 ̸= 1, then there exists j1 ̸= j0 such that
e = [χH ′ , λi0 ,j1 ]H ′ = [χH ′ , λi0 ,j0 ]H ′ = e + δ ′
which is not possible. Then ti0 = 1 and hence µ′ is H-invariant. This yields that
αi0 ′ = µ′ = λi0 ,1 . In particular αi0 is a linear character of H and so mi0 = |H/H ′ |.
H
Furthermore we have
{
nαi0 mi0
if i0 < b
e + δ′ =
.
nαb mb + δ if i0 = b
Now |H0 /H ′ | divides the greatest common divisor of e and mi0 which forces that
i0 = b as H0 /H ′ is nontrivial. Furthermore if δ ̸= δ ′ we have |H0 /H ′ | = 2, which
implies by Lemma 2.1 that H/H ′ is a 2-group. This contradiction shows that δ = δ ′
m −δ
= nαb is an integer.
and hence nαb mb = e by the above formula. In particular p |H|
On the other hand we also have e = [χH ′ , λi,1 ]H ′ = nαi mi if i < b.
H
ri
′
Set next ri = |Ti /H ′ | . As T\
i /H = ⟨ϑ|T ⟩ we obtain Ti ≤ Ker ϑ . As αi = ξi
i
for some ξi of Ti and Ti is normal in H, we observe that αi (x) = 0 for any x ∈
/
Ti . Combining these two observations we get ϑri αi = αi . Thus mi divides ri
and hence |H/H ′ | = ri ti = mi ci ti for some positive integer ci . It follows now that
m −δ
m −δ
m −δ
ci αi (1) ≥ p |H|
αi (1). Thus p |H|
ρ + δµ
nαi mi ci ti = eci αi (1) and hence nαi = p |H|
occurs in χH . As the degrees of these characters are the same we see that they are
equal. This completes the proof of the theorem.
The next example shows that the hypothesis about the structure of H can not
be avoided.
Example Let V be the GF (3)-space GF (34 ). We define the map
(·|·) : V × V −→ GF (3) by (·|·)(x, y) = Tr (d · (xy 9 − x9 y)) for x, y ∈ V , where
d is an element of order 16 in GF (34 )∗ . One can check that (·|·) is a nonsingular
symplectic form on V.
Let b ∈ GF (34 )∗ be an element of order 5 and c ∈ GF (34 )∗ be an element of
order 4. We define τb : V −→ V by τb (x) = b · x and σ : V −→ V by σ(x) = c · x9 .
Then H = ⟨τb , σ⟩ is a subgroup of GL(4, 3) preserving the symplectic form, with
|H| = 20, H ′ = ⟨τb ⟩ of order 5, and F (H) = H ′ × ⟨σ 2 ⟩ of order 10. Furthermore
h(v) = v for some 0 ̸= v ∈ V and h ∈ H implies that h = 1. So if P is the
extraspecial group of order 35 and exponent 3, then it admits H as a subgroup of
automorphisms of P, centralizing Z(P ) and satisfying [P, h] = P for any nonidentity
element h ∈ H. Let χ be any irreducible character of the group P H which does
2 −δ
not contain Z(P ) in its kernel. Clearly, we have χH ̸= 3|H|
ρ + δµ for the regular
H-character ρ and any δ ∈ {−1, 1} and µ ∈ Irr (H), because
7
32 −δ
|H|
is not an integer.
References
[1] B. Huppert, Endliche Gruppen, Springer-Verlag, Berlin-New York 1967.
[2] G. Ercan, İ. Ş. Güloğlu: ”Action of a Frobenius-like Group” 2014, will appear
in J. Algebra.
8
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
The Classification of Groups via Capability; A Reality
to Dream
Saeed Kayvanfar
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
[email protected]; [email protected]
Abstract
This talk is a survey article on the classification of groups.
The classification of prime power groups of order at most p6
was done using the notion of isoclinism and invoking a fundamental instrument, namely the capability of groups. Since
the basic concepts of this classification i.e., isoclinism and
capability were generalized to any variety of groups, therefore this talk intends to propound a basic question whether
it is possible to define some suitable varieties that could play
the key role for classifying some other families of groups.
1
Introduction
One of the most important problem in group theory is the classification of groups.
The problem which has been always studied along with the age of group theory.
There have been also various approaches to face the problem. Among several different approaches, one of the most classical notions is the concept of isomorphism
between groups. However, this notion is too strong in many cases. For this reason,
P. Hall in 1940 [5] introduced the notion of isoclinism between two groups (which
is weaker than isomorphism) and could classify some groups of prime power order.
Using his method, his student Easterfield [6] and then M. Hall and J. Senior in 1964
[6] and later R. James in 1988 [8] completed the classification of groups of order at
most pn , where p is a prime and n is at most 6.
2010 Mathematics Subject Classification. Primary 20E10; Secondary 20D15.
Key words and phrases. Classification, isologism, capability.
9
We also know that the notion of isoclinism was generalized by P. Hall [5] to
isologism. Isologism is in fact isoclinism with respect to a certain variety of groups.
If one takes the variety of all trivial groups, one gets the notion of isomorphism back.
The variety of all abelian groups yields isoclinism.
On the other hand, we know that the main strong tool in the P. Hall’s classification is the notion of capability and this notion also was simultaneously generalized
to varietal capability by J. Burns and G. Ellis [3] and a joint paper of the author [9]
in 1997.
Now, using all these facts in hand, we just intend to propound a fundamental
question; Is it possible to define some suitable varieties so that invoking them the
classification of some other families of groups happens? Indeed, in the face of these
facts, it seems that the classification of some other suitable families of groups will not
be so far! By this sentence, we mean that it might probably exist some special kind
of varieties such that their obtained classes caused by isologim can be considered
as the first step of screening in the classification, though they might be so broad.
Although we do understand that as usual, the stating such a problem is so easy
while finding the answer may not!
2
Reality; The Classification of some Prime Power Groups
There are different approaches that can be considered for the description of finite
p-groups. We used the word “description” rather than “classification” because we
know that classifying p-groups is notoriously open and difficult. Some of the approaches are, for instance, the order, the coclass and the class of nilpotency. But
each of them has some restrictions and difficulties in this way so that P. Hall in 1940
[5] introduced the notion of isoclinism for the classification of all groups, though he
could classify only some prime power groups.
Definition 2.1. Two groups G1 and G2 are said to be isoclinic provided that there
exist two isomorphisms α : G1 /Z(G1 ) → G2 /Z(G2 ) and β : γ2 (G1 ) → γ2 (G2 )
such that if α(a1 Z(G1 )) = a2 Z(G2 ) and α(b1 Z(G1 )) = b2 Z(G2 ), then β([a1 , b1 ]) =
[a2 , b2 ]. This notion is written by G1 ∼ G2 .
In the P. Hall’s classification, regular group which was defined by him, plays the
key role. There are many equivalent ways to define regular p-groups. One is that if
a and b are any elements of the group, then
r
r
r
r
r
(ab)p = ap bp c1 p . . . cpt ,
where ci are elements of the commutator subgroup of < a, b >. In fact, a group is
regular if the operation of taking pth powers interacts “well” with taking commutators. P. Hall [5] showed that in a regular p-group, one can define “type invariants”
10
which are similar to the invariant factors for finite abelian groups. Though they do
not completely determine the groups the way the invariant factors do for abelian
groups, they are usually a very good first reduction towards the analysis. Note that
if p ≥ n, then a group of order pn is necessarily regular (more generally, if the group
is of class c and p > c, then the group is regular, in particular, since a group of
order pn is of class at most n − 1, the observation just made follows). In fact, P.
Hall mentioned in [4] that if we fix n, then “most” groups of order pn are regular
(since only those with p < n may fail to be regular). This leads, classically, to a
separation of p-groups into those of “small class” (when the class is smaller than p),
and “the rest”. In other words, this means that when classifying groups of order
pn , the analysis usually breaks into two different cases: when the group is regular
(which includes all p ≥ n), and when the group is irregular. The latter case leads
to a case-by-case analysis for small primes. This occurs, for example, in the classification of groups of order p3 (in which odd primes and 2 should be considered
separately). Likewise Burnside’s work on group of order p4 . Or similar to the work
of R. James [8] in the classification of groups of order p6 and E. A. O’Brien and M.
R. Vaughan-Lee [10] for p7 . The latter separates p ≥ 7 with the groups of order 37
and 57 . Note that the latest work uses Lie rings and algebras as a starting point.
There are algorithms that are known to produce and check isomorphism types (see
[10]).
The above comments explains that how groups of order pn for all primes p have
been fully classified for n ≤ 7, first (most of the times) by isoclinism and then up to
isomorphism. Note, in particular, that the number of isomorphism classes increases
with p when n ≥ 5. One may know that the Higman PORC conjecture (polynomial
on residue classes) is that, for each n, this number is a polynomial function of p and
of p (mod k) for a finite collection of values of k. Although a family of examples
constructed by M. D. Sautoy and M. R. Vaughan-Lee [12] of order p10 show while
not actually disproving the conjecture, suggests that it is very unlikely indeed that
it is true. We should remind also that their construction depends on the geometry
of elliptic curves. It will probably illustrate that how describing all groups of order
p10 would be complicated.
Let us comeback to the isoclinism. The notion of isoclinism defines an equivalence
relation on the class of all groups and has this trait that some other properties
of groups are invariant under isoclinism. For instance, it is proved in [2] that,
restricting ourselves to finite groups, we have the following hierarchy of classes of
groups, invariant under isoclinism: abelian < nilpotent < supersolvable < strongly
monomial < monomial < solvable. For charactering the families of isoclinism, P.
Hall [5] tried to find some properties which are invariant in each family. Accordingly,
any quantity depending on a variable group and which is the same for any two groups
11
of the same family is called a family invariant. For instance, it is easy to see that the
members of the derived series and the central quotient groups are family invariants.
It follows that the groups belonging to the same family have the same derived length
and nilpotency class. Note that the commutator quotient group and the center are
not family invariants, as is the minimal number of generators.
Now, the importance of central quotient groups for this classification may be
seen. Such groups are called capable. More precisely;
Definition 2.2. A group G is called capable if there exists a group E such that
E
G∼
.
= Z(E)
Among many different points on capability, there are two important tools for
recognition the capability of a group. One of them is a necessary condition which
was established by P. Hall [5]. He considered a generating system J for a group G
and defined ∆J = ∩x∈J < x > and denoted the join of all subgroups ∆J , where J
varies over all generating systems of G, by ∆(G) and gave a necessary condition for
the capability of G in such a way that a capable group G must satisfy ∆(G) = 1.
The other tool for characterizing the capability is a criterion that was introduced
by F. R. Beyl, U. Felgner and P. Schmid [1]. They showed that every group G
possesses a uniquely determined central subgroup Z ∗ (G) which is minimal subject
to being the image in G of the center of some central extension of G. This Z ∗ (G) is
characteristic in G and is the image of the center of every stem cover of G.
Definition 2.3. The intersection of all subgroups of the form ψ(Z(E)), where ψ :
E → G is a surjective homomorphism with kerψ ⊆ Z(E), is called the precise center
subgroup of G and denoted by Z ∗ (G).
F. R. Beyl et al. [1] proved that Z ∗ (G) is the smallest central subgroup of G
whose factor group is capable. Now using this fact the criterion for capability can
be given as follows.
Theorem 2.4. (F. R. Beyl, U. Felgner and P. Schmid 1979 [1]) G is capable if and
only if Z ∗ (G) = 1.
3
Dream; The Classification of Other Classes of Groups!
The works and attempts which were explained as the “Reality” in Section 1 for the
classification of some prime power groups evidence that this way of classification
might not be quite simple. In other words, determining finite p-groups of order pn
up to isomorphism, will be so complicated while n becomes large and larger. This
shows the necessity of primary screening not only for p-groups, but also for other
families of groups.
12
On the other hand, we should not forget that whatever, for instance, R. James
[8] did was heavily indebted to the work of P. Hall [5] on isoclinism. That is why,
we believe that as isoclinism could help us as the first step of classification of prime
power groups of order pn for small n, paying attention to its generalization may be
helpful as the first step of classification of some other classes of groups! Therefore in
this section, we remind all the notions and the tools in Section 1 which were generalized to any variety of groups. In fact, the primary definitions and the preliminary
statements which provide the context of determining of the equivalence classes in
isoclinism were generalized in two steps. First, the “isoclinism” and “capability”
were transformed to “c-isoclinism” and “c−capability and then they were generalized to “isologism” and “varietal capability”, respectively. In the following, the
general case is provided.
Definition 3.1. Let V be a variety of groups defined by the set of laws V . Two
groups G and H are V- isologic if there exist isomorphisms
α:
G
V
∗ (G)
→
H
V
∗ (H)
and β : V (G) → V (H),
such that β(v(g1 , g2 , . . . , gn )) = v(h1 , h2 , . . . , hn ), where gi ∈ G, hi ∈ α(gi V ∗ (G))
for each 1 ≤ i ≤ n. In this case, we write G ∼
V H. The pair (α, β) is said to be a
V-isologism between G and H.
Likewise the isoclinism, for each variety V, isologism gives an equivalence relation
on the class of all groups. The larger variety implies the weaker equivalence relation.
If V is the variety of all abelian groups, V-isologism coincides with isoclinism. The
groups in a variety V fall into one single equivalence class, they are actually Visologic to the trivial group. (For more information about V-isologism see [7]).
On the other hand, it is observed from the definition of isologism that the
marginal factor group can play an important role in this system of classification.
Such a group is called varietal capable with respect to the variety V, or briefly
V-capable [9].
Definition 3.2. Let V be a variety of groups defined by the set of laws V . A group
G is said to be V-capable , if there exists a group E such that G ∼
= E/V ∗ (E) .
Some properties of varietal capability are given in [9]. Specially, finding a criterion for recognition of varietal capability is illustrated in [9]. More precisely, it is
shown that every group G possesses a uniquely determined subgroup (V ∗ )∗ (G) of
the marginal subgroup V ∗ (G) , which is minimal subject to being the image in G of
the marginal subgroup of some V-marginal extension of G. In fact, if ψ : E → G is
a surjective homomorphism with kerψ ⊆ V ∗ (E), then (V ∗ )∗ (G) is defined to be the
intersection of all subgroups of the form ψ(V ∗ (E)).
13
If V is the variety of abelian groups then the subgroup (V ∗ )∗ (G) is Z ∗ (G) and
the V-capability coincides with the usual capability. If one takes V to be the variety
of nilpotent groups of class at most c, c ≥ 1, then one gets (V ∗ )∗ (G) to be Zc∗ (G) as
J. Burns and G. Ellis introduced in [3] and the V-capability will be their c-capability.
(V ∗ )∗ (G) is characteristic and it is also proved in [9] that (V ∗ )∗ (G) is the smallest subgroup contained in the marginal subgroup of G for which the factor group
G/(V ∗ )∗ (G) is V-capable. In other words;
Theorem 3.3. (M. R. R. Moghaddam and S. Kayvanfar [9]) G is V-capable if and
only if (V ∗ )∗ (G) = 1.
The above comments explain that for a classification of a family of groups, the
notion of isologism might be helpful as a primary screening. The most important
tool that can be considered for characterizing the families of isologism is the Vcapable group. On the other hand, there are a few statements for recognition the
varietal capability (for instances, see [3] and [9]) invoking them one can find out
the V-capability of some groups. There is also another helpful tool for recognizing
the V-capability; the Baer invariant of groups, which has been calculated for many
varieties. Invoking the Baer invariant, the varietal capability of some types of groups
has been characterized for some special kind of varieties (for example see [11]).
All these facts motivate us to think more to the dream of classification of groups
by varietal isologism via the varietal capability.
Acknowledgement
The author would like to thank Dr. A. Kaheni with whom I have had many useful
conversations on the classification of groups.
References
[1] F. R. Beyl, U. Felgner and P. Schmid, On groups occuring as centre factor
groups, J. Algebra 61 (1979), 161–177.
[2] J. C. Bioch and R. W. van der Waall, Monomiality and isoclinism of groups, J.
reine ang. Math. 298 (1978), 74–88.
[3] J. Burns and G. Ellis, On the nilpotent multipliers of a group, Math. Z. 226
(1997), 405–28.
[4] P. Hall, A contribution to the theory of groups of prime power order, Proc.
London Math. Soc. (series 2) 36 (1934), no.1, 29–95.
14
[5] P. Hall, The classification of prime-power groups, J. reine ang. Math.182 (1940),
130–141.
[6] M. Hall, Jr., and J. K. Senior, The groups of order 2n (n ≤ 6), Macmillan, New
York, 1964.
[7] N. S. Hekster, Varieties of groups and isologism, J. Austral. Math. Soc. (series
A) 46 (1989), 22–60.
[8] R. James, The groups of order p6 (p an odd prime), Math. of Computation 34
no. 150 (1980), 613–637.
[9] M. R. R. Moghaddam and S. Kayvanfar, A new notion derived from varieties
of groups, Algebra Colloquium 4:1 (1997), 1–11.
[10] E. A. O’Brien and M. R. Vaughan-Lee, The groups with order p7 for odd prime
p, J. Algebra 292 (2005), 243–258.
[11] M. Parvizi, B. Mashayekhy and S. Kayvanfar, Polynilpotent capability of finitely
generated abelian groups, Journal of Advanced Research in Pure Mathematics
2(3) (2010), 81 – 86.
[12] M. D. Sautoy and M. R. Vaughan-Lee, Non-PORC behaviour of a class of
descendant p-groups, arXiv:1106.5530, 30 Jan, 2013.
15
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Direct Limits of Finitary Symmetric Groups
Mahmut Kuzucuoğlu
Middle East Technical University Department of Mathematics Ankara, Turkey
[email protected]
Abstract
We describe the construction of a new class of simple locally
finite groups as a direct limit of finitary symmetric groups.
Moreover we investigate the structure of the centralizers of
elements in these groups.
1
Introduction
A group is called a locally finite group if every finitely generated subgroup is a
finite group. One of the natural construction of locally finite groups is by taking the
direct limit of finite groups. Using this method uncountably many, simple locally
finite groups of countably infinite order is constructed as a direct limit of finite
symmetric groups in Kegel-Wehfritz [2, Chapter 6], (1976). Then the classification of
these groups by using Steinitz numbers is done by N. V. Kroshko-V. I. Sushchansky
in [3], (1998). We now describe this construction.
(
)
1... n
Let α be the permutation defined by α =
. Then the permutation
i1 . . . i n
(
dr (α) =
1
i1
...
...
n
in
n+1
n + i1
...
...
...
...
2n
n + in
(r − 1)n + 1
(r − 1)n + i1
...
...
rn
(r − 1)n + in
)
is called a homogeneous r-spreading of the permutation α.
Let Π be the set of sequences consisting of prime numbers and ξ ∈ Π. So ξ =
(p1 , p2 , . . .) is a sequence consisting of not necessarily distinct primes pi . We obtain
direct systems by using homogeneous pi -spreading from the following embeddings
dp1
dp2
dp3
dp4
{1} → Sn1 → Sn2 → Sn3 → . . .
2010 Mathematics Subject Classification. Primary:20F50; Secondary: 20E32.
Key words and phrases. Centralizers, Direct Limit groups, Simple groups.
16
dp1
dp2
dp3
dp4
{1} → An1 → An2 → An3 → . . .
where n0 = 1, n1 = p1 , ni = ni−1 pi , i = 2, 3 . . . and Sni is the symmetric group
on ni letters, Ani is the alternating group on ni letters. The direct limit groups
obtained from the above direct systems are of strictly diagonal type and denoted by
S(ξ) and A(ξ), respectively. Observe that S(ξ) ≤ Sym(N).
It is proved that such groups satisfy the followings:
• If the prime 2 appears infinitely often in the sequence ξ, then the limit group
S(ξ) is a simple non-linear locally finite group.
• If a prime p appears infinitely often, then S(ξ) contains an isomorphic copy of
the locally cyclic p-group Cp∞ .
• For each sequence ξ, we define Char(ξ) = pr11 pr22 . . . where ri is the number of
times that prime pi repeat in ξ. If it repeats infinitely often, then we write
p∞
i . Therefore for each ξ there corresponds a Steinitz number Char(ξ). For a
group S(ξ) obtained from the sequence ξ we define Char(S(ξ)) = Char(ξ)
Two groups S(ξ1 ) and S(ξ2 ) are isomorphic if and only if Char(S(ξ1 )) =
Char(S(ξ2 )).
• There are uncountably many non-isomorphic simple locally finite groups of
this type.
We will discuss some of the results for the centralizers of elements in S(ξ), in
particular the following Theorem.
Theorem 1.1. (Güven, Kegel, Kuzucuoğlu [1]) Let ξ be an infinite sequence, g ∈
S(ξ) and the type of principal beginning g0 ∈ Snk be t(g0 ) = (r1 , r2 , . . . , rnk ). Then
nk
CS(ξ) (g) ∼
=Dr Ci (Ci¯≀ S(ξi ))
i=1
where Char(ξi ) = Char(ξ)
ri
nk
corresponding factor is {1}.
for i = 1, . . . , nk . If ri = 0, then we assume that
Let Ω be an arbitrary set and Sym(Ω) be the symmetric group on the set Ω.
Let g ∈ Sym(Ω). We define Supp(g) = {α ∈ Ω | g(α) ̸= α } the set of elements of Ω which are moved by the permutation g. Then F Sym(Ω) = {g ∈
Sym(Ω)| |Supp(g)| < ∞ }. Then F Sym(Ω) is a locally finite, normal subgroup of
Sym(Ω) with the same cardinality of Ω for an infinite set Ω.
Let κ be an arbitrary infinite cardinal number. Let F Sym(κ) denote the finitary
symmetric group and Alt(κ) denote the alternating group on the set κ. As before, let
17
Π be the set of sequences of prime numbers and ξ ∈ Π. Then ξ is a sequence of not
necessarily distinct primes. Let α ∈ F Sym(κ), ( Alt(κ) ). For a natural number
p
p ∈ N a permutation dp (α) ∈ F Sym(κp) defined by (κs + i)d (α) = κs + iα , i ∈ κ
and 0 ≤ s ≤ p − 1 is called a homogeneous p-spreading of the permutation α.
We divide the ordinal κp into p equal parts and on(each part
) we repeat the permu1... n
tation diagonally as in the finite case. So if α =
∈ F Sym(κ), then the
i1 . . . i n
homogeneous p−spreading of the permutation α is
(
dp (α) =
1
i1
...
...
n
in
κ+1
κ + i1
...
...
κ+n
κ + in
...
...
κ(p − 1) + 1
κ(p − 1) + i1
...
...
κ(p − 1) + n
κ(p − 1) + in
)
with the assumption that the elements in κp \ supp(dp (α)) are fixed.
We continue to take the embeddings using homogeneous p-spreadings with respect to the given sequence of primes in ξ. From the given sequence of embeddings,
we have direct systems and hence direct limit groups F Sym(κ)(ξ), (Alt(κ)(ξ)). Observe that F Sym(κ)(ξ) and Alt(κ)(ξ) are subgroups of Sym(κω).
The principal beginning α0 of an element α ∈ F Sym(κ)(ξ) is defined to be the
smallest positive integer nj ∈ N such that α0 ∈ F Sym(κnj ) and α0 is not obtained
as a sequence of embeddings dpi for any pi ∈ ξ.
Theorem 1.2. (Güven, Kegel, Kuzucuoğlu [1]) Let ξ be an infinite sequence. If
α ∈ F Sym(κ)(ξ) with principal beginning α0 ∈ F Sym(κni ), t(α0 ) = (r1 , . . . , rn ),
and |supp(α0 )| = n. Then
n
CF Sym(κ)(ξ) (α) ∼
= (Dr Ci (Ci ≀ S(ξi ))) × F Sym(κ)(ξ ′ )
i=1
where Char(ξi ) = Char(ξ)
ri and Char(ξ ′ ) = Char(ξ)
. If ri = 0, then we assume that
ni
ni
the corresponding factor in the direct product is {1}.
References
[1] Güven Ü. B., Kegel O. H., Kuzucuoğlu M.; Centralizers of subgroups in direct limits of symmetric groups with strictly diagonal embedding, To appear in
Communications in Algebra .
[2] Kegel O. H., Wehrfritz B. A. F.; Locally Finite Groups, North-Holland Publishing Company - Amsterdam, 1973.
18
[3] Kroshko N. V.; Sushchansky V. I.; Direct Limits of symmetric and alternating
groups with strictly diagonal embeddings, Arch. Math. 71, 173–182, (1998).
19
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Algebraically closed groups and embedding theorems
M. Shahryari
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz,
Tabriz, Iran
[email protected]
Abstract
Using the notion of algebraically closed structures, we obtain
new embedding theorems for groups and Lie algebras. We
also prove the existence of some groups and Lie algebras
with prescribed properties.
The group Z2 is the only finite group which has just two conjugacy classes. Is there
any infinite group with the same property? Denis Osin [7], proved that there is a
finitely generated infinite group with exactly two conjugacy classes. His method is
based on the small cancellation theory over relatively hyperbolic groups. Another
example of such groups is obtained by Higman, Neumann and Neumann using their
well-known embedding methods, [3]. In this article, we are dealing with problems
like this. Using the concept of algebraically closed groups, we prove that any R-group
can be embedded in a Q-group which has only two conjugacy classes. Remember
that a group G is called R-group, if for any integer m, the equality xm = y m implies
x = y. We also give a generalization of these groups and define an Rπ -group to
be a group G such that the equality xm = y m implies x = y, whenever m is a π ′ number. We prove that any Rπ -group can be embedded in a simple Qπ -group, whose
elements of the same order are conjugate. We study many interesting properties of
such groups and then we obtain similar results for Lie algebras. By the notion of
algebraically closed Lie algebras, we show that any Lie algebra L can be embedded
in a simple Lie algebra L∗ which has many rare properties; for any non-zero elements
a and b, there is x such that [x, a] = b, and so for all non-zero x the derivation ad x
is not nilpotent. In the case of finite fields, every finite dimensional Lie algebra can
2010 Mathematics Subject Classification. Primary 20E45, Secondary 20E06.
Key words and phrases. Conjugacy classes, R-groups; Rπ -groups, algebraically closed groups;
HNN-extension; Q-groups; Lie algebras; derivations; embedding theorems, Monsters
20
be embedded in L∗ and it is possible to describe the derivation algebra of any finite
dimensional algebra A as the quotient algebra NG∗ (A)/CG∗ (A). We also prove the
existence of some groups and Lie algebras with prescribed properties. Our main
tool in this work is HNN-extensions of groups and Lie algebras. The first one is
well-known and the reader can consult any book on combinatorial group theory
(for example [3]) to see definition and properties of HNN-extensions of groups. The
HNN-extensions of Lie algebras are not so popular and it seems that there are only
two articles ever published in the subject, [2] and [11].
Although, we are dealing just with groups and Lie algebras in this article, it is
useful to give a general definition of algebraically closed structures in the frames of
the universal algebra. Let L be an algebraic language and A be an algebra of type
L. We extend L to a new language LA by adding new constant symbols ca for any
a ∈ A. Let TA (X) be the term algebra of LA with variables from a countable set X.
If p(x1 , . . . , xn ) and q(x1 , . . . , xn ) are elements of this term algebra, then we call the
expression p(x1 , . . . , xn ) = q(x1 , . . . , xn ) an equation with coefficients from A. An
inequation is the negation of an equation. A system of equations and inequations
over A (or a system over A) is a finite set consisting equations and inequations. We
say that A is algebraically closed (a.c. for short), if and only if any system over A
having a solution in an extension B of A, has already a solution in A. In this article,
we consider the case of groups and Lie algebras, but some of the theory here, can be
generalized to arbitrary algebraic structures. Note that in the case of a group G, we
may assume that an equation has the form w(x1 , . . . , xn ) = 1, where w is an element
of the free product G ∗ F (X). Here F (X) is the free group on the set X. Similarly,
if L is a Lie algebra (or any non-associative algebra), then an equation over L has
the form w(x1 , . . . , xn ) = 0, with w and element of the free product L ∗ F (X), where
F (X) is the free Lie algebra over X.
1
Algebraically closed groups
A group G is called algebraically closed, if any finite consistent system of equations
and inequations with coefficients from G has a solution in G. A system
S = {wi (x1 , . . . , xn ) = 1; (1 ≤ i ≤ r), wj (x1 , . . . , xn ) ̸= 1; (r + 1 ≤ j ≤ s)} (I)
with coefficients in G is called consistent, if there is a group K containing G, such
that S has a solution in K. One can generalize this definition to an arbitrary class
of groups: Let X be a class of groups. A group G ∈ X is called algebraically closed in
the class X, if every X-consistent system S has a solution in G. Here, X-consistency
means that there exists a group K ∈ X which contains G and S has a solution in K.
21
Recall that a class of groups is called inductive, if it contains the union of any
chain its elements. We will use the next theorem to obtain our embedding results
on groups.
Theorem 1.1. Let X be an inductive class of groups which is closed under the
operation of taking subgroups. Let G ∈ X. Then, there exists a group G∗ ∈ X, with
the following properties,
1- G is a subgroup of G∗ .
2- G∗ is algebraically closed in the class X.
3- |G∗ | ≤ max{ℵ0 , |G|}.
We use the concept of algebraically closed groups and above theorem to embed
R-groups in Q-groups of the same cardinality with just two conjugacy classes. Then,
we will extend our results to the class of Rπ -groups for any set of primes π. Our
main tool in this section is HNN-extensions of groups and normal forms of elements
in such groups, [3]. Recall that an R-group is group for which xm = y m implies
x = y for any non-zero integer m. A divisible R-group is a rational exponential
group or a Q-group in other words. The reader can consult [4] or [5] for a theory of
exponential groups.
Theorem 1.2. Let G be an R-group. Then G can be embedded in a Q-group G∗
with only two conjugacy classes. Further |G∗ | = |G|.
We can generalize the embedding theorem of R-groups to a more general classes
of groups. Let π be a set of prime numbers. We consider the ring
Qπ = {
m
∈ Q : n is a π ′ − number}.
n
If π consists of a single element p, then we denote this ring by Qp . A group G is an
Rπ -group, iff for any x and y and any π ′ -number n, we have the implication
xn = y n ⇒ x = y.
The order of any element of such a group is infinite or a π-number. We say that G
is π-divisible, iff for any x ∈ G and any π ′ -number n, there exists y ∈ G, such that
y n = x. Note that if a group G is both Rπ -group and π-divisible, then there is a
unique y satisfying y n = x, for a given π ′ -number n. So, we denote this element by
1
x n . Now, a group which is both Rπ -group and π-divisible, can be regarded as an
m
1
exponential group over the ring Qπ (or Qπ -group for short) via x n = (x n )m . The
reader most consult [4] and [5] for the theory of exponential groups. Note that any
Qπ -group is also Rπ -group and π-divisible.
22
Theorem 1.3. Let G be an Rπ -group. Then there exists a Qπ -group G∗ containing
G and with same cardinality as G, such that
1- G∗ is simple,
2- element of the same order in G∗ are conjugate,
3- G∗ is not finitely generated,
4- every finite π-group embeds in G∗ .
5- every finitely presented π-group can be residually embedded in G∗ ,
N ∗ (A)
6- for any finite π-group A, we have Aut(A) ∼
= CGG∗ (A) .
2
Algebraically closed algebras
Our next mission is to find similar embedding theorems for Lie algebras. But there
are two major differences between groups and Lie algebras. First, we don’t have
any suitable definition of torsion in the case of Lie algebras so, in advance, we don’t
have a parallel concept of R-Lie algebra and so on. Instead, we can express our
theorems in terms of arbitrary Lie algebras. The second main difference is related
to HNN-extensions of Lie algebras. Here, an HNN-extension comes from a Lie
algebra and a derivation of some subalgebra, despite groups where HNN-extensions
are always defined by groups and isomorphisms between subgroups. We will give
a brief summary of HNN-extensions of Lie algebras in the next section. In this
section, we give the analogue of Theorem 1.1 for Lie algebras, in fact since it can be
formulated for arbitrary non-associative algebras, we prove it in the most general
form.
Theorem 2.1. Let X be an inductive class of (not necessarily associative) algebras
over a field K. Suppose X is closed under subalgebra and L ∈ X. Then there exists
an algebra L∗ ∈ X with the following properties,
1- L is a subalgebra of L∗ .
2- L∗ is algebraically closed in the class X.
3- dim L∗ ≤ max{ℵ0 , dim L, |K|}.
In [2] and [11], the concept of the HNN-extension is defined for Lie algebras.
Suppose L is a Lie algebra over a filed K and A be a subalgebra. Let δ : A → L be
a derivation. Define a Lie algebra Lδ with the presentation
Lδ = ⟨L, t : [t, a] = δ(a); (a ∈ A)⟩.
The properties of this HNN-extension is studied in [2] and [11]. It is proved that L
is a subalgebra of Lδ . Similar constructions are also introduced for Lie p-algebras
and rings in [2]. In this section, using this HNN-extension, the theorem above, and
the notion of algebraically closed Lie algebras, we obtain a new embedding theorem.
23
Theorem 2.2. Let L be a Lie algebra over a field K. Then there exists a Lie algebra
L∗ having the following properties,
1- L is a subalgebra of L∗ ,
2- for any non-zero a, b ∈ L∗ , there exists x ∈ L∗ such that [x, a] = b,
3- for any non-zero x ∈ L∗ the derivation ad x is never nilpotent,
4- L∗ is simple,
5- dim L∗ ≤ max{ℵ0 , dim L, |K|},
6- L∗ is not finitely generated,
7- every finite dimensional simple Lie algebra over K embeds in L∗ ,
8- every finitely presented Lie algebra over K embeds residually in L∗ ,
9- if K is finite, then every finite dimensional Lie algebra over K embeds in L∗ ,
10- if K is finite and A is finite dimensional Lie algebra over K, then we have
Der(A) ∼
=
3
NL∗ (A)
.
CL∗ (A)
Some Olshanskii like groups
In mid twenties, Alfred Tarski asked about the existence of infinite groups all proper
non-trivial subgroups of which are of fixed prime order p. In 1982, A. Yu. Olshanskii
[6], constructed an uncountable family of such groups using his geometric method of
graded diagrams over groups, for all primes p > 1075 . The groups constructed are
called Tarskii monsters since then. These groups are two-generator simple groups
and hence are countable. In this section, for any fixed prime p, we give a quite elementary proof for existence of countable non-abelian simple groups with the property
that their all non-trivial finite subgroups are cyclic of order p.
We will consider two special classes of groups in this section. The first one
consists of groups all finite subgroups in which are cyclic. We will denote this class
by Xf c . The second class which will be denoted by Xp , is the class of all groups
in which their non-trivial finite subgroups are of order p, for a fixed prime p. Note
that both classes are inductive and closed under subgroup. Clearly the Monsters
constructed by Olshanskii satisfy the requirements of the next theorem, but we don’t
use that monsters, since we have a very elementary proof for our claims. What we
need is the theorem 1.2 and some facts about finite subgroups of HNN-extensions
(and also those of free products). It is known that (see [3], page 212) every finite
subgroup of any HNN-extension
G = ⟨A, t : tF t−1 = ϕ(F )⟩
is contained in some conjugate of A. Also, every finite subgroup of any free product
A ∗ B is contained in some conjugate of A or some conjugate of B.
24
Theorem 3.1. There exists a countable non-abelian simple group M such that all
finite subgroups of M are cyclic and for any prime p, the group M has an element
of order p.
A similar result can be obtained if we use the class Xp .
Theorem 3.2. Let p be a fixed prime. Then there exists a countable non-abelian
simple group M (which is not torsion free) such that any finite non-trivial subgroup
of M is cyclic of order p.
References
[1] Higman, G., Scott, E. L. Existentially closed groups, Clarendon Press, 1988.
[2] Lichtman, A. I., Shirvani, M. HNN-extensions for Lie algebras, Proc. AMS, Vol.
125, No. 12, pp. 3501-3508, 1997.
[3] Lyndon, R. C., Schupp, P. E. Combinatorial group theory, Springer-Verlag,
2001.
[4] Myasnikov A. G., Remeslennikov V. N. Exponential groups I: fundations of the
theory and tensor completions, Siberian Math. J. Vol 35, No. 5, pp. 986-996,
1994.
[5] Myasnikov A. G., Remeslennikov V. N. Exponential groups II: extensions of
centralizers and tensor completion of CSA-groups, International J. Algebra and
Computation, Vol. 6, No. 6, pp. 687-711, 1996.
[6] Olshanskii, A. Y. Geometry of defining relations in groups, Kluwer Academic
Publishers, 1991.
[7] Osin, D. Small cancellations over relatively hyperbolic groups and embedding
theorems, Ann. of Math. (2), 172 (1), pp. 1-39, 2010.
[8] Scott, W. R. Algebraically closed groups, Proc. of AMS, No. 2, pp. 118-121,
1951.
[9] Shahryari, M. A note on derivations of Lie algebras, Bull. Aust. Math. Soc.
Vol. 84, pp. 444-446, 2011.
[10] Shahryari, M. Embeddings coming from algebraically closed groups, submitted.
[11] Wasserman, A. A derivation HNN construction for Lie algebras , Israil J. Math.
Vol. 106, pp. 76-92, 1998.
25
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
On the cover-avoiding properties in finite groups
Kar Ping Shum
The Chinese University of Hong Kong
.
Abstract
In this talk, I will talk about the process of the investigation
on cover-avoiding properties. Mainly talk about:
• Cover-avoiding properties and the structure of finite
groups
• Semi cover-avoiding properties and the structure of finite groups
• Further investigations
All groups mentioned here are finite.
26
Talks
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Some open problems in non-commuting graphs of
groups
Alireza Abdollahi
Department of Mathematics, University of Isfahan, Isfahan 81746-73441, Iran
and School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O.Box: 19395-5746, Tehran, Iran
[email protected]
Abstract
Let G be a non-abelian group. The non-commuting graph of
G, denoted by ΓG , is the graph whose vertex set is G and two
vertices are adjacent if they do not commute. In this talk,
we briefly review some open problems about non-commuting
graphs of finite groups.
1
Introduction
Let G be a non-abelian group. The non-commuting graph of G, denoted by ΓG ,
is the graph whose vertex set is G and two vertices are adjacent if they do not
commute. The non-commuting graph is studied in [2]. Here we briefly review some
open problems about non-commuting graphs of finite groups.
2
Order Conjecture
It is conjectured in [2] that if two finite non-abelian groups G and H have the same
non-commuting graph, then |G| = |H|. This conjecture is refuted by an example
due to I. M. Isaacs given in [4]. M. R. Darafsheh [4] has proved that the conjecture
is valid whenever one the groups G or H are simple. Abdollahi et al. [1] showed
the validity of the conjecture whenever one the groups G or H has prime power
2010 Mathematics Subject Classification. Primary 20D60; Secondary 20F99.
Key words and phrases. Non-commuting graph; Nilpotent group; Finite group.
28
order. In [3] it is proved that the conjecture holds if the non-commuting graphs of
the groups are irregular. Note that in the example given by Isaacs, groups have the
same regular non-commuting graph.
3
Nilpotent Conjecture
It is not known if G is a finite non-abelian nilpotent group such that ΓG ∼
= ΓH for
some H, then H is also nilpotent. In [2] it is noted that by using a result of [4] the
latter is true if |G| = |H|.
4
Nilpotency class conjecture
It is not known if two finite non-abelian nilpotent groups with same non-commuting
graphs have the same nilpotency class. The least example of two nilpotent nonisomorphic groups with the same non-commuting graphs are dihedral group of order
8 and the quaternion group of order 8, where these two latter groups have the same
nilpotency class 2. By examples given in [4] one can see that for every prime p ≥ 3
there are groups G1 and G2 of order p5 such that ΓG1 ∼
= ΓG2 such that the nilpotency
class of G1 is 2 and the nilpotency class of G2 is 3. Let us bring the details from [4].
Let G be a non-abelian finite p-group possessing an abelian maximal subgroup M
(necessarily normal of index p). Observe that, for x ∈ M \ Z(G), CG (x) = M
and if x ∈ G \ M , then CG (x) = Z(G)⟨x⟩. It follows that all proper centralizers
of G are abelian and their orders are |G| or |G|/p or p|Z(G)|. Therefore the noncommuting graph of G is the complete multipartite graph K|G|/p,p|Z(G)|,...,p|Z(G)| ,
|G|
. Hence, if G1 and G2
where the number of parts of order p|Z(G)| is equal to p|Z(G)|
are two non-abelian finite p-groups possessing abelian maximal subgroups such that
|G1 | = |G2 | and |Z(G1 )| = |Z(G2 )|, then ΓG1 ∼
= ΓG2 .
Back to the non-abelian p-group G possessing an abelian maximal subgroup M
and suppose further that the subgroup M is elementary abelian, so that we can
regard it as a vector space over Fp , the field with p elements. The element x induces
some linear transformation on this space, and if t is the number of blocks of the
Jordan form of this transformation, then |Z(G)| = pt . The class of G, on the other
hand, is the maximal size of these blocks. Now each Jordan block has size at most
p, and so if p = 2, the partition into blocks is uniquely determined by their number
and the Fp -dimension of M . However, if p ≥ 3 and dim(M ) = 4, it is easy to
construct examples of linear transformations x and x′ of M with the same number
of blocks but having different maximal sizes. (For instance, when dim(M ) = 4, the
linear transformation x could have two blocks of size 2 while x′ has a block of size 1
and a block of size 3.) Then the semidirect products G1 = M ⟨x⟩ and G2 = M ⟨x′ ⟩
29
have the same non-commuting graphs. In particular, there is a pair of such groups
of order p5 for any p ≥ 3.
5
Solvable conjecture
We do not know if G is a finite non-abelian solvable group such that ΓG ∼
= ΓH for
some H, then H is also solvable.
Acknowledgement
This research was in part supported by a grant from IPM (No. 92050219).
References
[1] A. Abdollahi, S. Akbari, H. Dorbidi and H. Shahverdi, Commutativity pattern
of finite non-abelian p-groups determine their orders, Comm. Algebra, 41 No. 2
(2013) 451-461.
[2] A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group,
J. Algebra, 298 (2006) 468-492.
[3] A. Abdollahi and H. Shahverdi, Non-commuting graphs of nilpotent groups, to
appear in Comm. Algebra.
[4] J. Cossey, T. Hawkes, A. Mann, A criterion for a group to be nilpotent, Bull.
London Math. Soc., 24 (1992) 327-332.
[5] M. R. Darafsheh, Groups with the same non-commuting graph, Discrete Applied
Math., 157 No. 4 (2009) 833-837.
[6] A. R. Moghaaddamfar, About noncommuting graphs, Siberian Math. J., 47 No.
5 (2006) 911-914.
30
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
The relative n−th nilpotency degree of two subgroups
of a finite group
Muhanizah Abdul Hamid1∗ , Nor Muhainiah Mohd Ali2 , Nor Haniza
Sarmin3 and Ahmad Erfanian4
1,2,3
4
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi
Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
[email protected] , [email protected] , [email protected]
Department of Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of
Mashhad, Mashhad, Iran
[email protected]
Abstract
The commutativity degree of a group is the probability that
two randomly chosen elements of G commute. The concept
of commutativity degree is then extended to the relative
commutativity degree of a group, which is defined as the
probability that two arbitrary elements one in H and another in G commute. Similarly, we can extend it to two
arbitrary elements one in H and another in K, where H
and K are two subgroups of G. In this research, the relative commutativity degree concept is further extended to
the relative n-th nilpotency degree of two subgroups of a
group G which is defined as the probability that the commutator of two arbitrary elements h ∈ H and k ∈ K belong
to Zn (G), where Zn (G) is the n−th central series of G. We
give some upper and lower bounds for the about probability
and compute it for some known groups.
∗
Speaker
2010 Mathematics Subject Classification. Primary 16U80; Secondary 20F99, 20D15.
Key words and phrases. Commutativity degree, relative commutativity degree, relative n-th
nilpotency degree of two subgroups.
31
1
Introduction
The theory of commutativity degree in group theory is one of the oldest areas in
group theory and plays a major role in determining the abelianness of the group.
It has been attracted by many researchers and it is studied in various directions.
Many papers give explicit formulas of the commutativity degree of G denoted by
P (G) for some particular finite groups G. The concept of commutativity degree
can be generalized and modified in many directions. For instance, commuting of an
element of a subgroup H with an element of G or even commuting of two elements
on in subgroup H and another in subgroup K of G ; or by changing the role of
commuting the first element in subgroup H to the n-th power of such element. One
different way of generalization of commutativity degree is to replace the notion of
commutativity by nilpotency class n. It is clear that if n = 0 then two concepts of
nilpotency class 0 and commutative are concide. So, we may associate parameter n
to this kind of generalization of commutativity degree. First, let us remind that if
G is a finite group, then the commutativity degree of G, denoted by P (G), is the
probability that two randomly chosen element of G commute. The first appearance
of this concept was in 1944 by Miller [5]. Then, the idea to compute P (G) for
symmetric groups has been introduced by Erdos and Turan [3] at the end of the 60s.
For any finite group G and its subgroup H, the relative commutativity degree
of G denoted by PG (H, G), is defined as the probability for an element of H and an
element of G commute. This concept was first introduced by Erfanian et al. in [3].
Similarly, if K is another subgroup of G then we may extend it as the probability
for an element of H commute to an element of K which is denoted by PG (H, K).
This probability is called the relative commutativity degree of two subgroups H and
K of a group G which can be written as
PG (H, K) =
|{(h, k) ∈ H × K|hk = kh}|
.
|H| |K|
In 2011, Erfanian et al. [4] defined the relative n-th commutativity degree denoted by Pn (H, G) as the probability that the n-th power of a random element of H
commutes with a random element of G. Now, we are going to give a generalization
of commutativity degree, relative commutativity degree of a subgroup and relative
commutativity degree of two subgroups in a different way as the following. For any
group G and two subgroups H and K of G define
Pnil(G) (n, H, K) =
|{(h, k) ∈ H × K|[h, k] ∈ Zn−1 (G)}|
,
|H| |K|
32
which is called the relative n-th nilpotency degree of two subgroups H and K in G.
If K = G then is denoted by Pnil(G) (n, H, G) and called the relative n-th nilpotency
degree of subgroup H in G and if H = K = G then is denoted by Pnil(G) (n, G) and
called the n-th nilpotency degree of G. It is clear that if n = 1 then Pnil(G) (1, G) =
P (G), Pnil(G) (1, H, G) = PG (H, G) and Pnil(G) (1, H, K) = PG (H, K). Morover,
Pnil(G) (n, G) = 1 if and only if G is nilpotent of class n. Also, if H ≤ Zn (G) then
Pnil(G) (n, H, G) = 1.
2
Main results
The following theorems give a lower and upper bound for the above probabilties.
Theorem 2.1. Let G be a finite group, H be a subgroup and p is the smallest prime
number dividing the order of G. Then for every n ≥ 1
|Zn (G)| p(|G| − |Zn (G)|)
|H| + |Zn (G)|
+
.
≤ Pnil(G) (n, G) ≤
2
|G|
2 |H|
|G|
We can also improve it for Pnil(G) (n, H, G) and Pnil(G) (n, H, K).
Theorem 2.2. If G is a finite group and H is a subgroup of G. Then for every
n≥1
|Zn (G) ∩ H| p(|H| − |Zn (G) ∩ H|)
|H| + |Zn (G) ∩ H|
+
≤ Pnil(G) (n, H, G) ≤
,
2
|G|
2 |H|
|G|
where p is the smallest prime number dividing the order of G.
The following theorem gives a comparison of the probability of G and
G
N.
Theorem 2.3. Let G be a finite group, H and N be subgroups of G such that
N ≤ H. If N is normal then for every n ≥ 1
Pnil(G) (n, H, G) ≤ Pnil( G ) (n,
N
H G
, )P
(n, N ).
N N nil(N )
Finally, we state some evaluations ofthe above probabilities for some small groups
and small values of n through the group theory package GAP.
33
Acknowledgement
The authors would like to thank Universiti Teknologi Malaysia (UTM) for the financial funding through the Research University Grant (RUG) Vote No. 04H13
and UTM Mobility Program. The first author would also like to thank Ministry of
Education (MOE) Malaysia for her MyPhD Scholarship.
References
[1] D. S. Dummit and R. M. Forte, Abstract Algebra, Third Edition, USA. John
Wiley and Sons, Inc., 2004.
[2] P. Erdos, and P. Turan, On some problems of a statistical group theory, IV, Acta
Math. Acad Sci. Hungaricae, 19 (1968), 413-435.
[3] A. Erfanian, B. Tolue and N. H. Sarmin, Some considerations on the n-th commutativity degrees of finite groups, Ars Combinatorial Journal. Press, 2011.
[4] A. Erfanian and B. Tolue, Relative non nill-n graphs of finite groups, ScienceAsia
38 (2012), no 1, 201 - 206.
[5] A. Erfanian, R. Rezaei and P. Lescot, On the relative commutativity degree of a
subgroup of a finite group, Communications in Algebra, 35 (2007), 4183-4197.
[6] G. Miller, A relative number of non-invariant operators in a group, Proc. Nat.
Acad. Sci. USA, 30 (1944), no. 2, 25-28.
34
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Generalize commutator on polygroups and hypergroups
Gholamhossien Aghabozorgi1∗ , Morteza Jafarpour1 and Bijan Davvaz2
1
Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran
[email protected], [email protected]
2
Department of Mathematics, Yazd University, Yazd, Iran
[email protected]
Abstract
The purpose of this paper is to provide a detailed structure
description of derived subpolygroups of polygroups. We introduce the concept of perfect and solvable polygroups and
we give some results in this respect.
1
Introduction
Hyperstructure theory was born in 1934 at the 8th congress of Scandinavian Mathematicions, where Marty introduced the hypergroup notion as a generalization of
groups and after, he proved its utility in solving some problems of groups, algebraic
functions and rational fractions. Surveys of the theory can be found in the books
of Corsini [3], Davvaz [4], Corsini and Leoreanu [4]. In the following we generalize
Commutator and defined derived subpolygroup. We recall here some basic notions
of hypergroup theory.
Let H be a non-empty set and P ∗ (H) be the set of all non-empty subsets of
H. Let · be a hyperoperation (or join operation) on H, that is, · is a function from
H × H into P ∗ (H). If (a, b) ∈ H × H, its image under · in P ∗ (H) is denoted by
a · b. The join operation is extended to subsets of H in a natural way, that is, for
non-empty subsets A, B of H, A · B = ∪{ab | a ∈ A, b ∈ B}. The notation a · A is
∗
Speaker
2010 Mathematics Subject Classification. 20N20.
Key words and phrases. Hypergroup, Polygroup, derived subpolygroup, solvable polygroup,
Commutator.
35
used for {a} · A and A · a for A · {a}. Generally, the singleton {a} is identified with
its member a. The structure (H, ·) is called a semihypergroup if a · (b · c) = (a · b) · c
for all a, b, c ∈ H, which means that
∪
∪
u·z =
x · v,
u∈x·y
v∈y·z
and is called a hypergroup if it is a semihypergroup and a·H = H·a = H for all a ∈ H.
A hypergroup P is called polygroup and is denoted by ⟨P, ·, e, −1⟩ if the following
conditions hold:
(1) P has a scalar identity e (i.e., e · x = x · e = x, for every x ∈ P );
(2) every element x of P has a unique inverse x−1 in P ;
(3) x ∈ y · z implies y ∈ x · z −1 and z ∈ y −1 · x.
A non-empty subset K of a polygroup ⟨P, ·, e, −1⟩ is a subpolygroup of P if x, y ∈ K
implies x · y ∈ K, and x ∈ K implies x−1 ∈ K. A subpolygroup N of a polygroup
⟨P, ·, e, −1⟩ is normal in P if x−1 · N · x ⊆ N , for all x ∈ P.
2
Derived subhypergroups
In this section, we introduce and analyze a new definition for derivative of a hypergroup H.
Definition 2.1. Let H be a hypergroup. We define
(1) [x, y]r = {h ∈ H |x · y ∩ y · x · h ̸= ∅} ;
(2) [x, y]l = {h ∈ H |x · y ∩ h · y · x ̸= ∅} ;
(3) [x, y] = [x, y]r ∪ [x, y]l .
From now on we call [x, y]r , [x, y]l and [x, y] right commutator x and y, left
commutator x and y and commutator x and y, respectively. Also, we will denote
[H, H]r , [H, H]l and [H, H] the set of all right commutators, left commutators and
commutators, respectively.
,
= [x, y]r = [x−1 , y −1 ]l = [y −1 , x−1 ]−1
Proposition 2.2. If H be a group, then [y, x]−1
r
l
for every x, y in H.
36
Example 2.3. Suppose that H = {e, a, b}. Consider the hypegroup (H, ·), where ·
is defined on H as follows:
·
e
a
b
e a, b e
e
a e
a
b
b e a, b a, b
It is easy to see that {a} = [a, a]r ̸= [a, a]l = {a, b} = [a−1 , a−1 ]l , where a−1 is
the inverse of a in H.
Proposition 2.4. If H is a commutative hypergroup, then [x, y]r = [x, y]l = [x, y],
for all (x, y) ∈ H 2 .
Let X be a nonempty subset of a polygroup ⟨P, ·, e, −1⟩. Let {Ai | i ∈ J} be
the family of all subpolygroups of P in which contain X. Then ∩i∈J Ai is called the
subpolygroup generated by X. This subpolygroup is denoted by < X > and we
have < X >= ∪{xε11 · . . . · xεkk | xi ∈ X, k ∈ N, εi ∈ {−1, 1}}. If X = {x1 , x2 , . . . , xn },
then the subpolygroup < X > is denoted < x1 , x2 , . . . , xn >. In a special case
< [P, P ]r >, < [P, P ]l > and < [P, P ] > are shown by Pr′ , Pl′ and P ′ , respectively.
Proposition 2.5. Let ⟨P, ·, e, −1⟩ be a polygroup (x, y) ∈ P 2 . Then,
(1) [x, y]r = [x−1 , y −1 ]l ;
(2) P ′ = Pr′ = Pl′ ;
(3) x ∈ P ′ ⇒ x−1 ∈ P ′ .
Corollary 2.6. If ⟨P, ·, e, −1⟩ is a polygroup, then P ′ is a subpolygroup of P.
From now on we call P ′ the derived subpolygroup of P.
Proposition 2.7. Let ⟨P, ·, e, −1⟩ be a polygroup. Then, P ′ = {e} if and only if P
be an abelian group.
Definition 2.8. A polygroup P is called perfect if and only if P ′ = P .
Definition 2.9. A polygroup P is called solvable if and only if P (n) = ωP , for some
n ∈ N, where P (1) = P ′ , P (n+1) = (P (n) )′ and ωP is heart of polygroup P .
Proposition 2.10. Every non-trivial perfect group is not solvable.
In the following, we show that the above proposition is not true for the class of
polygroups.
37
Example 2.11. Suppose that P = {e, a, b, c}. Consider the commutative polygroup
⟨P, ·, e, −1⟩, where · is defined on P as follows:
·
e
a
b
c
e
a
b
c
e
a
b
c
a
P
a, b, c a, b, c
b a, b, c
P
a, b, c
c a, b, c a, b, c
P
We can easily see that P is a perfect and solvable polygroup. Notice that P ′ = P =
ωP .
Example 2.12. Suppose that P = {e, a, b, c}. Consider the non-commutative polygroup ⟨P, ·, e, −1⟩, where · is defined on P as follows:
·
e
a
b
c
e
a
b
e
a
b
a
a
P
b e, a, b b
c a, c
c
c
c
c
b, c
P
In this case, we can see that P ′ = P = ωP .
Example 2.13. (Double coset algebra) Suppose that H is a subgroup of a group
G. Define a system
G//H = ⟨{HxH|x ∈ G}, ∗, H, −I⟩,
where (HxH)−I = Hx−1 H and (HxH) ∗ (HyH) = {HxhyH|h ∈ H}. The algebra
of double cosets G//H is a polygroup.
Theorem 2.14. Let (G, ·) be a group and H be a subgroup of G. We set HG′ H =
{HgH|g ∈ G′ }. Then,
(1) HG′ H ⊆ (G//H)′ ;
(2) If G′ · H = G then (G//H) is a perfect polygroup;
(3) If HG′ H = (G//H) then G′ · H = G.
References
[1] H. Aghabozorgi, B. Davvaz and M. Jafarpour, Solvable polygroups and derived
subpolygroups, Comm. Algebra, 41(8)(2013) 3098-3107.
38
[2] H. Aghabozorgi, B. Davvaz and M. Jafarpour, Nilpotent groups derived from
hypergroups. J. Algebra 382 (2013) 177-184.
[3] P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editore, Tricesimo, 1993.
[4] P. Corsini and V. Leoreanu, Applications of Hyperstructure Theory, Kluwer Academical Publications, Dordrecht, 2003.
[5] B. Davvaz, Polygroup Theory and Related Systems, World Scientific, 2013.
39
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Some solved and unsolved problems in loop theory
Karim Ahmadidelir
Department of Mathematics, College of Basic Sciences, Tabriz Branch, Islamic Azad
University, Tabriz, Iran
[email protected], k [email protected]
Abstract
In this talk, we consider some solved and unsolved problems
in the theory of loops and quasigroups and disscuss about
recent progresses and advances or improvements of them.
Some of them are long-standing and well-known problems
that newly have been solved and developed the theory and
some of them are the existing theorems in group theory that
have been generalized to some special kinds and classes of
loops. On the other hand, there have been achievements and
improvements in some of unsolved problems in recent years
that opened new horizons in the theory.
1
Introduction
A set Q with one binary operation is a quasigroup if the equation xy = z has a unique
solution in Q whenever two of the three elements x, y, z ∈ Q are specified. Loop is
a quasigroup with a neutral element 1 satisfying 1x = x1 = x for every x. Moufang
loops are loops in which any of the (equivalent) Moufang identities ((xy)x)z =
x(y(xz)), x(y(zy)) = ((xy)z)y, (xy)(zx) = x((yz)x), (xy)(zx) = (x(yz))x holds.
Moufang loops are certainly the most studied loops. They arise naturally in algebra (as the multiplicative loop of octonions), and in projective geometry (Moufang
planes), for example. Although Moufang loops are generally nonassociative, they
retain many properties of groups that we know and love. For instance: (i) every x
2010 Mathematics Subject Classification. 20N05.
Key words and phrases. Theory of loops and quasigroups, Moufang loops, Bol loops, Bruck
loops, A-loops.
40
is accompanied by its two-sided inverse x−1 such that xx−1 = x−1 x = 1, (ii) any
two elements generate a subgroup (this property is called diassociativity), (iii) in
finite Moufang loops, the order of an element divides the order of the loop, and it
has been shown recently in [4] that the order of a subloop divides the order of the
loop, (iv) every finite Moufang loop of odd order is solvable.
On the other hand, many essential tools of group theory are not available for
Moufang loops. The lack of associativity makes presentations very awkward and
hard to calculate, and permutation representations in the usual sense impossible.
The other most studied classes of loops are: Bol loops, Bruck loops, Osborn loops,
RS-loops, A-loops. In this talk, we consider some solved and unsolved problems
about these special classes of loops and disscuss about recent progresses and advances
or improvements of them.
2
Basic concepts
Let Q be a loop with neutral element 1. We define:
• Left multiplcation operator by:
Lx : Q →→ Q; y 7→ x · y;
• Right multiplcation operator by:
(Lx and Rx are bijections of Q)
Rx : Q →→ Q; y 7→ y · x,
• Commutator of x and y by: = [x, y] :
• Associator of x, y and z by: = [x, y, z] :
xy = (yx) · [x, y];
(xy)z = x(yz) · [x, y, z];
• Commutant of a subset S of Q by: {x ∈ Q | xs = sx, ∀s ∈ S};
• Center of Q by:
Z(Q) = {x ∈ Q | [x, y] = [x, y, z] = [y, x, z] = 1};
• Multiplication group of Q by:
M lt(Q) = ⟨Lx , Rx | x ∈ Q⟩;
• Inner mapping group of Q by:
Inn(Q) = {f ∈ M lt(Q) | f (1) = 1};
• A subloop S ≤ Q is normal if f (S) = S for every f ∈ Inn(Q);
• The nucleus Q by: {x | x(yz) = (xy)z, y(xz) = (yx)z, y(zx) = (yz)x, ∀y, z ∈
Q;
• A loop Q is solvable if there is a series 1 = Q0 E Q1 E · · · E Qm = Q such that
Qi+1
Qi is an abelian group for every i;
• p-loops by: Loops of order pk , p a prime;
41
• A finite Moufang loop Q is a p-loop if and only if every element of Q has order
that is a power of p;
Q
• A loop Q is (centrally) nilpotent if the sequence Q, Z(Q)
,
Q
Z(Q)
Q
Z( Z(Q)
)
, . . . eventually
yields the trivial loop;
• Let Q be a centrally nilpotent p-loop. The Frattini subloop ϕ(Q) of Q is the
intersection of all maximal subloops of Q;
• An isotopism of loops Q1 , Q2 is a triple (α, β, γ) of bijections Q1 → Q2 such
that α(x)β(y) = γ(xy) holds for every x, y ∈ Q1 ;
• G-loop: all isotopes of a loop Q are isomorphic to Q;
• Right Bol loops: the loops are given by the right Bol identity x((yz)y) =
((xy)z)y (Left Bol loops are defined analogously);
• Bruck loop or K−loop: a Bol loop satisfying the automorphic inverse property,
(ab)−1 = a−1 b−1 for all a, b in Q.
We refer the reader to [1] and [6] for a systematic introduction to the theory of
loops.
3
Some open problems about Moufang loops
Problem: Let p and q be distinct odd primes. If q is not congruent to 1 modulo p,
are all Moufang loops of order p2 q 3 groups? What about pq 4 ? (Proposed by Andrew
Rajah at Loops ’99, Prague 1999)
Comments: The former has been solved by Rajah and Chee (2011) where they
showed that for distinct odd primes p1 < · · · < pm < q < r1 < · · · < rn , all Moufang
loops of order p21 · · · p2m q 3 r12 · · · rn2 are groups if and only if q is not congruent to 1
modulo pi for each i.
Phillips’ problem: (Odd order Moufang loop with trivial nucleus) Is there a Moufang loop of odd order with trivial nucleus? (Proposed by Andrew Rajah at Loops
’03, Prague 2003)
Problem: (Presentations for finite simple Moufang loops) Find presentations for
all nonassociative finite simple Moufang loops in the variety of Moufang loops. (Proposed by P. Vojtěchovský at Loops ’03, Prague 2003)
Comments: It is shown in (Vojtěchovský, 2003) that every nonassociative finite
simple Moufang loop is generated by 3 elements, with explicit formulas for the
generators.
42
Conjecture: (The restricted Burnside problem for Moufang loops) Let M be a
finite Moufang loop of exponent n with m generators. Then there exists a function
f (n, m) such that |M | < f (n, m). (Proposed by Alexander Grishkov at Loops ’11,
Tret 2011)
Comments: In the case when n is a prime different from 3 the conjecture was
proved by Grishkov. If p = 3 and M is commutative, it was proved by Bruck. The
general case for p = 3 was proved by G. Nagy. The case n = pm holds by the
Grishkov-Zelmanov Theorem.
Conjecture: (The Sanov and M. Hall theorems for Moufang loops) Let L be a
finitely generated Moufang loop of exponent 4 or 6. Then L is finite. (Proposed by
Alexander Grishkov at Loops ’11, Tret 2011)
Conjecture: Let L be a finite Moufang loop and ϕ(L) Frattini subloop of L. Then
ϕ(L) is a normal nilpotent subloop of L. (Proposed by Alexander Grishkov at Loops
’11, Trest 2011)
Conjecture: (Torsion in free Moufang loops) Let M Fn be the free Moufang loop
with n generators. M F3 is torsion free but M Fn with n ≥ 4 is not. (Proposed by
Alexander Grishkov at Loops ’03, Prague 2003)
Problem: (Minimal presentations for loops M (G, 2)) Find a minimal presentation
for the Moufang loop M (G, 2) with respect to a presentation for G. (Proposed by P.
Vojtěchovský at Loops ’03, Prague 2003)
Comments: Chein showed in (Chein, 1974) that M (G, 2) is a Moufang loop that is
nonassociative if and only if G is nonabelian. Vojtěchovský (2003) found a minimal
presentation for M (G, 2) when G is a 2−generated group.
4
Some solved and Open problems about Bol loops
Remark 4.1. We have the following implications:
Right Bol loops ⇐= Right and left Bol loops = Moufang loops ⇐= groups.
Problem: (Existence of a finite simple Bol loop) Is there a finite simple Bol loop
that is not Moufang? (Proposed at: Loops ’99, Prague 1999, Solved by: Gbor P.
Nagy, 2007)
Solution: A simple Bol loop that is not Moufang will be called proper. There are
several families of proper simple Bol loops. A smallest proper simple Bol loop is of
order 24 [Nagy 2008]. There is also a proper simple Bol loop of exponent 2 [Nagy
2009], and a proper simple Bol loop of odd order [Nagy 2008].
Comments: The above constructions solved two additional open problems:
Problem: Is there a finite simple Bruck loop that is not Moufang? Yes, since any
proper simple Bol loop of exponent 2 is Bruck.
43
Problem: Is every Bol loop of odd order solvable? No, as witnessed by any proper
simple Bol loop of odd order.
Problem: (Left Bol loop with trivial right nucleus) Is there a finite non-Moufang
left Bol loop with trivial right nucleus? (Proposed at Milehigh conference on quasigroups, loops, and nonassociative systems, Denver 2005, Solved by: Gbor P. Nagy,
2007)
Solution: There is a finite simple left Bol loop of exponent 2 of order 96 with
trivial right nucleus. Also, using an exact factorization of the Mathieu group M24 ,
it is possible to construct a non-Moufang simple Bol loop which is a G−loop.
Problem: (Nilpotency degree of the left multiplication group of a left Bol loop)
For a left Bol loop Q, find some relation between the nilpotency degree of the left
multiplication group of Q and the structure of Q. (Proposed at Milehigh conference
on quasigroups, loops, and nonassociative systems, Denver 2005)
A loop is universally flexible if every one of its loop isotopes is flexible, that is,
satisfies (xy)x = x(yx). A loop is middle Bol if every one of its loop isotopes has
the antiautomorphic inverse property, that is, satisfies (xy)−1 = y −1 x−1 .
Problem: (Universally flexible loop that is not middle Bol) Is there a finite, universally flexible loop that is not middle Bol? (Proposed by Michael Kinyon at Loops
’03, Prague 2003)
Problem: (Finite simple Bol loop with nontrivial conjugacy classes) Is there a
finite simple nonassociative Bol loop with nontrivial conjugacy classes? (Proposed
by Kenneth W. Johnson and Jonathan D. H. Smith at the 2nd Mile High Conference
on Nonassociative Mathematics, Denver 2009)
5
Other problems
Much more problems about special kinds of loops have been solved or modified
recently and published in various journals those will be presented and discussed in
this talk, such as: a problem about Enumerating Nilpotent Loops up to Isotopy, by L.
Clavier (2012); a problem about differences and similarities between Bruck loops and
finite groups: Do finite Bruck loops behave like groups?, by B. Baumeister (2012);
a problem about On RS-loops by P. Sclifos (2011); a problem about Necessary
conditions for the existence of the finite Osborn loop with trivial nucleus by T.G.
Jaiyeola, J.O. Adeniran and A.R.T. Solari, (2011); a problem about Universality of
Osborn loops by M. Kinyon (2005); a problem about Associativity of automorphic
loops of order p2 by P. Csörgő (2013).
44
References
[1] R.H. Bruck, A survey of binary systems, Springer-Verlag, 1958.
[2] O. Chein, A. Rajah, Possible orders of nonassociative Moufang loops, Comment.
Math. Univ. Carolin., 41, (2000) 237-244.
[3] S. M. Gagola III, Halls theorem for Moufang loops, J. Algebra, 323, N12 (2010)
3252-3262.
[4] A.N. Grishkov, A.V. Zavarnitsine, Lagranges theorem for Moufang loops, Math.
Proc. Camb. Phil. Soc., 139, N1 (2005) 4157.
[5] A.N. Grishkov, A.V. Zavarnitsine, Sylows theorem for Moufang loops, J. Algebra,
321, No. 7, (2009) 18131825.
[6] H.O. Pflugfelder, Quasigroups and loops: Introduction, Heldermann Verlag,
Berlin, 1990.
45
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Inequality for the number of generators of the
c−nilpotent multiplier
Mahboubeh Alizadeh Sanati1 and Zahra Mahdipour2∗
1
Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan, Iran
[email protected]
2
Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan, Iran
[email protected]
Abstract
In this paper, we are going to prove that an inequality for
minimal number generators of c−nilpotent multiplier, to the
following form
(
) n2
d Nc M (G)) >
− n,
4
where, n is the minimal number of generators of G.
1
Introduction
I. Schur [5], in 1904, using projective representation theory of groups, introduced the
notion of multiplier of a finite group. It was known later that the Schur multiplier
had a relation with homology and cohomology of groups. In fact, if G is a finite
group, then
∼ H2 (G, C∗ ) and M (G) =
∼ H 2 (G, Z),
M (G) =
where M (G) is Schur multiplier of G, H2 (G, C∗ ) is the second cohomology of G with
coefficient in C∗ (for more details see [2]). Let
1 −→ R −→ F −→ G −→ 1 ,
∗
Speaker
2010 Mathematics Subject Classification. Primary 20E10; Secondary 20F18.
Key words and phrases. c−nilpotent multiplier, The number of generators of the c−nilpotent
multiplier.
46
be a free presentation for G, in which F is free group, Then
∩
M (G) ∼
= (R F ′ )/[R, F ],
this quotient group is independent from the choice of the free presentation of G.
Definition 1.1. Let F∞ be free group on a countably infinite set {x1 , x2 , . . .} and
V be an arbitrary subset of F . Suppose that v = xil11 . . . xilrr ∈ V where lj = ±1,
1 ≤ j ≤ r and y1 , . . . , yt be distinct elements in xl1 , · · · , xlr . Consider t arbitrary
distinct elements g1 , · · · , gt ∈ G. By uniformly placement gi instead of xij , obtaine
some element of G that is said to be the value of the word v at (g1 , . . . , gt ). The
subgroup of G generated by all values in G of words in V is called the verbal subgroup
of G determined by V (G),
V (G) = ⟨v (g1 , g2 , . . . , gn ) | gi ∈ G, 1 ≤ i ≤ n, n ∈ N, v ∈ V ⟩ .
Also the class of groups with respect to V , i.e V = {G|V (G) = 1}, is called the
variety, (see [4] for more details).
R. Baer [4], in 1945 generalized the notion of Schur multiplier to Baer-invariant
as follows
∩
R V (F )
,
VM (G) =
[RV ∗ F ]
where V (F ) is the verbal subgroup of F with respect to V, and
[RV ∗ F ] =< v(f1 , . . . , fi−1 , fi r, fi+1 , . . . , fn )v(f1 , . . . , fi , . . . , fn )−1 |
r ∈ R , fi ∈ F, v ∈ V , 1 ≤ i ≤ n , n ∈ N > .
It can be proved that the Baer-invariant of a group G is independent of the
choice of the presentation of G and it is always an abelian group [3]. In particular,
if V = [x, y] and
∩ V is the variety of abelian groups, Ab , then the Baer-invariant of
G will be (R F ′ )/[R, F ], the Schur-multiplier of G.
In special case, for any c ∈ N, when V = [x1 , · · · , xc+1 ], V is the variety of
nilpotent groups of the class at most c ≥ 1, Nc , then the Baer-invariant of G with
respect to [x1 , · · · , xc+1 ] which is called the c-nilpotent multiplier of G will be
∩
Nc M (G) = R γc+1 (F )/[R,c F ],
where γc+1 (F ) is the (c + 1)-st term of the lower central series of F and [R,1 F ] =
[R, F ], [R,c F ] = [[R,c−1 F ], F ], inductively.
47
2
Main results
Theorem 2.1. Suppose that G be a p-group and d, be the minimal number of
⟨
⟩
d2
generators of G and G = g1 , ..., gd . G is generated by k relations, thus k > .
4
R
, where R̃ is K[F ]- or K(G)-rightmodul. Also
R′ Rp
′
p
f
f
′
p
′
p
g
(rR R ) = r R R or (rR R ) = rf R′ Rp , for each g = Rf ∈ G and r ∈ R.
Let r′ is minimal number of generators of R̃. Thus k ≥ r′ .
Theorem 2.2. Let R̃ =
By theorems we can conclude that r′ >
d2
. ([1])
4
Proposition 2.3. (I. Schur [5]) Let G be a group, F be a free group of rank n and
G ≃ F/R. Also, the following relations hold
D = F/[R,c F ]
and
H = R/[R,c F ].
Then H = S × Nc M (G), such that D/S is V-covering of G, where V is the variety
of nilpotent groups of the class at most c ≥ 1.
Proof. H = S × Nc M (G). Set K = D/S and T = H/S. So
(i) T ≤ Zn (K) because
[R, F ][R,c F ]
F
⊆
≃ G,
[R,c F ]
R
also
T = H/S = Nc M (G)S/S ≤ γc+1 (D)S/S = γc+1 (K),
thus T ≤ γc+1 (K) ∩ Zn (K).
(ii) K/T ≃ D/H ≃ G.
(iii) T ≃ Nc M (G).
Hence K is V-covering of G, where V is the variety of nilpotent groups of the
class at most c ≥ 1.
Theorem 2.4. Let G be a finite group. Then
(
) d(G)2
d Nc M (G) >
− d(G).
4
48
Proof. Suppose that G be a group, generated by elements g1 , · · · , gn and relations
r1 , · · · , rk . If F be free group by free generators f1 , · · · , fn , then G ≃ F/R, where
R
] is generated by elements
R = ⟨rif i = 1, · · · , k, f ∈ F ⟩. Abelian group H = [
R,c F
[
]
ri R,c F , (i = 1, · · · , k).
By proposition 2.3, we can write H = Nc M (G) × S, where S is free abelian group
of rank n. Assume that d, be minimal number generators. Hence d + n ≤ k or
d ≤ k − n, therefore n ≤ k. Now, consider G be a p−group and n = d(G) be
minimal number generators of G. So Nc M (G) is a p−group, also and we have
Rp
[
H
R
N M (G)
S
]≃ p ≃( c
)p × p ,
H
S
R,c F
Nc M (G)
Thus, for number generators, we can write
(
)
(
)
Nc M (G)
S
R
)p ) + d( p ) = d Nc M (G) + n.
= d( (
d p
R [R,c F ]
S
Nc M (G)
R
as G-module. So by theorem
Rp R′
(
)
n2
R
] and so
2.1 and by theorem 2.2 we have r′ >
. Also, we have r′ ≤ d p [
4
R R,c F
we prove as follows for n ≥ 4
We consider r′ , be minimal number generators
(
)
n2
− n ≥ 0.
d Nc M (G)) ≥ r′ − n >
4
References
[1] B. Huppert, Endliche Gruppen I. Springer-Verlag, Berlin-Heidelberg-New York.
1967.
[2] G. Karpilovsky, The Schur Multiplier, London Math. Soc. Monographs, New
Series no. 2, 1987.
[3] C. R. Leedham-Green and S. McKay, Baer-invariant, isologism, varietal laws
and homology. Acta Math. 137 (1976), 99 -150.
[4] H. Neumann, Varieties of Groups, Springer-Verlag, Berlin, Heidelberg, New
York, 1967.
[5] I. Schur, Über die darstellung der endlichen gruppen durch gebrochene lineare
substitutionen, J. Für Math. 127 (1904), 20-50.
49
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Characterization of 2 Dn (2) by the set of orders of
maximal abelian subgroups
Bahareh Asadian1∗ and Neda Ahanjideh2
Department of Mathematics, Faculty of Mathematical Sciences, Shahrekord University,
Shahrekord, Iran
1
[email protected]
2 [email protected]
Abstract
Let G be a finite group and M (G) be the set of orders of
maximal abelian subgroups of G. Let n ≥ 15. In this talk,
we prove that if M (G) = M (2 Dn (2)), then G ∼
= 2 Dn (2).
1
Introduction
Throughout this paper, p is a prime number, q = pk and G is a finite group. We
denote by π(G) the set of prime divisors of the order of G. The prime graph
GK(G) of G is the graph with vertex set π(G), where two distinct primes r and s
are joined by an edge (we write (r, s) ∈ GK(G)) if G contains an element of order
rs. An independent set in a graph Γ is a set of pairwise non-adjacent vertices.
ρ(G) (ρ(r, G)) denotes the independent set in GK(G) (containing a prime r) with
a maximal number of vertices. Let t(G) = |ρ(G)| and t(r, G) = |ρ(r, G)|. We use
a | n when a is a divisor of n and use |n|a = ae , when ae || n, i.e., ae | n but
ae+1 - n. If a is a natural number, r is an odd prime number and gcd(r, a) = 1,
then by e(r, a) we denote the minimal natural number n with an ≡ 1 (mod r).
The prime r with e(r, a) = m is called a primitive prime divisor of am − 1. We
denote by rm (a) some primitive prime divisor of am − 1. If a is odd, then let
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D06; Secondary 20D20.
Key words and phrases. Maximal abelian subgroup, prime graph, maximal independent set.
50
e(2, a) = 1 if a ≡ 1 (mod 4) and let e(2, a) = 2 if a ≡ −1 (mod 4). By Fermat’s
little theorem, e(r, a) | r − 1. Also, if an ≡ 1 (mod r), then e(r, a) | n. Put M (G) =
{|H| : H is a maximal abelian subgroup of G}. By a(G) and ar (G), we denote the
maximum number in M (G) and the order of the largest abelian subgroup of r-Sylow
subgroup of G, respectively. A finite group G is called characterizable by the set of
orders of maximal abelian subgroups, if each finite group H with M (G) = M (H) is
isomorphic to G. For instance, alternating group An , where n and n − 2 are primes
or n ≤ 10, is characterizable by the set of orders of its maximal abelian subgroups
(see [6]). In this paper, we prove that:
Main Theorem. Let G be a finite group and let n ≥ 15. If M (G) = M (2 Dn (2)),
then G ∼
= 2 Dn (2).
2
Main results
In the following, we have brought some useful lemmas which will be used during the
proof of the main theorem:
Lemma 2.1. [6] Let G and K be two finite groups such that M (G) = M (K). Then
the prime graph of G and the prime graph of K are same.
Lemma 2.2. [3, Theorem 1] Let G be a finite group with t(G) ≥ 3 and t(2, G) ≥ 2.
Then the following hold:
1. There exists a finite non-abelian simple group S such that S ≤ Ḡ = G/K ≤
Aut(S) for the maximal normal soluble subgroup K of G.
2. For every independent subset ρ of π(G) with |ρ| ≥ 3 at most one prime in ρ
divides the product |K|.|Ḡ/S|. In particular, t(S) ≥ t(G) − 1.
3. One of the following holds:
(a) every prime r ∈ π(G) non-adjacent to 2 in GK(G) does not divide the
product |K|.|Ḡ/S|; in particular, t(2, S) ≥ t(2, G);
(b) there exists a prime r ∈ π(K) non-adjacent to 2 in GK(G); in which case
t(G) = 3, t(2, G) = 2, and S ∼
= A7 or A1 (q) for some odd q.
For a finite integer n, we define the following function which will be used in some
following lemmas:
{
n if n is odd;
η(n) =
n
2 otherwise.
51
Lemma 2.3. [4] Let G = Dnϵ (q) be a finite simple group of Lie type over a field
of characteristic p. Suppose r and s are odd prime and r, s ∈ π(Dnϵ (q))\{p}. Put
k = e(r, q), l = e(s, q), and 1 ≤ η(k) ≤ η(l). Then r and s are non-adjacent if and
only if 2.η(k) + 2.η(l) > 2n − (1 − ϵ(−1)k+1 ) and l/k is not an odd integer. Moreover,
if ϵ = +, the chain of equalities n = l = 2η(l) = 2η(k) = 2k is not true as well.
Lemma 2.4. [4, Proposition 3.1] Let G = Dnϵ (q) , r ∈ π(G) and r ̸= p. Then
(r, p) ̸∈ GK(G) if and only if η(e(r, q)) > n − 2.
Proof of the main theorem. According to our assumption, M (G) = M (2 Dn (2))
and hence, Lemma 2.1 shows that GK(G) = GK(2 Dn (2)). Thus {2, r2n } ⊂ ρ(2,2 Dn (2)) =
ρ(2, G), considering Lemma 2.4. Since t(G) ≥ 3 and t(2, G) = t(2,2 Dn (2)) ≥ 2,
Lemma 2.2(1) guarantees the existence of maximal solvable subgroup K of G and
a finite non-abelian simple group S such that S ≤ G = G/K ≤ Aut(S) and
t(S) ≥ t(G) − 1. In [2], the authors show that S ∼
= 2 Dn (2). In the following,
we are going to show that K = 1. Let
ρ = {r2n (2), r2(n−1) (2), r2(n−2) (2)}.
By Lemma 2.3, ρ is an independent set and hence, by Lemma 2.2(2), there exists
t ∈ {r2(n−1) (2), r2(n−2) (2)} ∩ π(2 Dn (2))
such that t ̸∈ π(K). Let S1 ∈ Sylt (2 Dn (2)). Since r2n (2) ∈ ρ(2,2 Dn (2)), Lemma
2.2(3) implies that r2n (2) ∈ π(2 Dn (2)) and r2n (2) ̸∈ π(K). Let S2 ∈ Sylr2n (2) (2 Dn (2)).
Obviously, S1 and S2 act co-primely on K. We claim that |K|2 = 1. If not, then there
exist S1 and S2 -invariant 2-Sylow subgroups P1 and P2 of K. Therefore, Z(P1 )S1
and Z(P2 )S2 are subgroups of G. Applying Lemma 2.4 shows that (2, r2n (2)) ̸∈
GK(G) = GK(2 Dn (2)) and hence, r2n (2) | 2m − 1, where |Z(P1 )| = |Z(P2 )| = 2m .
So, 2n | m and hence, there exists a natural number l such that m = 2nl. If
|CZ(P2 ) (S1 )| = 2e , then there is α ∈ M (G) = M (2 Dn (2)) such that t2e | α. Considering the elements of M (2 Dn (2)) shows that if t = r2(n−1) (2), then 2e = 1 and if t =
r2(n−2) (2), then 2e ≤ 22 . Thus 2(n−u) | 2nl−e = 2(n−u)l+2ul−e, where u ∈ {1, 2}.
This forces 2(n − u) < 2ul and hence, m ≥ n(n − 2)/2. Therefore, 2n(n−2)/2 ≤ 2m .
Since Z(p2 ) is an abelian subgroup of G, |Z(P2 )| = pm ≤ a(G) = a(2 Dn (2)). But [5,
(n−1)(n−2)
+2
2
, so (n−1)(n−2)
Table 2] shows that a(2 Dn (2)) = 2
+ 2 ≥ n(n − 2)/2, which
2
is a contradiction. We claim that |K| = 1. If not, then there is r ∈ π(K) − {2}.
Since M (G) = M (2 Dn (2)), a(2 Dn (2)) = a(G) is a power of 2. Thus there exists
an abelian 2-subgroup P of G such that |P | = a(G). We proved |K|2 = 1, so P
acts co-primely on K and hence, we can see that K has a P -invariant r-Sylow subgroup R. Let |Z(R)| = rβ . Without loss of generality, we can assume that Z(R)
52
is an r-elementary abelian group. Also, P is abelian with |P | = a(G) and hence,
CP Z(R) (Z(R)) is abelian. Therefore, |CP Z(R) (Z(R))| = rβ .pγ < |P |. Also,
NP Z(R) (Z(R))
≤ Aut(Z(R)) = GLβ (r).
CP Z(R) (Z(R))
But considering the 2-Sylow subgroups of GLβ (r) leads us to see that a2 (GLβ (r)) <
|
β
rβ , so |P
pγ < r , which is a contradiction. This forces |K| = 1, as desired. This
−
implies that 2 Dn (2) ≤ G ≤ Aut(2 Dn (2)). But Aut(2 Dn (2)) ∼
(2). Thus
= P SO2n
−
−
2
∼
∼
either G ∼
P
SO
(2)
or
G
D
(2).
If
G
P
SO
(2),
then
we
can
see that
=
=
=
n
2n
2n
n−1
2
2
2(2
+ 1) ∈ M (G) = M ( Dn (2)). This forces (2, r2(n−1) (2)) ∈ GK( Dn (2)),
which is a contradiction with Lemma 2.4. Thus G ∼
= 2 Dn (2), so theorem follows.
References
[1] G.Y. Chen, A characterization of alternating groups by the set of orders of their
maximal abelian subgroups , Siberian Math. J. 47 (2006), no. 3, 300–306.
[2] B. Khosravi, H. Moradi, Quasirecognition by prime graph of some orthogonal
groups over the binary fields, J. Algebra Appl. DOI: 10.1142/S0219498812500569.
[3] A.V. Vasil, ev, I.B. Gorshkov, On recognition of simple groups with connected
prime graph, Siberian Math. J. 50 (2009), no. 2, 233–238.
[4] A.V. Vasil, ev, E.P. Vdovin, An adjacency criterian for the prime graph of a finite
simple group, Algebra and Logic. 44 (2005), no. 6, 381–406.
[5] E.P. Vdovin, Maximal orders of abelian subgroups in finite Chevalley groups,
Mathematicheskie zametki. 69 (2001), no. 4, 524–549.
53
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
A characterization of Sz(8) by nse
Soleyman Asgary1∗ and Neda Ahanjideh2
Department of pure Mathematics, Faculty of Mathematical Sciences, Shahrekord
University, Shahrekord, Iran
1
[email protected], 2 [email protected]
Abstract
Let G be a group and πe (G) be the set of element orders of G.
Suppose that k ∈ πe (G) and mk is the number of elements of
order k in G. Set nse(G) := {mk : k ∈ πe (G)}. In this paper,
we prove that if G is a group with nse(Sz(8)) = nse(G), then
G∼
= Sz(8).
1
Introduction
Denote by π(G) the set of prime divisors of the order of G and the set of element
orders of G is denoted by πe (G). A finite group G is called a simple Kn -group, if G is
a simple group with |π(G)| = n. For a group G and i ∈ πe (G), set mi (G) = |{g ∈ G :
the order of g is i}|. In fact, mi (G) is the number of elements of order i in G and
nse(G) := {mi (G) : i ∈ πe (G)}. For a finite group H and p ∈ π(H), let np (H) denote
the number of Sylow p-subgroups of H. We say that the group G is characterizable
by the set of nse if every group H with nse(G) = nse(H) is isomorphic to G. In [4],
it has been shown that the finite simple group A5 is characterizable by its nse. In
this paper, we show that the finite simple group Sz(8), which is a simple K4 -group
of order 26 .5.7.13 is characterizable by nse. In fact, the main result of this paper is
the following theorem:
Main theorem. If G is a group such that nse(G) = nse(Sz(8)), then G ∼
= Sz(8).
Lemma 1.1. [1] Let G be a finite group and m be a positive integer dividing |G|.
If Lm (G) = {g ∈ G | g m = 1}, then m | |Lm (G)|.
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D05; Secondary 20D06, 20D20.
Key words and phrases. Set of the numbers of elements of the same order, element order, simple
Kn -groups.
54
Lemma 1.2. [4] Let G be a group containing more than two elements. If the
maximal number s of elements of the same order in G is finite, then G is finite and
|G| ≤ s(s2 − 1).
Lemma 1.3. [3] Let G be a finite solvable group and |G| = mn, where m = pα1 1 ...pαr r
and (m, n) = 1. Let π = {p1 , ..., pr } and hm be the number of Hall π-subgroups of
G. Then hm = q1β1 ...qsβs satisfies the following conditions for all i ∈ {1, 2, ...s}:
(1) qiβi ≡ 1 (mod pj ) for some pj ;
(2) the order of some chief factor of G is divided by qiβi .
Lemma 1.4. [5] Let G be a simple K4 -group. Then G is isomorphic to one of the
following groups:
(1) A7 , A8 , A9 , A10 ;
(2) M11 , M12 , J2 ;
(3) one of the following simple groups:
(a) P SL(2, r), where r is a prime and satisfies r2 −1 = 2a .3b .v c with a, b, c ≥ 1
and v > 3 is a prime;
(b) P SL(2, 2m ), where m ≥ 2 satisfies 2m − 1 = u and 2m + 1 = 3tb , with u, t
are primes, t > 3 and b ≥ 1;
(c) P SL(2, 3m ), where m ≥ 2 satisfies either 3m + 1 = 4t and 3m − 1 = 2uc
or 3m + 1 = 4tb and 3m − 1 = 2u, with u, t are odd primes and b, c ≥ 1;
(4) one of the following 28 simple groups:
P SL(2, 16), P SL(2, 25), P SL(2, 49), P SL(2, 81), P SL(3, 4), P SL(3, 5), P SL(3, 7),
P SL(3, 8), P SL(3, 17), P SL(4, 3), P Sp(4, 4), P Sp(4, 5), P Sp(4, 7), P Sp(4, 9),
P Sp(6, 2), O8+ (2), G2 (3), P SU (3, 4), P SU (3, 5), P SU (3, 7), P SU (3, 8), P SU (3, 9),
P SU (4, 3), P SU (5, 2), Sz(8), Sz(32), 3 D4 (2), 2 F4 (2)′ .
Lemma 1.5. Let G be a finite simple K4 -group such that |G| | 26 .5.7.13, then
G∼
= Sz(8).
Proof. The proof is easy by Lemma 1.7.
Remark 1.6. Let G be a group with nse(G) = nse(Sz(8)). By Lemma 2.4, we can
see that G is finite. For n ∈ πe (G), it is known that mn = kϕ(n), where k is the
number of cyclic subgroups of order n in G and if n > 2, then ϕ(n) is even, so mn
55
is even. Thus if mn ∈ nse(G) is odd, then n = 2. If n ∈ πe (G), then by Lemma 1.10
and the above notation, we have:


 ϕ(n) | mn
∑
(1)
.

n
|
md

d|n
2
Main results
Theorem 2.1. If G is a group such that nse(G) = nse(Sz(8)), then G ∼
= Sz(8).
Proof. By Remark 1.23, G is a finite group and by [2], we can see that
nse(G) = nse(Sz(8)) = {1, 455, 3640, 5824, 12480, 6720}.
(2)
We are going to prove that π(G) ⊆ {2, 5, 7, 13}. Since 455 ∈ nse(G), by Remark
1.23, 2 ∈ π(G) and m2 = 455. Furthermore, if n ∈ πe (G) ∩ {4, 5, 7, 13}, then we
obtain
(n, mn (G)) ∈ {(4, 3640), (5, 5824), (7, 12480), (13, 6720)},
by (1). Let 2 ̸= p ∈ π(G). Then by (1), p | (1 + mp ) and (p − 1) | mp , so checking
the elements of nse(G) in (2) implies that p ∈ {5, 7, 11, 13, 3641, 6721, 12481}. If
11 ∈ π(G), then by (1), m11 ∈ {3640, 6720}. If 11r ∈ πe (G), then ϕ(11r ) = 10.11r−1 |
m11r . So we can see that r = 1 and hence, exp(P11 ) = 11, which shows that
|P11 ||(1 + m11 ). Therefore |P11 | = 11. Let m11 = 3640. It is easy to see that
2.11 ̸∈ πe (G). This shows that the Sylow 11-subgroup P11 of G acts fixed point
freely on the set of the elements of order 2, so |P11 | | m2 = 455, which is impossible.
Thus 11 ̸∈ π(G). If m11 = 6720, then similar to the above we can get a contradiction.
Therefore 11 ̸∈ π(G). Similarly we can see that 3641, 6721, 12481 ̸∈ πe (G).
1) If 2i ∈ πe (G), then ϕ(2i ) = 2i−1 | m2i and hence, checking the elements of (2)
shows that 1 ≤ i ≤ 6. Since 2 ∈ π(G) and exp(P2 ) = 2i = 2, 4, 8, 16, 32, 64, Lemma
1.10 leads us to see that |P2 ||212 .
2) If 5j ∈ πe (G), then ϕ(5j ) = 4.5j−1 | m5j and hence, checking the elements of
nse(G) in (2) shows that j = 1 and m5 = 5824. So |P5 ||(1 + m5 ) and hence, |P5 | | 52 .
3) If 7k , 13l ∈ πe (G), then similar to the above, we can see that l = k = 1 and hence,
|P7 | = 7 and |P13 | = 13.
In the following cases, we are going to prove that π(G) = {2, 5, 7, 13}.
Case a. Let π(G) = {2}. Since 27 ̸∈ πe (G), πe (G) = {1, 2, ..., 26 }. On the other
hand, |G| = 2m = 29120+3640k1 +5824k2 +12480k3 +6720k4 , where
∑ k1 , k 2 , k 3 , k 4 , m
are nonnegative integer numbers. Hence, |πe (G)| = 7 and 0 ≤ 4i=1 ki ≤ 1. But
this equation has no solution. Therefore this case is impossible.
56
Case b. Let π(G) = {2, 5}. By (2), |P5 | | 52 and exp(P5 ) = 5. If |P5 | = 5, then
n5 (G) = m5 (G)/ϕ(5) = 5824/4 = 1456. Thus 7 ∈ π(G), which is a contradiction. If
|P5 | = 25, then |G| = 2m .52 = 29120 + 3640k1 + 5824k2 + 12480k3 + 6720k4 , where
k1 , k2 , k3 , k4 , m are nonnegative integer numbers. Using a computer calculation we
can see that the equation has no solution in N.
If π(G) = {2, 7}, {2, 13}, {5, 7}, {5, 13}, {7, 13}, then applying the argument given
for Case b leads us to get a contradiction.
Case c. Let π(G) = {2, 5, 7}. By (3), |P7 | = 7 and so n7 (G) = m7 (G)/ϕ(7) =
12480/6 = 2080. Thus 13 ∈ π(G), which is a contradiction. If π(G) = {2, 5, 13}
or {2, 7, 13}, then similar to the above, we can get a contradiction. Thus π(G) =
{2, 5, 7, 13}.
By (3), we know that |P7 | = 7. We can see that 5.7, 5.13 ̸∈ πe (G). Therefore
|P5 | = 5. We can see that 2.7 ̸∈ πe (G), so |P2 | | m7 and hence, |P2 | | 26 . Also
the Sylow 7-subgroup of G is cyclic, so n7 = m7 /ϕ(7) = 12480/6 = 2080 and
hence, 2080 | |G|. This forces 25 × 5 × 13 | |G|. On the other hand, 7 | |G| and
hence, 25 .5.7.13 | |G| and |G| | 26 .5.7.13. Therefore, |G| = 26 .5.7.13 or |G| =
25 .5.7.13,
∑ considering the order of the Sylow subgroups of G obtained above. Since
|G| > m∈nse(G) m, we deduce by (2) that |G| > 14560 = 25 .5.7.13. Thus |G| =
26 .5.7.13 = |Sz(8)|. By Lemma 1.12, it is easy to see that G is a non-solvable group.
Therefore, G has a normal series 1 E N E M E G such that N is a maximal solvable
normal subgroup of G. Considering the order of G shows that M/N is not simple
K3 -group. Thus M/N is a simple K4 -group satisfying Lemma 1.16. Thus by Lemma
1.16, M/N ∼
= Sz(8). Let A/N := CG/N (M/N ). We can see that G/A ∼
= Sz(8) or
∼
G/A = Aut(Sz(8)). But 3 - |G|, so G/A ∼
= Sz(8). On the other hand, |G| = |Sz(8)|,
so A = 1 and G ∼
= Sz(8), as desired. References
[1] G. Frobenius, Verallgemeinerung des Sylowschen Satze, Berliner Sitz,
(1985) 981-993.
[2] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson,
Atlas of finite groups, Clarendon Press, New York, 1985.
[3] M. Hall, The theory of groups, Macmilan, 1959.
[4] R. Shen, C.G. Shao, Q.H. Jiang, W.J. Shi and V. Mazurov, A new characterization of A5 , Monatsh Math. 160 (2010), 337-341.
[5] W.J. Shi, On simple K4 -group, Chin. Sci. Bul. 36 (1991), 1281-1283.
57
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Symmetry classes of polynomials with respect to
product of groups
Esmaeil Babaei1∗ and Yousef Zamani1
1
Faculty of Sciences, Sahand University of Technology, Tabriz, Iran
e− [email protected], [email protected]
Abstract
Let Gi be subgroups of Smi , 1 ≤ i ≤ k respectively. We
consider G = G1 × · · · × Gk as subgroup of Sm , where m =
m1 + · · · + mk . In this paper, we give a formula for the
dimension of Symmetry classes of polynomials with respect
to G and its irreducible character, then invastagete exsitence
of an o-basis of these classes.
1
Introduction
The relative symmetric polynomials as a generalization of symmetric polynomials
are introduced in [1]. In [3, 4], the authors studied the space of relative symmetric
polynomials (symmetry class of polynomials) with respect to the irreducible characters of the some groups. We first give a review of this notion (for more details, see
[1]).
Let Hd [x1 , . . . , xm ] be the complex space of homogeneous polynomials of degree d
with independent commuting variables x1 , . . . , xm . Let Γ+
m,d
∑be the set of all mtuples of non-negative integers α = (α1 , . . . , αm ), such that m
i=1 αi = d. For any
α1 α2
+
α
α
m
α ∈ Γm,d , let X be the monomial x1 x2 . . . xm . Then the set {X α | α ∈ Γ+
m,d } is
a basis of Hd [x1 , . . . , xm ]. An inner product on Hd [x1 , . . . , xm ] is defined by
⟨X α , X β ⟩ = δα,β .
∗
Speaker
2010 Mathematics Subject Classification. Primary 05E05; Secondary 15A69.
Key words and phrases. Symmetry class of polynomials, permutaion groups, Irreducible characters.
58
Suppose G is a subgroup of the full symmetric group Sm . G acts on Hd [x1 , . . . , xm ]
by
(σq)(x1 , . . . , xm ) = q(xσ−1 (1) , . . . , xσ−1 (m) ),
and this action is extended linearly to the group algebra CG. Let χ be an irreducible
complex character of G. Consider the idempotent operator
T (G, χ) =
χ(1) ∑
χ(σ)σ,
|G|
σ∈G
in the group algebra CG. The image of Hd [x1 , . . . , xm ] under the map T (G, χ) is
called the symmetry class of polynomials of degree d with respect to G and χ and it is
denoted by Hd (G, χ). If G = Sm and χ = 1Sm , where 1Sm is the principal character
of Sm , we obtain the space of ordinary symmetric homogeneous polynomials of
degree d, and if χ = ε, where ε is alternating character of Sm , we obtain the space of
anti-symmetric homogeneous polynomials of degree d. For any q ∈ Hd [x1 , . . . , xm ],
q ∗ = T (G, χ)(q)
is called a symmetrized polynomial with respect to G and χ. For α ∈ Γ+
m,d , we denote
α
∗
α,∗
the symmetrized monomial (X ) by X . So
Hd (G, χ) = ⟨X α,∗ | α ∈ Γ+
m,d ⟩.
The group G also acts on Γ+
m,d by
ασ = (ασ(1) , . . . , ασ(m) ).
Let ∆+ be a set of representatives of orbits of Γ+
m,d under the action G.
+
For any α ∈ Γm,d , we have
∥X α,∗ ∥2 = χ(1)
[χ, 1]Gα
,
|G : Gα |
(1)
where Gα is the stabilizer subgroup of α under the action of G. Hence, X α,∗ ̸= 0 if
and only if [χ, 1]Gα ̸= 0.
Let Ω+ be the set of all α ∈ Γ+
m,d with [χ, 1]Gα ̸= 0 and suppose
+
+
+
¯
∆ = ∆ ∩ Ω . We have
∑
dim Hd (G, χ) = χ(1)
[χ, 1]Gα .
(2)
¯+
α∈∆
An orthogonal basis of Hd (G, χ) of the form {X α,∗ | α ∈ S}, where S is a subset of
α,∗ | α ∈ ∆
¯ + } is
Γ+
m,d is called an o-basis of Hd (G, χ). If χ is linear, then the set {X
59
an o-basis of Hd (G, χ). If χ is not linear, then Hd (G, χ) may have no o-basis.
In this paper, we consider G = G1 × · · · × Gk , where Gi are permutation geroups and
give a formula for the dimension of Hd (G, χ) so invastagete exsitence of an o-basis
of these classes. A similar result has been obtained for symmetry classes of tensors
in [2].
2
Main results
By a partition of m we mean a non-increasing finite sequence of positive integers
whose sum is m. If π is a partition of m we denote π ⊢ m. We know that there is a
”standard” one-to- one correspondence between the partitions of m and the complex
irreducible characters of Sm . If π is a partition of m, we denote the corresponding
irreducible character by χπ , so the set of all irreducible complex characters of Sm
is Irr(Sm ) = {χπ |π ⊢ m}. A partition π = (π1 , . . . , πk ) is usually represented by
a collection of m boxes arranged in k rows such that the number of boxes of row
i is equal to πi , i = 1, . . . , k. This collection is called Young diagram (or Ferrers
diagram) associated with π and denoted by [π]. A Young tableau of shape π or
π-tableau, tπ , is an array t obtained by replacing the boxes of the Ferrers diagram
of π with the numbers 1, . . . , m, bijectively. For any 1 ≤ i ≤ k, suppose
Λi = {t :
i−1
∑
πj < t ≤
j=1
i
∑
πj }.
j=1
The corresponding Young subgroup is defined by
Sπ = SΛ1 × · · · × SΛk ,
where SΛi is the symmetric group on Λi .
In general, Sπ and Sπ1 × · · · × Sπk are isomorphic.
Now let Gi be subgroup of Sπi , 1 ≤ i ≤ k respectively, and suppose m = π1 +· · ·+πk ,
without any losses, we can consider π = (π1 , · · · , πk ) ⊢ m. Let G = G1 × · · · × Gk ,
so G ≤ Sπ , specially we can consider G as a subgroup
of full symmetric group Sm .
∏
If χ is an irreducible character of G, then χ = ki=1 χi , where χi is an irreducible
character of Gi , 1 ≤ i ≤ k.
∑
i
Let Γ+
j∈Λi αj =
Λi ,ρi be the set of all πi -tuples, α , of non-negative integers such that
+
+
ρi , and suppose ∆Λi ,ρi is a set of representatives of orbits of ΓΛi ,ρi under the action
+
i
Gi . We denote Ω+
Λi ,ρi by the set of all α ∈ ΓΛi ,ρi with [χ, 1](Gi )αi ̸= 0 and suppose
¯ + = Ω+ ∩ ∆+ . Summing up, we have the following result.
∆
Λi ,ρi
Λi ,ρi
Λi ,ρi
60
Theorem 2.1. We have,
⊎
¯+ =
∆
k
∏
¯+ .
∆
Λi ,ρi
i=1
ρ=(ρ1 ,...,ρk )∈Γ+
k,d
By considering previous notions, we have a formula for dimenson of Hd (G, χ) as
follow,
Theorem 2.2. we have
∑
k
∏
(ρ1 ,...,ρk )∈Γ+
k,d
i=1
dim Hd (G, χ) =
dim Hρi (Gi , χi ).
Corollary 2.3. Let G = G1 × G2 and χ = χ1 × χ2 where χi ∈ Irr(Gi ), i = 1, 2,
then
dim Hd (G, χ) =
d
∑
dim Hi (G1 , χ1 ) dim Hd−i (G2 , χ2 ).
i=0
Proof. Since Γ+
2,d = {(i, d − i), 0 ≤ i ≤ d}, then result is obtained by Theorem
3.4.
+
Let Dk,d
be the set of all (ρ1 , . . . , ρk ) ∈ Γ+
k,d such that Hρi (Gi , χi ) ̸= 0 for any
1 ≤ i ≤ k.
∏
Theorem 2.4. Let G = G1 × · · · × Gk , and χ = ki=1 χi ∈ Irr(G), then we have
⊕
Hd (G, χ) =
Hρ1 (G1 , χ1 ) · · · Hρk (Gk , χk )
+
(ρ1 ,...,ρk )∈Dk,d
+
Theorem 2.5. Hd (G, χ) has an o-basis if and only if for any ρ ∈ Dk,d
and for any
1 ≤ i ≤ k, Hρi (Gi , χi ) has an o-basis.
References
[1] M. Shahryari, Relative symmetric polynomials, Linear Algebra Appl, 433, (2010)
1410–1421.
[2] M. Shahryari and Y. Zamani, Symmetry classes of tensors associated with Young
subgroups, Asian-Eur. J. Math, 04, (2011), 179–185.
61
[3] Y. Zamani and E. Babaei, The dimensions of cyclic symmetry classes of polynomials, J. Algebra Appl. 13 (2014), no. 2, 3318–3321.
[4] Y. Zamani and E. Babaei, Symmetry classes of polynomials associated with the
dicyclic group, Asian-Eur. J. Math. 6 (2013), no. 2.
62
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
(Strongly) Gorenstein homological dimension of groups
Abdolnaser Bahlekeh
Department of Mathematics, Faculty of Sciences, Gonbad Kavous University, Gonbad
Kavous, Iran
[email protected]
Abstract
Inspired by the theory of Gorenstein homological algebra,
which goes back to the Auslander-Bridger theory of Gorenstein modules, we assign some numerical invariants to any
group Γ. Precisely, we consider the Gorenstein flat and
strongly Gorenstein flat dimension of the trivial ZΓ-module
Z that will be called Gorenstein homological and strongly
Gorenstein homological dimension of Γ, respectively. It is
shown that these invariants have enough potential in reflecting the properties of the underlying group. We also provide
a version of Serre’s theorem on cohomological dimension of
groups for these invariants.
1
Introduction
For a two-sided noetherian ring R, Auslander and Bridger [2] introduced the Gdimension, G-dimR M , for every finitely generated R-module M , which is a refinement of projective dimension.
Several decades later, Enochs and Jenda [4] introduced the notion of Gorenstein
projective dimension as an extension of G-dimension to modules that are not necessarily finitely generated over any associative ring, and the Gorenstein injective
dimension as a dual notion of Gorenstein projective dimension. To complete the
analogy with the classical homological dimension, Enochs, Jenda and Torrecillas [5]
2010 Mathematics Subject Classification. Primary 20J05; Secondary 20J06 , 16D40
Key words and phrases. Gorenstein homological dimension of groups, strongly Gorenstein
homological dimension of groups, Gorenstein projective modules, group rings.
63
introduced the Gorenstein flat dimension. Recall that a (left) R-module M is said
to be Gorenstein projective, if M is a syzygy of a complete projective resolution,
i.e., if there exists an acyclic complex of projective (left) R-modules;
δn+1
δ
n
P• : · · · −→ Pn+1 −→ Pn −→
Pn−1 −→ · · · ,
which remains acyclic when applying the functor HomR (−, P ) for any projective
R-module P , such that M = Imδ0 . This definition can be dualized and allows one
to define the class of Gorenstein injective modules. A (left) R-module M is called
Gorenstein flat, if there exists an exact sequence of flat (left) R-modules;
δn+1
δ
n
F• : · · · −→ Fn+1 −→ Fn −→
Fn−1 −→ · · · ,
such that I ⊗R F• is exact for any right injective R-module I and M = Imδ0 .
The notion of strongly Gorenstein flat modules is introduced and studied by
Ding et al. in [3]. A left R-module M is said to be strongly Gorenstein flat if there
exists an exact sequence of projective R-modules;
δn+1
δ
n
Pn−1 −→ · · · ,
P• : · · · −→ Pn+1 −→ Pn −→
with M = Imδ0 and such that the functor HomR (−, F ), whereas F is a flat Rmodule, leaves the sequence exact. Since every projective module is flat, one may
deduce that every strongly Gorenstein flat module is Gorenstein projective. On
the other hand, there are examples of the strongly Gorenstein flat modules which
is not projective (flat), and (Gorenstein) flat module. It is proved in [3] that over
coherent rings, strongly Gorenstein flat modules lie strictly between projective and
Gorenstein flat modules.
The Gorenstein projective, Gorenstein flat and strongly Gorenstein flat dimensions are defined in terms of Gorenstein projective, Gorenstein flat and strongly
Gorenstein flat resolutions, respectively, and denoted by GpdR (−), GfdR (−) and
SGfdR (−), respectively.
Let Γ be an arbitrary group. The Gorenstein projective dimension of modules
over the group ring ZΓ, has been studied extensively in [1]. Among other results,
they have proved that if the Gorenstein projective dimension of the trivial ZΓmodule Z is finite, then every module over ZΓ has finite projective dimension. In
fact, for any ZΓ-module M , one has the inequality GpdZΓ M ≤ GcdΓ + 1, where
GcdΓ := GpdZΓ Z; see [1]. It is known that the Gorenstein cohomological dimension of a group Γ, GcdΓ, plays a central role in the cohomology of groups. So,
it is natural to investigate other Gorenstein homological invariants in this context.
In this direction, this note aims to study Gorenstien flat and strongly Gorenstein
64
flat dimension of modules over group rings. In particular, we consider Gorenstien
flat and strongly Gorenstein flat dimension of the trivial ZΓ-module Z, which are
called Gorenstein homological and strongly Gorenstein homological dimension of Γ,
respectively. It turns out that these invariants, which will be denoted by GhdΓ and
SGhdΓ, respectively, have enough potential in reflecting the properties of the underlying group. Among others, we have shown that for a given group Γ, GhdΓ = 0
if and only if SGhdΓ = 0 if and only if Γ is a finite group. We also deduce a version
of Serre’s theorem on cohomological dimension of group for these invariants from
more general context.
2
Main results
Proposition 2.1. Let Γ be a group and Γ′ be a subgroup of Γ of finite index. Let
M be a ZΓ-module which is flat over ZΓ′ . Then M is a Gorenstein flat ZΓ-module.
Example 2.2. Let Γ be a finite group and Γ′ be its trivial subgroup. It is known
that the ZΓ-module Q, with trivial action, is flat over ZΓ′ . So the above proposition
yields that Q is a Gorenstein flat ZΓ-module.
Theorem 2.3. Let Γ′ be a subgroup of Γ of finite index and M be a ZΓ-module.
Then M is Gorenstein flat if and only if it is Gorenstein flat as a ZΓ′ -module.
Proposition 2.4. Let Γ be a group such that ZΓ be a coherent ring. Let Γ′ be a
subgroup of Γ of finite index. Then for a given ZΓ-module M , GfdZΓ M = GfdZΓ′ M .
In particular, GhdΓ = GhdΓ′ .
Theorem 2.5. For any group Γ, the following are equivalent:
(1) Γ is a finite group.
(2) GhdΓ = 0.
(3) SGhdΓ = 0.
Theorem 2.6. Let Γ′ be a subgroup of Γ of finite index and M be a ZΓ-module.
Then SGfdZΓ M = SGfdZΓ′ M .
As a direct consequence of the above theorem, we include the sequel result which
provides a version of Serre’s theorem for SGhd.
Corollary 2.7. Let Γ be a group and Γ′ be a subgroup of Γ of finite index. Then
SGhdΓ′ = SGhdΓ.
65
Lemma 2.8. Let Γ be a finite group and M be a finitely generated ZΓ-module.
Then the following are equivalent:
(1) GpdZΓ M = 0.
(2) GfdZΓ M = 0.
(3) SGfdZΓ M = 0.
Proposition 2.9. Let Γ be a group of finite Gorenstein cohomological dimension.
Then for any ZΓ-module M , the equality GpdZΓ M = SGfdZΓ M holds true.
References
[1] J. Asadollahi, A. Bahlekeh and Sh. Salarian, On the hierarchy of cohomological
dimensions of groups, J. Pure Appl. Algebra, 213 (2009), 1795-1803.
[2] M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc.,
94 (1969).
[3] N. Ding, Y. Li and L. Mao, Strongly Gorenstein flat modules, J. Aust. Math.
Soc., 86 (3) 2009, 323-338.
[4] E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules,
Math. Z., 220 (1995), 611-633.
[5] E. E. Enochs, O. M. G. Jenda, and B. Torrecillas, Gorenstein flat modules,
Nanjing Daxue Xuebao Shuxue Bannian Kan, 10 (1993), 1-9.
66
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
OD-Characterization of the simple group G2 (p), where
p < 100
Masoumeh Bibak1∗ , Masoumeh Sajjadi2 and Gholamreza Rezaeezadeh3
1,2
3
Department of Mathematics, Payame Noor University, Iran
[email protected], [email protected]
Department of Mathematics, Faculty of Mathematical Sciences, Shahrekord University,
Shahrekord, Iran
[email protected]
Abstract
Let G be a finite group and D(G) be the degree pattern
of G. The group G is called k-fold OD-characterizable if
there exist exactly knon-isomorphic groups H satisfying conditions (1) |G| = |H| and (2) D(G) = D(H). Moreover,
a one-fold OD-characterizable group is simply called ODcharacterizable group. In this paper, as the main result, we
prove that G2 (p), where p < 100 and p ̸= 67, 73, 79, 97 is
OD-characterizable.
1
Introduction
For a finite group G, we denote by ω(G) the set of orders of its elements and by
π(G) the set of prime divisors of |G|. We associate to π(G) a simple graph called
the prime graph of G, denoted by Γ(G). The vertex set of this graph is π(G) where
two distinct vertices p and q are adjacent by an edge if pq ∈ ω(G), in which case, we
write p ∼ q. Obviously, Γ(G) is uniquely determined by ω(G), while the spectrum
itself is in turn reconstructed from the set µ(G) of maximal elements of ω(G) with
respect to divisibility. Let t(G) be the number of connected components of Γ(G)
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D10; Secondary 20E26 , 20F17, 20J99.
Key words and phrases. Prime graph, Degree pattern, OD-characterizable.
67
and π1 , π2 ,..., πt(G) be the connected components of Γ(G). If 2 ∈ π(G) we always
suppose that 2 ∈ π1 .
The degree deg(p) of a vertex p ∈ π(G) is the number of edges incident on
p. If π(G) = {p1 , p2 , ..., pk } with p1 < p2 < ... < pk , then we define D(G) :=
(deg(p1 ), deg(p2 ), ..., deg(pk )), which is called the degree pattern of G, and leads a
following definition.
Definition 1.1. A finite group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |G| = |H| and D(G) = D(H).
In particular, a 1-fold OD-characterizable group is simply called OD-characterizable.
The degree pattern of a finite group G associated with its prime graph has
been introduced by M. R. Darafsheh and et. all in 2005 and it is proved that the
following simple groups are uniquely determined by their order and degree patterns:
All sporadic simple groups, the alternating Ap with p and p − 2 primes and some
simple groups of Lie type. In [4] the characterization by order and degree pattern
of L2 (q), where q ≥ 4 is an odd prime power is proved. The authors in [2] it is
proved that the automorphism groups of simple K3 -groups except A6 and U4 (2) are
OD-characterizable (we recall that a finite group possessing exactly n prime divisors
is called Kn -group). Also in [3], it is proved that all finite simple K4 -groups except
A10 are OD-characterizable.
In this talk, we consider the simple group G2 (p), where p < 100 and p ̸=
67, 73, 79, 97 and prove that this group is characterizable by order and degree patterns.
Throughout this note, all groups are finite and by simple groups we mean nonabelian simple groups. All further unexplained notations are standard and refer to
[5].
2
Main results
Theorem 2.1. Let G be a finite group such that |G|=|G2 (p)| and D(G)=D(G2 (p)),
where p < 100 and p ̸= 67, 73, 79, 97. Then G ∼
= G2 (p).
As a consequence of theorem 2.1, we have the following corollaries:
Corollary 2.2. Let G be a finite group and let p be a prime number where p < 100
and p ̸= 67, 73, 79, 97. If |G|=|G2 (p)| and Γ(G) = Γ(G2 (p)), then G ∼
= G2 (p).
Remark 2.3. Shi and Bi in [3] put forward the following conjecture:
Conjecture 2.4. Let G be a finite group and M be a finite simple group. Then
G∼
= M if and only if
68
(i) |G| = |M |;
(ii) ω(G) = ω(M ).
This conjecture is valid for all finite simple groups. As a consequence of theorem
2.1, by a new proof the validity of this conjecture is obtained for the groups under
study.
Corollary 2.5. Let G be a finite group and let p be a prime number where p < 100
and p ̸= 67, 73, 79, 97. If |G|=|G2 (p)| and ω(G) = ω(G2 (p)), then G ∼
= G2 (p).
References
[1] J. H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, Atlas of
finite groups, Clarendon Press (Oxford), London - New York, 1985.
[2] W. J. Shi and X. J. Bi, A characteristic property for each finite projective special
linear group, Lecture Note in Math 1456: 171-180.
[3] L.C. Zhang and W.J. Shi, OD-characterization of all simple groups whose orders
are less than 108 , Front. Math. China 3(3): 461-474 2008.
[4] L. C. Zhang and W.J. Shi, OD-Characterization of the projective special linear
groups L2 (q), Algebra Colloq. 19(3): 509-524 (2012).
[5] H. Xu. Yan, G. Chen and L. He, OD-Characterization of the automorphism
groups of simple K3 -group, Journal of Inequalities and Applications. 95 2013.
69
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Combinatorial conditions on groups
Asadollah Faramarzi Salles1∗ and Hassan Khosravi2
1
2
Department of Mathematics and Computer Science, Damghan University, Damghan,
Iran
[email protected]
Department of Mathematics, Gonbad-e Kavous University , Gonbad-e Kavous, Iran
hassan [email protected]
Abstract
Let Bn and An be the varieties of groups defined by the
laws (xy)n (yx)−n = 1 and xn y(yxn )−1 = 1, respectively.
Let Bn∗ and (An )∗ be the classes of groups in which for any
infinite subset X, Y there exist x ∈ X and y ∈ Y such that
n )−1 = 1, respectively. In this
(xy)n (yx)−n = 1 and xn y(yx∪
paper we prove
that Bn∗ = Bn F and whit certain condition
∪
∗
An = An F, where F is the class of finite groups.
1
Introduction
Let V be a variety of groups defined by the law w(x1 , . . . , xn ) = 1. A group G
is said to be a V ∗ -group if for any infinite subsets X1 , . . . , X∪n of G there exist
x1 ∈ X1 , . . . , xn ∈ Xn such that w(x1 , . . . , xn ) = 1. Clearly V F ⊆ V ∗ , where F
is the class of all finite groups.
It is known that for many varieties V and for many
∪
words w the equality V F = V ∗ holds (see, for instance,[1], [5]). However, it is
still an open question whether this is true for all varieties V and for all words w.
Let n be a positive integer and An and Bn be the varieties of groups generated
by the laws (xy)n (yx)−n = 1 and xn y(yxn )−1 = 1, respectively. It is easy to see
thatAn = Bn for all n ∈ N.
∗
Speaker
2010 Mathematics Subject Classification. Primary 20E10; Secondary 20F26.
Key words and phrases. Variety of groups, infinite groups, class of groups, combinatorial
conditions.
70
2
Main results
In this section we give a positive answer to the question.
Lemma 2.1 ([1]). Let G be an infinite Bn∗ -group. Then CG (an ) is infinite for all
a ∈ G.
Lemma 2.2. Let G be an infinite Bn∗ -group, x an element of G. If CG (x) is infinite,
then [y, xn ] = 1, for all y ∈ G.
Theorem 2.3. Let G be an infinite group in Bn∗ . Then G ∈ Bn .
Proof. Let x be an element of G. We can assume that CG (x) finite, by Lemma 2.2.
It follows from lemma 2.1 that T = CG (xn ) is infinite, and then xT and yT are
infinite subsets of G. Since G ∈ Bn∗ , there exist distinct elements t1 and t2 of T such
that [(xn )t1 , yt2 ] = 1 and therefore [xn , y] = 1.
Lemma 2.4 ([1]). Let G be an infinite A∗n -group. If A is an infinite abelian subgroup
of G, then An ≤ Z(G).
Lemma 2.5. Let G be an infinite A∗n -group with some elements of infinite order.
Then G is an An -group.
Lemma 2.6. Let G be an infinite locally finite A∗n -group. If G has an element g
with finite centralizer CG (g), then G is an An -group.
Let An denote the variety of groups generated by the law (xy)n y −n x−n = 1.
Now we have the following results.
∩
Lemma 2.7. Let G be a group in A∗n (An )∗ . Then G is in Bn∗ .
∪
∩
Corollary 2.8. Let n ∈ {3, 6} {2k : k ∈ N} and G be a group in A∗n (An )∗ .
Then G is in An .
References
[1] A. Abdollahi and B. Taeri, Some conditions on infinite subsets of infinite groups,
Bull. Malaysian Math. Soc. (Second Series) 22 (1999) 87-93.
[2] J. L. Alperin. A classification of n-abelian groups, Canad. J. Math. 21 (1969),
1238-1244.
71
[3] C. Delizia and C. Nicotera, Groups with conditions on infinite subsets, In Ischia
group theory 2006: Proc. conf. in honor of Akbar Rhemtulla (World Scientific
Publishing, 2007), 46-55.
[4] G. Endimioni, On a combinatorial problems in varieties of groups, Comm. Algebra, 23 (1995), 5297-5307.
[5] A. Faramarzi Salles and H. Khosravi, A combinatorial property of Burnside
variety of groups of finite exponent, J. Algebra and Its Applications Vol. 8, No.
6 (2009) 845-853
72
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
On the number of elements of a given order
M. Farrokhi D. G.1∗ and F. Saeedi2
1
2
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
[email protected]
Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
[email protected]
Abstract
We study all finite groups in which the number of elements
of each order is almost the same.
1
Introduction
Counting the number of elements of a given order had been an old problem in the
theory of finite groups. The celebrated theorem of Frobenius (see Theorem 1.5) is
the most strongest result in this area, which is also strengthened by many authors
in special cases, see [1, 4] for example.
Let wd (G) be the number of all elements of order d in a finite group G. Freud
and Pálfy [2] studied the quantities wd (G) and showed that the set of all wd (G)
when d ranges over all natural numbers and G ranges over all finite groups coincides
with the set of all natural numbers. Clearly, φ(d) divides wd (G) for all finite groups
G and natural numbers d. Freud and Pálfy showed that wd (G) can take any odd
value when d = 2 and take all values divisible by φ(d) whenever d = 6 or 4d. Also,
they give partial results for other values of d, namely d = 3.
In this talk we shall present the structure of those finite groups G such that
almost all the numbers wd (G) are the same.
Definition 1.1. The Spectrum of a group G is the set of all orders of elements of G
and it is denoted by w(G). The number of elements of a given order d is denoted by
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D60; Secondary 20E99.
Key words and phrases. Finite group, spectrum, order.
73
wd . Also, the set of all frequencies of elements of each order is denoted by w∗ (G),
that is, w∗ (G) = {wd : d ∈ w(G)}.
The following results will be used in the proof of our main theorems.
Lemma 1.2. Let G be a finite group and d ∈ w(G). Then φ(d) divides wd .
Corollary 1.3. Let G be a finite group and d ∈ w(G) \ {1}. If wd is odd, then
d = 2.
∑
w .
Lemma 1.4. Let G be a finite group of order k. Then |G| =
d k d
Theorem 1.5 (Frobenius,
[3]). Let G be a finite group whose order is divisible by
∑
a number n. Then
wd is divisible by n.
d n
2
Main results
Clearly, the only finite groups G with |w∗ (G)| = 1 are the trivial group and the
cyclic group of order 2. In what follows, we obtain the classification of all finite
groups with |w∗ (G)| = 2 or 3.
Theorem 2.1. Let G be a finite group. Then |w∗ (G)| = 2 if and only if G is
isomorphic with one of the following groups:
(1) Z4 ,
(2) Q8 ,
(3) a group of exponent p different from Z2 ,
(4) H × Z2 , where H is a group of exponent p > 2.
Theorem 2.2. Let G be a finite group. If |w∗ (G)| = 3, then
(1) G is a p-gorup,
(2) G is a Frobenius group whose kernel is a p-group of exponent p and complements are cyclic q-groups of order q,
(3) G = Opqp (G) is a 3-step group, Opq (G) = Op (G) ⋊ Zq is a Frobenius group,
G/Op (G) ∼
= Zq ⋊ Zp is a Frobenius group, exp(P ) = p and Q ∼
= Zq ,
(4) G/Z(G) is a Frobenius group, Z(G) ∼
= Z2 and either G ∼
= Z2 × (Zkp ⋊ Z2 ) or
G∼
= Zk2 ⋊ Zp
for some odd primes p and q.
74
References
[1] Y. Berkovich, On the number of elements of given order in a finite p-group, Israel
J. Math. 73(1) (1991), 107–112.
[2] R. Freud and P. P. Pálfy, On the possible number of elements of given order in
a finite group, Israel J. Math. 93 (1996), 345–358.
[3] G. Frobenius, Über einen Fundamentalsatz der Gruppentheorie, Sitz. Ber.
Königl. Preuss. Akad. Wiss. Berlin, 1903, 987–991.
[4] M. Herzog, Counting group elements of order p modulo p2 , Proc. Amer. Math.
Soc. 66(2) (1977), 247–250.
75
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
On the lower autocentral series of groups
Ali Gholamian1,2∗ and Mohammad Mehdi Nasrabadi1
1
Department of Mathematics,University of Birjand, Birjand, Iran
[email protected]
2
Department of Mathematics, Birjand Education, Birjand, Iran
[email protected]
Abstract
The concept of the lower autocentral series of groups was
introduced and studied by Moghaddam(2009). Using this
concept, we introduce the notion of A-nilpotent groups and
investigate their properties. Also we prove an analogue of
Robinson Theorem for the lower autocentral series of groups
and next we present sufficient conditions under which a
group can be finite or finite exponent.
1
Introduction
Let G be a group and A = Aut(G) denote the group of automorphisms of G. As in
[2], if g ∈ G and α ∈ A, then the element [g, α] = g −1 α(g) is an autocommutator
of g and α. Now following [3] one may define the autocommutator of weight m+1
(m ≥ 2) inductively as:
[g, α1 , α2 , ..., αm ] = [[g, α1 , ..., αm−1 ], αm ],
for all α1 , α2 , ..., αm ∈ A.
Now put K0 (G) = G and for any natural number m
Km (G) = [G, A, ..., A] = ⟨[g, α1 , α2 , ..., αm ]|g ∈ G, α1 , ..., αm ∈ A⟩,
| {z }
m−times
∗
Speaker
2010 Mathematics Subject Classification. 20D45, 20D25, 20E36.
Key words and phrases. Automorphism, lower autocentral series, Abelian groups.
76
which is called the mth -autocommutator subgroup of G. Hence, we obtain a descending chain of autocommutator subgroups of G as follows,
G = K0 (G) ⊇ K1 (G) ⊇ K2 (G) ⊇ ... ⊇ Km (G) ⊇ ...,
which is called the lower autocentral series of G.
In [3] some properties of autocommutator subgroups of a finite abelian group are
studied. Now, we introduce the new notion of A-nilpotent groups.
Definition 1.1. A group G is called as A-nilpotent, if the lower autocentral series
ends in the identity subgroup after a finite number of steps.
If G is an A-nilpotent group, then the length of the lower autocentral series of
G is called the A-class of G.
If a group G is A-nilpotent, then A is the stability group of the lower autocentral
series of G. In [1], Hall proved that the stability group is nilpotent. Hence, a group
with non-nilpotent automorphism group can not be A-nilpotent.
Throughout this paper, we adopt additive notation for all abelian groups.
Example 1.2. Let n be a natural number and G = Z2n . Then Km (G) = 2m G, for
any natural number m. Hence, G is an A-nilpotent group.
Example 1.3. Let G = D8 , dihedral group of order 8. Then K1 (G) = Z4 , K2 (G) =
Z2 and K3 (G) is trivial. Hence, G is an A-nilpotent group.
A-nilpotent groups are nilpotent, but the converse is not true in general.
Example 1.4. Let G = Q8 , generalized quaternion group of order 8 or G = Z9 .
Then G is a nilpotent group, but is not A-nilpotent. Since, Km (G) = G, for any
natural number m.
Remark 1.5. The set of elements L(G) = {g ∈ G | [g, α] = 1, ∀α ∈ A} is called
the autocentre of G. Clearly, it is a characteristic subgroup of G. Now as in [3] the
upper autocentral series of G is defined in following way:
m (G)
⟨e⟩ = L0 (G) ⊆ L1 (G) = L(G) ⊆ L2 (G) ⊆ ... ⊆ Lm (G) ⊆ ..., where LLm−1
(G) =
G
L( Lm−1
(G) ). In [3], a group G is said to be autonilpotent group, if the upper autocentral series ends in the group G after a finite number of steps. It is easy to check
that any autonilpotent group is A-nilpotent group but the converse is not true in
general. The dihedral group of order 8, D8 , is A-nilpotent, but is not autonilpotent.
Robinson [4, 5.2.5] showed that how the first lower central factor Gab = GG′
exerts a very strong influence on subsequent lower central factors of a group G. An
analogue of Robinson Theorem for lower autocentral factors shall be stated. Some
sufficient conditions under which a group can be finite or finite exponent shall be
described.
77
2
Main results
In this paper the floor function of x , the largest integer less than or equal to x, is
written by the symbol ⌊x⌋.
Lemma 2.1. i) Let H and T be two arbitrary groups. Then for any natural number
m, Km (H) × Km (T ) ⊆ Km (H × T ).
ii) Let H and T be finite groups such that (|H|, |T |) = 1. Then for any natural
number m, Km (H) × Km (T ) = Km (H × T ).
Corollary 2.2. If H or T is not A-nilpotent group, then so is not H × T .
Lemma 2.3. For all natural numbers n1 > n2 , if α ∈ Aut(Z2n1 ⊕ Z2n2 ), then for
some k, r ∈ {0, 1, 2, ..., 2n1 − 1} and k ′ , r′ ∈ {0, 1, 2, ..., 2n2 − 1},
α((a, b)) = ((2k + 1)a + 2n1 −n2 rb, k ′ a + (2r′ + 1)b)
for all (a, b) ∈ Z2n1 ⊕ Z2n2 .
Theorem 2.4. Let n1 , n2 , m be natural numbers and G = Z2n1 ⊕ Z2n2 . Thus
i) if n1 = n2 , then
Km (G) = G.
ii) If n1 − n2 = 1, then
Km (G) = 2⌊
m+1
⌋
2
Z2n1 ⊕ 2⌊ 2 ⌋ Z2n2 .
m
iii) If n1 − n2 ≥ 2, then
Km (G) = 2m Z2n1 ⊕ 2m−1 Z2n2 .
Corollary 2.5. For all natural numbers n1 and n2 ,
i) if n1 = n2 , then
Km (Z2n1 ⊕ Z2n2 ) ̸= ⟨0⟩ for any natural number m,
hence G is not A-nilpotent.
ii) If n1 > n2 then
K2n1 −1 (Z2n1 ⊕ Z2n2 ) = ⟨0⟩,
hence G is A-nilpotent.
In following theorem we prove an analogue to the work of Robinson for lower
autocentral factors of a given group.
78
i (G)
Remark 2.6. Note that in the under theorem Fi = KKi+1
(G) and Aut(G) act trivially
on each other. Hence the non abelian tensor product Fi ⊗ Aut(G) is isomorphic to
their abelian tensor product.
Theorem 2.7. Let G be a group and Fi =
Ki (G)
Ki+1 (G)
for i ≥ 0. Then the map
Fi ⊗ Aut(G) −→ Fi+1
gKi+1 (G) ⊗ α 7−→ [g, α]Ki+2 (G)
is a well-defined epimorphism.
Corollary 2.8. If G is an A-nilpotent group such that
then so is G.
G
K1 (G)
and Aut(G) are finite,
Theorem 2.9. If G is a group such that Aut(G) is finite and
n. Then, for any natural number m,
Km (G)
Km+1 (G)
G
K1 (G)
has exponent
has exponent dividing n.
Corollary 2.10. If G is an A-nilpotent group of A-class c such that Aut(G) is finite
and K1G(G) has exponent m. Then G has finite exponent dividing mc .
The following example shows that the corollaries 2.8 and 2.10 may fail to hold
if G is not A-nilpotent, even if G is nilpotent.
Example 2.11. Let G = Z. Then, Aut(G) ≃ Z2 and one can easily check that
Km (G) = 2m G, for any non-negative integer m. Hence, Aut(G) and K1G(G) are
finite, but G is not finite.
References
[1] P, Hall, Some sufficient conditions for a group to be nilpotent, Illinois J. Math.
(1958) , no 2, 787-801.
[2] P. Hegarty, The absolute centre of a group, J. Algebra 169, (1994), no. 3 929-935.
[3] M. R. R. Moghaddam, Some properties of autocommutator groups, The first
Two-Days Group Theory Seminar in Iran, University of Isfahan, 12-13 March
2009,Isfahan.
[4] D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, 80. Springer-Verlag, New York-Berlin, 1982.
79
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
On a conjecture about automorphisms of finite p-groups
S. M. Ghoraishi
Department of Mathematics, Faculty of Mathematics and Computer Sciences, Shahid
Chamran University, Ahvaz, Iran
[email protected]
Abstract
A longstanding conjecture asserts every finite nonabelian pgroup has a noninner automorphism of order p. In this talk
the verification of the conjecture is reduced to the case of
p-groups G satisfying Z2⋆ (G) ≤ CG (Z2⋆ (G)) = Φ(G), where
Z2⋆ (G) is the preimage of Ω1 (Z2 (G)/Z(G)) in G. This improves Deaconescu and Silberberg’s reduction of the conjecture.
1
Introduction
Let p be a prime and G be a finite nonabelian p-group. A longstanding conjecture
asserts that G has a noninner automorphism of order p [5, Problem 4.13]. This
conjecture, is still open and has faced no counterexamples yet. In fact, the statement
of the conjecture is a sharpened version of a well-known and nontrivial property of
finite p-groups that with the exception of groups of order p, they always have a
noninner automorphism of p-power order [2].
The conjecture has been established for p-groups of class 2 and 3, for p-groups
of coclass 2, for regular p-groups, for p-groups G in which G/Z(G) is powerful, for
p-groups G in which (G, Z(G)) is a Camina pair and p ̸= 2, for 2-groups with a
cyclic commutator subgroup, and for p-groups of order pm and exponent pm−2 . It
is worth noting that most of the noninner automorphisms given in these results
leave either Φ(G) or Z(G) elementwise fixed. Also, Deaconescu and Silberberg have
proved if CG (Z(Φ(G))) ̸= Φ(G), then G has a noninner automorphism of order p
2010 Mathematics Subject Classification. Primary 20D15; Secondary 20D45.
Key words and phrases. Finite p-groups, automorphisms, noninner automorphisms.
80
leaving Φ(G) elementwise fixed [2]. Hence, they have reduced the verification of the
conjecture to the degenerate case in which
CG (Z(Φ(G))) = Φ(G).
(∗)
The main motivation of this talk is to reduce the verification of the conjecture
further.
2
Main results
We need the following preliminary lemmas.
Lemma 2.1. Let G be a finite p-group, M be a maximal subgroup of G and g ∈
G \ M . Let u ∈ Z(M ) such that (gu)p = g p . Then the map α given by g 7→ gu and
m 7→ m, for all m ∈ M , can be extended to an automorphism of order |u| that acts
trivially on M .
Let Z2⋆ (G) = Z(G) = Ω1 (Z2 (G) = Z(G)), where for a finite p-group H, Ω1 (H) =
⟨h ∈ H | hp = 1⟩.
Lemma 2.2. If G is a finite p-group, then [Z2⋆ (G), Φ(G)] = 1.
Our first main result is the following.
Theorem 2.3. Let p be a prime and G be a finite nonabelian p-group. If G fails to
fulfill the condition
Z2⋆ (G) ≤ CG (Z2⋆ (G)) = Φ(G),
(∗∗)
then G has a noninner automorphism of order p leaving the Frattini subgroup of G
elementwise fixed.
Proof. Assume that G has no noninner automorphism of order p leaving the Frattini
subgroup of G elementwise fixed. Then by [2, Theorem], we may assume that
CG (Z(Φ(G))) = Φ(G). Now Lemma 2.2, implies that Z2⋆ (G) ≤ Z(Φ(G)). Thus
Z2⋆ (G) ≤ CG (Z2⋆ (G)). Let M be a maximal subgroup of G and g ∈ G \ M . Let
u be an element of order p in Z(G) ∩ M . Then by Lemma 2.1 the map α given
by g 7→ gu and m 7→ m, for all m ∈ M , can be extended to an automorphism of
order p that leaves M elementwise fixed. Thus for some xM ∈ G, α = θx⊆M , the
inner automorphism induced by xM . Therefore xM ∈ Z2⋆ (G) and M = CG (xM ). By
Lemma 2.2, Φ(G) ≤ CG (Z2⋆ (G)). Hence
∩
∩
Φ(G) ≤ CG (Z2⋆ (G)) ≤
CG (xM ) =
M = Φ(G),
⊆M ∈M (G)
and the result follows.
81
⊆M ∈M (G)
Theorem 3.5 reduces the verification of the conjecture to the case of finite pgroups satisfying (∗∗). Let Gp∗ and Gp∗∗ denote the sets of all finite p-groups with
the properties (∗) and (∗∗), respectively.
Then, we have the following theorem.
Theorem 2.4. For every prime p, Gp∗∗ ⊆ Gp∗ and Gp∗ \ Gp∗∗ contains infinitely many
p-groups.
To prove Theorem 2.4, we use the following observation.
Lemma 2.5. If G1 belongs to Gp∗ \ Gp∗∗ , then so does G1 × G2 , for all G2 ∈ Gp∗ .
Proof. The result follows immediately from the following elementary facts. Let G1
and G2 be two finite p-groups. Let H1 ≤ G1 and H2 ≤ G2 . Set G = G1 × G2 and
H = H1 × H2 . Then Φ(G) = Φ(G1 ) × Φ(G2 ) and CG (H) = CG1 (H1 ) × CG2 (H2 ).
Proof of Theorem 2.4. Let G ∈ Gp∗∗ . Then by Lemma 2.2, Z2⋆ (G) ≤ Z(Φ(G)).
Therefore
Φ(G) = CG (Z2⋆ (G)) ≥ CG (Z(Φ(G))) ≥ Φ(G).
This proves the first part of the theorem. For the second part, by Lemma 2.5 it
suffices to show that for every prime p, Gp∗ \ Gp∗∗ ̸= ∅. First, assume that p > 3 and
let G be a group with the following power-commutator presentation.
G = Pc⟨g1 , g2 , g3 , g4 , g5 |g1p = g2p = g3p = g4p = g5p = 1,
g3 = [g2 , g1 ], g4 = [g3 , g1 ], g5 = [g4 , g1 ],
[g5 , g1 ] = 1, [g3 , g2 ] = g5 , [g4 , g2 ] = 1, [g5 , g2 ] = 1
[g4 , g3 ] = 1, [g5 , g3 ] = 1, [g5 , g4 ] = 1⟩,
Then it follows that G ∈ Gp∗ \ Gp∗∗ ̸= ∅. Next, if p ≤ 3, then one may find a group
G ∈ Gp∗ \ Gp∗∗ in GAP small groups library.
We refer the interested reader to [4], [1] and the bibliography therein.
References
[1] A. Abdollahi, S. M. Ghoraishi, Y. Guerboussa , M. Reguiat and B. Wilkens,
Noninner automorphisms of order p for finite p-groups of coclass 2, to appear in
J. Group Theory.
[2] M. Deaconescu and G. Silberberg, Noninner automorphisms of order p of finite
p-groups, J. Algebra 250 (2002), 283–287.
82
[3] W. Gaschütz, Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen, J.
Algebra 4 (1966), 1–2.
[4] S. M. Ghoraishi, On noninner automorphisms of finite nonabelian p-groups Bull.
Austral. Math. Soc. available on CJO2013. doi:10.1017/S0004972713000403.
[5] V. D. Mazurov and E. I. Khukhro (ed.), Unsolved problems in group theory, The
Kourovka Notebook, No. 16, Russian Academy of Sciences, Siberian Division,
Institue of Mathematics, Novosibirisk, 2006.
83
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Embeddings of borel subgroup of the Ree groups of
type 2 F4 (q 2 )
Maryam Ghorbany
Department of Mathematics, Iran University of Science and Technology, Emam, Behshahr,
Mazandaran, Iran
m [email protected]
Abstract
By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral
trace.Thus every permutation matrix over C is a quasipermutation matrix.For a given finite group G , let c(G)
denotes the minimal degree of a faithful representation of
G by quasi-permutation matrices over the complex numbers
and let r(G) denote the minimal degree of a faithful rational
valued complex character of G. The purpose of this paper
is to calculate c(G) and r(G) for the Borel Subgroup of the
Ree groups 2 F4 (q 2 ).
1
Introduction
If F is a subfield of the complex numbers C, then a square matrix over F with
non-negative integral trace is called a quasi-permutation matrix over F . Thus every
permutation matrix over C is a quasi-permutation matrix. For a given finite group
G, let c(G) be the minimal degree of a faithful representation of G by complex
quasi-permutation matrices.
It is easy to see that for a finite group G the following inequalities hold
r(G) ≤ c(G) ≤ q(G).
2010 Mathematics Subject Classification. 20C15
Key words and phrases. Character table, Quasi-permutation representation, Borel subgroup,
Parabolic subgroup, Ree group.
84
Finding the above quantities have been carried out in some papers, for example in
[2], [3], [4], [5] and [6] we found these for the groups GL(2, q), SU (3, q 2 ) ,P SU (3, q 2 )
, SP (4, q) , G2 (2n ) and K22 (q) respectively.
2
Notation and preliminary results
Let 2 F4 (q 2 ) be the simple Ree group with q 2 = 22n+1 and n a positive integer.
Let V be a Euclidean vector space with scalar product (., .), let e1 , e2 , e3 , e4 be an
orthonormal basis of V and let Φ be the set consisting of the 48 vectors
1
ei , ei + ej , (ei + ej + ek + el ),
2
where i, j, k, l ∈ ±1, ±2, ±3, ±4, |i|, |j|, |k|, |l| are different and e−i = −ei for all i.
The set Φ is a root system of type F4 and the set ∆ := r1 , r2 , r3 , r4 with the simple
roots
1
r1 := e2 − e3 , r2 := e3 − e4 , r3 := e4 , r4 := (e1 − e2 − e3 − e4 )
2
is a basis of Φ.
In the theory of algebraic groups, a Borel subgroup of an algebraic group G
is a maximal Zariski closed and connected solvable algebraic subgroup. All Borel
subgroups of a given group are conjugate. Any Borel group is connected and equal
to its own normalizer, and contains a unique Cartan subgroup. The intersection of
B with a maximal compact subgroup K of G is t torus of K are one of the two
key ingredients in understanding the structure of simple (more generally, reductive)
algebraic groups, in Jacques Tits, theory of groups with a (B, N ) pair. Herer the
group B is a Borel subgroup and N is the normalizer of a maximal torus contained
in B.See [7]
Now we give algorithms for calculation of r(G) and c(G) .
Definition 2.1. Let χ be a character of G such that, for all g ∈ G, χ(g) ∈ Q and
χ(g) ≥ 0. Then we say that χ is a non-negative rational valued character.
Let ηi for 0 ≤ i ≤ r be Galois conjugacy classes of irreducible complex characters
of G. For
∑ 0 ≤ i ≤ r let φi be a representative of the class ηi , with φo = 1G . Write
Ψi =
χi and mi = mQ (φi ) and Ki = kerφi . We know that Ki = kerΨi . For
χi ∈ηi
I ⊆ {0, 1, 2, · · · , r} , put KI =
∩
Ki . By definition of r(G) and c(G) and using
i∈I
above notations we have: r(G) = min{ξ(1) : ξ =
r
∑
i=1
85
ni Ψi , ni ≥ 0, KI = 1 f or I =
{i, i ̸= 0, ni > 0}}
c(G) = min{ξ(1) : ξ =
r
∑
ni Ψi , ni ≥ 0, KI = 1 f or I = {i, i ̸= 0, ni > 0}}
i=0
where n0 = − min{ξ(g)|g ∈ G} in the case of c(G).
In [1] we defined d(χ), m(χ) and c(χ) [See Definition 3.4]. Here we can redefine
it as follows:
Definition 2.2. Let χ be a complex charater of G, such that ker χ = 1 and χ =
χ1 + · · · + χn for some χi ∈ Irr(G). Then define
n
∑
(1)d(χ) =
|Γi (χi )|χi (1),
i=1

(2) m(χ) =
(3) c(χ) =
if χ = 1G ,
 0
∑
∑n
α
| min{ i=1
χi (g) : g ∈ G}| otherwise,

n
∑
∑
α∈Γi (χi )
χαi + m(χ)1G .
i=1 α∈Γi (χi )
So
r(G) = min{d(χ) : ker χ = 1},
and
c(G) = min{c(χ)(1) : ker χ = 1}.
We can see all the following statements in [1] .
∑
Corollary 2.3. Let χ ∈ Irr(G) ,then α∈Γ(χ) χα is a rational valued character
of G . Moreover c(χ) is a non-negative rational valued character of G and c(χ) =
d(χ) + m(χ).
Lemma 2.4. Let χ ∈ Irr(G), χ ̸= 1G . Then c(χ)(1) ≥ d(χ) + 1 ≥ χ(1) + 1 .
Lemma 2.5. Let χ ∈ Irr(G). Then
(1) c(χ)(1) ≥ d(χ) ≥ χ(1) ;
(2) c(χ)(1) ≤ 2d(χ) . Equality occurs if and only if Z(χ)/kerχ is of even order .
3
Quasi-permutation representations
In this section we will calculate r(G) and c(G) for Borel subgroup of the Ree groups
2 F (q 2 ) .
4
86
Theorem √
3.1. Let G be a Borel subgroup B of√the Ree groups 2 F4 (q 2 ), then
1) r(B) = 22 |Γ(Bχ51 (k))|q 9 (q 2 − 1) 2) c(B) = 22 |Γ(Bχ51 (k))|q 11
Proof. In order to calculate r(G) and c(G) , ∩
we need to determine d(χ), m(χ) and
c(χ)(1) for all characters that are faithful or χ Kerχ = 1 ,so, since the degrees of
faithful characters are minimal , therefore, we consider only the faithful characters
, and by Lemmas 2.3 and 2.4 and table A6 of [7]we will prove ......
References
[1] H. Behravesh, Quasi-Permutation representations of p-groups of class 2, J. London Math. Soc. 55 (1997), 251–260.
[2] M.R. Darafsheh , M. Ghorbany , A. Daneshkhah and H. Behravesh Quasipermutation representation of the group GL(2, q), Journal of Algebra 243 (2001),
142–167.
[3] M.R. Darafsheh and M.Ghorbany, Quasi-permutation representations of the
groups SU (3, q 2 ) and P SU (3, q 2 ), Southest Asian Bulletin of Mathemetics 26
(2002), 295–406.
[4] M.R. Darafsheh and M. Ghorbany, Special representations of the group SP (4, q)
with minimal degrees, Acta Math. Hungar. 102 (2004), 287–296.
[5] M.Ghorbany, Special representations of the group G2 (2n ) with minimal degrees,
Southest Asian Bulletin of Mathemetics 30 (2006) 663–670.
[6] M.Ghorbany, Quasi-permutation representations for the group K22 (q), Italian
Journal of Pure and Applied Mathematics N. 24 (2008), 157–168.
[7] F. Himstedt and S.C Huang, Character tables of Borel subgroup of the Ree groups
2 F (q 2 ), LMS J. Comput. Math. 12 (2009), 1–53.
4
87
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
The commutativity degree of a polygroup
Azam Hokmabadi1 , Fahimeh Mohammadzadeh2 and Elaheh
Mohammadzadeh3∗ ,
Department of Mathematics, Faculty Of Science,Payame Noor University, P.O. Box
19395-3697, Tehran ,Iran
1
2
[email protected], [email protected], 3 [email protected]
Abstract
In this paper, we define the commutativity degree of a polygroup and then calculate the commutativity degree of the
direct product of polygroups. We also give a lower bound
for commutativity degree of a polygroup under some condition.
1
Introduction
The commutativity degree of a group is defined to be the probability that two elements in the group commute, denoted by P (G). In the past twenty years and
particularly during the last decade there has been growing interest in the use of
probability in finite group (see[1]). Erdios and Turan [2] introduced this idea in
1968, which explored this concept for symmetric group.
The concept of polygroup is introduced by S.D. Comer as a special hypergroup.
He pointed out that polygroups have application in color schemes and also developed
the algebraic theory for polygroups([3, 4]).
In this paper, we define the commutativity degree of a poly group and then
calculate the commutativity degree of the direct product of polygroups. We also
give a lower bound for commutativity degree of a polygroup under some condition.
∗
Speaker
2010 Mathematics Subject Classification. Primary 20N20; Secondary 20P05.
Key words and phrases. Polygroup, Commutativity degree.
88
2
Main Results
A poly group is a system ⟨P, ., e,−1 ⟩, where e ∈ P , −1 is a unitary operation on P , .
maps P × P into the non-empty subset of P , and the following axioms hold for all
x, y, z in P :
(i) (x.y).z = x.(y.z);
(ii) e.x = x.e = x;
(iii) x ∈ y.z implies y ∈ x.z −1 and z ∈ y −1 .x.
∪
Note that if A and B are non-empty subsets of P , then we have A.B = a∈A,b∈B a.b.
The following elementary facts about polygroups follow easily from the axioms:
e ∈ x.x−1 ∩ x−1 .x, e−1 = e, (x−1 )−1 = x, and (x.y)−1 = y −1 .x−1 where A−1 =
{a−1 |a ∈ A}. For more details see [5]
Definition 2.1. A non-empty subset K of a polygroup P is said to be a subpolygroup of P if K itself forms a polygroup, under the hyperoperation in P .
Definition 2.2. The subpolygroup N of P is normal in P if and only if a−1 N a ⊆ N
for all a ∈ P .
For a subpolygroup K of P and x ∈ P , denote the right coset of K by Kx and
let P/K is the set of all right cosets of K in P . If K E P then xK = Kx and the
set of all cosets of K in P is a partition of P (see [5]).
Let P be a polygroup. The relation β ∗ is defined as the smallest equivalence
relation on P such that the quotient P/β ∗ , the set of all equivalence classes, is a
group. In this case β ∗ is called the fundamental equivalence relation on P and P/β ∗
is called the fundamental group. The unite of P/β ∗ is β ∗ (e) and is denoted by ω. It
is easy to see that ω E P (see [5]).
Now we define the commutativity degree of a polygroup. Hereafter, we suppose
that the polygroup P is finite.
Definition 2.3. We define the commutativity degree of a polygroup P , denoted
by d(P ), in the following way,
d(P ) =
|{(x, y) ∈ P × P ; x.y.ω = y.x.ω}|
.
|P |2
A polygroup P is called commutative if x.y = y.x for all x, y ∈ P . Therefore
d(P ) = 1, if P is a commutative polygroup.
89
Let H and K be two poly groups. One can check that P = H × K is a polygroup
with the operation (x, y).(z, t) = {(r, s); r ∈ x.z, s ∈ y.t} (see [5]).
Theorem 2.4. Let H and K be two poly groups and P = H × K. Then
d(P ) = d(H)d(K).
Proof. By Theorem 4.3.3 in [5] for all h ∈ H, k ∈ K we have β ∗ (h, k) = (β1∗ (h), β2∗ (k))
where β1∗ and β2∗ are the fundamental equivalence relations on H and K, respectively.
Using this identity and Definition 2.3, one can obtain the assertion.
Lemma 2.5. Let P be polygroup and H be a normal subpolygroup of P . If β ∗ (e) =
{e} then |Hx| = |H|, for all x ∈ P .
Proof. Using the identity β ∗ (e) = {e} one can prove that a1 = a2 if and only if
β ∗ (a1 ) = β ∗ (a2 ). This fact implies that the map h 7→ h.x is a bijection from H to
Hx and so the result follows.
The following corollary is a useful consequence of Lemma 2.5.
Corollary 2.6. Let P be polygroup such that β ∗ (e) = {e}. Then |P/H| = |P |/|H|,
for any normal subpolygroup H of P .
Theorem 2.7. Let P be polygroup such that β ∗ (e) = {e} and H be a normal
subpolygroup of P . Then
d(H)
≤ d(P ).
|P/H|2
Proof. Let x ∈ P and CP (x) = {y ∈ P ; x.y.ω = y.x.ω} be the centralizer of x in P .
Then by Definition 2.3 and Corollary 2.6 we have
∑
∑
x∈P |CP (x)|
x∈H |CH (x)|
d(P ) =
≥
|P |2
|P |2
d(H)
=
.
|P/H|2
Definition 2.8. Let P be a polygrpoup. Then the center of P is defined to be
Z(P ) = {x|x.y.ω = y.x.ω, ∀y ∈ P }.
It is easy to see that Z(P ) E P and ω ⊆ Z(P ).
The following theorem gives a lower bound for the commutativity degree of a
polygroup under some condition.
90
Theorem 2.9. Let P be polygroup such that β ∗ (e) = {e}. If |P/Z(P )| = l, then
d(P ) ≥
2l − l
.
l2
Proof. Put Z = Z(P ). Since Z E P and ω ⊆ Z, thus |P | = l|Z|, by Corollary 2.6.
Then we have
d(P )|P |2 = |{(x, y); x.y.ω = y.x.ω}|
≥ |{(x, y); x ∈ Z or y ∈ Z}|
= |{(x, y); x ∈ Z, y ∈ P }| + |{(x, y); x ∈ P, y ∈ Z}| − |{(x, y); x, y ∈ Z}|
= |P × Z| + |Z × P | − |Z × Z|
= (2l − 1)|Z|2 .
This implies that d(P ) ≥ 2l − 1/l2 .
References
[1] J.D. Dixon . Probabilistic group theory. Carleton University (September 27,
2004) http://www.math.carleton.ca/ jdixon/Prgrpth.pdf.
[2] P. Erdos and P. Turan, (1968). On some problems of a statistical group-theory
iv, Acta Mathematica Academiae Scientiarum Hungaricae Tomus, 19 (1968),
413-435.
[3] S.D. Comer, Hyperstructures associated with character algebra and color
schemes, New Frontiers in Hyperstructures, Hadronic Press, (1996) 49-66.
[4] S.D. Comer, Extension of polygroups by polygroups and their representations
using colour schemes, Lecture notes in Meth., No 1004, Universal Algebra and
Lattice Theory (1982), 91-103.
[5] B.Davvaz, Polygroup theory and related systems, World scientific, 2013.
91
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Finite p-groups whose order of their Schur multiplier is
given(t=6)
S. Hadi Jafari
Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
s.hadi− [email protected]
Abstract
We classify finite p-groups by the order of their Schur multiplier.
1
Introduction
Study on the Schur multiplier of groups which has influence in other scopes of group
theory returns to 1904. To achieve the order of it, there are obtained several bounds.
For instance when G is a finite p-group of order pn , Green [3] has shown that the
1
order of M(G), the Schur multiplier of G is at most p 2 n(n−1) . One of the interesting
subjects is to classify p-groups when the order of their Schur multiplier is determined.
1
More exactly, if |M(G)| = p 2 n(n−1)−t , for some integer t ≥ 0, the characterization
of p-groups for 0 ≤ t ≤ 5 is given ([1], [2], [1], [5]). We will continue that when
t = 6. Another concept which is studied independently from 1987 is the non-abelian
tensor square of groups was introduced by R. Brown et al. Both of these concepts
effect on each other and we will use these effects to prove our results.
2
Main results
Lemma 2.1. If Z is a central subgroup of a group G contained in G′ , then the
following sequence is exact:
Z ⊗ G −→ G ⊗ G −→ G/Z ⊗ G/Z −→ 1
2010 Mathematics Subject Classification. Primary 20D99; Secondary 19C09.
Key words and phrases. Schur multiplier, Non-abelian tensor square.
92
(∗)
Lemma 2.2. Let G be a d-generator group of order pn , the derived factor GG′ of
G
be a δ-generator group, then
order pm with exponent pe and the central factor Z(G)
|M(G)| ≤ pd(m−e)/2+(δ−1)(n−m)−max{0,δ−2} .
Assume that G is a d-generator p-group of order pn with
1
|M(G)| = p 2 n(n−1)−t
(1)
G
We also assume that Z(G)
is a δ-generator group, |G′ | = pc and the Frattini subgroup Φ(G), is of order pa and so a = n − d. Ellis [2] proved that |M(G)| ≤
1
p{ 2 d(2n−2c−d−1)+2(δ−1)c} . So
2(t − c) ≥ a2 − a
(2)
The classification of p-groups when 0 ≤ t ≤ 5 is given as follows.
Theorem 2.3. Let G be a p-group of order pn . Then
(i) (Ya. G. Berkovich [1]) t = 0 if and only if G is elementary abelian;
(ii) (Ya. G. Berkovich [1]) t = 1 if and only if G ∼
= Cp2 or G ∼
= Ep13 ;
(iii) (X. Zhou [5]) t = 2 if and only if G ∼
= Cp × Cp2 , G ∼
= D8 or Ep13 × Cp ;
(iv) (G. Ellis [2]) t = 3 if and only if G ∼
= Cp3 , (Cp )2 × Cp2 , G ∼
= D8 × C2 , Ep13 ×
(2)
Cp , Q8 or Ep23 ;
(v) (F. Saeedi et al. [1]) t = 4 if and only if G ∼
= (Cp2 )2 , (Cp )3 × Cp2 , G ∼
=
(2)
(3)
Q8 × C2 , D8 × C2 , T1 , T4 , Ep13 × Cp , Ep23 × Cp , X1 , X4 , X5 , X6 or Y3 ;
(vi) (F. Saeedi et al. [1]) t = 5 if and only if G ∼
= Cp × Cp3 , (Cp )4 × Cp2 ,
(2)
(3)
G ∼
= Q8 × C2 , D8 × C2 , T4 × C2 , T2 , D8 ∗ D8 , D8 ∗ Q8 , D16 , Ep13 ×
(4)
(2)
Cp , Ep23 × Cp , Ep15 , Ep25 , X4 × Cp , X2 , X7 , X8 or X9 .
Our main result is the following Theorem:
Theorem 2.4. Let G be a p-group of order pn . Then t = 6 if and only if G ∼
= (Cp )5 ×
(3)
(4)
Cp2 , Cp × (Cp2 )2 , Cp4 , T12 , ϕ5 (214 )b, Q8 × C2 , D8 × C2 , T4 × (C2 )2 , D8 ∗ D8 ×
(5)
(3)
C2 , D8 ∗Q8 ×C2 , Ep13 ×Cp , Ep23 ×Cp , Ep15 ×Cp , Ep25 ×Cp , X4 ×(Cp )2 , X3 , T3 , Ep13 ×
Cp2 , X1 × Cp , T1 × C2 , Q16 , QD16 , T10 , T11 , Y4 , Y3 × Cp or X6 × Cp .
where Dn , Qn , and QDn are Dihedral, Quaternion and QuasiDihedral groups of
order n, respectively. Also the groups Ti, s, Xi, s and Y3 are described in table I.
93
Proof. Suppose t = 6. Then (2) implies that (c, a) = (0, 0), (0, 1), (0, 2),
(0, 3), (0, 4), (1, 1), (1, 2), (1, 3), (2, 2), (2, 3) or (3, 3).
It is readily verified the abelian cases. So assume that G is non-abelian. We use
the notations of James Classification of p-groups of order at most p6 , p is odd.
Suppose (c, a) = (1, 1). If Z(G) is cyclic, then G is generalized extra special,
|G| = p6 and G = T12 for p = 2 and G = ϕ5 (214 )b when p ̸= 2. If Z(G) is non-cyclic,
then G ∼
= M × Cp where M is a maximal subgroup of G and t(M ) = t(G) − 1. So
the desired groups obtain.
Suppose (c, a) = (1, 2). Then Gab ∼
= Cp2 × (Cp )n−3 and Lemma 2.2 implies that
2n − 4 ≤ 6, hence n = 4, 5. If n = 4, then G ∼
= X1 = ϕ2 (31) for odd prime p and for
p = 2, G = T3 . If n = 5 then it follows that G ∼
= Y1 = ϕ2 (2111)c or Y2 = ϕ2 (2111)d.
If p = 2 then G = T1 × C2 by GAP.
The case (c, a) = (1, 3) contradicts our condition. Suppose (c, a) = (2, 2). Then
d = δ or δ + 1.
Case 1) If d = δ then by Lemma 2.2, n ≤ 7. If n = 7 then d = 5 and G is of
exponent p. The exact sequence (*) implies that (1) can not hold. If n = 6 then G
belongs to one of families ϕ12 , ϕ13 , ϕ15 or ϕ22 and there is no group in any case.
Let n = 5. Then G may be in the families ϕ4 or ϕ7 . At the first family put
Z = Z(G) and see that |(Z ⊗ G) × (G ⊗ Z)| ≤ p2 and |G ⊗ G| ≤ p11 , a contradiction.
In other groups of this family the same result holds and exceptionally the order of
non-abelian tensor square of group ϕ4 (15 ) is equal to p14 .
In family ϕ7 , put Z = Z(G) and see that Z ⊗ G = 1 except ϕ7 (15 ). When
G = ϕ7 (15 ), |M(G)| = p4 . If n = 4 then G belongs to ϕ3 and there is no group
which satisfies (1).
Case 2) Suppose d = δ + 1, then by Lemma 2.2, n ≤ 6. Thus G belongs to one
of families ϕ3 , ϕ4 , ϕ7 . If d = 4 and n = 6, then by Lemma 2.2, G can belong to
ϕ4 or ϕ7 . If G ∼
= H × Cp then the order of H ⊗ H should be p14 and H ∼
= ϕ4 (15 ),
5
∼
so G = ϕ4 (1 ) × Cp . If d = 3 and n = 5, G should belongs to family ϕ3 . When G
has the direct product form, only the group ϕ3 (14 ) × Cp satisfies our condition and
there isn’t any other group. If p = 2 then G = Q16 , QD16 , T10 or T11 by GAP.
In other cases one can similarly show that there is no group to satisfies our
condition.
94
TABLE I
Name
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
T11
T12
Relations/Structure Description
a4 = b2 = c2 = 1, [a, c] = b, [a, b] = [b, c] = 1
a4 = b4 = 1, [a, b] = a2
a8 = b2 = 1, [a, b] = a4
a4 = b2 = c2 = 1, [b, c] = a2 , [a, b] = [a, c] = 1
a4 = b2 = c2 = 1, [a, c] = [b, c] = 1, (ba)2 = (ab)2 , ba2 = a2 b
a4 = b4 = c2 = 1, ba = ab−1 , [b, c] = [a, c] = 1
a4 = b2 = c4 = 1, ca−1 = a−1 c, [c, b] = 1, bc−1 a−1 = a−1 cb−1 , bca−1 = a−1 c−1 b−1 , a−1 b = ab−1 a−2
b2 = c2 = 1, [b, c] = 1, ca−1 = a−1 c, a−2 ba−1 = ab
b2 = c2 = 1, [b, c] = 1, ca−1 = a−1 c, a−2 ba−1 = ab
a2 = b2 = c2 = 1, [a, b] = 1, [a, c, b] = 1, [b, c, a] = 1, [b, c, b] = 1
a4 = b4 = c2 = 1, [a, b] = 1, [a, c] = a2 , [b, c] = b2
b2 = c2 = d2 = e2 = 1, [b, c] = 1, [d, c] = [e, b] = a2 , [a, b] = a, c] = [a, d] = [a, e] = [b, c] = [b, d] = [c, e] = [d, e] = 1
ap
X1
2
X3
= bp
p
=b
2
= 1, [a, b] = ap
2
= 1, [a, b] = ap
2
ap = bp = cp = 1, [b, c] = ap , [a, b] = [a, c] = 1
a9 = b3 = c3 = 1, [a, b] = 1, [a, c] = b, c−1 bc = a−3 b
ap = bp = cp = dp = 1, [c, d] = b, [b, d] = a, [a, b] = [a, d] = [b, c] = [a, c] = 1
2
ap
X7
= bp = cp = 1, [a, c] = b, [b, c] = 1, [a, b] = ap
p2
X8
Y3
Y4
Y5
Y6
Y7
Y8
Y9
2
p3
a
X4
X5
X6
X9
Y1
Y2
= bp = cp = 1, [a, c] = b, [a, b] = [b, c] = 1
ap
X2
2
ap
a
= bp = 1, cp = ap , [a, c] = b, [b, c] = 1, [a, b] = ap
= bp = 1, cp = aαp , [a, c] = b, [b, c] = 1, [a, b] = ap ( α ̸= 0, 1 and non residue mod p )
Cp × ((Cp2 × Cp ) o Cp )
Cp2 × ((Cp × Cp ) o Cp )
(4)
Cp o Cp
(Cp × ((Cp × Cp ) o Cp )) o Cp
Cp × (Cp2 o Cp2 )
(Cp2 × Cp2 ) o Cp
Cp2 × (Cp2 o Cp )
Cp × (Cp3 o Cp )
(Cp3 × Cp ) o Cp
References
[1] Ya. G. Berkovich, On the order of the commutator subgroups and the Schur
multiplier of a finite p-group, J. Algebra 144 (1991), 269-272.
[2] G. Ellis, On the Schur multipliers of p-groups, Comm. Algebra 27 (9) (1999),
4173-4177. (1999), 191–196.
[3] E. Khamseh, F. Saeedi and M. R. R. Moghaddam , Characterization of finite
p-groups by their Schur multiplier, Journal of Algebra and its applications 12 (5)
(2013), 1250035(9 pages).
[4] G. Karpilovsky, The Schur Multiplier, LMS Monogrphs New Series 2. New York:
Oxford University Press, 1987.
[5] X. Zhou, On the order of Schur multipliers of finite p-groups, Comm. Algebra 22
(1) (1994), 1–8.
95
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Investigating equality of edge and vertex connectivity
number in prime graph of alternative groups
Maryam Jahandideh1∗ , Hamid Kazemi Esfeh2 and Nabieh Farhami3
1
Department of Mathematics, Mahshahr Branch, Islamic Azad University, Mahshahr, Iran
[email protected]
2,3
Faculty of Chemical Engineering, Mahshahr Branch, Islamic Azad University,
Mahshahr, Iran
[email protected], [email protected]
Abstract
Let G be a finite group. We define the prime graph Γ(G) as
follows: The vertices are the primes numbers dividing the
order of G and two vertices p, q are joined by an edge if
there is an element in G of order pq. Suppose that G is a
finite graph, the edge connectivity number is the minimum
of edges λ(G) whose deletion from G disconnects G and
vertex connectivity number is the minimum of vertices k(G)
whose deletion from G the disconnects G. In this paper,
we will prove that the edge and vertex connectivity number
of Γ(An ) where An is the alternative group of degree n are
equal to each other or equal to the degree of the biggest
prime number of π1 (An ). We use the classification of finite
simple groups.
1
Introduction
Let An be an alternative group and Γ(An ) the set of prime numbers dividing the
order of An . We define the prime graph Γ(An ) as follows: The vertices are elements
∗
Speaker
2010 Mathematics Subject Classification. 05C12.
Key words and phrases. Symmetric group ,Alternative group , Prime graph, Edge and vertex
connectivity number.
96
of π(An ), and two distinct vertices p, q are joined by an edge. We write p ∼ q,
if there is an element of order pq of An . Note that p ∼ q if and only if there is
a cycle subgroup of An of order pq . All the studies and researches conducted on
the connected components of Γ(An ) can be found in reference [1 − 4].We denote
the connected components of Γ(An ) by π1 , π2 , . . . , πt , where t is the number of
connected components of Γ(An )[5] .Since the order of An is even , we take π1 to be
the component containing 2. Let d(p, q) be the distance between two vertices p, q
in An [8]. We can define the edge and vertex connectivity number in An as follows:
The edge connectivity number is the minimum of edges λΓ(An )) whose deletion
from Γ(An ) disconnects Γ(An ) and vertex connectivity number is the minimum of
verticesk (Γ(An )) whose deletion from Γ(An ) the disconnects Γ(An ). The purpose
of this paper is to prove:
Theorem 1.1. Let An be an alternative group. If p ∈ π1 (An ), d(2, p) = 2, then
λ(Γ(An )) = k(Γ(An )) = 1(inπ1 (An )).
Theorem 1.2. Let An be an alternative group. q is the biggest prime number in
π1 (An )(q ̸= 3), then λ(Γ(An )) = k(Γ(An )) = degree(q).(inπ1 (An ))
2
PRELIMINARIES
The following lemmas are fundamental to prove the theorems.
Lemma 2.1. Let p, q are odd prime factors of n! and p + q > n, then Sn (where Sn
is the symmetric group) dosn’t have element of order pq.
Proof. We prove this by contradiction. We assume Sn has a permutation like σ of
order pq.We know that every permutation in Sn is a cycle or a finite product of
separate cycles [9 − 10]. We consider the two following cases:
Case1. If σ be a cycle. So ,σ = (a1 a2 . . . ak ), σ pq = 1, then the length of σ is equal
pq. We know pq ≥ p + q > n. It is not possibility because the length of σ is bigger
than n.
Case 2. If σ be finite product of separate
cycles.
Thus, there exists i and j then: po(σi ), q o(σj ). σi and σj don’ t separate because
p + q > n. Therefore Sn doesn’t have an element of order pq.
Lemma 2.2. Let An be an alternative group. p is the prime factor
of n! p > 3 and p + 4 > n, then An doesn’t have element of order 2p.
Proof. We assume that An has a permutation like σ of order 2p.σ can’t be a cycle
of order 2p , because the number of transpositions is odd. Thus π is finite product
of separate cycles.
97
Thus there exists i ̸= j∃2o(σi ), po(σj ). It is not possible, because p + 4 > n.
Now,An doesn’t have two separate cycle of order p and 2.
3
THE MAIN RESULTS
Proof of Theorm 1.1. According to assumption of theorem, we have
that there exists a prime number as q that 2 ∼ q ∼ p and p + 4 > n.
that q is equal to 3.We have (p + q) < n and (p + 4) > n, then q = 3.
degree p is equal one. If degree of p is not one, there exists another
as r bigger than 3 such that p ∼ r, (p + r) < n.
d(2, p) = 2, so
Now, we show
We prove that
prime number
5 ≤ r → (5 + p) ≤ (r + p) < n → (p + 4) < (p + 5) ≤ p + r < n → d(2, p) = 1.
Thus we delete the edge between two vertices p and 3. The first connected components of Γ(An ) is disconnected or (λ(Γ(An )) = 1). Deleting vertex (3) of Γ(An ), the
first connected component of Γ(An ) is disconnected and
k(Γ(An )) = 1 .
Proof of Theorm 1.2. At first, we suppose that q is the biggest prime number in
π1 (An ) and degree of q is equal to m. Thus m edges pass from q, we define this edge
as the following:
q ∼ 2(e1 ), q ∼ 3(e2 ), . . . , q ∼ pm (em ) where (2 < 3 < . . . < pm ).
All of the prime numbers of this set {2, 3, . . . , pm } join to q. Thus these prime
numbers are joined by an edge to all prime numbers of π1 (An ). Now, we show that
λ(Γ(An )) = k(Γ(An )) = m. We know that degree (q) = m, thus by deleting m edges
passing from q, this graph is disconnected. Now, we prove that m is the smallest.
For proof, we use of contradiction, that isλ(Γ(An )) = t, t < m. In this prime graph,
all of the degrees of vertices are bigger than m; so that we can delete these t edges
from different vertices (a vertex may be repetitious). These t edges are defined by:
p1 ∼ q1 (f1 ), p2 ∼ q2 (f2 ), . . . , pt ∼ qt (ft ). For proof, we consider the following three
different cases:
1. The elements of two set {p1 , . . . , pt }, {q1 , . . . , qt } have been selected of this set
{2, 3, 5, . . . , pm }. We have t < m, so there exist
pi ∈ {2, 3, . . . , pm } − {p1 , . . . , pt }, pj ∈ {2, 3, . . . , pm } − {q1 , . . . , qt }
that pi ∼ pj and no edge delete of this two vertices. Because (pi +q) < n, (pj +q) < n.
Therefore this prime graph is connected.
2. The elements of two set {p1 , . . . , pt }, {q1 , . . . , qt } have been selected of this set
{2, 3, 5, . . . , pm }. If all elements of the two sets {p1 , . . . , pt }, {q1 , . . . , qt } are selected
98
from this set {pm+1 , . . . , q}, all of the vertices are connected to the vertices 2 and
3. There is an edge between the two vertices 2 and 3 too. Thus this prime graph is
connected.
3. If all of prime numbers of this set {p1 , . . . , pt } are selected from the elements of
the two sets {2, . . . , pm }, {pm+1 , . . . , q}. We have:
p i ∼ q1
.
∃pi ∈ {2, . . . , pm } − {p1 , . . . , pt } → ..
p i ∼ qt
In this case, prime graph is connected again. Thus the edge number connectivity
can’t be smaller than m and λ(Γ(An )) = m. Now, we show that k(Γ(An )) =
m. Degree (q) is m, so vertex q connects to m vertices. If we delete m vertices
from the prime graph, vertex q is alone and the prime graph is disconnected. For
proof, we use contradiction. Assume that, k(Γ(An )) = t and t < m, hence by
deleting t verticesp1 , . . . , pt this graph be disconnected. If vertices {p1 , . . . , pt } are
selected from the elements of the set {2, 3, . . . , pm }, there exists a vertex pi that pi ∈
{2, 3, . . . , pm }−{p1 , . . . , pt }. Because vertex pi is not deleted and it is connected to all
remainder vertices, thus the remainder graph is connected. If vertices {p1 , . . . , pt }
are selected from the elements of the set {pm+1 , . . . , q}, in this case vertex 2 is
connected to all other vertices, thus the remainder graph is connected.
Now, if vertices {p1 , . . . , pt } are selected from the elements of the sets
{2, 3, . . . , pm } and {pm+1 , . . . , q}.
Thus there exists pi ∈ {2, 3, . . . , pm } − {p1 , . . . , pt } that connects to all remainder
vertices and the remainder graph is connected and k(Γ(An )) = m.
Acknowledgements
Authors gratefully acknowledge the financial support provided by Islamic Azad University of Mahshahr, Iran to perform research project entitled: ’Investigating equality of edge and vertex connectivity number prime graph of alternative groups’.
References
[1] J. H. Conway, Atlas of finite groups : maximal subgroups and ordinary characters
for simple groups, Clarendon Press; New York, 1985.
[2] D. J. S. Robinson, A course in the theory of groups, Springer-Verlag, New York,
1982.
99
[3] J. J. Rotman, An introduction to the theory of groups, 4th ed., Springer-Verlag,
New York, 1995.
[4] J. S. Williams, “Prime graph components of finite groups”, Journal of Algebra,
69 (1981) 487–513.
[5] N. Iiyori, H. Yamaki, “Prime Graph Components of the Simple Groups of Lie
Type over the Field of Even Characteristic”, Journal of Algebra, 155 (1993)
335–343.
100
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
A note on the tensor and exterior center of a pair of Lie
algebras
Farangis Johari1∗ , Peyman Niroomand2 and Mohsen Parvizi3
1
2
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
[email protected]
School of Mathematics and Computer Science, Damghan University, Damghan, Iran
[email protected]
3
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
[email protected]
Abstract
In this talk using the notions of tensor and exterior centralizers for a pair of Lie algebras, we collect some facts concerning these notions for a pair of Lie algebras and study
the relations between them and capability of pairs.
1
Introduction and Motivation
Lemma 1.1. Let L be a Lie algebra, N be an ideal of L and l ∈ L. Then
Cl⊗ (N ) = {n ∈ N |0 = l ⊗ n ∈ L ⊗ N }
is a subalgebra of L which is called the (l, N )-tensor centralizer of l ∈ L, in fact it
is an ideal of CL (l).
Definition 1.2. Let L be a Lie algebra, N be an ideal of L and l ∈ L. Then
(i)
⊗ E(N, l)
= {n ∈ N |l ⊗ (x + n) = l ⊗ x, ∀x ∈ N },
∗
Speaker
Key words and phrases. Tensor and exterior centralizers of pair Lie algebras, Tensor center
of pair Lie algebras, Exterior center of pair Lie algebras, Capability, Relative Schur multiplier,
Non-abelian tensor product.
101
(ii) E ⊗ (N, l) = {n ∈ N |l ⊗ (n + x) = l ⊗ x, ∀x ∈ N }.
Lemma 1.3. Let L be a Lie algebra, N be an ideal and l ∈ L. Then ⊗ E(N, l) and
E ⊗ (N, l) are subalgebras of L. Also we have
(i)
⊗ E(N, l)
= Cl⊗ (N ).
(ii) E ⊗ (N, l) =⊗ E(N, l).
Corollary 1.4. Let L be a Lie algebra with ideal N . Then
⊗
⊗
Z(L)
(N ) = ∩l∈L E ⊗ (N, l) = ∩⊗
l∈L E(N, l) = ∩l∈L Cl (N ).
Proposition 1.5. Let L be a Lie algebra with ideal N , K be an ideal of L contained
in N and l ∈ L. Then the following sequences are exact
⊗
(i) 0 → Cl⊗ (N ) ∩ K → Cl⊗ (N ) → Cl+K
(N/K),
⊗
(ii) 0 → ZL⊗ (N ) ∩ K → ZL⊗ (N ) → ZL/K
(N/K).
Corollary 1.6. Let L be a Lie algebra with ideal N , K be an ideal of L contained in N and l ∈ L. According to Lemma 2, K ≤ Cl⊗ (N ) if and only if
⊗
Cl⊗ (N )/K = Cl+K
(N/K). In particular, we have K ≤ Cl⊗ (N ) for all l ∈ L if
⊗
and only if ZL⊗ (N )/K = ZL/K
(N/K).
Lemma 1.7. Let L be a Lie algebra, N be an ideal and l ∈ L. Then
Cl∧ (N ) = {n ∈ N |0 = l ∧ n ∈ L ∧ N }
is a subalgebra of L. It is called the (l, N )-tensor centralizer of l ∈ L. It is also an
ideal of CL (l).
Definition 1.8. Let L be a Lie algebra, N be an ideal of L and l ∈ L. Then
(i)
∧ E(N, l)
= {n ∈ N |l ∧ (x + n) = l ∧ x, ∀x ∈ N },
(ii) E ∧ (N, l) = {n ∈ N |l ∧ (n + x) = l ∧ x, ∀x ∈ N }.
Lemma 1.9. Let L be a Lie algebra, N be an ideal and l ∈ L. Then ∧ E(N, l) and
E ∧ (N, l) are subalgebras of L. Also we have
(i)
∧ E(N, l)
= Cl∧ (N ).
(ii) E ∧ (N, l) =∧ E(N, l).
102
Corollary 1.10. Let L be a Lie algebra with ideal N . Then
∧
∧
Z(L)
(N ) = ∩l∈L E ∧ (N, l) = ∩∧
l∈L E(N, l) = ∩l∈L Cl (N ).
Proposition 1.11. Let L be a Lie algebra with ideal N and K be an ideal contained
in N and l ∈ L. Then the following sequences are exact
∧ (N/K),
(i) 0 → Cl∧ (N ) ∩ K → Cl∧ (N ) → Cl+K
∧ (N/K).
(ii) 0 → ZL∧ (N ) ∩ K → ZL∧ (N ) → ZL/K
Corollary 1.12. Let L be a Lie algebra with ideal N and K be an ideal contained in
∧ (N/K). In
N and l ∈ L. By Lemma 2, K ≤ Cl∧ (N ) if and only if Cl∧ (N )/K = Cl+K
∧
∧
∧ (N/K).
particular, we have K ≤ Cl (N ) for all l ∈ L if and only if ZL (N )/K = ZL/K
Corollary 1.13. ZL∧ (N ) is the smallest subalgebra of L contained in N such that
the pair (L/ZL∧ (N ), N/ZL∧ (N )) is capable.
Proposition 1.14. Let L be a Lie algebra with ideal N and l ∈ L. Then CN (l)/Cl∧ (N )
is isomorphic to a subalgebra of M(L, N ).
Corollary 1.15. Let L be a Lie algebra with ideal N and l ∈ L such that CN (l) ̸=
Cl∧ (N ). Then M(L, N ) ̸= 0
Let T be defined as
T = ∩l∈Z(L) Cl∧ (N ).
It is easy to see that T is an ideal of L contained in N . In particular if L is an
abelian Lie algebra, then T = ZL∧ (N ).
Lemma 1.16. Let T = ∩l∈Z(L) Cl∧ (N ). Then [N, L] ≤ T .
Proposition 1.17. Let L be a Lie algebra with ideal N such that CL (x) = Cx∧ (N )
for all x out of Z(L). Then [N, L] ∩ Z(L) is a subalgebra of ZL∧ (N ). In particular if
(L, N ) is a capable pair then [N, L] ∩ Z(L) = 0.
Corollary 1.18. let (L, N ) be a nilpotent capable pair. Then there is no element
x in Z(L) such that CL (x) ̸= Cx∧ (N ).
Corollary 1.19. Let T = ZL∧ (N ) then (L, N ) is a nilpotent pair of class at most 2.
Furthermore if M(L, N ) = M(L/T, N/T ) then (L, N ) is an abelian pair.
Recall that the pair (L, N ) of Lie algebra is called unicentral if ZL∧ (N ) = Z(L) ∩
N.
103
Corollary 1.20. Let L be a Lie algebra with ideal N . Then T = N if and only if
(L, N ) is an unicentral pair.
Theorem 1.21. Assume that (L, N ) be a pair of Lie algebras with M(L, N ) of finite
dimension and L/ZL∧ (N ) is d-generator. Then
dim(Z(L) ∩ N )/ZL∧ (N ) ≤ d(dim M(L, N )).
Corollary 1.22. With the assumptions of Theorem 1.21, a pair (L, N ) with trivial
M(L, N ) has Cx∧ (N ) = CN (x) for all x ∈ L. In particular (Z(L) ∩ N ) = ZL∧ (N ).
Corollary 1.23. Let L be a finite dimensional Lie algebra with ideal N and [N, L] ≤
ZL∧ (N ) and L/ZL∧ (N ) is d-generator. Then
dim L ≤ d(dim M(L, N ) + dim[N, L]) + dim ZL∧ (N ).
Proposition 1.24. Let (L, N ) be a capable pair of Lie algebra introduced in Theorem 1.21. Then dim Z(L) ∩ N ≤ d(L/T )(dim M(L, N )).
References
[1] U. Haagerup, Solution of the similarity problem for cylic representations of
C ∗ -algebras, Ann. of Math. (2) 118 (1983), no. 2, 215–240.
[2] A.A. Jamshidian, Čebyshev
arXiv:1154.1464v2 (to appear).
inequality,
Linear
Multilinear
Algebra,
[3] Z. Rezaei, A.A. Jamshidian and Gh. Babaei Tehrani, Cyclic nilpotent groups,
Proc. Amer. Math. Soc. 145 (2008), no. 11, 3318–3321.
[4] G. Karpilovsky, The Schur multiplier, LMS Monogrphs New Series 2, Oxford
Univ. Press, 1987.
[5] J. Miĺovć, Lie symmetries of systems of second-order linear ordinary differential
equations, Nonlinear analysis, 239–267, Springer Optim. Appl., 65, Springer,
New York, 2011.
104
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Capability of finite nilpotent groups of class 2 with
cyclic Frattini subgroups
Azam Kaheni1∗ , Rasoul Hatamian2 and Saeed Kayvanfar3
1
Department of Mathematics, University of Birjand, Birjand, Iran
[email protected]
2
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
[email protected]
3
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
[email protected]
Abstract
A group is called capable if it is a central factor group. Let
N denote the set of all finite groups of nilpotency class 2
whose derived subgroups be cyclic and coincide with their
Frattini subgroups. This talk is organized to provide the
explicit structures of capable groups in N.
1
Introduction
A group G is said to be capable if it is isomorphic to the group of inner automorphisms of some group K. Baer [1] characterized the capable groups that are direct
sums of cyclic groups. The capability of very specific classes, such as metacyclic
groups, extra special p-groups, nilpotent products of cyclic p-groups of class less
than or equal to p is also investigated. Moreover, some numerical necessary and
sufficient conditions for capability of p-groups of class 2 and prime exponent are
known.
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D15; Secondary 19C09
Key words and phrases. Capable groups, Extra special p-groups, Nilpotent groups, Schur
multiplier.
105
Now, let N denote the set of all finite groups of nilpotency class 2 whose derived
subgroups be cyclic and coincide with their Frattini subgroups. In this talk, we will
determine the exact structures of capable groups in N. Since, a nilpotent group is
capable if and only if each of its Sylow subgroups is capable, we reduce the problem
to a restricted subclass. In other words, we shall first focus on p-groups with at least
two generators, then conclude the main result for finite nilpotent groups in N. Finite capable two generated p-groups of class 2 had been listed by Bacon, Kappe and
Magidin. Notice that the commutator subgroup of a finite two generated p-group
of class 2 is always cyclic. But a finite p-group of class 2 with cyclic commutator
subgroup may not be two generated such as extra special p-groups. For this reason,
Yadav [4] interested in studying finite capable p-groups of class 2 with cyclic commutator subgroups. He proved this problem for a group G for which Z(G) ⊆ Φ(G).
Accordingly, our result for p-groups gives an answer to Yadav’s problem [4] and the
main result also is the generalization of the work in [3].
2
Preliminaries
For a group G, recall from [2] that the epicenter of G is denoted by Z ∗ (G) and
defined to be the intersection of all φ(Z(E)), where (E, φ) is a central extension of
G. A relation between Z ∗ (G) and the notion of capability is provided by Beyl et al.
[2] as follows.
Theorem 2.1. Z ∗ (G) is the smallest central subgroup of G whose factor group is
capable. In particular, G is capable if and only if the epicenter of G is trivial.
Obviously, the class of all capable groups is neither subgroup closed nor under
homomorphic image. But this
under direct product (see [2, Proposi∏ class is closed
∏
tion 6.1]), and therefore Z ∗ ( i∈I Gi ) ⊆ i∈I Z ∗ (Gi ), for each family {Gi } of groups.
One should also note that the inclusion is proper in general. Beyl et al. [2] gave a
sufficient condition forcing equality as follows.
∏
Proposition 2.2. Let G = i∈I Gi . Assume that for i ̸= j the maps υi ⊗ 1 :
′
Z ∗ (Gi ) ⊗ Gj /G′j → Gi /G′i ⊗ G
zero, where υi is the natural map Z ∗ (Gi ) →
∏j /Gj are
′
∗
∗
Gi → Gi /Gi . Then Z (G) = i∈I Z (Gi ).
This certainly indicates that a finite nilpotent group is capable if and only if its
Sylow subgroups are capable.
The exact structures of capable extra special p-groups are given by Beyl et al.
[2] as follows.
106
Theorem 2.3. An extra special p-group is capable if and only if it is either D8 or
E1 , where D8 is the dihedral group of order 8 and E1 is the extra special p-group of
order p3 and exponent p (p > 2).
Beyl et al.[2] proved a necessary condition for a group to be capable as follows.
Lemma 2.4. [2, Proposition 1.2] If G is capable and the commutator factor group
G/G′ of G is of finite exponent, then also Z(G) is bounded and the exponent of
Z(G) divides that of G/G′ .
3
Main results
In view of Proposition 3.4, one can obtain some interesting results about nilpotent
capable groups. For example, it is easy to conclude that there is no nilpotent capable
group of square free order. Moreover, if the order of a nilpotent capable group G is
as pα1 1 pα2 2 . . . pαt t , then αi should be greater than 1, for each 1 ≤ i ≤ t. In this section,
the capability of some nilpotent groups of class 2 has been studied. Actually, the
explicit structures of nilpotent capable groups in N are determined.
Lemma 3.1. Let G be a p-group of nilpotency class 2. If G′ = Φ(G), then there
exists a subgroup H of G such that G = HZ(G) and H ′ = Z(H).
Let G be a finite p-group such that Φ(G) ⊆ Z(G). It is easy to see that the
derived subgroup of G is an elementary abelian group. Now under some conditions,
it can be deduced that the exponent of Z(G) is also p.
Lemma 3.2. If G is a capable p-group such that Φ(G) ⊆ G′ , then Z(G) is an
elementary abelian p-group.
Invoking Theorem 2.3 and Lemma 3.2 we have the following corollary.
Corollary 3.3. Let G be a finite capable p-group of nilpotency class 2. If G′ = Φ(G)
is a cyclic group, then G is of order p3 or Z(G) is not cyclic.
The following theorem shorten the proof of the next theorem.
Theorem 3.4. Let G be a finite p-group of nilpotency class 2 and G′ = Φ(G) be a
cyclic group. Then G is capable if and only if G ∼
= E1 × K or G ∼
= D8 × K, where
K is an elementary abelian p-subgroup of Z(G).
Certainly, the exact structure of a capable p-group of order pn which its derived
factor group is elementary abelian of rank pn−1 , is given by the above theorem. So,
the work of Niroomand and Parvizi [3] is generalized.
107
We are now ready to prove the main theorem as follows.
Suppose that G belongs to N. Then G ∼
= H1 × . . . × Ht , in which Hi ’s are Sylow
pi -subgroups of G. Let Hj be an abelian pj -subgroup for each 1 ≤ j ≤ r, and the
(s )
nilpotency class of other Hi ’s be 2. Let Gl = Zpl l , where sl is a natural number
grater than 1, and 1 ≤ l ≤ r. Also, let Gm = Epm × Kpm , in which Kpm is an
elementary abelian pm -subgroup of Z(Hm ) and Epm is an extra special pm -group of
order p3m and exponent pm (pm > 2) for each r < m ≤ t.
Theorem 3.5. Suppose that G belongs to N. Consider Hi ’s and Gi ’s the groups
introduced as above.
(i) If 2 /| |G| or the Sylow 2-subgroup of G is abelian, then G is capable if and only
t
∏
∼
if G =
Gi .
i=1
(ii) If the Sylow 2-subgroup of G is non-abelian, then G is capable if and only if
t
∏
∼
G = D8 × K2 × (
Gi ), where i0 is the index of Sylow 2-subgroup of nilpotency
i=1
i̸=i0
class 2 and K2 is an elementary abelian 2-subgroup of Z(Hi0 ) .
References
[1] R. Baer, Groups with preassigned central and central quotient group,Trans.
Amer. Math. Soc. 44 (1938), 387 – 412.
[2] F.R. Beyl, U. Felgner, P. Schmid, On groups occurring as center factor groups,
J. Algebra 61 (1979), 161 – 177.
[3] P. Niroomand, M. Parvizi, A remark on the capability of finite p-groups, J. Adv.
Res. Pure Math. 5 (2013), 91 – 94.
[4] M. Yadav, On finite capable p-groups of class 2 with cyclic commutator subgroups,
arXiv:1001.3779v1, 2010.
108
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Support enumerators for some permutation groups
Reza Kahkeshani1 and Masoomeh Yazdany Moghaddam1∗
1
Department of Pure Mathematics, Faculty of Mathematical Sciences, Kashan University,
Kashann, Iran
[email protected], [email protected]
Abstract
By defining a distance on a permutation group G, we can
consider it as a code such that the codewords are the elements of G. The support enumerator QG (x, y) is one of the
polynomials related to G. In this paper, we determine the
polynomial QG (x, y) for the dihedral group, some alternating groups and some projective special linear groups.
1
Introduction
Let G be a permutation group acting on Ω = {1, ..., n} and let π be the permutation
character of G (i.e., π(g) denotes the number of fixed points of g). A Hamming
distance on G is defined by dH (g, h) = n − π(gh−1 ), for any g, h ∈ G. So, we
can consider the permutation group G as a code such that the codewords are the
elements of G.
Definition 1.1. The distance enumerator of G is the polynomial
∑
∑
∆G (x) =
xdH (1,g) =
xn−π(g) .
g∈G
g∈G
Definition 1.2. The cycle index of G is the polynomial
1 ∑ ∏ ci (g)
Z(G; x1 , ..., xn ) =
xi ,
|G|
g∈G i≥1
∗
Speaker
Key words and phrases. Cycle index, Distance enumerator, Permutation group, Support enumerator.
109
where ci (g) is the number of cycles of g of lenght i. For simplicity, Z(G; x1 , ..., xn )
is sometimes shown by Z(G). Moreover,
n
1 ∑
1 ∑ π(G)
x
=
mi x i ,
PG (x) =
|G|
|G|
i=0
g∈G
where mi = |{g ∈ G|π(g) = i}|.
Definition 1.3. The homogeneous support enumenerator of G is defined by
1 ∑ π(g) n−π(g)
QG (x, y) =
x
y
.
|G|
g∈G
Obviously, we have the following relations:
1. ∆G (x) = |G|xn PG ( x1 ),
2. QG (x, y) = Z(G; x, y 2 , ..., y n ),
(∑
i>1 ici (g)
)
= n − π(g) ,
3. PG (x) = QG (x, 1),
4. ∆G (y) = |G|QG (1, y).
2
Some Identities
Let G and H be two groups acting on sets Ω and Γ, respectively, where |Ω| = n and
|Γ| = m.
Proposition 2.1. [3]
Z(G × H) = Z(G)Z(H).
Proposition 2.2. [3]
(
)
Z(G ≀ H) = Z H; Z(G; x1 , x2 , ...), Z(G; x2 , x4 , ...), ... .
Proposition 2.3. [2]
PG×H (x) = PG (x)PH (x).
Proposition 2.4. [2]
(
)
PG≀H (x) = PH PG (x) .
Proposition 2.5. [1]
QG×H (x, y) = QG (x, y)QH (x, y),
˙
where G × H is in the intransitive action on Ω∪Γ.
110
Proposition 2.6. [1]
QG×H (x, y) = QG (x, y) ◦ QH (x, y),
where G × H is in the product action on Ω × Γ and xa y b ◦ xc y d = xac y bc+ad+bd .
Proposition 2.7. [1]
(
)
QG≀H (x, y) = QH QG (x, y), y n .
3
results
In this section, the support enumerator of some groups are determined.
(
)
1. QZn (x, y) = n1 xn + (n − 1)y n (see [1]).
(n )
1 ∑n
k n−k , where d(i) is the number of derange2. QSn (x, y) = n!
k=0 k d(n − k)x y
ments of a set of size i (see [1]).
∑
i mn−i , where
3. QZm ≀Sn = mn1 n! mn
i=0 f (i)x y
{∑
f (i) =
n
n−k (m
k=0 m
i
− 1)k− m
(n)(
k
k
i/m
)
d(n − k),
if m | i,
if m - i,
0,
(see [1]).
{
4.
QD2n (x, y) =
1 n
n−1 ),
2 (y + xy
1
n
2 n−2 ),
4 (3y + x y
if n is odd,
if n is even.
Using a computer program in GAP [4], we determine the support enumerators for
some alternating groups and projective special linear groups. See tables 1 and 2.
References
[1] R. F. Bailey, J. P. Dixon, Distance enumerators for permutation groups, Comm.
Algebra 35 (2007) 3045–3051.
[2] N. Boston, W. Dabrowski, T. Foguel, P. J. Gies, J. Leavitt, D. T. Ose, D. A.
Jackson, The proportion of fixed-points-free elements of a transitive permutation
group, Comm. Algebra 21 (1993) 3259–3275.
111
n
4
5
6
7
8
9
10
Table 1: Support Enumerators of An ’s
QAn (x, y)
1 4
(x
+ 8xy 3 + 3y 4 )
4
1 5
2
3
4
5
5 (x + 20x y + 15xy + 24y )
1 6
3 3
2 4
5
6
6 (x + 40x y + 45x y + 144xy + 130y )
1 7
4
3
3
4
2
5
6
7
7 (x + 70x y + 105x y + 504x y + 910xy + 930y )
1 8
5 3
4 4
3 5
8 (x + 112x y + 210x y + 1344x y
2
6
7
8
+3640x y + 7440xy + 7413y )
1 9
(x
+ 168x6 y 3 + 378x5 y 4 + 3024x4 y 5
9
+10920x3 y 6 + 33480x2 y 7 + 66717xy 8 + 66752y 9 )
1
10
7 3
6 4
5 5
4 6
10 (x + 240x y + 630x y + 6048x y + 27300x y
+11600x3 y 7 + 333585x2 y 8 + 667520xy 9 + 667476y 10 )
Table 2: Support Enumerators of PSL(2,q)’s
q
QP SL(2,q) (x, y)
1 3
2
(x
+ 3xy 2 + 2y 3 )
3
1 4
3
4
3
4 (x + 8xy + 3y )
1 5
2
3
4
5
4
5 (x + x y + 15xy + 24y )
1 6
2
4
5
6
5
6 (x + 15x y + 24xy + 20y )
1 8
2 6
7
8
7
8 (x + 56x y + 48xy + 63y )
1 9
2
7
8
9
8
9 (x + 216x y + 63xy + 224y )
1
10
2
8
9
10
9
10 (x + 135x y + 80xy + 144y )
1
11 12
(x12 + 264x2 y 10 + 120xy 11 + 275y 12 )
[3] P. J. Cameron, Combinatorics: topics, techniques, algorithms, Cambridge University Press, 1994.
[4] The GAP Group, GAP–Groups, Algorithms and Programming, Version 4.4,
available at http://www.gap-system.org, 2004.
112
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
On semigroups generated by vector spaces
Ali Reza Khoddami
Department of Pure Mathematics, Shahrood University of Technology, Shahrood, Iran
[email protected]
Abstract
For an arbitrary vector space S over C and 0 ̸= λ ∈ S ∗
we equip S with a multiplication converting S into a semigroup, denoted by Sλ . The semigroup structure of Sλ are
investigated and in particular, the endomorphisms and automorphisms of Sλ are characterized.
1
Introduction
A semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup with an identity element is called a monoid.
A monoid in which every element has an inverse is called a group. A semigroup
homomorphism between two semigroups T and T ′ is a function φ : T −→ T ′ such
that the equation φ(ab) = φ(a)φ(b) is hold for all elements a, b ∈ T . A semigroup
homomorphism from T into itself is called an endomorphism. For semigroup T , a
bijective endomorphism of T is called a semigroup automorphism. The set of all
semigroup automorphisms of T is denoted by AutT . In the case where T is an
algebra, Hom(T ) is the set of all algebra endomorphisms of T .
In this paper let S be a non-zero vector space on C and let λ be a non-zero linear
functional on S. For each a, b ∈ S define a · b = λ(a)b. One can simply verify
that “ · ” converts S into an associative algebra. We denote (S, ·) by Sλ that is a
semigroup. Note that Sλ is not a monoid in general. Indeed Sλ is a monoid if and
only if dimS = 1. Also if dimS > 1 then Z(Sλ ) = {0}, where Z(Sλ ) is the center of
Sλ . It follows that Sλ is not abelian semigroup. Indeed Sλ is abelian if and only if
2010 Mathematics Subject Classification. Primary 20M15; Secondary 16B99, 15A03.
Key words and phrases. Semigroup, semigroup homomorphism, semigroup automorphism,
algebraic homomorphism, algebraic automorphism.
113
dimS = 1. Some basic properties of Sλ such as Arens regularity, n−weak amenability, minimal idempotents and ideal structure are investigated in [1] in the case where
S is a Banach Space. The number of roots of a polynomial equation with coefficients
in Sλ is investigated in [2] in the case where S is a vector space. In this note our
purpose is to characterize the semigroup endomorphisms and automorphisms of Sλ .
In particular we characterize the algebraic endomorphisms and automorphisms of
Sλ . It is worthwhile mentioning that the study of the semigroup endomorphisms
and semigroup automorphisms of Sλ is very interesting. The following examples are
some different endomorphisms of Sλ that are worthy of consideration.
1. φ : Sλ −→ Sλ ,
λ(e) = 1.
φ(a) = e, where e is a constant element of Sλ satisfying,
2. φ : Sλ −→ Sλ , φ(a) = an , for all n ∈ N(note that Sλ is not abelian and also
φ is not linear).
3. φ : Sλ −→ Sλ , φ(a) = λ(a)e, where e is a constant element of Sλ , satisfying,
λ(e) = 1.
4. φ : Sλ −→ Sλ , φ(a) = a + ϕ(a), where ϕ is a linear operator from S into
kerλ = {s ∈ S|λ(s) = 0}.
2
Main results
Theorem 2.1. Let S be a non-zero vector space and let 0 ̸= λ ∈ S ∗ be a non-zero
linear functional. If φ : Sλ −→ Sλ is a semigroup endomorphism then one of the
following statements is hold.
1. φ = 0.
2. φ(a) = e, (a ∈ Sλ ), where e is a constant element of Sλ satisfying, λ(e) = 1.
a
3. φ(0) = 0, λ = λ ◦ φ on kerλ and φ( λ(a)
)=
λ(a) = 1 implies λ ◦ φ(a) = 1.
114
φ(a)
λ◦φ(a)
on (kerλ)C . In particular
Corollary 2.2. Let S be a non-zero vector space and let 0 ̸= λ ∈ S ∗ be a non-zero
linear functional. If
φ : Sλ −→ Sλ is a semigroup automorphism then φ(0) = 0.
Theorem 2.3. Let S be a non-zero vector space, let 0 ̸= λ ∈ S ∗ be a non-zero linear
functional, and let φ : Sλ −→ Sλ be a non-zero linear map. Then the following
statements are equivalent.
1. φ is a semigroup endomorphism.
2. λ = λ ◦ φ.
3. there exists a linear operator ϕ : S −→ kerλ satisfying, φ(a) = a + ϕ(a), (a ∈
S).
Theorem 2.4. Let S be a non-zero vector space and let 0 ̸= λ ∈ S ∗ be a non-zero
linear functional. Then the linear map φ : Sλ −→ Sλ is a semigroup automorphism
if and only if φ(a) = a + λ(a)c, where c is a constant element of kerλ.
References
[1] A. R. Khoddami and H. R. Ebrahimi Vishki, The higher duals of a Banach
algebra induced by a bounded linear functional, Bull. Math. Anal. Appl., 3 (2011),
118–122.
[2] A. R. Khoddami, Polynomial equations with coefficients in an associative algebra,
Preprint.
115
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Engel degree and Isoclinism classes of finite groups
Hassan Khosravi1∗ and Mahdi Araskhan2
1
Department of Mathematics, Faculty of Sciences, Gonbad-e Kavous University, Gonbad,
Iran
hassan [email protected]
2
Department of Mathematics, Faculty of Sciences, Azad University, Yazd, Iran
[email protected]
Abstract
The aim of this article is to give the concept of Engel degree
of a finite group G, denote by E(G). We shall state some
results concerning the new concept which all mostly new
and similar to some result about d(G), the commutativity
degree of a finite group G. In particular, we prove that if G
and H are both isoclinic groups then E(G) = E(H).
1
Introduction
Let G be any group and n be a non-negative integer. For any two elements a and
b of G, we define inductively [a,n b] the n-Engel commutator of the pair (a, b), as
follows:
[a,0 b] := a, [a,1 b] := [a, b] = a−1 b−1 ab and [a,n b] = [[a,n−1 b], b]
for all n > 0.
An element g of G is called left Engel whenever for each a ∈ G there exists an
integer n = n(a, g) such that [a,n g] = 1. We denote by L(G), the set of all Left
Engel elements of G. The corresponding subset to L(G) which can be similarly
defined is R(G) the set of all right Engel elements of G. An element g of G is called
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D10; Secondary 20E26.
Key words and phrases. Engel degree, finite group, isoclinism classes.
116
right Engel whenever for each x ∈ G there exists an integer n = n(x, g) such that
[g,n x] = 1. For any group G the statements G = L(G) and G = R(G) are clearly
equivalent, and a group with this property is called Engel group. M. Zorn [4] has
shown that Engel condition in finite group equivalent with nilpotency condition.
Therefore if G is a locally finite group then L(G) equals the Hirsch-Plotkin radical
and R(G) is a subgroup of L(G). In particular, if G satisfies the maximal condition,
then R(G) coincide with the hypercenter, which equals Zm (G) for some finite m(see
[3] Baer’s Theorem p. 360). We define
E(G) =
1
|{(x, y) ∈ G × G| ∃ n ∈ N, [x,n y] = 1}|
|G|2
Obviously, G is an Engel group if and only if E(G) = 1. By Zorn’s Theorem we
can introduce E(G) as Engel degree or equivalently nilpotency degree for a group
G. Moghadam et. al. [1] introduce n-nilpotency degree of a group G as follow
d(n) (G) =
1
|{(x1 , ..., xn+1 ) ∈ Gn+1 | [x1 , ..., xn+1 ] = 1}|.
|G|n+1
It is clear that in this definition d(n) (G) is dependent on n, so E(G) is a more
perfect definition for probability of nilpotency of elements of a group G. T. Peng [2]
define the class of E-groups as the class of those groups G in which EG (x) := {g ∈
G| [g,n x] = 1 f or some n ∈ N} is a subgroup for every x ∈ G; and, for any prime p,
the class Ep of those groups in which EG (x) is a subgroup for every p-element x of
G. Also he has shown that G is an E-group iff it is an Ep -group for every prime p.
In this paper we are interested in finding some results as above for E(G). In fact
we prove that
Theorem 1.1. Let G be a non-Engel finite E-group. Then
1. E(G) ≤ 23 .
2. If E(G) =
2
2
3
then
G
L(G)
∼
= Z2 .
Main results
Lemma 2.1. Let G and H be finite groups. Then E(G × H) = E(G) × E(H).
Proof. Let EG (x) = {y ∈ G| ∃ n ∈ N, [y,n x] = 1}, for an arbitrary group G and x
117
in G. Then we have
1
|{((x, y), (x′ , y ′ )) ∈ (G × H)2 |
|G × H|2
∃n ∈ N, ([x,n x′ ], [y,n y ′ ]) = 1}|
∑
1
=
|E(G×H) (x, y)|
2
2
|G| |H|
E(G × H) =
(x,y)∈G×H
∑
=
1
2
|G| |H|2
=
1 ∑
1 ∑
|E
(x)|
|EH (y)|
G
|G|2
|H|2
|EG (x)||EH (y)|
(x,y)∈G×H
x∈G
y∈H
= E(G)E(H)
Theorem 2.2. Let G be a finite E-group and N be a normal subgroup of G such
G
that N ≤ R(G). Then E(G) = E( N
). In particular E(G) = E( Z∗G(G) ), that Z∗ (G)
is the hypercenter of group G.
In the end of this Section we prove that
Theorem 2.3. Let G and H be two isoclinic finite groups. Then E(G) = E(H).
References
[1] M. R. R. Moghaddam, K. Chiti, A. R. Salemkar, n-Isoclinism classes and nnilpotency degree of finite group, Algebra Colloquium 12(2)(2005) 255-261.
[2] T. A. Peng, Finite soluble groups with an Engel condition, J. Aljebra 14 (1969),
319-330.
[3] D. J. S. Robinson, A course in the theory of groups, Springer-Verlag,, New York,
1982.
[4] M. Zorn, Nilpotency of finite groups, Bull. Amer. Math. Soc. 42 (1936) 485-486.
118
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Burnside condition on some intersection subgroups
Hanieh Mirebrahimi1∗ and Fatemeh Ghanei2
1
Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
h− [email protected]
2
Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
[email protected]
Abstract
In this paper, first we present some preliminaries about
graphs, core graphs, and combinatorial algebraic topology.
Using these tools, and specially using immersions and covering maps, we establish our main theorem. Indeed, we can
prove the Burnside condition for the intersection of those
subgroups of free groups satisfying the Burnside condition.
1
Introduction
All our conceptions come from [1], [2] and [3]. A graph X consists of two sets E and
V (edges and vertices), with three functions −1 : E −→ E and s, t : E −→ V such
that (e−1 )−1 = e, e−1 ̸= e, s(e−1 ) = t(e) and t(e−1 ) = s(e). We say that the edge
e ∈ E has initial vertex s(e) and terminal vertex t(e). The edge e−1 is the reverse
of e.
A map of graphs f : X −→ Y is a function which maps edges to edges and vertices
to vertices. Also we have f (e−1 ) = f (e)−1 , f (s(e)) = s(f (e)) and f (t(e)) = t(f (e)).
A path p in X of length n = |p|, with initial vertex u and terminal vertex v, is an
n-tuple of edges of X of the form p = e1 ...en such that for i = 1, ..., n − 1, we have
t(ei ) = s(ei+1 ) and s(e1 ) = u and t(en ) = v. For n = 0, given any vertex v, there
∗
Speaker
2010 Mathematics Subject Classification. Primary 05E15 05E18; Secondary 55Q05.
Key words and phrases. Graphs, fundamental group, immersion and covering theory, Burnside
condition.
119
is a unique path Λv of length 0 whose initial and terminal vertices coincide and are
equal to v. A path p is called a circuit if its initial and terminal vertices coincide.
If p and q are paths in X and the terminal vertex of p equals the initial vertex
of q, they may be concatenated to form a path pq with |pq| = |p| + |q|, whose initial
vertex is that of p and whose terminal vertex is that of q.
A round-trip is a path of the form ee−1 . A reduced path is a path in X containing
no round-trip. An elementary reduction is insertion or deletion a round-trip in a
path. Two paths p and q are homotopic (written p ∼ q) iff there is a finite sequence
of elementary reductions taking one path to the other. Homotopic paths must have
the same start and terminal vertices and also, homotopy is an equivalence relation
on the set of paths in X. Moreover, any path in X is homotopic to a unique reduced
path in X.
Let v be a fix vertex in X, π1 (X, v) is defined to be the set of all homotopy
classes of closed paths with initial and terminal vertex v. Then π1 (X, v) together
with the product [p][q] := [pq] forms a group with identity [Λv ] and inverse element
[w]−1 = [w−1 ].
For a fix vertex v in X, the star of v in X is defined as follows:
St(v, X) = {e ∈ E : s(e) = v}.
A map f : X −→ Y yields, for each vertex v ∈ X, a function fv : St(v, X) −→
St(f (v), Y ). If for each vertex v ∈ X, fv is injective, we call f an immersion. If
each fv is bijective, we call f a covering.
The theory of coverings of graphs is almost completely analogous to the topological theory of coverings. Immersions have some of the properties of coverings. One
of them which is more important, and we also need it more, is the following one:
”For a given finite set of elements {α1 , ..., αn } ⊆ π1 (X, u), there is a connected
graph Y and an immersion f : Y −→ X such that f∗ (π1 (Y )) = S, in which S is the
subgroup of π1 (X, u) generated by {α1 , ..., αn }”.
If G is a group, a G-graph X is a graph with an action of G on the left on X by
maps of graphs, such that for all g ∈ G and every edge e, ge ̸= e−1 . In this case, the
quotient graph X/G, and the quotient map of graphs X → X/G can be defined. It
is easy to see that, in general X → X/G is locally surjective.
It is said that G acts freely on X when, whenever v is a vertex of X, g ∈ G,
and gv = v, then g = 1, the identity element of G. In this case, X → X/G is an
immersion, and hence is a covering.
A ttranslation of a map of graphs f : X → Y is a map g : X → X which is an
isomorphism of graphs and for which f g = f . The set of all translations of f forms
a group G(f ) which acts on X. If f is an immersion, and X is connected, then G(f )
acts freely on X.
120
The universal cover f : X̃ → X, of a connected graph X, is a covering with (X̃
is connected and π1 (X̃) trivial. In this case, G(f ) ∼
= π1 (X) which acts freely, by
covering translations, on X̃, and f is isomorphic to the quotient map X̃ → X̃.
Theorem 1.1. [4] Let
be a pullback diagram of graphs, where f1 and f2 are immersions. Let v1 and v2 be
vertices in Z1 and Z2 that f1 (v1 ) = f2 (v2 ) = w. Let v3 be corresponding vertex in
Z3 . Define f3 = f1 g1 = f2 g2 : Z3 → X, and Si = fi∗ (π1 (Zi , vi )), for i = 1, 2, 3. Then
S3 = S1 ∩ S2 .
Theorem 1.2. [4] Let f : X −→ Y be an immersion of graphs. Suppose that Y has
only one vertex and X has only finitely many vertices. Then there exists a graph
X́ containing X such that X́ − X consists only of edges, and there exists a map
f´ : X́ −→ Y extending f such that f´ is a covering.
2
Main results
In this section, we deduce our main result. before it, we recall some notes from [4]
which are essential in the proof of the main theorem. First, we note to the core
graphs whose roles are more important.
A cyclically reduced circuit in a graph X is a circuit p = e1 ...en , which is reduced
as a path and for which e1 ̸= e−1
n . A graph X is said to be a core-graph if X is
connected, has at least one edge and every edge belongs to at least one cyclically
reduced circuit.
If X is a connected graph with non-trivial fundamental group, an essential edge
of X is an edge belonging to some cyclically reduced circuit. The core of X consists
of all essential edges of X and all initial vertices of essential edges.
If X is a connected graph with non-trivial fundamental group and X́ is the core
of X, then X́ is a core-graph. If v is a vertex of X́, then the inclusion π1 (X́, v) −→
π1 (X, v) is an isomorphism.
Another notion, we are dealing with, is the Burnside condition for subgroups. If
S is a subgroup of a group G, we say that S ⊆ G satisfies the Burnside condition
when, for every g ∈ G, there exists some positive integer n such that g n ∈ S.
121
Lemma 2.1. [4] (a) Let f : X −→ Y be a finite-sheeted covering of connected
graphs, v a vertex of X. Then f∗ (π1 (X, v)) ⊆ π1 (Y, f (v)) satisfies the Burnside
condition.
(b) Let f : X −→ Y be an immersion of connected graphs. Suppose that Y
is a core-graph; v a vertex of X, f∗ (π1 (X, v)) ⊆ π1 (Y, f (v)) satisfies the Burnside
condition. Then f is a covering.
Finally, using all the above notes, we establish the following theorem, which is
our main result in this paper.
Theorem 2.2. Let S1 and S2 be finitely generated subgroups of a free group F .
Suppose that S1 ∩ S2 satisfies the Burnside condition both in S1 and S2 . Then
S1 ∩ S2 satisfies the Burnside condition in the join S1 ∨ S2 , the subgroup generated
by A ∪ B.
References
[1] B. Everitt, The geometry and topology of groups, Notes from lectures given at
the Universidad Autonama, Madrid, and the University of York (2003).
[2] B. Everitt, Galois theory, graphs and free groups, arXiv:0606326, (2006).
[3] B. Everitt, Graphs, free groups and the Hanna Neuman conjecture, Journal of
Group Theory 11 (2008), no. 6, 885–899.
[4] John R. Stallings, Topology of finite graphs, Invent. math. 71 (1983), 551–565.
122
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Some properties of centralizer and autocommutator
subgroup in auto-Engel groups
Mohammad Reza R. Moghaddam1 and M. Badrkhani Asl2∗
1
2
Department of Mathematics, Mashhad Branch, Islamic Azad University, and
Khayyam Higher Education Institute, Mashhad, Iran
[email protected]
Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
[email protected]
Abstract
In 1994, Hegarty introduced the notion of autocentral automorphism, L(G) and autocommutator subgroup of G,
K(G), and proved that if G/L(G) is finite, then K(G)
is finite. Also if K(G) and Aut(G) are both finite, he
showed that so is G/L(G), but did not construct an upper bound for G/L(G). In the present article, we show
that if K(G) is finite and Aut(G) is finitely generated, then
|G/L(G)| ≤ |K(G)|d , where d is the minimal number of generators of the automorphisms group of G. In 2012, Endimion
and Moravec considering polycyclic groups and proved that
the finiteness of L(G) implies that G/K(G) is also finite.
In this article we show that a similar result hold for the
second autocentral and autocommutator subgroups, where
introduced by the first author et. al in 2011.
∗
Speaker
2010 Mathematics Subject Classification. Primary 20E36; Secondary 20F28.
Key words and phrases. Polycyclic groups, auto-Engel, autocentral and auocommutator subgroups.
123
1
Introduction
In 1904, I. Schur showed that the finiteness of the central factor group of a given
group implies that the derived subgroup is also finite. B. H. Neumann in 1951 proved
that the converse of Schur’s result holds, for finitely generated groups. In 2005, K.
Podoski and B. Szegedy proved that for a group G, if [γ2 (G) : γ2 (G) ∩ Z(G)] = n,
n
then [G : Z(G)] ≤ n2 log2 . One notes that the converse of Schur’s theorem is not
true in general. For the counterexample, consider the infinte extra special p-group,
for any odd prime number p. Niroomand in 2010 generalized Neumann’s theorem
as follows: If γ2 (G) is finite and G/Z(G) is finitely generated, then [G : Z(G)] ≤
|γ2 (G)|d(G/Z(G)) , where d(X) is the minimal number of generators of a group X.
Also if the set of commutator elements S of a group G is finite and G/Z(G) is
finitely generated, then B. Sury in 2010 proved that γ2 (G) is finite and also he gave
an upper bound for the central factor group [G : Z(G)] ≤ |S|d(G/Z(G)) . D. Gumber
et.al , by assuming G/Z(G) = ⟨x1 Z(G), . . . , xn Z(G)⟩
∏nsuch that [xi , G] is finite, for
all 1 ≤ i ≤ n, then they showed that [G : Z(G)] ≤ i=1 |[xi , G]| and |γ2 (G)| < ∞.
If α is an automorphism of a group G and x ∈ G, then [x, α] = x−1 α(x) is the
autocommutator of x and α. So one may define
⟨
⟩
L(G) = x ∈ G|[x, α] = x−1 α(x) = 1, f or all α ∈ Aut(G) ,
and
K(G) = ⟨[x, α]|x ∈ G, α ∈ Aut(G)⟩,
which are called the autocenter and autocommutator subgroups of G, respectively.
Clearly both subgroups are characteristic so that L(G) is contained in the center of
G and K(G) contains the derived subgroup (see also [3]). In [2], P. Hegarty proved
that if G/L(G) is finite, then so is K(G). Also he showed that G/L(G) is finite,
whenever both K(G) and Aut(G) are finite. In this section, we construct an upper
bound for the autocentral factor group of G, assuming its autocommutator subgroup
is finite and the automorphism group is finitely generated. For any automorphism
α of the group G, we may define
CG (α) = {x ∈ G|[x, α] = 1},
[G, α] = ⟨[x, α]|x ∈ G⟩,
which are the centralizer of α in G and the commutator subgroup of α, which is a
subgroup of [G, Aut(G)] = K(G). Clearly, they are both normal subgroups of G so
that CG (α) contains the autocenter of G. The nth -autocommutator subgroup and
nth -autocenter of G can be defined, respectively as follows:
Kn (G) = ⟨[x, α1 , ..., αn ] = [[α1 , ..., αn−1 ], αn ]|x ∈ G, αi ∈ Aut(G)⟩,
124
Ln (G) = {x ∈ G|[x, α1 , ..., αn ] = 1, f or all αi ∈ Aut(G)}.
So we obtain the following series
G = K0 (G) ⊇ K(G) = K1 (G) ⊇ K2 (G) ⊇ · · · ⊇ Kn (G) ⊇ · · · ,
1 = L0 (G) ⊆ L1 (G) = L(G) ⊆ · · · ⊆ Ln (G) ⊆ · · · .
One observes that Kn (G) and Ln (G) are both characteristic subgroups in G such
that Kn (G) contains γn+1 (G) and Ln (G) is contained in Zn (G)(see [3] for more
information).
2
Main Results
As it was discussed in the introduction, Hegarty in [2] generalized Schur’s theorem.
Theorem 2.1.
Aut(G).
(a) [2, Theorem 1.1] If [G : L(G)] is finite, then so are K(G) and
(b) [2, Theorem 1.2] If K(G) and Aut(G) are both finite, then so is [G : L(G)].
In the following, for a given group G we construct an upper bound for the order
of G/L(G) when K(G) is finite and Aut(G) is finitely generated.
Theorem 2.2. Let G be an arbitrary group such that K(G) is finite and its automorphism group is finitely generated. Then |G/L(G)| ≤ |K(G)|d , where d is the
minimal number of generators of Aut(G).
Consider the cyclic group of order 2, Z2 , then clearly L(Z2 ) is trivial and
K(Z2 ) = Z2 . So in Theorem 2.2, the equality holds which shows that the upper bound is attained. Also one can easily check that the automorphism group of
the Dihedral group D8 is isomorphic to D8 , L(D8 ) = Z2 and K(D8 ) = Z4 which
gives |K(D8 )|d(D8 ) = 16, while |D8 /L(D8 )| = 4.
For a given group G, the element x ∈ G is called right n-auto-Engel if [x, α, . . . , α] =
| {z }
n−times
1, for all α ∈ Aut(G). The element x is said to be left n-auto-Engel, whenevere
[α, . . . , α, x] = 1. A group G is called n-auto-Engel if [x, α, . . . , α] = 1, for all x ∈ G
| {z }
n−times
and α ∈ Aut(G) (see also [4] for more details). For the automorphisms α1 and α2
of a given group G, we introduce the centralizer of {α1 , α2 } as follows:
CG ((α1 , α2 )) = {x ∈ G|[x, α1 , α2 ] = 1}.
Considering 2-auto-Engel groups, one can easily check that CG ((α1 , α2 )) and [G, α1 , α2 ] =
⟨[x, α1 , α2 ]|x ∈ G⟩ are both normal subgroups of G.
The proof of the following lemma can be seen immediately by the definition.
125
Lemma 2.3. Let N be a characterisctic subgroup of a 2-auto-Engel group G with
the automorphisms α1 and α2 . Then
1. CG (α1 ) ≤ CG ((α1 , α2 ));
2. |CG/N ((α1 , α2 ))| ≤ |CG ((α1 , α2 ))|, where G is considered to be finite.
Endiomio and Moravec [1], proved the following result.
Theorem 2.4. [1, Theorem 2] Let α be an automorphism of a polycyclic group G.
If CG (α) is finite, then so is G/[G, α].
In our terminology, we conclude the following corollary of the above theorem.
Corollary 2.5. In a polycyclic group G, if L(G) is finite then so is G/K(G).
In the above theorem, we impose an extra assumption so that the group G
is 2-auto-Engel. Then we prove that the finiteness of CG ((α1 , α2 )) implies that
G/[G, α1 , α2 ] is also finite. It is equivalent to saying that, if L2 (G) is finite, so is
G/K2 (G).
Theorem 2.6. Let G be a polycyclic 2-auto-Engel group with CG ((α1 , α2 )) is finite,
for α1 , α2 ∈ Aut(G). Then G/[G, α1 , α2 ] is also finite.
The above theorem has the following corollaries.
Corollary 2.7. If G is a polycyclic 2-auto-Engel group with finite L2 (G), Then
G/K2 (G) is also finite.
Corollary 2.8. If G is a polycyclic 2-auto-Engel group with finite L2 (G), Then
G/Z(G) is also finite.
References
[1] G. Endimioni. and P. Moravec, On the centralizer and the commutator subgroup
of an automorphism, Monatshefte fr Mathematik, 167, (2012), 165-174.
[2] P. V. Hegarty, Autocommutator subgroups of finite groups, J. Algebra,
190(1997), 556-562.
[3] M. R. R. Moghaddam, F. Parvaneh and M. Naghshineh, The lower autocentral
series of abelian groups, Bull. Korean Math. Soc.48(2011), 79-83.
[4] M. R. R. Moghaddam, H. Safa and M. Farrokhi D. G., Some properties of 2auto-Engle groups, to appear in Quaestiones Mathematicae.
126
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Counting centralizers in non-abelian n-dimensional Lie
algebras
M. R. R. Moghaddam1 , M. Hoseini Ravesh2 and S. Saffarnia2∗
1
Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of
Mashhad and Khayyam Higher Education Institute, Mashhad, Iran
[email protected]
2
International Campus, Faculty of Mathematical Sciences, Ferdowsi University of
Mashhad, Mashhad, Iran
[email protected], s saff[email protected]
Abstract
In this paper we study the number of centralizers of nonabelian n-dimensional Lie algebra. Given a Lie algebra L
and an element x ∈ L, the set CL (x) = {y ∈ L : [x, y] =
0} is called the centralizer of x in L. The set of all such
centralizers in L is denoted by Cent(L).
1
Introduction
Let x be a non-identity element of a given group G. Then the centralizer of x in G
is denoted by CG (x). Clearly a group is 1-centralizer if and only if it is abelian. In
1994 Belcastro and Sherman [3] proved the following results:
(i) There is no 2-centralizer and no 3-centralizer groups.
(ii) A finite group G is 4-centralizer if and only if
G
Z(G)
∼
= C2 × C2 .
(iii) A finite group G is 5-centralizer if and only if
G
Z(G)
∼
= C3 × C3 or S3 .
∗
Speaker
2010 Mathematics Subject Classification. Primary 20E06, 20E08; Secondary 20E70.
Key words and phrases. Lie algebra, centralizer, n-centralizer Lie algebras.
127
Let L be a finite dimension Lie algebra over the field F . Then for any element
x ∈ L, the set CL (x) = {y ∈ L | [x, y] = 0} is called the centralizer of x in L. The
set of all centralizers in L is denoted by Cent(L) and |Cent(L)| denotes the number
of distinct centralizers in L. A Lie algebra L is called n-centralizer if |Cent(L)| = n
and L is called primitive n-centralizer if |Cent(L/Z(L))| = |Cent(L)| = n, where
Z(L) denotes the centre of L. A subalgebra K of L is called a proper centralizer of
L if K = CL (x), for some x ∈ L \ Z(L).
Similar to group theory one can easily see that L is abelian if and only if
|Cent(L)| = 1.
Lemma 1.1. Let L1 and L2 be two Lie algebras, then
Cent(L1 ⊕ L2 ) = Cent(L1 ) ⊕ Cent(L2 ).
Lemma 1.2. Let ∩
L be a Lie algebra. Then Z(L) is the intersection of all centralizers
in L, i.e. Z(L) = x∈L CL (x), for all x ∈ L.
Lemma 1.3. If L is a Lie algebra,
then L is the union of centralizers of all non∪
central elements of L, i.e. L = x∈L−Z(L) CL (x).
Lemma 1.4. A Lie algebra L cannot be written as a union of two proper Lie
subalgebras.
Theorem 1.5. Let L be a non-abelian Lie algebra, then |Cent(L)| ≥ 4.
2
Main Results
In this section, we study the centralizers of low-dimensional Lie algebra over the
field of p elements Zp , for any prime number p.
Lemma 2.1. Suppose that Li is a finite dimension Lie
with |Cent(Li )| = ni ,
∏algebra
r
for i = 1, 2, ..., r. Then |Cent(L1 ⊕ L2 ⊕ ... ⊕ Lr )| = i=1 ni .
Proof. Suppose L = L1 ⊕ L2 ⊕ ... ⊕ Lr . Using Lemma 1.1, we have
CL (x1 , x2 , ..., xr ) = CL1 (x1 ) ⊕ CL2 (x2 ) ⊕ ... ⊕ CLr (xr ),
for all (x1 , x2 , ..., xr ) ∈ L. It follows that CL (x1 , x2 , ..., xr ) = CL (y1 , y2 , ..., yr ) if and
only
∏r if CLi (xi ) = CLi (yi ), for all 1 ≤ i ≤ r. This implies that |Cent(L1 ⊕ L2 ⊕ ... ⊕ Lr )| =
i=1 ni .
Lemma 2.2. Let L be a finite dimension Lie algebra and K a subalgebra of L.
Then |Cent(K)| ≤ |Cent(L)|.
128
Proof. Let CK (ki ) be the distinct centralizers in K and i = 1, 2, ..., m. On the other
hand CK (ki ) = K ∩ CL (ki ), then CL (ki ) ̸= CL (kj ), for all i ̸= j and hence we obtain
the claim.
Lemma 2.3. Let L be an n-centralizer Lie algebra and L′ ∩ Z(L) = 0. Then L is
a primitive n-centralizer Lie algebra.
Proof. Suppose that Cent(L) = {CL (x1 ), CL (x2 ), ..., CL (xn )} is the distinct centralizers in L. One can easily check that C(x + Z(L)) = C(x)/Z(L). Hence it is
enough to show that for any 1 ≤ i ̸= j ≤ n, CL (xi + Z(L)) ̸= CL (xj + Z(L)). So
assume there exist some 1 ≤ i ̸= j ≤ n such that CL (xi + Z(L)) = CL (xj + Z(L)).
Suppose y ∈ CL (xi ), then y + Z(L) ∈ CL (xi + Z(L)) = CL (xj + Z(L)) and by the
assumption we get [y, xj ] = 0, i.e., CL (xi ) ⊆ CL (xj ). Using similar argument, we
have CL (xj ) ⊆ CL (xi ) which gives a contradiction. Thus |Cent(L/Z(L))| = n and
hence L is a primitive n-centralizer.
In the following we determine the number of centralizer of 2-dimension nonabelian Lie algebras over the field of p elements.
Theorem
2.4. Let L be a 2-dimensional non-abelian Lie algebra over the field Zp .
Then Cent(L) = p + 2.
Definition 2.5. Let L be a non-abelian 3-dimensional Lie algebra over a field F ,
with L′ to be 1-dimension so that L′ is contained in Z(L). Such a Lie algebra is
known as the Heisenberg Lie algebra.
Theorem 2.6. Let L be the Heisenberg Lie algebra over the field Zp , then |Cent(L)| =
p + 2.
Example
2.7. n(3, Zp ) = ⟨e12 , e13 , e23 ⟩, [e12 , e23 ] = e13 and L′ = Z(L). Hence
Cent(n(3, Zp )) = p + 2.
As in Theorem 3.2[4], there exists a unique 3-dimensional Lie algebra over a field
F such that L′ is 1-dimensional and L′ ̸⊆ Z(L). Such a Lie algebra is the direct
sum of the 2-dimensional non-abelian Lie algebra with 1-dimensional Lie algebra.
Theorem 2.8. Let L be the 3-dimensional Lie algebra as above over the field Zp .
Then |Cent(L)| = p + 2.
Lemma 2.9. Let L be a Lie algebra such that dimL = 3 and dimL′ = 2. Then
(i) L′ is abelian.
(ii) The map adx : L′ → L′ is an isomorphism, for all x ∈ L − L′ .
129
Proof. See [4], Lemma 3.3.
Theorem 2.10. Let
L be a Lie2 algebra over the field Zp such that dimL = 3 and
′
dimL = 2. Then Cent(L) = p + 2.
The following example justifies the above theorem.
Example 2.11. Consider the 3-dimension Lie algebra as in Theorem 2.10 over
the field Z5 , then one may calculate all the centralizers of L and observes that
|Cent(L)| = 27.
On the other hand, by Theorem 2.10, the number of distinct centralizers must
be (5 − 1)2 + 2(5) + 1 = 27.
Theorem 2.12. Let
L be a Lie algebra over the field Zp such that dimL = 3 and
dimL′ = 3. Then Cent(L) = (p − 1)2 + 3p + 1.
The following example justifies the above theorem.
Example 2.13. The sets of distinct centralizers of the 3-dimension Lie algebra in
Theorem 2.12 over the field Z3 are as follows:
CL (x) = {0, x, 2x}, CL (y) = {0, y, 2y}, CL (z) = {0, z, 2z}, CL (0) = L,
CL (x + y) = {0, x + y, 2x + 2y, },
CL (2x + y) = {0, 2x + y, x + 2y, },
CL (x + z) = {0, x + z, 2x + 2z, },
CL (2x + z) = {0, 2x + z, x + 2z, },
CL (y + z) = {0, y + z, 2y + 2z, },
CL (2y + z) = {0, 2y + z, y + 2z, },
CL (x + y + z) = {0, x + y + z, 2x + 2y + 2z, },
CL (2x + y + z) = {0, 2x + y + z, x + 2y + 2z, },
CL (x + 2y + z) = {0, x + 2y + z, 2x + y + 2z, },
CL (x + y + 2z) = {0, x + y + z, 2x + 2y + 2z, }.
So |Cent(L) = 14 and using Theorem 2.12, we get the same number, i.e. |Cent(L)| =
(3 − 1)2 + 3(3) + 1 = 14.
References
[1] A. Abdollahi, S. M. J. Amiri and A. M. Hassanabadi, Groups with specific number
of centralizers, Houston J. Math., 33(1) (2007), 43-57.
[2] A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra
Colloq., 7(2) (2000), 139-146.
130
[3] S. M. Belcastro and G. J. Sherman, Counting centralizers in finite groups, Math.
Magazine, 67(5) (1994), 366-374.
[4] Karin Erdmann and Mark J. Wildon. Introduction to Lie algebras. Undergraduate Mathematics Series. Springer-Verlag, London Ltd., 2006.
131
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Some properties of 2-Engel transitive groups
Mohammad Reza R. Moghaddam1 and Amin Rostamyari2∗
1
Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of
Mashhad and Khayyam Higher Education Institute, Mashhad, Iran
[email protected]
2
International Campus, Faculty of Mathematical Sciences, Ferdowsi University of
Mashhad, Mashhad, Iran
[email protected]
Abstract
In 2013, Ciobanu, Fine and Rosenberger introduced and
studied the relationship among the notions of conjugately
separated abelian, commutative transitive and fully residually χ-groups. In this article we introduce the concept of
2-Engel transitive groups. Among other results we establish the relationship of 2-Engel transitive groups with conjugately separated 2-Engel and fully residually χ-groups.
1
Introduction
An element x of a group G is called a right Engel element, if for every y ∈ G,
there exists a natural number n = n(x, y) such that [x, n y] = 1. If n can be chosen
independent of y, then x is called a right n-Engel element or simply a bounded right
Engel element. We denote the sets of all right Engel elements and bounded right
Engel elements of G by R(G) and R(G), respectively.
An element x of G is called a left Engel element, if for every y ∈ G, there exists
a natural number n = n(x, y) such that [y, n x] = 1. If n can be chosen independent
of y, then x is called a left n-Engel element or simply a bounded left Engel element.
∗
Speaker
2010 Mathematics Subject Classification. Primary 20E06, 20E08; Secondary 20E70.
Key words and phrases. 2-ET group, CSE 2 -group, residually χ-group, fully residually χ-group.
132
We denote the sets of all left Engel elements and bounded left Engel elements of
G by L(G) and L(G), respectively. For any positive integer n, a group G is called
n-Engel group if [x, n y] = [y, n x] = 1, for all x, y ∈ G.
Let χ be a class of groups. Then a group G is residually χ if for every nontrivial
element g ∈ G, there is a homomorphism ϕ : G → H, where H is a χ-group such
that ϕ(g) ̸= 1. Also a group G is fully residually χ if for finitely many nontrivial
elements g1 , ..., gn in G there exists a homomorphism ϕ : G → H where H is a
χ-group such that ϕ(gi ) ̸= 1, for all i = 1, ..., n.
Definition 1.1. A subgroup H of a group G is called malnormal or conjugately
separated if H ∩ H x = 1, for all x ∈ G − H.
It is clear that the intersection of a family of malnormal subgroups of a given
group G is again malnormal, which allows us to define the malnormal closure of a
subgroup H of G. Clearly the intersection of all malnormal subgroups of G contains
H is malnormal.
2
Main Results
A group G is called a conjugately separated 2-Engel (henceforth CSE 2 -group) if all
of its maximal 2-Engel subgroups are malnormal. In the following, we introduce the
notion of 2-Engel transitive group and then give its relationship with CSE 2 -group
and fully residually χ-groups.
Definition 2.1. (a) A group G is 2-Engel transitive (henceforth 2-ET), if [x, y, y] =
1 and [y, z, z] = 1 implies that [x, z, z] = 1, for all nontrivial elements x, y, z in G.
2 (x) = {y ∈ G : [x, y, y] = 1, [y, x, x] = 1}
(b) For a given element x of G, we call EG
to be the set of 2-Engelizer of x in G. The set of all 2-Engelizers in G is denoted by
E 2 (G) and |E 2 (G)| denotes the number of distinct 2-Engelizers in G.
The following lemma is useful for our further investigations.
Lemma 2.2. For any nontrivial group G, the following statements are equivalent:
(i) G is 2-ET group;
(ii) 2-Engelizer of each nontrivial element of G is 2-Engel set.
We remark that the 2-Engelizer of each nontrivial element of a group G does not
form a subgroup, in general. However using Lemma 2.2, if G is a 2-ET group, then
each 2-Engelizer of nontrivial element of G is 2-Engel.
The following lemma is very usefull.
133
2 (x) and x ∈ G,
Lemma 2.3. Let G be a 2-ET group. Then for all y, z ∈ EG
−1
(i) [x, y, z] = [x, z, y] ;
(ii) [x, [y, z]] = [x, y, z]2 .
Corollary 2.4. If G is a 2-ET group, then the set of each 2-Engelizer of a nontrivial
element x in G forms a subgroup.
The proof of the following lemma is a routine argument by using Zorn’s Lemma.
Lemma 2.5. Every 2-Engel subgroup H of a given group G is contained in a
maximal 2-Engel subgroup.
The following lemma gives a connection between CSE 2 -groups and 2-Engel transitive groups.
Lemma 2.6. Every CSE 2 -group is a 2-ET group.
The following fact is needed in proving our main result.
Proposition 2.7. Let G be a CSE 2 -group, then every 2-Engel normal subgroup of
G is maximal.
Using the above proposition, we obtain the following useful result.
Corollary 2.8. Let G be a CSE 2 -group, then every 2-Engel normal subgroup of G
is equal to the second centre of G.
Lemma 2.9. Let χ be a class of groups such that each non-2-Engel group H ∈ χ
is CSE 2 -group. Let N be a 2-Engel normal subgroup of a non-2-Engel residually
χ-group G, then N is contained in the second centre of G.
Remark 2.10. Let G be a 2-ET and non 2-Engel group, then it is clear that
Z2 (G) = 1. So it follows from the above lemma that any normal 2-Engel subgroup
of G must be trivial.
Now we establish the relationship between the non 2-Engel CSE 2 , 2-ET and
fully residually χ-groups.
Theorem 2.11. Let χ be a class of groups such that each non 2-Engel χ-group is
CSE 2 and G be a non 2-Engel and residually χ-group. Then the following statements are equivalent:
(i) G is fully residually χ;
(ii) G is CSE 2 ;
(iii) G is 2-ET.
134
In 1967, B. Baumslag [1] introduced the notion of fully residually free groups
and proved that a residually free group is fully residually free if and only if it is
commutative transitive. A group G is commutative transitive, if [x, y] = 1 and
[y, z] = 1 implies that [x, z] = 1, for nontrivial elements x, y, z in G.
One can easily check that if G is commutative transitive then G is 2-Engel
transitive. Here we show that Baumslag’s theorem is also true in the case of 2-Engel
transitive groups.
Corollary 2.12. Let G be a residually free group. Then G is fully residually free
if and only if G is 2-Engel transitive.
References
[1] B. Baumslag, Residually free groups, Proc. London Math. Soc. 17 No.3 (1967)
402-418.
[2] L. Ciobanu, B. Fine and G. Rosenberger, Classes of groups generalizing a theorem of Benjamin Baumslag, Preprint.
[3] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Parts
1 and 2, Springer-Verlag, 1972.
135
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Embedding a special subgroup in n-autocentral
subgroups of a group
Mohammad Reza R. Moghaddam1 and Mohammad Javad Sadeghifard2∗
1
2
Department of Mathematics, Mashhad Branch, Islamic Azad University and Khayyam
Higher Education Institute, Mashhad, Iran
[email protected]
Department of Mathematics, Mashhad Branch, Islamic Azad University , Mashhad, Iran
[email protected]
Abstract
For a given group G and positive integer n, W.P. Kappe
in 2003 introduced and studied the subgroup Bn (G), which
contains all 2-Engel elements when n = 1. Also the first author et. al. in 2013 introduced the concept of 2-auto-Engel
elements of G. In this article, we introduce and study the
subgroup ABn (G), which contains all 2-auto-Engel elements
of G. Among other result, it will be shown that ABn (G) is
embedded in some subgroups of G.
1
Introduction and Preliminary results
We denote the automorphisms of a group G by A = Aut(G) and Inn(G) is the
inner automorphisms of G. In 2003, W.P. Kappe [1] defined the following subset of
elements of the group G
Bn (G) = {x ∈ G | [x, g, a1 , . . . , an , g] = 1, ∀g, a1 , . . . , an ∈ G}, n ≥ 1.
He then showed that the above set is a characteristic subgroup of G and B1 (G)
contains the set of all right 2-Engel elements of G. The first author et.al. [3]
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D45; Secondary 20F45, 20E07, 20F99.
Key words and phrases. 2-auto-Engel element, Engel element, autocentre.
136
introduced the set of all right n-auto-Engel elements of the group G which is denoted
by ARn (G) = {g ∈ G | [g,n α] = 1, ∀α ∈ A}. Here [g, α] = g −1 g α = g −1 .(g)α,
[g, α1 , . . . , αn ] = [[g, α1 , . . . , αn−1 ], αn ] and [g,n α] = [g, α, . . . , α]. It is shown in [3]
| {z }
n
that AR2 (G) is a subgroup of G.
In the present article, using the automorphisms group of G we introduce the
subgroup of G which is some how a generalization of AR2 (G), and will be denoted
by ABn (G).
Now, the following definition is vital in our further investigation.
Definition 1.1. Let G be any group. For a positive integer n, let
ABn (G) = {g ∈ G | [g, β, α1 , . . . , αn , β] = 1, ∀β, α1 , . . . , αn ∈ A};
ACn (G) = {g ∈ G | [gh, β, α1 , . . . , αn , β] = [h, β, α1 , . . . , αn , β],
∀h ∈ G, β, α1 , . . . , αn ∈ A}.
As observed above, ACn (G) ⊆ ABn (G). It will be shown that ACn (G) =
ABn (G), which implies that ABn (G) is a characteristic subgroup of G.
In the next section, we study the basic properties of ABn (G) and also show that
the set of AB1 (G) contains AR2 (G) and it is contained in B1 (G). Finally, we give
some sufficient conditions that when ABn (G) can be embedded in certain subgroups
of G.
In the following, we collect some of the basic facts, which are needed for the
proofs of our main results. The subgroup of right 2-auto-Engel elements of G,
AR2 (G) = {g ∈ G | [g, α, α] = 1, ∀α ∈ A} is a model and the fundamental tool for
the investigation of ABn (G).
The following result of [3] is very useful for our further studies.
Theorem 1.2. ([3] Lemma 3.2 and Theorem 3.3) Let G be a group, then for all
g, h ∈ AR2 (G) and α, β, γ ∈ A
(a) AR2 (G) is a characteristic subgroup of G;
(b) g A = ⟨g α : α ∈ A⟩ is abelian and its elements are right 2-auto-Engel elements;
(c) [g, α, β] = [g, β, α]−1 ;
(d) [g, [α, β]] = [g, α, β]2 ;
(e) [g, α, β, γ]2 = 1;
(f) [g, [α, β], γ] = 1.
137
If α and β are the automorphisms of a group G, we set g αβ = (g α )β , for all
g ∈ G. Using the above notation, and according to identities in [3] about the
autocommutators, we are able to prove the following lemmas which will be used
frequently in the next section.
Lemma 1.3. If [g, α1 , . . . , αm−1 , αm , β] = 1, for a fixed element g ∈ G, α1 , . . . , αm−1 , β ∈
A and all αm ∈ A, then
[[g, α1 , . . . , αm−1 , αm ]γ1 , β γ2 ] = 1, f or all γ1 , γ2 ∈ A.
Lemma 1.4. Let g be an element of ABn (G), then for all β, α1 , . . . , αn , γ0 , . . . , γn ∈
A,
(a) [[g, β, α1 , . . . , αn ]γ1 , β γ2 ] = 1;
(b) [[g, β, α1 , . . . , αn ]γ1 , φγg 2 ] = 1;
(c) [. . . [[g, β]γ0 , α1 ]γ1 , . . . , αn ]γn , β] = 1.
2
Basic properties of ABn (G)
The main goal of this section is to prove that ABn (G) is a characteristic subgroup
of G and we obtain some more properties of this subgroup. We recall that
L(G) = {g ∈ G | [g, α] = 1 or g α = g, ∀α ∈ A},
is the autocentre of G. Clearly, it is a characteristic subgroup of G and one notes
that if we take A = Inn(G) then L(G) = Z(G) is the centre of G. Also we may
define the nth autocentre of G, for n ≥ 1 as follows
Ln (G) = {g ∈ G | [g, α1 , . . . , αn ] = 1, ∀α1 , . . . , αn ∈ A}.
(See [2] for more details).
Theorem 2.1. Let G be a group, then for all g ∈ ABn (G), β, α1 , . . . , αn , γ, φ, ψ ∈ A
and n ≥ 1,
(a) ABn (G) = ACn (G) and hence ABn (G) is a characteristic subgroup of G;
(b) AR2 (G) ⊆ AB1 (G) and ABn (G) ⊆ ABn+1 (G);
(c) [g, β, α1 , . . . , αn , γ, γ] = 1; i.e., [g, β, α1 , . . . , αn ] ∈ AR2 (G).
(d) [g, β, β, α1 , . . . , αn , γ] = 1;
138
(e) [g, β, α1 , . . . , αn , γ, φ, ψ]2 = 1;
(f) [g, β, α1 , . . . , αn , γ] = [g, γ, α1 , . . . , αn , β]−1 .
Proof. (a) By the previous discussion and using induction
[gh, β, α1 , . . . , αn ] = [. . . [[g, β]w0 , α1 ]w1 , . . . , αn ]wn [h, β, α1 , . . . , αn ],
for all h ∈ G and suitable w0 , w1 , . . . , wn ∈ G. Since g ∈ ABn (G), [gh, β, α1 , . . . , αn , β] =
[h, β, α1 , . . . , αn , β], i.e., ABn (G) ⊆ ACn (G) and the result follows, as we already
have ACn (G) ⊆ ABn (G).
(b) Assume g ∈ AR2 (G), then by using Theorem1.2 (c)
[[g, β], α, β] = [[g, β], β, α]−1 = [1, α]−1 = 1,
which gives AR2 (G) ⊆ AB1 (G).
Clearly [g, β, α1 , . . . , αn+1 ] ∈ [g, β, α1 , . . . , αn ]A . Hence Lemma1.4 (a) yields ABn (G) ⊆
ABn+1 (G).
(c) By expansion of [g, βγ, α1 , . . . , αn ], we get
[g, βγ, α1 , . . . , αn ] = [[g, γ][g, β]γ , α1 , . . . , αn ] = y1 y2 ,
where y1 = [. . . [g, γ, α1 ]w1 , . . . , αn ]wn and y2 = [[g, β]γ , α1 , . . . , αn ], for suitable
w1 , . . . , wn ∈ G. Using Lemma1.4 (c), 1 = [y1 , β]γφy2 [y2 , γ]. Therefore [y2 , γ, γ] = 1.
By substitution of αiγ for αi finally gives
1 = [[g, β]γ , α1γ , . . . , αnγ , γ, γ] = [g, β, α1 , . . . , αn , γ, γ]γ ,
which proves the claim.
(d) Substitute [g, β] for g in 1 = [y1 , β]γφy2 [y2 , γ] and note that [y1 , β] = 1 by
Lemma1.4 (c). Thus 1 = [y2 , γ] = [[g, β, β]γ , α1 , . . . , αn , γ], for all αi ∈ A, which
proves part (d).
(e) This part follows from part (c) and Theorem1.2 (e).
(f ) Clearly, 1 = [g, βγ, α1 , . . . , αn , βγ]. As in the proof of part (c), we have
1 = [y1 , β]γφy2 [y2 , γ]. Now, as [g, β]γ ∈ g A we obtain y1 = [g, γ, α1 , . . . , αn ], by
using Lemma1.4 (b). Hence [y1 , β]γφy2 = [g, γ, α1 , . . . , αn , β], by Lemma1.4 (a).
To simplify [y2 , γ], Lemma1.4 (a) and (b) imply that [y2 , γ] = [g, β, α1 , . . . , αn , γ].
Altogether, we have 1 = [y1 , β]γφy2 [y2 , γ] = [g, γ, α1 , . . . , αn , β][g, β, α1 , . . . , αn , γ],
which gives the result.
Finally, in the following theorem we identify the embedding of ABn (G) in some
subgroups of G. Note that, for this purpose we impose some restriction on the
elements of order 2 in some subgroups of G.
139
Theorem 2.2. Let G be a group
(a) If [ABn (G),n+4 A] having no elements of order 2. Then ABn (G) ⊆ Ln+4 (G).
(b) If n is odd and [ABn (G),n+2 A] has no elements of order 2, then ABn (G) ⊆
Ln+2 (G).
References
[1] W.P. Kappe, Some subgroups defined by identities, Illinois J. Math. 47 (2003),
317-326.
[2] M.R.R. Moghaddam, M.A. Rostamyari, Autonipotent groups and their properties, to apppear in Southeast Asian Bulletin of Mathematics.
[3] M.R.R. Moghaddam, M. Farrokhi D.G and H. Safa, Some Properties of 2-auto
Engel groups, to appear in Quaestiones Mathematicae.
140
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Triangle-free commuting conjugacy classes graphs
A. Mohammadian1∗ , A. Erfanian2 and M. Farrokhi Derakhshandeh
Ghouchan3
1
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
[email protected]
2
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
[email protected]
3
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
[email protected]
Abstract
We study all finite groups whose commuting conjugacy class
graph are triangle-free.
1
Introduction
The character degree graph Γχ (G) of a group G is a graph whose vertices are nonlinear complex irreducible characters of G and two distinct characters are adjacent
whenever their degrees not coprime. Analogously, the conjugacy class graph ΓC (G)
of G is defined as graph whose vertices are non-central conjugacy classes of G and
two distinct vertices are adjacent whenever their sizes are not coprime. Also, the
prime graph ∆(G) on character degrees of G is defined as a graph whose vertices
is the set of all primes dividing some character degree of G such that two vertices
p, q ∈ π(G) are adjacent whenever pq divides some character degree of G.
The above mentioned graphs are studied in depth by many authors. For the case
where such a graph is triangle-free, we may refer the reader to
• Triangle-free character degree graph (Li, Liu and Song [3] and Wu and Zhang
[5]),
∗
Speaker
2010 Mathematics Subject Classification. Primary 05C25; Secondary 20E45.
Key words and phrases. Triangle-free, conjugacy class, graph.
141
• Triangle-free conjugacy class graph (Fang and Zhang [1]),
• Triangle-free prime graphs on character degrees (Tong-Viet [4]),
For a group G, let Γ(G), commuting conjugacy class graph of G, be a graph
associated with the non-central conjugacy classes of G in such a way that the vertices
of Γ(G) are the non-central conjugacy classes of G, and two distinct vertices C and
D are adjacent whenever there exist elements x ∈ C and y ∈ D such that xy = yx.
In 2009, Herzog, Longobardi and Maj [2] introduced and studied the graph Γ(G)
and proved that if G is a periodic non-abelian group, then Γ(G) has no edges if and
only if G is isomorphic to one of the groups S3 , D8 or Q8 .
In this talk, we discuss on the structure of those finite groups such that Γ(G) is
a triangle-free graph.
2
Main results
Throughout this section G stands for a non-abelian finite group G with triangle-free
commuting conjugacy class graph.
Theorem 2.1. If G is a group of odd order, then |G| = 21 or 27 .
Lemma 2.2. If G is group of even order which is not a 2-group, then either Z = 1
or Z ∼
= Z2 .
Theorem 2.3. Suppose G is a group of even order which is not a 2-group.
4 Then
Z ̸= 1 if and only if G is isomorphic to D12 or T12 , where T12 = ⟨a, ba = b3 =
1, ba = a−1 ⟩.
Theorem 2.4. G is a centerless non-soluble group if and only if G is isomorphic
to one of the groups P SL(2, 4), P SL(2, 7), P SL(2, 9), P SL(3, 4), or a group G of
order 960.
Theorem 2.5. G is a non-abelian soluble group with Z = 1 if and only if G is
isomorphic to one of the groups S3 , D10 , A4 , S4 , G1 , G2 or G3 , where G1 is a group
of order 72 and G2 and G3 are groups of order 192.
Theorem 2.6. If G is a 2-group, then cl(G) = 2 and both Z(G) and G/Z(G) are
elementary abelian 2-groups.
142
References
[1] M. Fang and P. Zhang, Finite groups with graphs containing no triangles, J.
Algebra 264 (2003), 613–619.
[2] M. Herzog, P. Longobardi and M. Maj, On a Commuting Graph on conjugacy
classes of groups, Comm. Algebra 37(10) (2009), 3369–3387.
[3] T. Li, Y. Liu and X. Song, Finite nonsolvable groups whose character graphs
have no triangles, J. Algebra 323(8) (2010), 2290–2300
[4] H. P. Tong-Viet,
arXiv:1303.3457v1.
Groups
whose
prime
graphs
have
no
triangles,
[5] Y.T.Wu and P. Zhang, Finite solvable groups whose character graphs are trees,
J. Algebra 308(2) (2007), 536–544.
143
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Isologism crossed modules
Hamid Mohammadzadeh
Department of Mathematics, Iran University of Science and Technology, Tehran, Iran
h [email protected]
Abstract
Let Aq be the variety of abelian crossed modules of exponent
q, where q is a square free positive integer. In this article,
we define the notions of Aq -covering crossed module and
Aq -isologism and, same as the work of Jones and Wiegold
[6], we show that all q-covering crossed modules of a given
crossed module is mutually Aq -isologic. Also, we present
the notion of an Aq -stem crossed module and we prove its
existence within an arbitrary Aq -isologism class.
1
Introduction and preliminaries results
A crossed module (M, G, δ) is a group homomorphism δ : M → G together with an
action of G on M , (g, m) 7→ g m, satisfying:
(i) The homomorphism δ is a precrossed module, i.e., δ(g m) = g δ(m), for all g ∈ G,
m ∈ M.
(ii) The Peiffer subgroup is trivial, i.e., δ(m) m′ = m m′ , for all m, m′ ∈ M .
We can consider any group as a crossed modules in two ways (G, G, id), (i, G, id).
A morphism of crossed modules α = (α1 , α2 ) : (M, G, δ) −→ (M ′ , G′ , δ ′ ) is a
pair of group homomorphisms α1 : M −→ M ′ and α2 : G −→ G′ such that δ ′ α1 =
α2 δ and α1 (g m) = α2 (g) α1 (m), for all g ∈ G, m ∈ M . We denote the resulting
category of crossed modules by CM. In this category the notions of (normal) crossed
submodule, commutator, center, product, etc. are defined in obvious ways. For more
details see [3, 13].
2010 Mathematics Subject Classification. 20E10, 20E34, 20N99.
Key words and phrases. Isologism, covering crossed module, stem crossed module, variety of
groups.
144
A crossed module (M, G, δ) is said to be finite if the groups M and G are finite.
In this case, we define |(M, G, δ)| to be the ordered pair (|M |, |G|). Clearly, a totally
order is defined on the class of all finite crossed modules by means of |(M, G, δ)| <
|(M ′ , G′ , δ ′ )| if and only if |M | < |M ′ |, or |M | = |M ′ | and |G| < |G′ |.
Let (M, G, δ) be a crossed module. For a nonnegative integer q, we define the
following G-invariant subgroups of M :
G♯q M = ⟨ g mm−1 , mq | m ∈ M, g ∈ G⟩,
MqG = {m ∈ M | mq = 1 and g m = m, for all g ∈ G}.
Clearly, if N is a normal subgroup of G, then the inclusion map i : N → G with the
action of G on N by conjugation is a simple example of a crossed module. In this
q
situation, G♯q N = [G, N ]N q and NqG , which is denoted it by ZG
(N ), is a central
subgroup of G whose elements have order dividing q.
Let (M, G, δ) be a crossed module and (N, K, δ) be a normal crossed submodule
of it. In [4] for nonnegative integer q, q-commutator crossed submodule, (M, G, δ)♯q (N, K, δ),
and q-center crossed submodule,
Z q (M, G, δ), are defined by ([K, M ]G♯q N, G♯q K, δ) and (MqG , Z q (G) ∩ stG (M ), δ),
respectively, where
Z q (G) = {g ∈ Z(G) | g q = 1},
StG (M ) = {g ∈ G |g m = m
f or
all
m ∈ M }.
In the following proposition we show that the category Aq -CM has enough projectives.
Proposition 1.1. (i) The category of q-abelian crossed modules, Aq -CM, has
enough projectives.
(ii) Let (Y, F, i) be a projective crossed module. Then any crossed submodule of
(Y, F, i)q.ab is Aq -projective.
Theorem 1.2. (Leedham-Green and Mackay [7]) Let ν be a variety of groups defined
by the set of laws V , then ν satisfies the following conditions:
(i) subgroups of ν-free groups are ν-splitting;
(ii) subgroups of the ν-marginal subgroup of a given group are normal, if and only
if the variety ν is the variety of all groups, the variety of abelian groups, or Aq , the
variety of abelian groups of exponent q, where q is a square free positive integer.
In sequel we assume that q is nonnegative square free integer unless stated otherwise. The following corollary plays an important role in our future investigations.
Corollary 1.3. Let (M, G, δ) be a q-abelian crossed module with a crossed submodule (S, H, δ). If (M, G, δ)/(S, H, δ) is a Aq -projective crossed module, then
(M, G, δ) = (S, H, δ) ×(U, N, δ), for some crossed submodule (U, N, δ) of (M, G, δ).
145
2
H2q (M, G, δ) and Aq -Covering Crossed Modules
In this section after the definition of Aq -covering crossed module, we examine the
connection between Aq -isologism and Aq -covering crossed module in one situation that the given crossed modules is finite. Let 1 → (V, R, µ) → (Y, F, µ) →
(M, G, δ) → 1 be a projective presentation of crossed module (M, G, δ). In [4]
Grandjean, et al. introduced the second homology crossed module modulo q of
(M, G, δ) by
R ∩ F ♯q F ∗
V ∩ F ♯q R
,
, µ ).
H2q (M, G, δ) = (
(R♯q R)(F ♯q V )
F ♯q R
In following, we convection that If 1 −→ (V, R, µ) −→ (Y, F, µ) −→ (M, G, δ) −→ 1
is a projective presentation of the crossed module (M, G, δ), then for every normal
crossed module (U, S, µ) of (Y, F, µ) satisfying the condition (Y, F, µ)♯q (V, R, µ) ⊆
(U, S, µ), we will denote the crossed module (U, S, µ)/(Y, F, µ)♯q (V, R, µ) by (Ū , S̄, µ̄).
In the next proposition we will show that the second homology modulo q of finite
crossed module is finite. But firstly we need to prove the following lemma.
Lemma 2.1. The following conditions in the category CM are equivalent:
(i) If (M, G, δ)/Z q (M, G, δ) is finite, then so is (M, G, δ)♯q (M, G, δ),
(ii) If (M, G, δ) is finite, then so is H2q (M, G, δ).
Proposition 2.2. (i) For any crossed module (M, G, δ), if
(M, G, δ)/Z q (M, G, δ)
is finite, then so is (M, G, δ)♯q (M, G, δ).
(ii) If (M, G, δ) is finite then so is H2q (M, G, δ).
Vieites and Casas in [14] generalized the notions of covering groups to the covering crossed modules, then Mohammadzadeh, et al. in [9] for any given crossed
module prove the existence covering crossed module and determined the structure
of it. Here we introduce the notion of q-covering crossed module as follows:
Let e : 1 → (A, B, δ ∗ ) → (M ∗ , G∗ , δ ∗ ) → (M, G, δ) → 1 be a q-central extension
of crossed module, i.e., (A, B, δ ∗ ) ⊆ Z q (M ∗ , G∗ , δ ∗ ), then e is called a stem cover if
q
(A, B, δ ∗ ) ⊆ (M ∗ , G∗ , δ ∗ )♯q (M ∗ , G∗ , δ ∗ ) and (A, B, δ ∗ ) ∼
= H2 (M, G, δ). In this case
∗
∗
∗
(M , G , δ ) named Aq -covering crossed module.
Proposition 2.3. Every finite crossed module admits at least one Aq -covering
crossed module.
Theorem 2.4. Let (M ∗ , G∗ , µ∗ ) be a covering crossed module of (M, G, δ) whose
second homology modulo q is finite. If
146
π
1 −→ (V, R, µ) −→ (Y, F, µ) −→ (M, G, δ) −→ 1
be a projective presentation of crossed module (M, G, δ). Then there exists an
epimorphism φ̄ = (φ̄1 , φ̄2 ) from (Ȳ , F̄ , µ̄) onto (M ∗ , G∗ , µ∗ ) such that Kerφ̄ is a
complement for H2q (M, G, δ) in (V̄ , R̄, µ̄).
References
[1] H. Behravesh, Quasi-Permutation representations of p-groups of class 2, J. London Math. Soc. 55 (1997), 251–260.
[2] D. Arias, M. Ladra, The precise center of a crossed module, J. Group Theory 12
(2009), 247-269.
[3] A.R. Grandjean, M. Ladra, On totally free crossed modules, Glasgow Math. J.
40(1998), 323-332.
[4] A. R. Grandjean, M. P. Lopez, H2 (T, G, ∂) and q-perfect crossed modules, Journal of Applied Categorical Structures. 11 (2003), 171-184.
[5] P. Hall, The classification of prime-power groups, Reine Angew. Math. 182
(1940), 130-141.
[6] M.R. Jones, J. Wiegold, Isoclinism and covering groups, Bull. Aust. Math. Soc.
11(1974), 71-76.
[7] C.R. Leedham-Green, S. Mckay, Baer-invariants, isologism, varietal laws ard
homology, Acta Math. 137 (1976), 99-150.
[8] H. Mohammadzadeh, A.R. Salemkar, S. Shahrokhi, Isoclinism of crossed modules, submitted.
[9] H. Mohammadzadeh, S. Shahrokhi, A.R. Salemkar, On the Schur multiplier of
crossed modules, submitted.
[10] M.R.R. Moghaddam, A.R. Salemkar, Characterization of varietal covering and
stem groups, Comm. Algebra 27(1999), 5575-5586.
[11] M.R.R. Moghaddam, A.R. Salemkar, Varietal isologism and covering groups,
Arch. Math. 75(2000), 8-15.
[12] B.H. Neumann, Groups with finite classes of conjugate subgroups, Math. Z. 63
(1955), 76-79.
147
[13] K.J. Norrie, Crossed modules and analogues of group theorems, Thesis, King’s
College, Univ. of London, London, 1987.
[14] A.M. Vieites, J.M. Casas, Some results on central extensions of crossed modules,
Homology, Homotopy Appl. 4(2002), 29-42.
148
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
The structure of non-solvable CTI-groups
Hamid Mousavi1 , Tahereh Rastgoo2∗ and Viktor Zenkov3
1
Department of Mathematics, Faculty of Sciences, Tabriz University, Tabriz, Iran
[email protected]
2
PhD graduate in Mathematics of Institute for Advanced Studies in Basic Sciences,
Zanjan, Iran
[email protected]
3
Institute of Mathematics and Mechanics of Ural Branch RAS,S. Kovalevskaya 16,
Ekaterinburg 620990, Russia
[email protected]
Abstract
A finite group G is called a CTI-group if for any cyclic subgroup H of G, H ∩ H g = 1 or H for all g ∈ G. This paper,
describes the structure of finite non-solvable CTI-groups.
1
Introduction
Throughout the following, G always denotes a finite group.
Let H be a subgroup of G. If for any g ∈ G we have H ∩ H g ∈ {1, H} then H
is called a TI-subgroup. Now if every subgroup of G is a TI-subgroup, then G is
called a TI-group. In [3], G. Walls classified the TI-groups :
Theorem 1.1. A finite group, G, is a TI-group if and only if one of the following
occurs:
(i) G is Hamiltonian,
(ii) G = H ⋆ Zpn , where H is a group of order p3 .
∗
Speaker
2010 Mathematics Subject Classification. 20D10; 20E34.
Key words and phrases. Solvable Group, TI-subgroup, CTI-group.
149
(iii) G = ⟨x, y|xp = y p = 1, y −1 xy = x1+p
n
n−1
⟩ for some n.
(iv) G = D8 ⋆Q8 , where D8 is the dihedral group of order 8 and Q8 is the quaternion
group of order 8.
(v) G = Zp ⋊ Zm , m|p − 1, Zm acts regularly on Zp .
(vi) G = (Zp × Zp ) ⋊ Zm , m|p + 1, Zm acts regularly on the subgroups of order p.
In [1], S. Li and X. Guo with P. Flavel determined the structure of groups whose
all Abelian subgroups are TI-subgroups. These groups are called ATI-groups.
Theorem 1.2. Let G be a finite ATI-group. Then one of the following holds:
(i) G is nilpotent.
(ii) G is a soluble Frobenius group. Let K be the Frobenius kernel and H a
complement. Then one of the following holds:
(a) Every H-chief factor of K is cyclic and for every p ∈ π(K), H is cyclic of
order dividing p − 1.
(b) K ∼
= Zp × Zp for some prime p, H acts irreducibly on K and either H is
cyclic or the direct product of a cyclic group of odd order and Q8
(iii) G ∼
= S4 .
(iv) G is isomorphic to one of the simple groups PSL2 (4), PSL2 (7) or PSL2 (9).
A group G is called a CTI-group if all of its cyclic subgroups are TI-subgroups.
Clearly, any ATI-group is a CTI-group; however, the converse is not true. This
paper, classifies the non-solvable CTI-groups.
Throughout this paper, F (G) is the Fitting subgroup of G, Z(G) is the center
of G; also Q8 and S4 are the quaternion group of order 8, and the symmetric group
of degree 4. All unexplained notation and terminology are standard.
2
Preliminaries
This section, states some results which will be useful in the proof of main results.
Theorem 2.1. Let G be a non-nilpotent CTI-group with non-trivial center. Then
Z(G) is an elementary abelian p-subgroup where p is smallest factor of |G|. More
specially, any p′ -subgroup of G is a normal.
150
Theorem 2.2. Let G be a non-nilpotent CTI-group such that Z(G) ̸= 1. Also
suppose that p divides |Z(G)| and let H be a Hall p′ -subgroup of G. Then H is
abelian and normal, and moreover G = HP is solvable, where P ∈ Syℓp (G). Also,
(i) if Z(G) ∩ G′ = 1 then G ∼
= K × (H ⋊ Zpi ), where p is the smallest divisor of
|G|, K = Z(G), P = Z(G) × Zpi and H = G′ ;
(ii) if Z(G) ∩ G′ ̸= 1 then p = 2 and P = K ⋊ Z2 , where K is an abelian normal
subgroup of G; also Z(G) = Ω1 (K), G′ = Hf1 (K) and Z2 inverts any element
of HK;
(iii) the Fitting subgroup F (G) = HK is abelian.
Theorem 2.3. Let G be a finite CTI-group with trivial center and V , its minimal
normal subgroup, be solvable. Then F (G) = CG (V ).
Theorem 2.4. Let G be a finite solvable CTI-group with trivial center. Then:
(i) If |F (G)| has more than one prime divisor, then G = F (G)H is a Frobenius
group with abelian kernel F (G) and complement H;
(ii) If F (G) is a p-group, then either G is isomorphic to S4 , or F (G) is a Sylow
p-subgroup of G and G is a Frobenius group with kernel F (G).
3
Mailn Result
This section, classifies non-solvable CTI-groups.
Theorem 3.1. A CTI-group G is solvable if and only if it has a solvable minimal
normal subgroup.
Let V be a minimal normal subgroup of a non-solvable CTI-group G. By Theorem 3.1, V cannot be solvable, since the centralizer of any element (in particular
any subgroup) of G is solvable, and so CG (V ) = 1. Therefore, V must be simple.
Also we have V ≤ G ↩→ Aut(V ) and G/V ↩→ Out(V ).
Theorem 3.2. Let G be a finite non-solvable CTI-group. Then G ∼
= PSL(2, q) or
PGL(2, q), where q > 3 is a prime power.
The inverses of Corollary 2.2 and Theorem 2.4 are simple: we just prove the
inverse of the non-solvable case. Before proving the inverse theorem, we consider
the simple fact that, if a non-normal subgroup ⟨x⟩ of G is normal in a non-normal
maximal subgroup M , then ⟨x⟩ ∩ ⟨x⟩g E G, where g ∈ G\M .
151
Theorem 3.3. Let G be isomorphic to K, where PSL(2, q) ≤ K ≤ PGL(2, q), q > 3
is a power of prime p. Then G is a CTI-group.
Proof. We can simply check by GAP that PSL(2, p) is CTI for p = 5, 7, 9, 11. Let x
be an element of G. If p | |x| then x must be a p-element, since by [4, Theorem 2.8.2
and Lemma 15.1.1] Sylow p-subgroups of G are elementary abelian and TI; therefore
|x| = p. If |x| | (q 2 − 1) and x is not a 2-element, then |x| | 2n m, where m is odd;
hence x = yz, where |z| > 1 is odd. In this case z belongs to the maximal subgroup
D2(q−1) or D2(q+1) by [2, Theorem 2.1 and Theorem 2.2]; since ⟨z⟩ is normal in these
groups, then NG (x) = NG (z) is a non-normal maximal subgroup of G. Therefore,
⟨x⟩ is normal in a non-normal maximal subgroup of G, and so is TI. Now, let x be a
2-element and |x| > 2; then p is an odd prime and again ⟨x⟩ belongs to the dihedral
group. Since ⟨x⟩ is normal in this group, then NG (x) is maximal in G. Hence ⟨x⟩ is
a TI-group. Therefore, G is a CTI-group.
References
[1] X. Guo, S. Li and P. Flavell, Finite groups whose abelian subgroups are TIsubgroup, J. Algebra. 307 (2007), 565–569.
[2] A. Lucchini and A. Maróti, On finite simple groups and Kneser graphs, J. Algebraic Combin. 30 (2009), No. 4, 549-566.
[3] G. Walls, Trivial intersection groups, Arch. Math. 32 (1979), 1–4.
[4] D. Gorenstein,Finite groups, Chelsea, New York (1980).
152
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
P -semisimple BCI-algebras and adjoint groups
Ardavan Najafi1∗ and Hamid Rasouli2
1
Department of Mathematics, Behbahan Branch, Islamic Azad University, Behbahan, Iran
najafi[email protected]
2
Department of Mathematics, Science and Research Branch, Islamic Azad University,
Tehran, Iran
[email protected]
Abstract
In this paper, we study the p-semisimple BCI-algebras and
their connection to Abelian groups. A ccording to the
structure of commutators in group theory, we introduce the
concept of pseudo-commutators and commutators in BCIalgebras, and obtain some related results.
1
Introduction
In 1966, Imai and Iseki defined a class of algebras of type (2,0) called BCK-algebra
which generalizes on one hand the notion of algebra of sets with the set subtraction
as the only fundamental non-nullary operation, and on the other hand the notion
of implication algebra. We can define an implication in each BCK-algebra by y →
x = x ∗ y. In the same year, Iseki introduced the concept of a BCI-algebra in as
follows. p-semisimple algebras are another special classes of BCI-algebras which
were touched by Lei in 1982. They play a basic role in the research of BCI-algebras
and have close contacts with Abelian groups.([1,2,3])
Definition 1.1. Let (G, ∗, 0) be an algebra of type (2, 0). Then G is said to be a
BCI-algebra if it satisfies the following properties:
∗
Speaker
2010 Mathematics Subject Classification. Primary 03G25; Secondary 08A05, 06F35.
Key words and phrases. Adjoint group, P -Semisimple, BCI-Algebra, Commutator.
153
(1)((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0,
(2)(x ∗ (x ∗ y)) ∗ y = 0,
(3)x ∗ x = 0,
(4) x ∗ y = y ∗ x = 0 implies x = y,
for all x, y, z ∈ G.
On any BCI-algebra (G, ∗, 0) we can define the natural order putting
(5) x ≤ y if and only if x ∗ y = 0. It is not difficult to verify that this order is partial.
A BCI-algebra G has the following properties:
(6) (x ∗ y) ≤ x,
(7) x ≤ y implies that x ∗ z ≤ y ∗ z and z ∗ y ≤ z ∗ x
(8) x ∗ 0 = x,
(9) x ∗ 0 = 0 implies that x = 0.
If a BCI-algebra G satisfies 0 ∗ x = 0 for all x ∈ G, then we say that G is a BCKalgebra. For elements x and y of a BCK-algebra G, we denote x ∧ y = y ∗ (y ∗ x).
A BCI-algebra G is called commutative if for any x, y ∈ G, x ∗ y = 0 implies
x = x ∧ y = y ∗ (y ∗ x). A non-empty subset S of a BCI-algebra G is called a
BCI-subalgebra of G if x ∗ y ∈ S whenever x, y ∈ S. A non-empty subset I of a
BCI-algebra G is called a BCI-ideal of G if it satisfies (i) 0 ∈ I, (ii) x ∗ y ∈ I and
y ∈ I imply that x ∈ I, for all x, y ∈ G. The set B = {x : 0∗x = 0} in a BCI-algebra
G is called the BCK-part of G. Obviously B is a subalgebra of G, and it is also
an ideal of G. If I is an ideal of G, and for every x in I, 0 ≤ x, then I is called
a p-ideal of G. Clearly, BCK-part B of G contains all p-ideals of G so that it is
a maximal p-ideal of G. If B = {0}, then G is called a p-semisimple BCI-algebra.
In the case, B is called a p-radical of G. Let G be a BCI-algebra. We choose an
element x0 ∈ G such that there does not exist any y ̸= x0 satisfying y ∗ x0 = 0 and
define A(x0 ) = {x ∈ G : x0 ∗ x = 0}. The element x0 is known as the initial element
of A(x0 ) as well as G. Let Ix denote the set of all initial elements of G. We call it
the center of G. A(0) = {x ∈ G : 0 ∗ x = 0} = B =the BCK-part of G. The center
Ix of a BCI-algebra G is p-semisimple.
Example 1.2. Let G = {0, 1, 2}. Define a binary operation ” ∗ ” on G by the
following table:
Table 3:
∗ 0 1
0 0 0
1 1 0
2 2 2
2
2
2
0
It is not difficult to verify that (G, ∗, 0) is a BCI-algebra, but G is not a p-semisimple
algebra, because the BCK-part of G is {0, 1}. The elements 0, 2 are initial elements
154
of G and A(0) = {0, 2}, A(2) = {2}. Therefore, Ix = {0, 2}. Also the set (Z, −, 0) is
a BCI-algebra. Since 0 − x = 0 gives that x = 0, the BCK-part of G equals B = {0}.
Hence, it is a p-semisimple algebra.
2
Main results
In this section we will present some results on a p-semisimple algebra. We use
Abelian group theory to study p-semisimple algebras. Also we will deal with the
relations between p-semisimple algebras and Abelian groups.
Lemma 2.1. Let G be a BCI-algebra and x, y ∈ G. Then the following are equivalent:
a) G is p-semisimple.
b) 0 ∗ x = 0 =⇒ x = 0.
c) 0 ∗ (0 ∗ x) = x.
d) x ∗ (0 ∗ y) = y ∗ (0 ∗ x).
Lemma 2.2. Let G be a semisimple algebra. If we define x⊕y = x∗(0∗y) = y∗(0∗x),
then (G, ⊕, 0) is an Abelian group.
Proof. Obviously, G is closed under the operation ⊕, and x ⊕ (y ⊕ z) = x ∗ (0 ∗ (y ∗
(0 ∗ z))) = (y ∗ (0 ∗ z)) ∗ (0 ∗ x) = (y ∗ (0 ∗ x)) ∗ (0 ∗ z) = (x ∗ (0 ∗ y)) ∗ (0 ∗ z) =
(x ⊕ y) ⊕ z But, x ⊕ y = x ∗ (0 ∗ y) = y ∗ (0 ∗ x) = y ⊕ x. Hence, the operation
⊕ is associative and commutative. Moreover, x ⊕ 0 = 0 ⊕ x = 0 ∗ (0 ∗ x) = x and
x ⊕ (0 ∗ x) = (0 ∗ x) ⊕ x = (0 ∗ x) ∗ (0 ∗ x) = 0. Therefore, 0 ∗ x is the inverse of x.
Thus G is an Abelian group with respect to ⊕.
Conversely, we can show that any Abelian group is a p-semisimple algebra. Suppose (G, ·, e) is an Abelian group with e as the unit element. Define a binary operation “∗” on G by putting x ∗ y = xy −1 . Then (G, ∗, e) is a BCI-algebra. We call
(G, ∗, e) the adjoint BCI-algebra of the Abelian group (G, ·, e).The Abelian group
induced by a p-semisimple algebra in above lemma is said adjoint group. In a psemisimple BCI-algebra G, the following hold:
(e) (x ∗ z) ∗ (y ∗ z) = x ∗ y;
(f ) a ∗ x = b ∗ x implies that a = b;
(g) x ∗ (0 ∗ y) = y ∗ (0 ∗ x);
(h) x ∗ y = 0 implies that x = y;
(i) x ∗ a = x ∗ b implies that a = b; (j) a ∗ (a ∗ x) = x.
Definition 2.3. Let (G, ∗, 0) be a BCI-algebra and let x, y be elements of G. Then
the pseudo-commutator [x, y] of x and y is defined as follows:
{
0
x=y
[x, y] =
(x ∗ (0 ∗ y)) ∗ (0 ∗ ((0 ∗ x) ∗ (0 ∗ (0 ∗ y)))) Otherwise.
If x ∗ y = [x, y] ∗ (y ∗ x), then [x, y] is called the commutator of x , y.
155
Lemma 2.4. Let (G, ., 0) be a BCI-algebra. Then for any x, y ∈ G,
i) if x ≤ y, then [x, y] = 0.
ii) [0, x] = [x, x] = 0.
Definition 2.5. The subset [G, G] = {[a, b]|a, b ∈ G} of G is called the derived
′
′
subalgebra of G and we will denote by G . In fact, G is the set all commutators of
elements of G.
Theorem 2.6. Let (G, ∗, 0) be a BCI-algebra. Then G is p-semisimple if and only
′
if G = {0}.
Theorem 2.7. Let (G, ∗, 0) be a BCI-algebra. Then
′
′
1) G is a subalgebra of G.
2) G is an ideal of G.
′
only if G is the BCK-part of G.
3) G is commutative if and
Example 2.8. Let G = {0, 1, a, b, c} and let the operation ” ∗ ” be given by the
following table:
∗
0
1
a
b
c
Table 4:
0 1 a
0 0 c
1 0 c
a a 0
b b a
c c b
b
b
b
c
0
a
c
a
a
b
c
0
It is not difficult to verify that (G, ∗, 0) is a BCI-algebra. In this algebra for all
′
x, y ∈ G we have x ∗ y = [x, y] ∗ (y ∗ x). Now we see that G = {0, 1} is a subalgebra
and also an ideal of G. The initial segments of G are of the form [0, 1] = {0, 1}.
′
Hence, G is an initial segment of G. Also Ix = {0, a, b, c}, which follows that
′
Ix ̸= G .
′
′
Theorem 2.9. If (G, ., 0) is a BCI-algebra and H is a subalgebra of G, then H ⊆ G .
References
[1] S. A. Bhatti, M. A. Chaudhry, Characterization of BCI-algebras of order 5,
(1992), no.25 , 99–121.
[2] Y. Imai and K. Iseki, On axiom system of propositional calculi XIV, Proc,Japan
Acad. 42 (1996), 19–22.
[3] K. Iseki, On BCI-algebras, Math. sem. Notes, 8, (1980), 125-130.
156
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
The finite π-solvable groups with three conjugacy class
sizes of primary and biprimary π-elements
Majid Najafi1∗ and Neda Ahanjideh2
Department of Mathematics, Faculty of Sciences, Shahrekord University, Shahrekord, Iran
1
majid− najafi[email protected]
2
[email protected]
Abstract
Let G be a finite π-solvable group, where π is a set of primes
with |π| ≥ 2. If the conjugacy class sizes of all π-elements of
primary and biprimary orders of G are {1, m, n} such that
m is a π -number and n is a π ′ -number, then we show that
there exists p ∈ π such that m = pα and G = K × P W ,
where P ∈ Sylp (G), K ≤ Z(G) is a Hall (π − {p})-subgroup
and W is a Hall π ′ -subgroup of G.
1
Introduction
Let G be a finite group. For x ∈ G, the conjugacy class of x is the set xG = {xg :
g ∈ G} and the size of this class is called the conjugacy class size of x in G and is
denoted by |clG (x)|. We say that x has primary order if its order is a prime power
and x has biprimary order if its order has exactly two distinct prime divisors.
We write π(G) for the set of prime divisors of order of G. Suppose that π is a set of
primes and π ′ is a complement of π. We say that g ∈ G is a π-element (π ′ -element)
if the prime divisors of the order of g are in π (π ′ ). There are many results on
the relationship between the structure of a group and its conjugacy class sizes. For
example, Ito shows in [2] that if the sizes of the conjugacy classes of a group G are
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D10; Secondary 20E26.
Key words and phrases. π-solvable group, conjugacy class sizes, elements of primary and
biprimary orders, Hall subgroup.
157
{1, m}, then G is nilpotent, m = pa for some prime p and G = P × A, where P is a
Sylow p-subgroup of G and A ≤ Z(G). Also, Ito in [3] shows that G is solvable if
the conjugacy class sizes of G are {1, m, n}.
In [4], the author find the structure of the finite p-solvable group which its conjugacy class sizes of its primary and biprimary elements are {1, pa , n}, where p is
prime and gcd(p, n) = 1. In this paper, we are going to prove that.
Main Theorem. Let π be non-empty set of prime numbers with |π| ≥ 2 and let G
be a π-solvable group. If 1, m and n are the only conjugacy class sizes of π-elements
of G of primary and biprimary orders, where m is a π-number and n is a π ′ -number,
then there exists p ∈ π such that m = pα and G = K × P W , where P ∈ Sylp (G), W
is a Hall π ′ -subgroup of G and K ≤ Z(G) is a Hall (π − {p})-subgroup of G.
2
Main results
Lemma 2.1. [1] Let G be a π-solvable group, where π is a non-empty subset of
π(G).
(a) |clG (x)| is a π-number for every π ′ -element x of primary order if and only if G
has an abelian Hall π ′ -subgroup.
(b) |clG (x)| is a π ′ -number for every π ′ -element x of primary order if and only if
G = Oπ (G) × Oπ′ (G).
The proof of the following lemma is straightforward and we are going to use it
in the proof of the main theorem.
Lemma 2.2. If G is a finite group and x, y ∈ G such that gcd(|x|, |y|) = 1 and
xy = yx, then CG (xy) = CG (x) ∩ CG (y).
Proof of the Main Theorem. We are going to show that the conjugacy class sizes
of π-elements of G of primary order are {1, m, n}. If not, then one of the following
holds:
i. For every π-element x of primary order, if conjugacy class size of x is 1 or m,
then since m is a π-number, Lemma 1 shows that
G = Oπ (G) × Oπ′ (G).
Thus for every y ∈ Oπ (G), Oπ′ (G) ≤ CG (y), so |clG (y)| is a π-number, which is
a contradiction, because by our assumption, there exists y ′ ∈ Oπ (G) such that
|clG (y ′ )| = n.
158
ii. For every π-element x of primary order, if conjugacy class size of x is 1 or n,
then by Lemma 1, G has an abelian Hall π-subgroup H. Thus for every x ∈ H,
H ≤ CG (x) and hence, for every π-element y in G, |clG (y)| is a π ′ -number, which is
contradiction.
Thus there exist π-elements x and w of primary orders in G such that |clG (x)| = n
|G|
and |clG (w)| = m. Since |clG (x)| = n is a π ′ -number and |clG (x)| = |CG
(x)| , we
deduce that CG (x) contains a Hall π-subgroup of G as H. We know that x is a
π-element of primary order, so for some p ∈ π, |x| = pα . Let q ∈ π − {p} and let y
be a q-element of G. Without loss of generality, we can assume that y ∈ H. Since
H ≤ CG (x), y ∈ CG (x). Thus we have
CG (xy) = CG (x) ∩ CG (y) ≤ CG (x).
This implies that |clG (x)| | |clG (xy)|. Thus n| |clG (xy)|. But xy is a π-element of
biprimary order. Thus by assumption of theorem, conjugacy class size of xy in G
is 1, n or m and since n divides the conjugacy class size of xy in G, |clG (xy)| = n.
Now since |clG (x)| = |clG (xy)| = n, CG (x) ≤ CG (y). Thus |clG (y)| | |clG (x)| = n.
Therefore |clG (y)| = 1 or n.
• If |clG (y)| = 1, then y ∈ Z(G).
• If |clG (y)| = n, then since |clG (x)| = n, applying the previous argument shows
that CG (xy) = CG (y), so CG (x) = CG (y). Also H ≤ CG (x), so H ≤ CG (y).
Thus y ∈ Z(H)
This shows that for every q ∈ (π − {p}) and Q ∈ Sylq (G), Q ≤ Z(H). Thus
H = P × K, where P ∈ Sylp (G), K ≤ Z(H) and K is a Hall (π − {p})-subgroup of
G. Therefore K is an abelian group.
Also, there exists a π-element w in G of primary order such that |clG (w)| = m.
Since H = P × K, K is abelian, w is of primary order and m is a π-number, we can
see that w ∈ P , where P ∈ Sylp (G). Thus K ≤ CG (w) and hence, |clG (w)| = pt ,
where t a positive integer.
Also, for every y ∈ K of primary order, since H = K × P and K is abelain,
H ≤ CG (y). Thus |clG (y)| is a π ′ -number. Therefore, |clG (y)| ∈ {1, n}. Also,
K ≤ CG (w) and (|w|, |y|) = 1. Thus
CG (yw) = CG (y) ∩ CG (w) ≤ CG (w).
Consequently, |clG (w)| | |clG (yw)| and hence, m | |clG (yw)|. On the other hand,
we assume that |clG (yw)| ∈ {1, m, n}, so |clG (yw)| = m. But CG (yw) = CG (y) ∩
159
CG (w) ≤ CG (y), so |clG (y)| | |clG (yw)| = m. But |clG (y)| = 1 or n, so |clG (y)| = 1
and hence, y ∈ Z(G).
So we show that K ≤ Z(G). Therefore, G = K × P W , where W is a Hall
′
π -subgroup of G and K ≤ Z(G). So theorem follows. References
[1] C. Shao, Q. Jiang, Finite groups with two conjugacy class sizes of π-elements of
primary and biprimary orders, Monatsh. Math. 169 (2013), 105-112.
[2] N. Ito, On finite groups with given conjugate types I, Nagoya Math. J. 6 (1953),
17-28.
[3] N. Ito, On finite group with given conjugate types II, Osaka J. Math. 7 (1970),
231-251.
[4] Q. Kong, Finite groups with three conjugacy class sizes of some elements, Proc.
Indian Acad. Sci. 122 (2012), no. 3, 335-337.
160
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
On the absolute center of some groups
Mohammad Mehdi Nasrabadi1∗ and Ali Gholamian2
1
Department of Mathematics,University of Birjand, Birjand, Iran
[email protected]
2
Department of Mathematics, Birjand Education, Birjand, Iran
[email protected]
Abstract
The concept of the absolute center of a group was introduced
and investigated by Hegarty in a series of papers. This paper
deals with the behavior of the absolute center of a group.
1
Introduction
Let G be a group and A(G) = Aut(G) denote the group of automorphisms of G. If
g ∈ G and α ∈ A(G), then the element [g, α] = g −1 α(g) is an autocommutator
element of g and α. In [2], Hegarty introduced the following subgroups,
K(G) = [G, Aut(G)] = ⟨[g, α]|g ∈ G, α ∈ A(G)⟩,
L(G) = {g ∈ G|[g, α] = 1, ∀α ∈ A(G)},
which are called the autocommutator subgroup and absolute center of G, respectively. Clearly, they are both characteristic subgroups in G so that K(G) contains the derived subgroup, G′ , and L(G) is contained in the center of G, Z(G). The
autocommutator subgroup and absolute center are already studied in [1-5].
We know that if G is a finite p-group, for some prime p, then Z(G) is non-trivial.
But there exist many p-groups, such that L(G) is trivial. In this paper we prove the
results in the absolute center of a group.
∗
Speaker
2010 Mathematics Subject Classification. 20D45, 20D25.
Key words and phrases. Automorphism, Autocommutator element, Absolute center.
161
2
Preliminary Results
We begin with some useful results that will be used in the proof of our main results.
Definition 2.1. Let H be a subgroup of a group G, h ∈ H and α ∈ A(G). Then
we define the sets
CA(G) (h) = {α ∈ A(G) | α(h) = h},
CA(G) (H) = {α ∈ A(G) | α(h) = h ∀h ∈ H},
NA(G) (H) = {α ∈ A(G) | α(H) = H},
CG (α) = {g ∈ G | α(g) = g}.
Clearly the sets CA(G) (h), CA(G) (H) and NA(G) (H) are subgroups of A(G) and
the set CG (α) is a subgroup of G.
Definition 2.2. Let H be a subgroup of a group G and x, y ∈ H. Then we say that
x is autoconjugate to y in H by A(G), if α(x) = y for some α ∈ A(G).
Remark 2.3. It is easy to check that the relation ”x is autoconjugate to y in H by
A(G)” is an equivalence relation on H. If H is a characteristic subgroup of G, then
this equivalence relation yields a partition of H and each cell in the partition arising
from an equivalence relation is equivalence class. The equivalence classes are called
autoconjugancy classes of H by A(G). The autoconjugancy class of x ∈ H by
A(G) is denoted by Acl(x).
Lemma 2.4. Let G be group and H be a subgroup of G. Then
i) NA(G) (H) ∩ NA(G) (K) ≤ NA(G) (H ∩ K).
ii) For any x ∈ G, NA(G) (H x ) = NA(G) (H)x̄ , where x̄ is an inner automorphism of
G.
iii) CA(G) (H) ▹ NA(G) (H).
iv) A(G) = CA(G) (H) if and only if H ≤ L(G).
Lemma 2.5. Let G be any group and H be a characteristic subgroup of G. Then
i) If H is finite, then
|H| = Σ|Acl(h)|,
such that sum is on distinct autoconjugate classes of H by A(G).
ii) If A(G) is finite, then for any h ∈ H
|Acl(h)| = [A(G) : CA(G) (h)].
162
Corollary 2.6. Let G be a finite group, then
i)
|G| = Σ[A(G) : CA(G) (g)],
such that each autoconjugate class is represented once and only once in the summation.
ii)
|G| = |L(G)| + Σg∈L(G)
[A(G) : CA(G) (g)],
/
such that each autoconjugate class is represented once and only once in the summation.
3
Main Results
We start with the following Lemma.
Lemma 3.1. Let G be an abelian group. Then L(G) is an elementary abelian
2-group.
Theorem 3.2. Let G be a finite abelian group. Then
L(G) = ⟨e⟩ or C2 ,
where C2 denotes the cyclic group of order 2.
The following corollary is the immediate result of the above theorem.
Corollary 3.3. Let G be a finite abelian group. Then
L(G) = C2 if and only if in the decomposition of the sylow 2-subgroup of G there is
unique cyclic direct factor of maximal order and L(G) = ⟨e⟩ otherwise.
Theorem 3.4. Let G be a group and H be a subgroup of G. Then
isomorphic to a subgroup of A(H).
NA(G) (H)
CA(G) (H)
is
Corollary 3.5. Let G be a finite group and H be a characteristic subgroup of prime
order p such that p be the smallest prime divisor of |A(G)|, then
H ≤ L(G).
Corollary 3.6. Let H be a cyclic characteristic subgroup of G. Then
[H, A(G)′ ] = ⟨e⟩.
Recall: A group G is called perfect if G = G′ .
163
Corollary 3.7. For a group G let A(G) be a perfect group and H be a cyclic
characteristic subgroup of G. Then
H ≤ L(G).
References
[1] C. Chiš, M. Chiš, and G. Silberberg, Abelian groups as autocommutator groups,
Arch. Math. (Basel). 90 no. 6 (2008) 490–492.
[2] P. Hegarty, The absolute centre of a group, Journal of Algebra 169 no. 3 (1994)
929–935.
[3] P. Hegarty,Autocommutator subgroups of finite groups, Journal of Algebra 190
(1997) 556–562.
[4] M. R. R. Moghaddam,Some properties of autocommutator groups, The first TwoDays Group Theory Seminar in Iran, University of Isfahan, 12-13 March 2009,
Isfahan.
[5] M. M. Nasrabadi and A. Gholamian, On finite A-perfect abelian groups, Int. J.
Group Theory. Vol. 01 No.3 (2012) 11–14.
164
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
On countability of homotopy groups
Tayyebe Nasri1∗ , Behrooz Mashayekhi2 and Hanieh Mirebrahimi3
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic
Structures, Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran
1
ta [email protected]
2
3
[email protected]
h [email protected]
Abstract
In this talk we intend to generalize some results of Conner
and Lamereaux on the countability of π1 (X, x). For this,
we show that some properties of topological spaces can be
transferred from X to the loop space Ωn (X, x), for some
x ∈ X. Finally, we intend to give some conditions in which
the space X is semilocally n-connected.
1
Introduction
Conner and Lamereaux [2] proved that several results concerning the existence of
universal covering spaces for separable metric spaces. They defined several homotopy theoretic conditions which are equivalent to the existence of a universal covering
space. For instance they proved that every connected, locally path connected separable metric space whose fundamental group is a free group admits a universal
covering space. As an application of these results, they proved that a connected,
locally path connected subset of the Euclidean plane, E 2 , admits a universal covering space if and only if its fundamental group is free, if and only if its fundamental
group is countable. In this talk we intend to generalize some of this results on the
∗
Speaker
2010 Mathematics Subject Classification. Primary 55Q05; Secondary 55P35, 20k20.
Key words and phrases. Homotopy group, Loop space, Free group, Countable group.
165
countability of π1 (X, x). For this, we show that some properties of topological spaces
can be transferred from X to the loop space Ωn (X, x), for some x ∈ X. Finally, we
intend to give some conditions in which the space X is semilocally n-connected.
2
Main results
In this section, we intend to generalize some results of Conner and Lamereaux [2] on
the countability of π1 (X, x). For this, we show that some properties of topological
spaces can be transferred from X to the loop space Ωn (X, x), for some x ∈ X.
Lemma 2.1. Let X be locally (n-1)-connected space. If Ω(n−1) (X, x) is semilocally
simply connected at the constant (n-1)-loop ex , then X is semilocally n-connected
at x, for all n ≥ 2.
Note that the converse of this fact has been shown by Wada [5, Remark]
Conner and Lamereaux [2] proved that the fundamental group of a path connected, locally connected, separable metric space which admits a universal cover
is countable. Using this result, Lemma 2.1, the fact that the separability, metrizability, locally connected properties of space X can be transferred to the loop space
Ωn (X, x), for some x ∈ X and applying the group isomorphism π1 (Ω(n−1) (X, x), ex ) ∼
=
πn (X, x) we conclude the following result. The following result is the first result on
countability of homotopy groups.
Theorem 2.2. Let X be an (n-1)-connected, locally (n-1)-connected, semilocally
n-connected and separable metric space. Then πn (X, x) is countable.
A space X is called n-homotopically Hausdorff at x ∈ X if for any essential nloop α based at x, there is an open neighborhood U of x for which α is not homotopic
(rel I˙n ) to any n-loop lying entirely in U . X is said to be n-homotopically Hausdorff
if it is n-homotopically Hausdorff at any x ∈ X (see [4]).
Consider Ωn (X, x) as the space of homotopy classes rel I˙n of n-loops at x in X.
If p is an n-loop at x, and U is an open neighborhood of x, then we define On (p, U )
to be the collection of homotopy classes of n-loops rel I˙n containing n-loops of the
form p ∗ α, where α is an n-loop in U at x. It is routine to check that the collection
On (p, U ) is a basis for Ωn (X, x). In the following, we show that Ωn (X, x) is Hausdorff
if and only if X is n-homotopically Hausdorff at x. ( see also [1] for the case n = 1.)
The following lemmas are essential to prove of the second result on countability
of homotopy groups.
Lemma 2.3. Ωn (X, x) is Hausdorff if and only if X is n-homotopically Hausdorff
at x.
166
Lemma 2.4. Ωn (X, x) with the above topology is homeomorphic to Ω(Ω(n−1) (X, x), ex ),
where Ω(n−1) (X, x) is equipped with the compact-open topology, for all n ≥ 2.
Lemma 2.5. Let n ≥ 2. Then a space X is n-homotopically Hausdorff at x if and
only if Ω(n−1) (X, x) is homotopically Hausdorff at ex , for any x ∈ X.
We shall also need the following well-known result of Dugundji [3].
Theorem 2.6. If X is second countable and Y is locally compact and second countable, then the function space X Y is second countable. In particular, if X is second
countable then Ωn (X, x) is also second countable, for all x ∈ X.
Now, the second result on countability of homotopy groups is as follows.
Theorem 2.7. Suppose that X is a second countable, locally (n-1)-connected and
n-homotopically Hausdorff space at x which is not semilocally n-connected at this
point. Then πn (X, x) is uncountable.
The following corollary is a consequence of Theorems 2.2 and 2.7.
Corollary 2.8. If X is an (n-1)-connected, locally (n-1)-connected, separable metric space, then the following statements are equivalent.
(i) X is semilocally n-connected.
(ii) X is n-homotopically Hausdorff and πn (X) is countable.
One of the main conditions of Theorem 2.2 is assuming that X is semilocally
n-connected. In the follow, we intend to give some conditions in which the space X
is semilocally n-connected.
Conner and Lamereaux [2] proved that if X is a connected, locally path connected separable metric space with a fundamental group which is a free group then
X admits a universal covering space. Using this result, the fact that the separability, metrizability, locally connected properties of space X can be transferred
to the loop space Ωn (X, x), for some x ∈ X and applying the group isomorphism
π1 (Ω(n−1) (X, x), ex ) ∼
= πn (X, x) we conclude the following result.
Proposition 2.9. Let X be an (n-1)-connected, locally (n-1)-connected, separable
metric space in which πn (X, x) is free. Then X is semilocally n-connected at x.
The following results are the extension of some results of Conner and Lamereaux
[2].
167
Definition 2.10. Let i : X → Y be an embedding of one path connected space into
another. Then we say that X is a πn -retract of Y if there exists a homomorphism r :
πn (Y ) → πn (X) such that the composition ri∗ : πn (X) → πn (X) is an isomorphism.
In this case the homomorphism r is called a πn -retraction for X in Y. Also, X is called
a πn -neighborhood retract in Y if X is a πn -retract of one of its open neighborhoods
in Y .
Definition 2.11. A separable metric space X is called a πn -absolute neighborhood
retract (πn -AN R) if whenever X is a subspace of a separable metric space Y , then
X is a πn -neighborhood retract in Y .
Lemma 2.12. Let Y be locally (n-1)-connected and semilocally n-connected and
X be a πn -retract of Y . Then X is semilocally n-connected.
Corollary 2.13. Let X be a separable metric space. If X is πn -AN R, then it is
semilocally n-connected.
References
[1] J.W. Cannon and G.R. Conner, On the fundamental groups of one-dimensional
spaces, Topology and its Applications, 153 (2006) 2648–2672.
[2] G.R. Conner, J.W. Lamoreaux, On the existence of universal covering spaces for
metric spaces and subsets of Euclidean plane, Fund. Math., 187 (2005), 95-110.
[3] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.
[4] H. Ghane, Z. Hamed, n-homotopically Hausdorff spaces, The 5th Seminar on
Geometry and Topology proceeding, 2009.
[5] H. Wada, Local connectivity of mapping space, it Duke Math. J., 22:3 (1955),
419-425.
168
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
The schur multiplier of pairs for some finite groups
Adnin Afifi Nawi1∗ , Nor Muhainiah Mohd Ali2 , Nor Haniza Sarmin3 ,
Samad Rashid4 and Rosita Zainal5
1,2,3,5
4
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi
Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
adnin− afifi@yahoo.com1 , [email protected] , [email protected] ,
[email protected]
Department of Mathematics, , Faculty of Science , Shahr-e-Rey Branch, Islamic Azad
University, Tehran, Iran
[email protected]
Abstract
The Schur multiplier of a group, M (G) is the second homology group of G with integer coefficients. Let (G, N ) be
an arbitrary pair of finite groups, then the Schur multiplier
of pairs of a group, M (G, N ) is a finite abelian group with
exponent dividing the order of G. Theoretically, Schur multiplier of a group and Schur multiplier of pairs of a group
are related to each other. In this research, we determine the
Schur multiplier of pairs of groups of order p3 and p2 q where
p and q are prime numbers.
1
Introduction
Let G be a group with a free presentation 1 → R → F → G → 1, then the Schur
multiplier of G, M (G) is isomorphic to R ∩ F ′ /[R, F ] in which F is any free group
and G ∼
= F/R is any presentation of G. Schur computed M (G) for many types of
groups and Karpilovsky compiled all the computations in [3].
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D15; Secondary 20F99, 20J99.
Key words and phrases. Groups of order p3 , Groups of order p2 q, Schur multiplier of pairs of
groups.
169
Let (G, N ) be an arbitrary pair of finite groups and N is a normal subgroup of
G, then the Schur multiplier of a pair of groups, M (G, N ) is said to be a functorial
abelian groups whose principal feature is the following exact sequence;
H3 (G) → H3 (G/N ) → M (G, N ) → M (G) → M (G/N ) → N/[N, G] → (G)ab →
(G/N )ab → 0
in which H3 (G) is the third homology of G with integer coefficients. Ellis [2] deduced
several basic results for M (G, N ) by assuming the existence of the above natural
sequence and the existence of a certain transfer homomorphism.
In this research, M (G, N ) for groups of order p3 where p is an odd prime is
determined.
In the following theorems,, some preliminary results that are necessary in proving
our main theorem are stated.
Theorem 1.1. [1] Let G be a group of order p3 , p an odd prime. Then exactly one
of the following holds:
1. G ∼
= Zp3 ,
2. G ∼
= Zp2 × Zp ,
3. G ∼
= (Zp )3 ,
4. G ∼
= Zp2 ⋊ Zp ,
5. G ∼
= (Zp × Zp ) ⋊ Zp ,
where Zn is cyclic group of order n.
Theorem 1.2. [3] Let G be a finite group. Then
1. M (G) is a finite group whose elements have order dividing the order of G.
2. M (G) = 1 if G is cyclic.
Theorem 1.3. [3] Let G be an extra-special p-group of order p2n+1 . Suppose that
|G| = p3 and p is odd. Then
{
Zp × Zp ; if G is of exponent p
M (G) =
1
; if G is of exponent p2 .
Theorem 1.4. [3] If G1 and G2 are finite groups, then
M (G1 × G2 ) = M (G1 ) × M (G2 ) × (G1 ⊗ G2 ).
170
Theorem 1.5. [2] Let (G, N ) be a pair of groups and Q be the complement of N
in G. Then |M (G, N )| = |M (N )| × |N ab ⊗ Qab |.
Theorem 1.6. [2] Let (G, N ) be a pair of groups and N = {e} then M (G, N ) =
M (G, {e}) = 1.
Theorem 1.7. [2] Let (G, N ) be a pair of groups and N = G then M (G, N ) =
M (G, G) = M (G).
Theorem 1.8. [2] Supppose that N and Q are subgroups of G, with G = N Q,
N ∩ Q = 1 and N normal. In other words, suppose that G is a semi direct product
of N and Q. Then M (G) ∼
= M (G, N ) × M (Q).
Theorem 1.9. [5] Let G be a cyclic group and H a subgroup of G. Then M (G, N ) =
1.
2
Main results
In the following theorems, the Schur multipler of pairs of groups of order p2 and pq
are stated.
Theorem 2.1. Let G be a group of order p2 where p is a prime number and let
(G, N ) be an arbitrary pair of finite groups where N is a normal subgroup of G.
Then M (G, N ) = 1 or Zp .
Theorem 2.2. Let G be a group of order pq where p and q are prime number and
p < q. Let (G, N ) be an arbitrary pair of finite groups where N is a normal subgroup
of G. Then M (G, N ) = 1.
In the following theorem, the Schur multipler of pairs of groups of order p3 is
stated.
Theorem 2.3. Let G be a group of order p3 , p an odd prime. Then exactly one of
the following holds:

1
; if G ∼
= Zp3 and G ∼
= Zp2 ⋊ Zp ,



∼
1 or Zp
; if G = Zp2 × Zp ,
M (G, N ) =
; if G ∼
1 or Z2p
= (Zp ⋊ Zp ) × Zp ,



3
2
1, Zp or Zp ; if G ∼
= Z3p .
Proof. In order to compute the Schur multiplier of pairs of groups of order p3 , the
classification in Theorem 1.1 is referred.
171
Firstly, lets consider the case where G ∼
= Zp3 . Since G is cyclic, we have M (G, N ) =
1. The result follows from Theorem 1.9.
For a group G ∼
= Zp2 × Zp , we have the following cases:
Case 1: If N = {e}, then by Theorem 1.6, M (G, N ) = M (G, {e}) = 1.
Case 2: If N = Zp2 then M (G, N ) = M (G, Zp2 ). ByTheorem 1.5, the computation
is as follows;
|M (G, Zp2 )| = |M (Zp2 )|×|(Zp2 )ab ⊗(Zp )ab |. Since Zp2 is cyclic, M (Zp2 ) = 1 (refer to
Theorem 1.2). We have (Zn )ab = Zn , thus it is clear that Zp2 ⊗ (Zp ) ∼
= Z(p2 ,p) = Zp .
Thus, M (G, Zp2 ) = Zp .
Case 3: If N = Zp then M (G, N ) = M (G, Zp ). The computation is similar as in
Case 2.
Case 4: If N = G then by Theorem 1.7 M (G, N ) = M (G, G) = M (G). Then by
Theorem 1.4, we have the following computation:
M (Zp2 × Zp ) = M (Zp2 ) × M (Zp ) × (Zp2 ⊗ Zp ). By Theorem 1.2, M (G) = Zp .
By using a similar method as the group G ∼
= Zp2 × Zp , the computation of M (G, N )
for G ∼
= Z3p is given as follows:
Case 1: If N = {e}, then M (G, N ) = M (G, {e}) = 1.
Case 2: If N = Zp × Zp then M (G, N ) = M (G, Zp × Zp ) = Z3p .
Case 3: If N = Zp then M (G, N ) = M (G, Zp ) = Z2p .
Case 4: If N = G then M (G, N ) = M (G, G) = M (G) = Z3p .
For the group G ∼
= Zp2 ⋊ Zp , we consider the following cases:
Case 1: If N = {e}, then by Theorem 1.6, M (G, N ) = M (G, {e}) = 1.
Case 2: If N = Zp2 then M (G, N ) = M (G, Zp2 ). By Theorem 1.8, M (G) ∼
=
M (G, Zp2 ) × M (Zp ). M (G) = 1 is computed in [3] and M (G) = 1 (refer to Theorem
1.3). Hence, it is clear that M (G, Zp2 ) = 1.
Case 3: If N = G then M (G, N ) = M (G, G) = M (G). Thus, M (G) = 1 as given in
Theorem 1.3.
For the group G ∼
= (Zp × Zp ) ⋊ Zp the following cases are considered:
Case 1: If N = {e}, then by Theorem 1.6 M (G, N ) = M (G, {e}) = 1.
Case 2: If N = Zp ×Zp then M (G, N ) = M (G, Zp ×Zp ). M (G) is computed in [3] and
M (G) = (Zp )2 (refer to Theorem 1.3). Hence, by Theorem 1.8 M (G, Zp ×Zp ) = Z2p .
Case 3: If N = G then by Theorem 1.7 M (G, N ) = M (G, G) = M (G). Thus,
M (G) = Z2p as given in Theorem 1.3.
Lastly, in the following theorem, we state the Schur multipler of pairs of groups
of order p2 q.
172
Theorem 2.4. Let G be a group of order p2 q where p and q are distinct primes
and (G, N ) be an arbitrary pair of finite groups where N is a normal subgroup of
G. Then M (G, N ) = 1 or Zp .
Acknowledgement
The authors would like to thank Universiti Teknologi Malaysia (UTM) for the financial funding through the Research University Grant (RUG) Vote No. 04H13
and UTM Mobility Program. The first author would also like to thank Ministry of
Education (MOE) Malaysia for her MyPhD Scholarship.
References
[1] D. S. Dummit and R. M. Foote Abstract Algebra, Third Edition, USA. John
Wiley and Sons, Inc., 2004.
[2] G. Ellis, The Schur multiplier of a pair of groups, Applied Categorical Structures,
6 (1998), 355–371.
[3] G. Karpilovsky, The Schur multiplier, LMS Monogrphs New Series 2, Oxford
Univ. Press, 1987.
[4] F. Mohammadzadeh, A. Hokmabadi and B. Mashayekhy, On the order of the
Schur multiplier of a pair of finite p-groups II, International Journal of Group
Theory, 2 (2013), no. 3, 1–8.
[5] A. A. Nawi, N. M. Mohd Ali, N. H. Sarmin and S. Rashid On The Second
Homology of Pairs of Groups of Finite Order, AIP Conf. Proc. (to appear).
173
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Separation properties of topological fundamental groups
Ali Pakdaman1∗ , Behrooz Mashayekhi2 and Hamid Torabi2
1
Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan, Iran
[email protected]
2
Department of Pure Mathematics,
Center of Excellence in Analysis on Algebraic Structures,
Ferdowsi University of Mashhad,
P.O.Box 1159-91775, Mashhad, Iran
[email protected] and hamid− [email protected]
Abstract
In this talk, we discuss on the topological properties of the
Spanier groups which are new subgroups of fundamental
group, constructed by an inverse limit method. Also, by
using these results, we establish an algebraic criteria for the
Hausdorffness of topological fundamental groups.
1
Introduction
In 2002, a work of Biss initiated the development of a theory in which the familiar
fundamental group π1 (X, x) of a topological space X becomes a topological space
denoted by π1top (X, x) by endowing it with the quotient topology inherited from the
path components of based loops in X with the compact-open topology. Among other
things, Biss claimed that π1top (X, x) is a topological group. However, there is a gap
in his proof. Brazas discovered some interesting counterexamples for continuity of
multiplication in π1top (X, x) (for more details, see [1]).
In fact, π1top (X, x) was a quasitopological group, that is, a group with a topology
such that inversion and all translations are continuous. After this obstacle, Brazas
[1] by removing some open sets of π1top (X, x), make it a topological group and denote
∗
Speaker
2010 Mathematics Subject Classification. Primary 20F38; Secondary 20k45 , 57M10.
Key words and phrases. Topological fundamental group, Spanier group, Hausdorff topology.
174
it by π1τ (X, x). Indeed, the functor π1τ removes the smallest number of open sets
from the topology of π1top (X, x) so as to make it a topological group. Here, by the
topological fundamental group we mean π1τ (X, x).
In the sequel, we study the topology of topological fundamental group from some
separation axioms viewpoint. The main idea is working with the Spanier groups
with respect to open covers of a given space X which have been introduced in [2]
and named in [2]. The importance of these groups and their intersection which is
named Spanier group, π1sp (X, x), is studied by H. Fischer et al. in [2] in order to
modification of the definition of semi-locally simply connectedness.
Throughout this article, all the homotopies between two paths are relative to
end points, X is a connected and locally path connected space with the base point
e −→ X is a covering of X with x̃ ∈ p−1 ({x}) as the base point of X.
e
x ∈ X, and p : X
eH we mean a covering space of X such
For a space X and any H ≤ π1 (X, x), by X
−1
e
eH −→ X is the corresponding
that p∗ π1 (X, tildex) = H, where x̃ ∈ p (x) and p : X
covering map.
2
Definitions and preliminaries
E.H. Spanier [2, §2.5] classified path connected covering spaces of a space X using
some subgroups of the fundamental group of X, recently named Spanier groups (see
[3]). If U is an open cover of X, then the subgroup of π1 (X, x) consisting of all
homotopy classes of loops that can be represented by a product of the following
type
n
∏
αj ∗ βj ∗ αj−1 ,
j=1
where the αj ’s are arbitrary paths starting at the base point x and each βj is a loop
inside one of the neighborhoods Ui ∈ U , is called the Spanier group with respect to
U, and denoted by π(U, x) [2]. For two open covers U , V of X, we say that V refines
U if for every V ∈ V, there exists U ∈ U such that V ⊆ U .
Definition 2.1. [2] The Spanier group of a topological space X, denoted by π1sp (X, x)
for an x ∈ X is
∩
π1sp (X, x) =
π(U , x).
open covers U
Also, we can obtain the Spanier groups as follows: Let U, V be open coverings
of X, and let U be a refinement of V. Then since π(U, x) ⊆ π(V, x), there exists
an inverse limit of these Spanier groups, defined via the directed system of all open
covers with respect to refinement and it is π1sp (X, x).
175
Definition 2.2. [3] We call a topological space X the Spanier space if π1 (X, x) =
π1sp (X, x), for an arbitrary point x ∈ X.
e −→ X is a covering and x ∈ X, then π sp (X, x) ≤
Proposition 2.3. [3] If p : X
1
e tildex).
p∗ π1 (X,
Theorem 2.4. ([2, §2.5 Theorems 12,13]) Let X be a connected, locally path cone −→ X
nected space and H ≤ π1 (X, x), for x ∈ X. Then there exists a covering p : X
e tildex) = H if and only if there exists an open cover U of X in
such that p∗ π1 (X,
which π(U, x) ≤ H.
eπ(U ,x) exists.
We immediately deduce that for every open cover U of X, X
3
Main results
By [5, Theorem 3.7], the connected coverings of a connected and locally path connected space X are classified by conjugacy classes of subgroups of π1qtop (X, x) with
open core, and since π1τ (X, x) and π1qtop (X, x) have the same open subgroups [1,
Corollary 3.9], we have the same result for open subgroups of π1τ (X, x). Using this
fact and Theorem 1.2, for every open cover U of X, π(U, x) is an open subgroup
of π1τ (X, x) and since π1τ (X, x) is a topological group, π(U, x) is a closed subgroup,
which implies that π1sp (X, x) is a closed subgroup of π1τ (X, x). Hence we have the
following proposition.
Proposition 3.1. For a connected and locally path connected space X, π1sp (X, x)
is a closed subgroup of π1τ (X, x), for every x ∈ X.
Using the above proposition, the Spanier group of a connected and locally path
connected space contains the closure of the trivial element of the topological fundamental group. Hence we have the following corollary.
Corollary 3.2. Let X be a connected and locally path connected space and x ∈ X.
If π1τ (X, x) has an indiscrete topology, then X is a Spanier space.
Corollary 3.3. Let X be a connected and locally path connected space and x ∈ X.
If π1sp (X, x), then π1τ (X, x) has the T1 topology.
We recall that a space X is called shape injective if the natural homomorphism
φ : π1 (X, x) −→ π̌1 (X, x) is injective, where π̌1 (X, x) is the first shape group of
(X, x). Also, an open cover U of X is called normal if it admits a partition of unity
subordinated to U. Also, every open cover of a paracompact space is normal. (see [3]
for further details). In [1, Proposition 3.25] it is proved that the topological fundamental groups of shape injective spaces are Hausdorff. Since the spaces with a trivial
176
Spanier group are not necessarily shape injective, the triviality of Spanier groups
can not certify the Hausdorffness of topological fundamental groups in general. In
the following, we provide the conditions that guarantee this.
Definition 3.4. [3] A space X is small loop homotopically Hausdorff if for each
x ∈ X and each loop α based at x, if for each normal open cover U of X, [α] ∈ π(U, x),
then [α] = 1.
Proposition 3.5. Suppose X is a connected, locally path connected and paracompact space. Then π1sp (X, x) if and only if X is shape injective.
Corollary 3.6 (A criterion for the Hausdorffness of π1τ (X, x)). Let X be a connected, locally path connected and paracompact space. If π1sp (X, x), then π1τ (X, x)
is Hausdorff.
Note that since π1τ (X, x) is a topological group, by the assumption of the above
corollary π1τ (X, x) is regular.
References
[1] J. Brazas, The fundamental group as a topological group, Topology Appl. 160
(2013), no. 1, 170–188.
[2] H. Fischer, D. Repovš, Ž. Virk, and A. Zastrow, On semilocally simply connected
spaces, Topology Appl. 158 (2011), no. 3, 397–408.
[3] B. Mashayekhy, A. Pakdaman, H. Torabi, Spanier spaces and covering theory
of non-homotopically path Hausdorff spaces, Georgian Mathematical Journal, 20
(2013) 303–317.
[4] E. H. Spanier, Algebraic Topology, McGraw-Hill Book Co., New York-Toronto,
Ont.-London 1966
[5] H. Torabi, A. Pakdaman, B. Mashayekhy, On the Spanier Groups and Covering
and Semicovering Spaces, arXiv:1207.4394v1.
177
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
On the characterizations of finite groups
Hosein Parvizi Mosaed1∗ , Ali Iranmanesh2 and Mahnaz Foroudi
Ghasemabadi3
1
Department of Mathematics, Science and Research Branch, Islamic Azad University,
Tehran, Iran
[email protected]
2
Department of Mathematics, Tarbiat Modares University, Tehran, Iran
[email protected]
3
Department of Mathematics, Tarbiat Modares University, Tehran, Iran
[email protected]
Abstract
Let G be a finite group and π(G) be the set of all prime
divisors of |G|. Let πe (G) be the set of element orders of G
and k(G) be the largest element order of G. Suppose that
mk be the number of elements of order k in G and nse(G) =
{mk | k ∈ πe (G)}. In this talk, we first prove that the finite
simple k4 -groups Sz(8), Sz(32), L2 (2m ), L2 (3m ), L2 (p2 ),
where p ∈ {5, 7} are characterizable by nse(G). Then we
show that the sporadic simple groups except F i22 and He
are characterizable by nse(G) and π(G). Finally the characterization of the finite groups P GL2 (p) and L2 (3m ) by |G|
and k(G) are investigated.
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D60; Secondary 20D06.
Key words and phrases. Element order, the largest element order, simple group, sporadic simple
group.
178
1
Introduction
During the classification of the finite simple groups, it has been observed that some
of the known simple groups are characterizable by some of their properties and
up to now, different characterizations are investigated for the finite simple groups.
For instance, in [4], motivated by one of the Thompson’s problem, the authors
introduced a new characterization for finite simple groups by nse(G) and |G|. For
a finite group G, denote by πe (G) the set of element orders of G. If k ∈ πe (G), then
mk denotes the number of elements of order k in G. Let nse(G) = {mk | k ∈ πe (G)}.
In fact, the authors in [4] proved that all finite simple K4 -groups can be uniquely
determined by nse(G) and |G|. Let π(G) be the set of all prime divisors of the
order of a finite groups G. The finite simple group G is called simple Kn -group if
|π(G)| = n. Following this result, in [3] and [5], the authors prove that the group
L2 (q), where q ∈ {3, 4, 5, 7, 8, 9, 11, 13} is determined only by nse(G) and up to the
present time, it has been proved that some other simple groups can be characterized
by nse(G) and |G| or only by nse(G) (see for instance [2]-[3]). In this talk, we
try to extend the results in [4]. In fact, we prove that the finite simple k4 -groups
Sz(8), Sz(32), L2 (2m ), L2 (3m ), L2 (p2 ), where p ∈ {5, 7} are uniquely determined
by nse(G). Also, we show that the sporadic simple groups except F i22 and He are
characterizable by nse(G) and π(G).
It is a well-known conjecture by Shi which states that every finite simple group
G is uniquely determined by πe (G) and |G| in the class of all group. A series
of papers proved that this conjecture is true for most of finite simple groups and
finally Mazurov, et al. completed the proof of the validity of Shi’s conjecture for all
finite simple groups. Now the authors try to characterize simple groups by using
less quantities. For instance, in [1], it has been proved that the linear simple group
L2 (q) with q = pn < 125 is determined by group order, the largest, the second
largest and the third largest element orders. Let k(G) denote the largest element
order of a finite group G. Our another goal in this talk is to investigate that the
group P GL2 (p), where p > 3 is a prime number and the simple K4 -group L2 (3m )
are characterizable by |G| and k(G).
2
Main results
Theorem 2.1. Let G be a group and S be one of the following simple groups:
1. L2 (2m ), where m, 2m − 1 and (2m + 1)/3 are primes greater than 3;
2. L2 (3m ), where m, (3m − 1)/2 are odd primes and (3m + 1)/4 is either a prime
or 112 for (m = 5);
179
3. Sz(8) and Sz(32);
4. L2 (p2 )), where p ∈ {5, 7}.
If nse(G) = nse(S), then G ∼
= S.
Theorem 2.2. Let G be a group and S be a sporadic simple groups except F i22
and He. If nse(G) = nse(S) and π(G) = π(S), then G ∼
= S.
Theorem 2.3. Let G be a group and S be one of the following groups:
1. L2 (3m ), where m, (3m − 1)/2 are odd primes and (3m + 1)/4 is either a prime
or 112 for (m = 5);
2. P GL2 (p), where p > 3 is a prime.
If |G| = |S| and k(G) = k(S), then G ∼
= S.
References
[1] He, L.G., Chen, G.Y.: A new characterization of simple L2 (q) where q = pn <
125. Italian Journal of Pure and Applied Mathematics, 28:127-136, 2011.
[2] Khalili Asboei, A.R., Salehi Amiri, S.S., Iranmanesh, A., Tehranian, A.: A
characterization of sporadic simple groups by NSE and order. J. Algebra Appl.
12, 1250158 (2013)
[3] Khatami, M., Khosravi, B., Akhlaghi, Z.: A new characterization for some linear
groups, Monatsh. Math. 163, 39-50 (2011)
[4] Shao, C.G., Shi, W.J., Jiang, Q.H.: Characterization of simple K4 -groups, Front.
Math. China 3, 355-370 (2008)
[5] Shen, R., Shao, C., Jiang, Q., Shi, W., Mazurov, V.: A new characterization of
A5 , Monatsh. Math. 160, 337-341 (2010)
180
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
An approach to c-imperfect groups
Azam Pourmirzaei1 and Mitra Hassanzadeh2∗
1
Department of Mathematics, Hakim Sabzevari University, P. O. Box 96179-76487,
Sabzevar, Iran
[email protected]
2
Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box
1159-91775, Mashhad, Iran
[email protected]
Abstract
We call a group G, c-imperfect if it has no nontrivial cperfect quotient group. In this paper, some equivalent conditions for a group to be c-imperfect is obtained. Furthermore
we show that every countable residually c-imperfect group
is c-imperfect. Finally this fact that the class of countable
c-imperfect groups forms a variety is concluded.
1
Introduction
Groups that coincide with their derived subgroups which are called perfect groups,
are extensively studied in group theory. It is very clear that every simple nonabelian
group is perfect. The smallest nontrivial finite perfect group must be simple, hence
is the alternating group A5 = P SL(n, K), for n ≥ 3 or n = 2 where |K| > 3 (see [1]).
A type of groups that are far removed from the domain of perfect groups are
called imperfect groups which is introduced by Robinson in [3]. A group is said to be
imperfect if it has no nontrivial perfect quotient group. Soluble and finite symmetric
groups are some examples of imperfect groups.
∗
Speaker
2010 Mathematics Subject Classification. Primary 20E34; Secondary 20E22, 20F05.
Key words and phrases. Perfect group, Imperfect group, Variety.
181
In this work we introduce a new notion “c-imperfect” groups, that is, a group
with no nontrivial c-perfect quotient group. By a c-perfect group G we mean
γc+1 (G) = G. For c = 1, we have imperfect groups, also it can be easily seen
that every imperfect group is c-imperfect. We emphasize that this generalization
yields to have better result for imperfect groups.
2
Main results
We begin this section by some basic properties of the class of c-imperfect groups.
• The class of c-imperfect groups is closed with respect to homomorphic images.
• The class of c-imperfect groups is closed with respect to extension.
Theorem 2.1. If M and N are c-imperfect normal subgroups of a group G, then
M N is c-imperfect.
Proof. By the hypothesis as M N/N is c-imperfect, we have the result.
The following lemma is essential for Theorem 2.3
Lemma 2.2. If G is a nilpotent group of class c, then G is c-imperfect.
Theorem 2.3. Suppose that M and N are normal subgroups of a group G. If G/M
and G/N are c-imperfect, then so is G/M ∩ N .
Proof. Without lose of generality we can assume that M ∩ N = 1. Consider a cperfect quotient of G, G/L. The c-perfect subgroup LM/M of G/M implies that
G = LM . Similarly G = LN . In the same vein we assume L ∩ M = 1. Then
[M,c G] = 1, hence M ≤ Zc (G). By Lemma 2.2, G = L.
Lemma 2.2 and Theorem 2.3 have the following nice but easy conclusion.
Corollary 2.4. Let G/Zc (G) be a c-imperfect group. Then G/ϕ(G) is c-imperfect.
Theorem 2.5. A direct product of c-imperfect groups is c-imperfect.
∏
Proof. Let G = λ∈Λ Gλ , where Gλ is a c-imperfect group. Consider the c-perfect
factor G/L of G. Similar to the previous argument let Gλ ∩ L = 1, for every λ ∈ Λ.
Therefore L ≤ Zc (G) and hence by the assumption Gλ = Zc (Gλ ). This shows that
G is nilpotent of class c and so L = G.
The following theorem asserts a description of c-imperfect groups by mapping
properties.
182
Theorem 2.6. For a group G the following conditions are equivalent:
(i) G is c-imperfect.
(ii) If φ : H −→ G is a homomorphism inducing an epimorphism φc : H/γc+1 (H) −→
G/γc+1 (G), then G =< Imφc >G .
(iii) If N E G and the natural map N/γc+1 (N ) −→ G/γc+1 (G) is surjective, then
N = G.
Proof. (i) ⇒ (ii): By contrary let < Imφc >G < G. Then since G is c-imperfect,
γc+1 (G) < Imφc >G < G. On the other hand φc is an epimorphism. These facts
force the subgroup < Imφc >G to be equal to G which is a contradiction.
(ii) ⇒ (iii): It is clear.
(iii) ⇒ (i): Let G/T be a c-perfect quotient of G. Consider the natural map
φ : T ↩→ G. Since γc+1 (G)T = G, φc : T /γc+1 (T ) −→ G/γc+1 (G) is an epimorphism.
Therefore G = T .
In sequel we explore that the class of all countable c-imperfect groups forms
a variety of groups. This determines that the property of being c- imperfect in a
countable group is inherited by its subgroup. Moreover we deal with residually cimperfect groups. As free groups are residually nilpotent so residually c-imperfect
but not necessarily c-imperfect ([4]). Now the following theorem may be interesting.
Theorem 2.7. If G is a countable residually c-imperfect group, then G is c-imperfect.
Proof. From the definition of residually X -group ([2]) and Theorem 2.3, the result
holds.
Corollary 2.8. A subcartesian product of c-imperfect groups is c-imperfect.
Proof. A group G is residually c-imperfect if and only if it is a subcartesian product
of c-imperfect groups ([2]). The previous theorem implies the result.
A class of groups which is closed with respect to forming homomorphic images
and subcartesian product of its members is a variety ([2]). Furthermore every variety
is closed with respect to forming subgroups, images and subcartesian product ([2]).
These facts help us to achieve our goals.
Corollary 2.9. The class of countable c-imperfect groups is a variety.
Corollary 2.10. Every subgroup of a countable c-imperfect group is c-imperfect.
183
References
[1] R. Beyl, J. Tappe, Group extensions, representations and Schur multiplicator,
Lecture notes in math, vol. 958, Springer-verlag, Berlin, 1982.
[2] D. J. S. Robinson, A course in the theory of groups, Speringer-Verlage, New
York, 1982.
[3] D. J. S. Robinson, Finiteness conditions and generalized soluble groups,
Speringer, Berlin, 1972.
[4] D. J. S. Robinson, Imperfect groups, Journal of Pure and Applied Algebra 88
(1993) 3-22.
184
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
The structure of Permutation Groups with
t = 1/3(6m − 2)
Behnam Razzaghmaneshi
Department of Mathematics, Faculty of Sciences, Islamic Azad University, Talesh Branch,
Talesh, Iran
b [email protected]
Abstract
Let G be a permutation group on a set Ω with no fixed
points in Ω. If t the number of G-orbits in Ω. Then in this
paper we classified all of groups for t = 1/3(6m − 2).
1
Introduction
Let G be a permutation group on a set Ω with no fixed points in Ω and let m be a
positive integer. If for a subset Γ of Ω the size |Γg \ Γ| is bounded, for g ∈ G , we define the
movement of Γ as move(Γ) = maxg∈G |Γg \ Γ|. If move(Γ) ≤ m for all Γ ⊆ Ω,then G is said
to have bounded movement and the movement of G is define as the maximum of move(Γ)
over all subsets Γ, that is,
m := move(G) := sup{|Γg \ Γ||Γ ⊆ Ω, g ∈ G}.
This notion was introduced in [3]. By [3,Theorem 1],if G has bounded movement m,then
Ω is finite. Moreover both the number of G-orbits in Ω and the length of each G-orbit
are bounded above by linear functions of m.In particular it was shown that the number of
G-orbits is at most 2m-1. In this paper we will improve this to 13(6m − 2), if the lengths of
all orbits not equal to 2. If m=1, then t = 1, |Ω| = 2 and G is Z2 or S2 . So in this paper we
suppose that m greater than 1. We present here a classification of all groups for which the
bound 13(6m − 2) is attained. We shall say that an orbit of permutation group is nontrivial
if its length is greater than 1.The main result is the following theorem.
Theorem 1.1. Let m be a positive integer and suppose that G is a permutation group on
a set Ω such that G has no fixed points in Ω, and G has bounded movement equal to m .
2010 Mathematics Subject Classification. 20B25.
Key words and phrases. permutation group, bounded movement, orbits.
185
If the lengths of all orbits not equal to 2. Then the number t of G-orbits in Ω is at most
13(6m − 2). And also if t = 13(6m − 2), then m is a power of 2, and G is order 2m, all
G-orbits have length 2 or 3, and the pointwise stabilizers of the G-orbits are precisely the
13(6m − 2) distinct subgroups of G of index 2 or 3 .
Note that an orbit of a permutation group is non trivial if its length is greater than 1.
The groups described below are examples of permutation groups with bounded movement
equal to m which have exactly 13(6m − 2) nontrivial orbits.
2
Examples and Preliminaries
Lemma 2.1. [5, Lemma 2.1]. Let G be a∑permutation group on a set Ω and suppose that
t
Γ ⊆ Ω . Then for each g ∈ G, |Γg \ Γ| ≤
i=1 ⌊li /2⌋, where li is the length of the ith cycle
of g and t is the number of nontrivial cycles of g in its disjoint cycle representation . This
upper bound is attained for Γ = Γ(g) defined above .
Now we will show that there certainly is an infinite family of 3-groups for which the maximum bound obtained in Theorem 1.1 holds .
Example 2.2. For a positive integer d and a prime number 3, let G1 := ⟨(123)⟩ ∼
= Z3 be
a permutation group on Ω1 := 1, 2, 3. Moreover, suppose that G2 := Z2d , and H1 , ..., Ht be
∪2d −1
all subgroups of index 2 in G on Ω2 := i=1 Ω2i , where Ω2i denotes the set of two cosets
of Hi in G2 , 1 ≤ i ≤ t = 2d − 1. Then G2 has movement equal 2d − 1 and also (2d − 1)
nontrivial orbits in Ω2 . Now we consider the direct product G := G2 × G2 as a permutation
on Ω which is the disjoint union of Ω1 and Ω2 , and G1 and G2 act trivially on Ω2 and Ω1 ,
respectively . Then G has movement 1 + 2d − 1 . The set Ω splits into 2d = 2m − 2 orbits
under G, which are Ω1 and 2d − 1 orbits of length 2 in Ω2 . In particular , none of them is
trivial . Furthermore , 4m − 3 = 4(1 + 2d+1 ) − 3 = 3 + 2(2d − 1) = |Ω1 | + |Ω2 | = |Ω| .
Example 2.3. Let d , G2 and Ω2 have the same meaning as Example 2.2 . Suppose that
the permutation group G1 := Z3 Z2d on Ω1 of length 3 is a Ferobenius group defined as above
of order 32d where 2d |2 for some positive integer d, let x be a generator of a complement
isomorphic to Z2d . Then G := G1 × G2 is a permutation group on Ω := Ω1 ∪ Ω2 ( as in
Example 2.2) with bounded movement m = 1 + 2d−1 and (2m − 2) − 1 = 2m − 3 orbits of
lengths 2 in Ω2 as well as single orbit in Ω1 . In particular, |Ω| = 2(2m − 3) + 3 = 4m − 3,
which meets the upper bound for the latter inequality in Theorem 1.1.
Example 2.4. Let r be a positive integer , let G := Z3 Z2r , let t := 2m − 2 , and let the
lengths of all orbits are not equal to 2, and H1 , ..., Ht be an enumeration
∪ of∪the subgroups of
index 2 in G. Define Ωi to be the coset space of Hi in G and Ω := Ω1 ... Ωt . If g ∈ G \ 1
then g lies in 13(3.2r+1 − 1) of the groups Hi and therefore acts on Ω as a permutation with
13(3.2r+1 − 2) = m − 1 fixed points and 2r disjoint 2-cycles . Taking one point from each
of these 2-cycles to form a set Γ we see that m(G) ≥ 2r ,and it is not hard to prove that
in fact m(G) = 2r . Thus n = 2t = 23(3.2r+1 − 4) = 23(6m − 2) . This proves bound of
G − orbits of Theorem 1.1 . It follows that G has bounded movement equal to m, and G
has 13(6m − 2) nontrivial orbits in Ω .
When m > 1, the classification in Theorem 1.1 follows immediately from the following
186
theorem about subsets with movement m.
Definition Let G be a permutation group on a set with orbits i , for i ∈ I. We shall say
that a subset G cuts across each G-orbit if Gi := G∩i ∈
/ {,i }, for every i ∈ I.
Theorem 2.3. Let GlSym(Ω) be a permutation group with t orbits for positive integer
t, such that the lengths all of orbits are not to 2. Moreover suppose that GΩ such that
move (G) = m > 1, and G cuts across each G-orbit. Then t ≤ 13(6m − 2) and moreover, if
t = 13(6m − 2) , then:
(1) G is an 2-group and all G-orbits of G has size 2 ;
(2) If the rank of the group G is r then r2, t = 13(3.2r+1 − 2) and m = 2r−1 ;
(3)If one of the G-orbits is 3, then The tdif f erentG − orbits are (isomorphic to) the coset
spaces of the 13(3.2r+1 − 2) − 1 = 13(3.2r+1 − 5) different subgroups of index 2inG.
3
Proof of Theorem 2.3.
From now on we may and shall assume that each |Gi | = 1. Let Gi = {i }. Further we
may assume that n1 ln2 l...lnt . For g ∈ G let c(g) denote the number of integers I such that
ωig = ωi . Note that since move (G) = m, we have c(g) > t − m = 2m − 23 − m = m − 23
and also c(1G ) = t > m − 23.
Lemma 3.2. If one of the orbits of G has length equal to 3, then the rest orbits of G has
size 2.
Proof∑
: Let X denote the number of pairs (g,i) such that g ∈ G, 1 ≤ i ≤ t , and ωig = ωi .
Then X = g∈G c(g), and by our observations, X > |G|.(m−1). On the other hand, for each
∑t
i, the number of elements of G which fix ωi is |Gωi | = |G|ni , and hence X = |G| i=1 n−1
i
If all the ni 2, and one of ni ≥ 3 then X ≤ |G|.t2 = |G|(m − 13) ≤ |G|.(m − 1) (since m3 )
which is a contradiction. Hence n=2.
A similar argument to this enables us to show that except one the rest of ni is ni = 2,
and hence that G is an 2 − group.
Lemma 3.3. The group G = Z3 Z2r for some r2. Moreover for each ni = 2, except one
, the stabilizers Gωi (2lilt) are pair wise distinct subgroups of index 2 in G, and for each
g ̸= 1, c(g) = (m − 12).
Proof: By Lemma 3.2, except one of ni the rest of ni is ni = 2. Thus H := Gωi is a subgroup
of index 2. This time we compute the number Y of pairs (g, i) such that g ∈ G H, 2lilt ,
g
and ωig = ωi . For each
∑ such g, ω1 ̸= ω1 and hence there are c(g) of these pairs with first
entry g. Thus Y = g∈G\H c(g)|G\H|(m − 1) = |G|(m − 12).
On the other hand, for each i2, the number of elements of G, which fix ωi is |Gωi \H|.
If H = Gωi then |Gωi \H| = 0, while if Gωi ̸= H, then |Gωi \H| = |Gωi |2 = |G|2ni l|G|4.
Hence
∑t
∑t
Y = i=2 |Gωi \H|l]|G|2 i=2 1ni l|G|2(13 + t − 12)
= |G|2(3t − 16) < |G|(m − 12)
It follows that equality holds in both of the displayed approximations for Y . This means
in particular that each ni = 2, Whence G = Z3r for some r. Further, for each i3, Gωi ̸= H
187
and so r2. Arguing in the same way with H replaced by Gωi , for some i2, we see that
Gωi ̸= Gωj if j ̸= i, and also if g ∈ Gωi then c(g) = (m − 12). Thus the stabilizers Gωi (1lilt)
are pairwise distinct , and if gl1 then c(g) = (m − 12). Finally we determine m.
Lemma 3.4.. m = 2r−1
Proof: We use the information in lemma 3.3 to determine precise the quantity X =
∑
r−1
− 1)(m − 23). On the other
g∈G c(g) : X = t + (|G| − 1).(m − 23) = 13(6m − 2)(3.2
hand, from the proof of lemma 3.2,
X = |G|
t
∑
n−1
= |G|(13 + t − 12) = 3.2r .(m − (3t − 1)6) = 2r−1 (3t − 1).
i
i=1
Thus implies that m = 2r−1 .
The proof of theorem 2 now follows from lemmas 3.2 − 3.4.
References
[1] L.Brailovsky, Structure of quasi-invariant sets, Arch.Math.,59 (1992),322-326.
[2] L.Brailovsky, D.Pasechnix , C.E.Praeger, Subsets close to invarianr subset of quasiinvariant subsets for group actions ,,Proc.Amer. Math.Soc. ,123(1995),2283-2295.
[3] C.E.Praeger,On permutation groups with bounded movement,J.Algebra ,144(1991),436442.
[4] C.E.Praeger, The separation theorem for group actions, in ”ordered Groups and Infinite
Groups”(W.charles Holland, Ed.), Kluwer Academic, Dordrecht/ Boston/ Lond, 1995.
[5] A.Hassani,M.Khayaty,E.I.Khukhro and C.E.Praeger, Transitive permutation groups
with bounded movement having maximum degree.J. Algebra,214(1999),317-337.
[6] J.R.Cho, P.S.Kim, and C.E.Praeger, The maximal number of orbits of a permutation
Group with Bounded Movement, J.Algebra,214 (1999),625-630.
[7] P.M.Neumann, The structure of finitary Permutation groups, Arch. Math. (Basel)
27(1976),3-17.
[8] B.H.Neumann, Groups covered by permutable subsets, J. London Math soc., 29(1954),
236-248.
[9] P.M.Neumann, C.E.Praeger, On the Movement of permutation Group, J.Algebra, 214,
(1999)631-635.
188
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Permutation Groups with Three Constant Orbits
Behnam Razzaghmaneshi
Department of Mathematics, Faculty of Sciences, Islamic Azad University, Talesh Branch,
Talesh, Iran
b [email protected]
Abstract
Let G be a permutation group on a set Ω with no fixed
points in Ω. If t the number of G-orbits in Ω. Then in this
paper we determined the maximum bound |Ω| = 3m + 2,
for t = 2, will be classified. We show that there are not no
group and no example for case |Ω| = 3m + t − 1,when t > 2.
1
Introduction
Let G be a permutation group on a set Ω with no fixed points in Ω and let m
be a positive integer. If each G-orbit has length at most 3m, t ≤ 2m − 1 and
n := |Ω| ≤ 3m + t − 1 ≤ 5m − 2, where t is the number of G-orbits on Ω. In [2]
it was shown that n = 5m − 2 if and only if n = 3 and G is transitive. But in [4],
this bound was refined further and it was shown that n ≤ (9m − 3)/2. Moreover, if
n = (9m − 3)/2 then either n = 3 and G = S3 or G is an elementary abelian 3-group
and all its orbits have length 3. Also this upper bound was improved to the bound
n ≤ 4m − p in [1, Theorem 1.1], where p ≥ 5 is the least odd prime dividing |G|.
Throughout this paper m is a positive integer and G is an intransitive permutation group on a set Ω of size n with movement m and t(≥ 2) orbits, such that
n = 3m + t − 1. The purpose of this paper is to classify all intransitive permutation
groups of degree n = 3m + t − 1, for t = 2. In order to classify such groups, we first
consider the upper bound n ≤ ⌊(9m − 3)/2⌋. In continue the groups which attains
the upper bound will be classified, that we use for the classification of permutation
groups with maximum degree n = 3m + 1. In [5] Praeger asked in Question 1.5
2010 Mathematics Subject Classification. 20B25.
Key words and phrases. Permutation group, bounded movement, orbits.
189
that the upper bound of |Ω| by 3m + t − 1 is sharp for any values of t greater than
1. In this paper we will give positive partial answer to this question for t = 2 and
n = 3m + 1. We note that for x ∈ R, ⌊x⌋ is the integer part of x.
THEOREM 1.1
Let G be a permutation group on a set Ω with t(≥ 3) orbits which
have no fixed points in Ω. Suppose further that m is a positive integer such that
move(G) = m and n = 3m + t − 1. Then
I) n ≤ ⌊(9m − 3)/2⌋, and the equality holds if and only if G is the semidirect product
of Z13 and Z3 with normal subgroup Z13 .
2
EXAMPLES AND PRELIMINARIES
′′
′′
′′
EXAMPLE 2.4. Let Ω1 = {1, ..., 13}, Ω2 = {(1′ , 2′ , 3′ )}, Ω2 = {1 , 2 , 3 }, , Z13 =
′′
′′
′′
⟨1...13⟩ and Z3 = ⟨(1′ 2′ 3′ )(1 2 3 )(123)(456)(789)(101112)⟩. Then G := Z13 ⋊ Z3 is
a permutation group on a set Ω := Ω1 ∪ Ω2 ∪ Ω3 of size 19, in which every 1 ̸= g ∈ G
has constant movement 6. Therefore the permutation group G has 3 orbits and degree 3m + 1 = 19 with move(G) = 6.
3
THE PROOF OF THEOREM 1.1
To prove Theorem 1.1 we introduce the following notation:
r3 :=number of G-orbits of length 3 on which G acts as Z3 ;
r3′ :=number of G-orbits of length 3 on which G acts as Sym(3);
and
r2 :=number of G-orbits of length 2;
r4 :=number of G-orbits of length 4;
s:=number of G-orbits of length ≥ 5.
The orbits are labelled accordingly: thus Ω1 ,...,Ωr3 are those of length 3 on which
G acts as Z3 ; Ωr3 +1 ,...,Ωr3 +r3′ are those of length 3 on which G acts as Sym(3);
Ωr3 +r3′ +1 ,...,Ωr3 +r3′ +r2 are those of length 2, etc. Define t := r3 + r3′ + r2 + r4 + s,
∑
∪ 0 +r4
∑
∪
t0 := r3 + r3′ + r2 , 4 := ti=t
Ωi , and 5 := ti=t0 +r4 +1 Ωi and so |Ω| = n =
0 +1
3r3 + 3r3′ + 2r2 + 4r4 + |Σ5 |.
190
With the above notation by [4, Lemma 3] we have
9
3
1
5
1
n < m − ( r3′ + r2 + r4 + (|Σ5 | − 3s)).
2
4
4
4
2
(1)
With some simple calculation, the Inequality (1) can be simplified as n ≤ ⌊(9m − 3)/2⌋.
The proof of the part (I) of Theorem 1.1 is complete now.
We now consider the maximum bound n = 3m + t − 1, for t = 3, that is n = 3m + 2.
Suppose that n = 3m + 2. Then we first have the following lemma, which follows
easily by 2η > 3r3′ + r2 + 5r4 + 2(|Σ5 | − 3s) ≥ 0.
LEMMA 3.1 If t = 3, then m ≥ 3.
Let Ω1 , Ω2 , ..., Ωt be t orbits of G of lenghs n1 , n2 , ..., nt . Then by [6], we know
that if ni ≥ 2, for i = 1, ..., t, then t ≤ 2m − 1. Now if ni ≥ 3, for i = 1, ..., t, then
t ≤ 2m − 2, and |Ω| ≤ t + 3m − 2, which for t = 3, |Ω| ≤ 3m + 1. Therefore if t = 3,
then |Ω| = 3m + 2, it follows that whose if all ni ≥ 3, then |Ω| = 3m + 1, which is a
contradiction. So the exist r2 ̸= 0, such that G can be classified.
Now from the equality t := r3 + r3′ + r2 + r4 + s = 2, it follows that there are only
the several possibilities:
Since t ≤ 2m − 2, then m ≥ (t + 2)/2 therefore if t ≤ 3, then m ≥ 3. Therefore |Ω| ≥ 11.
It follows easily from the above lemma and the equality n = |Ω| = 3r3 + 3r3′ + 2r2 +
4r4 + |Σ5 | = 3m + 2 that, the only six following cases will be remained:
(I) r3 = r3′ = s = 1, r4 = r2 = 0,
(II) r2 = r4 = s = 1, r3 = r3′ = 0,
(III) r3 = 1, r4 = 2, r3′ = r2 = s = 0,
(IV) r3′ = 1, r4 = 2, r3 = r4 = s = 0,
(V) r4 = 1, s = 2, r3 = r3′ = r2 = 0,
(VI) s = 1, r3 = 2, r3′ = r2 = r4 = 0,
(VII) s = 1, r3′ = 2, r3 = r2 = r4 = 0,
191
In the cases (I) and (II), |Ω| is equal to 11 and so m is equal to 3. Therefore
in these cases n = 3m + 2 = ⌊(13m − 4)/3⌋ = 11. But it was proved that the no
group satisfying in this equality, since in this case m ≥ 4. In the cases (III) and
(IV), since r4 = 2, therefore there are not any group whose satisfied in this cases.
In the cases (V) and (VI), |Ω| = 11, n = 6 + |Σ5 | = 3m + 2. With respect to equality
|Σ5 | = 3m − 4, it is easy to see that the movement m cannot be greater than 5. So
no group is not satisfies in this cases. We now show that case (VII) does not occur.
Suppose therefore that r4 = 1, s = 2, r3 = r3′ = r2 = 0, so |Ω| is equal to 14 and
hence m is equal to 4. Suppose that Ω := ∆1 ∪ ∆2 ∪ ∆3 , in which ∆1 is orbit of
length 4 and ∆2 , ∆3 are two orbits of length 5. Then these three orbits of lengths
4, 5, 5 respectively must be appeared in the cycle representation of some elements.
Therefore moveΩ (g) = move∆1 (g1 ) + move∆2 (g2 ) + move∆3 (g2 ) = 2 + 2 + 2 = 6, for
some g ∈ G. Thus m = 6 and this proves that the case (VII) can not arises. Now
the proof of Theorem 1.1 is complete.
References
[1] M. Alaeiyan and S. Yoshiara, Permutation groups of minimal movement, Arch.
Math. 85 (2005), 211-226.
[2] J. R. Cho, P. S. Kim, and C. E. Praeger, The maximal number of orbits of a
permutation groups with bounded movement, J. Algebra 214 (1999), 625–630.
[3] A. Hassani, M. Khayaty (Alaeiyan), E. I. Khukhro and C. E. Praeger, Transitive
permutation groups with bounded movement having maximal degree, J. Algebra
214 (1999), 317–337.
[4] P. M. Neumann and C. E. Praeger, On the movement of a permutation group,
J. Algebra 214 (1999), 631–635.
[5] C. E. Praeger, Movement and separation of subsets of points under group actions,
J. London Math. Soc. (2) 56 (1997), 519-528.
[6] C. E. Praeger, On permutation group with bounded movement, J. Algebra 144
(1991), 436–442.
192
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
On minimal non PST-groups
Gholamreza Rezaeezadeh1 and Zahra Aghajari2∗
1
Department of Mathematics, Faculty of Sciences, Shahrekord University, Shahrekord,
Iran
[email protected]
2
Department of Mathematics, Faculty of Sciences, Shahrekord University, Shahrekord,
Iran
[email protected]
Abstract
A finite group G is said to be a PST-group if every subnormal subgroup of G is S-permutable in G. In this paper, we
determinate if G be minimal non PST-group then the number of distinct prime divisor of |G| is two. In the end, a
theorem from Derek J. S. Robinson is brought that characterized minimal non PST-groups.
1
Introduction
All group considered in this paper will be finite. A group G is said to be a T-group if
every subnormal subgroup of G is normal in G. A group G is said to be a PT-group
if every subnormal subgroup of G is permutable in G. A subgroup H of a group G
is said to be S-permutable (or S-quasinormal) in G if it permutes with every Sylow
subgroups of G. A group G is said to be a PST-group if every subnormal subgroup
of G is S-permutable in G.
The structure of solvable PST-group was determined by Agrawal; also A. BallesterBolinches and R. Esteban-Romero[3].
if P is a property of group, a group G is said to be a minimal non P-group if
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D10; Secondary 20E26 , 20F17, 20J99.
Key words and phrases. Finite group, PST-group, Minimal non PST-group.
193
G does not have P but all its proper subgroups do. M. M. Al-mosa Al-shomrani [2]
determined the structure of minimal non-PT groups (non PT-groups all of its proper
subgroups are PT-groups) and Derek J. S. Robinson [5] characterised the structure
of minimal non-PST groups(non PST-groups all of its proper subgroups are PSTgroups).
In addition |π(G)| will denote the number of distinct prime divisors of |G|.
2
Main results
Lemma 2.1. If a group G possesses three solvable PST-group whose indices are
pairwise relatively prime, then G is a solvable PST-group.
Lemma 2.2. Let G be a solvable group. If G is a minimal non PST-group, then
|π(G)| = 2.
Lemma 2.3. If G is a minimal non PST-group, then |π(G)| = 2.
Theorem 2.4. If G is a minimal non-PST-group, then it is one of the types described in I to IV below. Let p and q be distinct primes.
TypeI : Let p and q be primes such that p ≡ 1 (mod q f ) where q f > 1. let i be
the least positive primitive q f -th root of unity modulo p. Put j = 1 + kq f −1 where
0 < k < q. Define
G1 = X ⋉ A
where X =< x > has order q r with r ≥ f , A =< a, b > is elementary abelian of
j
order p2 , and ax = ai , bx = bi .
TypeII : Let p and q be distinct primes such that p ̸≡ 1 (mod q f ). Let z be a
primitive q-th root of unity modulo p and denote by F the field Zp (z). Define
G2 = X ⋉ F +
where X =< x > has order q r > 1 and x act on F + via multiplication by z.
TypeIII : Let p and q be primes such that p ≡ 1 (mod q), and write q f for the
highest power of q dividing p-1. Let A be an elementary abelian p-group with basis
{a0 , a1 , ..., aq−1 } and let X =< x > have order q r where r > f . Set i equal to the
least positive primitive q f -th root of unity modulo p. Define
G3 =< x > ⋉A
where axi = ai+1 for 0 ≤ j < q − 1 and axq−1 = ai0 .
TypeIV : G4 = X ⋉ Q where Q is a quaternion group of order 8, X =< x > has
194
order 3r and x permutes cyclically the three subgroups of Q with order 4.
TypeV : Let p and q be distinct primes such that the exponent of p modulo q
is even, say 2m. Let P be a (non-abelian) special p-group of rank 2m which can
be generated by elements of order p. Take X =< x > to have order q r > 1 and
let x induce an automorphism in P such that P/P ′ is a simple Zp X-module and
[P ′, X] = 1. Define
G5 = X ⋉ P.
References
[1] M. J. Alejandre, A. Ballester-Bolinches, and M. C. Pedraza-Aguilera. Finite
soluble groups with permutable subnormal subgroups.J. Algebra, 240(2001),705721 .
[2] M. M. Al-Mosa Al-Shomrani. On minimal non PT-groups. International Mathematical Forum, 3, 2008, no. 10, 495-501.
[3] A. Ballester-Bolinches and R. Esteban-Romero. On finite soluble groups in which
Sylow permutability is a transitive relation. Acta Math. Hungar., 101(3):193-202,
2003.
[4] D. J. S. Robinson, A course in the theory of groups, 2nd ed., Springer-Verlage,
New York, 1996.
[5] D. J. S. Robinson. Minimality and Sylow-permutability in locally finite groups.
Ukr. Math. J., 54(2002), 1038-1049.
195
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
The structure of SS-semipermutable groups
Gholamreza Rezaeezadeh1 and Sayed Ebrahim Mirdamadi2∗
1
Department of Mathematics, Faculty of Sciences, Shahrekord University, Shahrekord,
Iran
[email protected]
2
Department of Mathematics, Faculty of Sciences, Shahrekord University, Shahrekord,
Iran
[email protected]
Abstract
A subgroup H of a finite group G is said to be SSsemipermutable in G if H has a supplement K in G such
that H permutes with every Sylow subgroup X of G such
that (|X|, |H|) = 1. The structure of SS-semipermutable
groups is investigated in this paper.
1
Introduction
Through in this paper all groups considered finite and for a group G, Let Π(G)
denotes the set of prime divisor of |G|. Given a finite group G, two subgroups H
and K are said to permute if HK = KH. A subgroup H of a group G is said to
be permutable (resp. S-permutable) in G if H permutes with all the subgroups (resp.
Sylow subgroups) of G. A group G is called a T-group(resp. PT-group, PST-group)
if normality (resp. permutability, S-permutability) is a transitive relation.
By [3], a group G is PST-group if anf only if every subnormal subgroup of G is
S-permutable in G. A subgroup H of a group G is said to be τ -quasinormal in G if
HGp = Gp H for every Gp ∈ Sylp (G) such that (|H|, p) = 1 and (|H|, |(Gp )G |) ̸= 1.
A subgroup H of a group G is said to be SS-permutable in G if H has a supplement
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D10; Secondary 20E26 , 20F17, 20J99.
Key words and phrases. PST-groups, SS-semipermutable groups.
196
K in G such that H permutes with every Sylow subgroup of K. In this case K is
called an SS-permutable supplement of H in G. Recall that a subgroup H of a group
G is said to be SS-semipermutable in G if H has a supplement K in G such that H
permutes with every Sylow subgroup X of K such that (|X|, |H|) = 1. In this case
K is called an SS-semipermutable supplement of H in G.
2
Main results
Lemma 2.1. Suppose that a subgroup H of a group G is SS-semipermutable in G
with an SS-semipermutable supplement K, L ≤ G and N E G. Then
(1) If H ≤ L, then H is SS-semipermutable in G.
(2) If H is a p-group, Where p ∈ Π(G), then (HN )/N is SS-semipermutable in G/N .
(3) If N ≤ L and L/N is SS-semipermutable in G/N , then L is SS-semipermutable
in G.
(4) Every conjugate of K in G is SS-semipermutable supplement of H in G.
(5) If N is nilpotent, then N K is an SS-semipermutable supplement of H in G.
(6) If H is a p-subgroup, where p ∈ Π(G), and H ≤ F (G), then H is S-semipermutable
in G.
Note that F (G) denotes the Fitting subgroup of G.
Lemma 2.2. Let G be a group. Then every subgroup of F ∗ (G) is τ -quasinormal
in G if and only if G is a solvable PST-group.
Note that F ∗ (G) denotes the generalized Fitting subgroup of G. That is the product
of all normal quasinilpotent subgroup of G.
Lemma 2.3. Let T and S are SS-semipermutable in a solvable group G such that
(|T |, |S|) = 1. Then ⟨T, S⟩ is SS-semipermutable in G.
Theorem 2.4. Let G be a group. Then the following statement are equivalent:
(1) G is solvable and every subnormal subgroup of G is SS-semipermutable in G.
(2) Every subgroup of F ∗ (G) is SS-semipermutable in G.
(3) G is a solvable PST-group.
Theorem 2.5. Let G be a group. Then the following statement are equivalent:
(1) Whenever H ≤ K are two p-subgroups of G with p ∈ Π(G), H is SS-semipermutable
in NG (K).
(2) G is a solvable PST-group.
197
References
[1] A. Ballester-Bolinches and R. Esteban-Romero. Sylow permutable subnormal
subgroups of finite groups. J. Algebra, 251:727-738, 2002.
[2] B. Huppert and N. Blackburn.
Berline/Heidelberg, 1982.
Finite
group
III.
Springer-Verlage,
[3] O. H. Kegel. Sylowgrouppen und subnormalteiler endlicher gruppen. Math. Z.
78:205-221. 1962.
[4] V. O. Lukyanenko and A. N. Skiba. On weakly τ -quasinormality is a transitive
relation. Rend. Semin. Math. Univ. Podova, 124:231-246, 2010.
[5] P. Shemid. Subgroup permutable with all Sylow subgroups. J. Algebra, 207:285293,1998.
198
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Movement of permutation groups with two orbits
Mehdi Rezaei
Buein Zahra Technical University, Buein Zahra, Qazvin, Iran
m [email protected]
Abstract
Let G be a permutation group with bounded movement,
move(G) = m. It was shown by Praeger that |Ω| 6 3m+t−1,
where t is the number of G-orbits on Ω. In this paper we
classify all permutation groups G with bounded movement
m attaining the maximum bound |Ω| = 3m + 1. Indeed, we
will give a positive partial answer to Praeger question that
whether the upper bound |Ω| = 3m + t − 1, is sharp for t > 1.
1
Introduction
Let G be a permutation group on a set Ω with no fixed points in Ω and let m be a
positive integer. If for a subset Γ of Ω the size |Γg − Γ| is bounded, for g ∈ G, we
define the movement of Γ as
move(Γ) := supg∈G |Γg − Γ|.
If move(Γ) ≤ m for all Γ ⊆ Ω, then G is said to have bounded movement m and the
movement of G is defined as the
move(G) := sup |Γg − Γ|.
Γ
This notion was introduced in [3, 6]. By [6, Theorem 1], if G has bounded movement
m, then Ω is finite. Moreover both the number of G-orbits in Ω and the length of
each G-orbit are bounded above by linear functions of m. In particular each G-orbit
2010 Mathematics Subject Classification. Primary 05C25; Secondary 20B25.
Key words and phrases. Intransitive permutation groups, orbits, bounded movement.
199
has length at most 3m, t ≤ 2m − 1 and n := |Ω| ≤ 3m + t − 1 ≤ 5m − 2, where t
is the number of G-orbits on Ω. In [2] it was shown that n = 5m − 2 if and only
if n = 3 and G is transitive. But in [4], this bound was refined further and it was
shown that n ≤ (9m − 3)/2. Moreover, if n = (9m − 3)/2 then either n = 3 and
G = S3 or G is an elementary abelian 3-group and all its orbits have length 3. Also
this upper bound was improved to the bound n ≤ 4m − p in [1, Theorem 1.1], where
p ≥ 5 is the least odd prime dividing |G|.
Throughout this paper m is a positive integer and G is an intransitive permutation
group on a set Ω of size n = 3m+1 with movement m and t = 2 orbits. The purpose
of this paper is to classify all intransitive permutation groups of degree n = 3m + 1.
In [5] Praeger asked in Question 1.5 that the upper bound of |Ω| by 3m + t − 1 is
sharp for any values of t greater than 1. In this paper we will give positive partial
answer to this question for t = 2 and n = 3m + 1. We note that for x ∈ R, ⌊x⌋ is
the integer part of x and K ⋊ H is a semi-direct product of K by H with normal
subgroup K.
2
Main results
Let 1 ̸= g ∈ G and suppose that g in its disjoint cycle representation has t (t is a
positive integer) nontrivial cycles of lengths l1 , ..., lt , say. We might represent g as
g = (a1 a2 ...al1 )(b1 b2 ...bl2 )...(z1 z2 ...zlt ).
Let Γ(g) denote a subset of Ω consisting of ⌊li /2⌋ points from the ith cycle, for each
i, chosen in such a way that Γ(g)g ∩ Γ(g) = ∅. For example, we could choose
Γ(g) = {a2 , a4 , . . . , ak1 , b2 , b4 , . . . , bk2 , ..., z2 , z4 , . . . , zkt },
where ki = li − 1 if li is odd and ki = li if li is even. Note that Γ(g) is not uniquely
determined as it depends on the way each cycle is written. For any set Γ(g) of this
kind, we say that Γ(g) consists of every second point of every cycle of g. From the
definition of Γ(g) we see that
|Γ(g)g \Γ(g)| = |Γ(g)| =
t
∑
⌊li /2⌋.
i=1
The next lemma shows that this quantity is an upper bound for |Γg \Γ| for an arbitrary
subset Γ of Ω.
Lemma 2.1. Let G be a permutation group on a set Ω and suppose that Γ ⊆ Ω.
Then for each g ∈ G, |Γg \Γ| ≤ Σti=1 ⌊li /2⌋, where li is the length of the ith cycle of
200
g and t is the number of nontrivial cycles of g in its disjoint cycle representation.
This upper bound is attained for Γ = Γ(g) defined above.
Let g be an element of a permutation group G on a set Ω. Assume that the
set Ω is the disjoint union of G-invariant sets Ω1 and Ω2 . Then every subset Γ
of Ω is a disjoint union of subsets Γi = Γ ∩ Ωi for i = 1, 2. Let gi be the permutation on Ωi induced by g for i = 1, 2. Since |Γg −Γ| = |Γg11 −Γ1 |+|Γg22 −Γ2 |, we have
moveΩ (g) =
∑2
gi
i=1 max{|Γi
\ Γi ||Γi ⊆ Ωi } = moveΩ1 (g1 ) + moveΩ2 (g2 ).
Now we introduce some examples of groups with bounded movement m satisfying
in |Ω| = 3m + 1.
Example 2.2. Let G1 := Z22 and G2 := Z3 be permutation groups on the sets
Ω1 = {1, 2, 3, 4} and Ω2 = {1′ , 2′ , 3′ } respectively, where Z22 ∼
=< (12)(34), (13)(24) >
and Z3 ∼
=< (123)(1′ 2′ 3′ ) >. Set Ω := Ω1 ∪ Ω2 . Then G := G1 ⋊ G2 is a permutation
group on Ω which has t = 2 orbits, and since each non-identity element of G has
two cycles of length 2 or two cycles of length 3, so m =move(G) = 2. It follows that
n = 3m + 1 = 7.
Example 2.3. Let Ω1 = {1, ..., 7}, Ω2 = {1′ , 2′ , 3′ }, α = (1...7) and β = (1′ 2′ 3′ )(235)(476).
Then G :=< α, β >∼
= Z7 ⋊ Z3 is a permutation group with t = 2 orbits on a set
Ω := Ω1 ∪Ω2 of size 10, in which every 1 ̸= g ∈ G has 3 cycles of size 3 or is a cycle of
size 7 and therefore in both cases have movement 3. It follows that n = 3m+1 = 10.
Theorem 2.4. Let G be a permutation group on a set Ω with t(≥ 2) orbits which
have no fixed points in Ω. Suppose further that m is a positive integer such that
move(G) = m and n = 3m + 1, which is the maximum bound for t = 2. Then G is
one of the following:
(a) G is the semidirect product of Z22 and Z3 with normal subgroup Z22 .
(b) G is the semidirect product of Z7 and Z3 with normal subgroup Z7 .
References
[1]
[2] M. Alaeiyan and S. Yoshiara, Permutation groups of minimal movement, Arch.
Math. 85 (2005), 211-226.
201
[3] J. R. Cho, P. S. Kim, and C. E. Praeger, The maximal number of orbits of a
permutation groups with bounded movement, J. Algebra. 214 (1999), 625-630.
[4] A. Hassani, M. Khayaty (Alaeiyan), E. I. Khukhro and C. E. Praeger, Transitive
permutation groups with bounded movement having maximal degree, J. Algebra.
214 (1999), 317-337.
[5] P. M. Neumann and C. E. Praeger, On the movement of a permutation group,
J. Algebra. 214 (1999), 631-635.
[6] C. E. Praeger, Movement and separation of subsets of points under group actions,
J. London Math. Soc. (2) 56 (1997), 519-528.
[7] C. E. Praeger, On permutation group with bounded movement, J. Algebra. 144
(1991), 436-442.
202
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Irreducible characters and conjugacy classes in finite
groups
Sajjad Mahmood Robati
Department of Mathematics, Imam Khomeini International University, Qazvin, Iran
[email protected]
Abstract
Let G be a finite group. If A and B are two conjugacy classes
in G, then AB is a union of conjugacy classes in G and
η(AB) denotes the number of distinct conjugacy classes of
G contained in AB. If χ and ψ are two complex irreducible
characters of G, then χψ is a character of G and again we let
η(χψ) be the number of irreducible characters of G appearing
as constituents of χψ. In this paper our aim is to study the
product of conjugacy classes in a finite group and obtain an
upper bound for η in general. Then we study similar results
related to the product of two irreducible characters.
1
Introduction
Let G be a finite group.For a ∈ G let Cl(a) = {g −1 ag|g ∈ G} denote the conjugacy
class of a in G. A subset X of G is called G-invariant if X G = {g −1 xg|x ∈ X, g ∈
G} = X, and in this case X is the union of some conjugacy classes of G. By
definition η(X) is the number of distinct conjugacy classes of G contained in G. If
Cl(a) and Cl(b) are two conjugacy classes of G, then it is clear that Cl(a)Cl(b) =
{ax by |x, y ∈ G} is a G-invariant subset of G, hence Cl(a)Cl(b) contains at least one
conjugacy class of G, i.e. η(Cl(a)Cl(b)) ≥ 1. If we know η(AB) for some conjugacy
classes A and B of G, then we may ask about the structure of G as well as the
conjugacy classes A and B. Similar concept can be defined for complex characters of
2010 Mathematics Subject Classification. Primary 20E45, 20C15.
Key words and phrases. Conjugacy classes, irreducible characters, products.
203
a finite group G. By Irr(G) we mean the set of all the complex irreducible
characters
∑
of G. It is known that for any character χ of G we have χ = ki=1 mi χi , where
mi ∈ N and χi ∈ Irr(G). The irreducible characters χi are called the constituents
of χ and similar to the case of product of conjugacy classes we set η(χ) = k. In
particular if χ and φ are irreducible characters of G, then χφ is a character of G,
hence η(χφ) ≥ 1. Similar questions as in the case of conjugacy classes may be asked,
for instance if we know η(χφ) for some irreducible characters χ and φ, then we may
ask about the structure of G as well as the irreducible characters χ and φ.
2
main theorems
For a finite group G it is important to find the η-function on the product of two
conjugacy classes of G. In [1], Adan-Bante shows that if G is a finite p-group and
a ∈ G and |Cl(a)| = pn , then η(Cl(a)Cl(a−1 )) ≥ n(p − 1) + 1. Let [a, G] be the set
of all commutators [a, g] = a−1 g −1 ag where g ∈ G. In [2], she proved the following
theorem about products of irreducible character of p-groups:
Theorem 2.1. Let G be a finite p-group, where p is a prime number.Let χ, φ ∈
Irr(G) be faithful characters. then either η(χφ) = 1 or η(χφ) ≤ (p + 1)/2.
Our first result is the following:
Proposition 2.2. Let G be a finite group and a, b ∈ G and Cl(a)Cl(b) ∩ Z(G) ̸= ∅.
Then [a, G] is a subgroup of G if and only if |Cl(a)Cl(b)| = |Cl(a)|.
The next proposition is an analogue of the previous proposition in character
theory.
Proposition 2.3. Let G be a finite group and χ, ψ ∈ Irr(G) with Irr(χψ) ∩
c = ρG/Z(G) , where
Lin(G) ̸= ∅. Then χ vanishes on G − Z(G) if and only if χψ
c
ρG/Z(G) is the regular character of G/Z(G) and χψ(gZ(G))
:= χψ(g) for all g ∈ G.
Proposition 2.4. Let G be a finite group and a ∈ G and the order of a be an odd
number. Then [a, G] is a subgroup of G if and only if η(Cl(a)Cl(a)) = 1.
At this point we prove the following Theorem which is an analogue of Proposition
2.4.
Theorem 2.5. Let G be a finite group of odd order and χ ∈ Irr(G). Then χ
vanishes on G − Z(χ) if and only if η(χ2 ) = 1.
204
Theorem 2.6. Let G be a finite group and a, b ∈ G and [a, G] be a subset of Z(G).
Then
(i) If Cl(a)Cl(b) ∩ Z(G) ̸= ∅, then η(Cl(a)Cl(b)) = |Cl(a)|;
(ii) If |Cl(a)| is an odd number, then η(Cl(a)Cl(a)) = 1;
(iii) If |Cl(a)| is an even number, then η(Cl(a)Cl(a)) = 2n (where n is the number
of cyclic direct factors in the decomposition of the Sylow 2-subgroup of [a, G]).
Concerning irreducible constituents of the square of an irreducible character the
following results are proved in [5].
Theorem 2.7. Let G be a finite group and χ, φ ∈ Irr(G) such that χ vanishes on
G − Z(χ). Then
(i) If χφ ∩ Lin(G) ̸= ∅, then η(χψ)) = |Irr(G/Z(G))|;
(ii) If χ(1) is an odd number, then η(χ2 ) = 1.
3
Conjugacy calsses and Linear groups
If P is a Sylow p-subgroup of GL4 (q), q = pm , p prime, then |P | = q 6 = p6m and in
[4] it is proved:
Theorem 3.1. If A and B denote two conjugacy classes of the group P , then
η(AB) = 1, q, 2q − 1 or q 2 + q − 1.
If Q is a Sylow p-subgroup of SP4 (q), q = pm , p prime, then |Q| = q 6 = p6m and
in [4] we have proved:
Theorem 3.2. If A and B denote two conjugacy classes of the group Q, then
η(AB) = 1, q, 2q − 1 or q 2 + q − 1 if p is odd and η(AB) = 1, q or q 2 if p = 2.
The following results are proved in [3] for GLn (q) and SLn (q).
Theorem 3.3. If A and B are non-scalar matrices in GLn (q), then η(Cl(A)Cl(B)) ≥
q − 1.
Theorem 3.4. If A and B are non-scalar matrices in SLn (q), then η(Cl(A)Cl(B)) ≥
[q/2].
References
[1] E. Adan-Bante, Conjugacy classes and finite p-group, Arch.Math, 85,(2005),
297-303.
205
[2] E. Adan-Bante, Products of characters and finite p-groups II, Arch.Math, 82,
No4, (2004), 289-297.
[3] E. Adan-bante, J. M. Harris, On conjugacy classes of GL(n, q) and SL(n, q),
ArXive: 0904.2152v1.
[4] M. R. Darafsheh, S. M. Robati, Products of Conjugacy Classes in certain pgroups, International Journal of Mathematics, Game Theory and Algebra, 22,
No3, (2013).
[5] M. R. Darafsheh, S. M. Robati, Products of Conjugacy Classes and Products of
Irreducible Characters in Finite Groups, Turk. J. Math., 37, (2013), 607-616.
206
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
The classification of some nilpotent Leibniz 4-algebras
Farshid Saeedi1 and Seyyedeh Nafiseh Akbarossadat2∗
1
Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad, Iran
[email protected]
2
[email protected]
Abstract
In this paper, we recall the definition of Leibniz n-algebras
and then classify the two steps strongly nilpotent Leibniz 4algebras with dimension 3.
1
Introduction
In 1993, Loday [1] introduced a non-skew-symmetric version of Lie algebras, the
so-called Leibniz algebras as follow
A Leibniz n-algebra is a vector space L over a field F equipped with an n-ary
linear operation [−, −, . . . , −] : L ⊗n −→ L satisfying in following identity, for all
x1 , . . . , xn , y2 , . . . , yn ∈ L
[[x1 , x2 , . . . , xn ], y2 , . . . , yn ] =
n
∑
[x1 , . . . , xi−1 , [xi , y2 , . . . , yn ], xi+1 , . . . , xn ]
(1)
i=1
If n = 2, L is called Leibniz algebra.
Then in 2006, S. Albeverio, B. A. Omirov and I. S. Rakhimov classified 4dimensional nilpotent Leibniz algebras.
Also, in 2002,[5] the n-ary version of Leibniz algebras was introduced, which is
Lie n-algebras, so that its bracket is symmetric. Also, it is Leibniz algebras when
n = 2. In [2] described the representations of Leibniz n-algebras by means of a universal enveloping algebra and established a PBW type theorem. Also, 3-dimensional
∗
Speaker
2010 Mathematics Subject Classification. Primary 17A32; Secondary 17B30.
Key words and phrases. Leibniz n-algebra, nilpotent, strongly nilpotent.
207
strongly-nilpotent Leibniz 3-algebras and two-step nilpotent 3-dimensional Leibniz 3algebras over an algebraic closed field of characteristic zero have been classified in
[4]. In fact they give eleven classes of nilpotent 3-dimensional Leibniz 3-algebras.
Now, our aim in this paper is the classification of some strongly nilpotent Leibniz
4-algebras which are introduced in next section.
Definition 1.1. A Leibniz n-algebra L is calleb nilpotent, if L r = 0 for some
r ≥ 0, where
L0 = L
L s+1 = [L s , L , . . . , L ],
| {z }
&
s ≥ 0.
(n−1)−times
A nilpotent Leibniz n-algebra L is called two-step nilpotent, if L satisfies L 1 ̸= 0
and L 2 = 0[4].
Abelian Leibniz n-algebras are the examples of nilpotent Leibniz n-algebras.
Definition 1.2. Let L be a Leibniz n-algebra and S ⊆ L . Then S is said
subalgebra of L , if it satisfis the following condition
[x1 , x2 , . . . , xn ] ∈ S ,
∀x1 , x2 , . . . , xn ∈ S
Also, the subalgebra I of L is called an ideal, when we have [y, x2 , . . . , xn ] ∈ I ,
for all y ∈ I and x2 , . . . , xn ∈ L .
The subalgebra L 1 = [L , L , . . . , L ] generated by the elements [x1 , x2 , . . . , xn ],
|
{z
}
n−times
for each x1 , x2 , . . . , xn ∈ L , is called derived algebra of L . If L 1 = 0, then we say
that L is abelian Leibniz n-algebra.
2
Classification of Some Leibniz n-Algebras
In this section, we classify some m-dimensional nilpotent Leibniz n-algebras for m =
2, 3, 4.
Theorem 2.1. Let L be a 2-dimensional two-step nilpotent Leibniz 3-algebra with
basis {e1 , e2 }, and L 1 = Fe1 . Then up to isomorphisms L is as following
[e2 , e1 , e2 ] = −e1
,
[e2 , e2 , e1 ] = e1
,
[e2 , e2 , e2 ] = e2
Theorem 2.2. [4] Let L be a 3-dimensional two-step nilpotent Leibniz 3-algebra
with basis e1 , e2 , e3 , and L 1 = Fe1 . Then up to isomorphisms one and only one of
the following possibilities holds:
208
1. [ei , ej , ek ] = αijk e1 , αijk ∈ F, 2 ≤ i, j, k ≤ 3, and at least one αijk ̸= 0. The
′
Leibniz 3-algebra L ′ of case (3.4) with coefficients αijk
is isomorphic to L if
(
)
a22 a23
and only if there exists a matrix A =
such that det A ̸= 0 and
a32 a33
αijk =
∑
′
ail ajt aks αlts
,
2 ≤ i, j, k ≤ 3.
(2)
l,t,s=2,3

 [e2 , e1 , e2 ] = e1
[e2 , e2 , e1 ] = −e1
2.
where αijk ∈ F and 2 ≤ i, j, k ≤ 3.

[ei , ej , ek ] = αijk e1
Theorem 2.3. [4] Let L be a 3-dimensional two-step nilpotent Leibniz 3-algebra
with a basis {e1 , e2 , e3 }, and dim L 1 = 2, then, up to isomorphisms, one and only
one of the following possibilities holds

[e3 , e1 , e3 ] = −e1




 [e3 , e2 , e3 ] = −α1 e2
[e3 , e3 , e1 ] = e1
αi ∈ F, i = 1, 2, 3, α1 ̸= 0.
(c1 ).


[e
,
e
,
e
]
=
α
e

1 2

 3 3 2
[e3 , e3 , e3 ] = α2 e1 + α3 e2

[e3 , e1 , e3 ] = −e1




 [e3 , e2 , e3 ] = −e1 − e2
[e3 , e3 , e1 ] = e1
αi ∈ F, i = 1, 2, 3.
(c2 ).


[e , e , e ] = e1 + e2


 3 3 2
[e3 , e3 , e3 ] = α1 e1 + α2 e2

 [e3 , e1 , e3 ] = e2
[e3 , e3 , e1 ] = −e2
(c3 ).

[e3 , e3 , e3 ] = e1 .

 [e3 , e2 , e3 ] = e2
[e3 , e3 , e2 ] = −e2
(c4 ).

[e3 , e3 , e3 ] = e1 + e2 .

 [e3 , e2 , e3 ] = e2
5
[e3 , e3 , e2 ] = −e2
(c ).

[e3 , e3 , e3 ] = e1 .
209
3
Classification of Some Nilpotent Leibniz 4-Algebras
Notation: Let L be a Leibniz 4-algebra. Denote [L , L 1 , L , L ], [L , L , L 1 , L ]
and [L , L , L , L 1 ] with M2 , M3 and M4 , respectively.
Proposition 3.1. Let L be a 3-dimensional two-step nilpotent Leibniz 4-algebra
with basis {e1 , e2 , e3 }. Then Mi = 0 if and only if Mj + Mk = 0, for i, j, k = 2, 3, 4
and i ̸= j ̸= k.
Theorem 3.2. Let L be a 3-dimensional two-step nilpotent Leibniz 4-algebra with
basis e1 , e2 , e3 , and L 1 = Fe1 . Then up to isomorphisms one and only one of the
following possibilities holds
1. If M4 = 0, then the multiplication table of L is as follow
{
[ei , ej , ek , el ] = αijkl e1
2 ≤ i, j, k, l ≤ 3
[ei , ej , e1 , el ] = − [ei , e1 , ej , el ] 2 ≤ i ≤ l ≤ 3,
2≤j≤3
2. If M4 ̸= 0, then the these conditions are hold

[ei , e1 , e1 , e1 ] = 0,




 [e1 , ej , ek , el ] = 0,
αi11l = −αi1l1 = −αil11 ,


α
+ αi1jj + αij1j = 0,


 ijj1
αi123 + αi132 + αi213 + αi231 + αi321 + αi312 = 0,
4
2≤i≤3
1 ≤ j, k, l ≤ 3
2 ≤ i, l ≤ 3
2 ≤ i, j ≤ 3
2≤i≤3
Classification of Some Strongly Nilpotent Leibniz 4Algebras
Let L be a Leibniz n-algebra. L is called s-step strongly nilpotent, if there exist
s ̸= 0 such that Ls = 0 and Ls−1 ̸= 0, where
L0 = L
&
Li =
n
∑
[L , ..., L , Li−1 , L , ..., L ]
| {z }
j=1 j−1−times
Proposition 4.1. Every strongly nilpotent Leibniz n-algebra is nilpotent.
Theorem 4.2. Let L be a 2-step strongly nilpotent Leibniz 4-algebra with dimension 3 and basis {e1 , e2 , e3 }. Then the conditions in Theorem ?? are hold.
210
Theorem 4.3. Let L be a 3-dimensional strongly nilpotent Leibniz 4-algebra with
basis {e1 , e2 , e3 } such that dim L1 = 2. Then L is one of the following (up to the
isomorphism)
⟨[e3 , e2 , e3 , e3 ] = αe1 ,
[e3 , e3 , e3 , e3 ] = βe1 ,
[e3 , e3 , e3 , e2 ] = −(α + β)e1 , [e3 , e3 , e3 , e3 ] = e2 ⟩
⟨[e2 , e3 , e3 , e3 ] = e1 ,
[e3 , e3 , e2 , e3 ] = αe1 , [e3 , e3 , e3 , e2 ] = −αe1 ,
(B) :
[e3 , e3 , e3 , e3 ] = e2 ⟩
(C) : ⟨[e2 , e3 , e3 , e3 ] = e1 , [e3 , e3 , e3 , e3 ] = e2 ⟩
(D) : ⟨[e3 , e3 , e2 , e3 ] = αe1 , [e3 , e3 , e3 , e2 ] = −αe1 , [e3 , e3 , e3 , e3 ] = e2 ⟩
(A) :
Acknowledgement
Acknowledgements could be placed at the end of the text but precede the references.
References
[1] J. M. Casas, J. L. Loday and T. Pirashvili, Leibniz n-algebras, Forum Math,
14, (2002), 189-207.
[2] J. Casas, M. Insua and M. Ladra, Poincare-Birkhoff-Witt theorem for Leibniz
n-algebras, J. Symbolic Comput., 42, (2007), 1052-1065.
[3] J. L. Loday, Une version non commutative des algebres de Lie: Les algebres de
Leibniz, Enseign Math, 3, (1993), 269-293.
[4] B. Ruipu and Z. Jie, The Classification of Nilpotent Leibniz 3-Algebras,Acta
Mathematica scientia, 31B(5), (2011), 1997-2006.
211
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Finite groups with a given number of relative
centralizers
F. Saeedi1∗ and M. Farrokhi D. G.2
1
Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad, Iran
[email protected]
2
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
[email protected]
Abstract
We will study all finite groups and their subgroups with at
most four relative centralizers.
1
Introduction
Counting centralizers in finite groups was initiated by Belcastro and Sherman [4]
in 1994 by characterizing all finite groups with at most 5 centralizers. Let Cent(G)
denotes the set of all centralizers of elements of a groups G and cent(G) be the size of
Cent(G). Belcastro and Sherman proved that cent(G) = 1 if and only if G is abelian
and that there is no finite groups with 2 or 3 centralizers. In addition, cent(G) = 4
if and only if G/Z(G) ∼
= Z2 × Z2 and cent(G) = 5 if and only if G/Z(G) ∼
= Z3 × Z3
or S3 . Later, Ashrafi [2, 3] proved that a group G with cent(G) = 6 must satisfy
G/Z(G) ∼
= D8 , A4 , Z2 × Z2 × Z2 or Z2 × Z2 × Z2 × Z2 . Moreover, if G/Z(G) ∼
= A4 ,
then cent(G) = 6 or 8. ( see also [1])
Let G be a finite group and H be a subgroup of G. A relative centralizer with
respect to G and H is a centralizer of the form CG (h) or CH (g), where g ∈ G and
h ∈ H. The set of all relative centralizers of H in G is defined as
CentG (H) = {CG (h) : h ∈ H}
∗
Speaker
2010 Mathematics Subject Classification. Primary 20F99; Secondary 20D99.
Key words and phrases. Centralizer, relative centralizer, cover, eccentralizerity, equicentralizer.
212
and the set of all relative centralizers of G in H is defined similarly as
CentH (G) = {CH (g) : g ∈ G}.
Let centG (H) and centH (G) denote the size of the sets CentG (H) and CentH (G),
respectively. Clearly, CentG (H) = CentH (G) = Cent(G), whenever H = G. The
aim of this paper is to study those finite groups and their subgroups which admit
few relative centralizers. Despite the fact that the central factor of finite groups with
a given number of centralizers have finitely many possible structures, there is no
specific structural result for finite groups and their subgroups with a given number
relative centralizers. Hence, we shall study the two sets CentG (H) and CentH (G) of
relative centralizers and their relationships. In sections 2 and 3, we show that for
n ≤ 3, the conditions centG (H) = n and centH (G) = n on a finite group G and its
subgroups H are equivalent. Also, we show that if centG (H) = 4, then centH (G) = 4
or 5, and if centH (G) = 4, then centG (H) = 4 or 5. Some examples presented to
support each case in our theorems.
In the last section, we shall prove that the quantity centH (G) − centG (H) can
take any integer value when G ranges over all finite groups and H ranges over
all subgroups of G. However, our computations indicates that probably the above
quantity is under control of both numbers centH (G) and centG (H), that is,
−centH (G) ≤ centH (G) − centG (H) ≤ centG (H).
2
main results
In this section, we obtain connections between two quantities centG (H) and centH (G)
for a finite group G and its subgroup H when they take small values. Clearly, if H
is a central subgroup of G, then CentG (H) = {G} and CentH (G) = {H}. Hence,
the following result follows easily.
Theorem 2.1. Let G be a finite group and H be a subgroup of G. Then centG (H) =
1 if and only if centH (G) = 1. In this case, H ⊆ Z(G), CentG (H) = {G} and
CentH (G) = {H}.
Theorem 2.2. Let G be a finite group and H be a subgroup of G. Then centG (H) =
2 if and only if centH (G) = 2. In this case, H is abelian, CentG (H) = {G, CG (H)}
and CentH (G) = {H, H ∩ Z(G)}.
Theorem 2.3. Let G be a finite group and H be a subgroup of G. Then centG (H) =
3 if and only if centH (G) = 3. In this case, H is abelian, CentG (H) = {G, P, CG (H)}
and CentH (G) = {H, K, H ∩ Z(G)}, where CG (H) < P < G, H ∩ Z(G) < K < H,
P = CG (K) and K = CH (P ).
213
Theorem 2.4. Let G be a finite group and H be a subgroup of G. Then
(i) if centG (H) = 4, then
(i) CentG (H) = {G, P, Q, CG (H)}, H is an abelian group, CentH (G) =
{H, CH (Q), CH (P ), H ∩ Z(G)} and centH (G) = 4, or
(ii) CentG (H) = {G, P, Q, R}, where P, Q, R ̸= CG (H), P ∩ Q = Q ∩ R =
R ∩ P = P ∩ Q ∩ R = CG (H), H/H ∩ Z(G) ∼
= Z2 × Z2 , and
(i) G = P ∪Q∪R, CentH (G) = {H, CH (R), CH (Q), CH (P )} and centH (G) =
4,
(ii) G ̸= P ∪ Q ∪ R,
CentH (G) = {H, CH (R), CH (Q), CH (P ), H ∩ Z(G)}
and centH (G) = 5
(ii) if centH (G) = 4, then
(i) CentH (G) = {H, K, L, H ∩ Z(G)}, H is an abelian group, CentG (H) =
{G, CG (L), CG (K), CG (H)} and centG (H) = 4, or
(ii) CentH (G) = {H, K, L, M }, where K, L, M ̸= H ∩ Z(G), H is abelian,
H/H ∩ Z(G) ∼
= Z2 × Z2 , or H/H ∩ Z(G) ∼
= Z2 × Z2 × Z2 and H/Z(H) ∼
=
Z2 × Z2 , and
(i) H = K ∪ L ∪ M ,
CentG (H) = {G, CG (M ), CG (L), CG (K)}
and centG (H) = 4, or
(ii) H ̸= K ∪ L ∪ M ,
K ∩ L = L ∩ M = M ∩ K9 = K ∩ L ∩ M = H ∩ Z(G),
CentG (H) = {G, CG (M ), CG (L), CG (K), CG (H)} and centG (H) = 5.
Now, we present some examples which supports Theorems 2.1, 2.2, 2.3 and 2.4.
m
n
m−k
+1 ⟩, where k ≥ 0, m > k,
Example 2.5. Let G = ⟨x, y : xp = y p = 1, xy = xp
k
n ≥ k and p is a prime. If H = ⟨x⟩, then CentG (H) = {G, HGp , . . . , HGp } and
k
CentH (G) = {H, H p , . . . , H p }. Therefore, centG (H) = centH (G) = k + 1.
The above example when k = 0, 1, 2, 3 provides pairs of groups satisfying Theorems 2.1, 2.2, 2.3 and 2.4(i.i and ii.i), respectively.
The following example gives another pair of groups with two relative centralizers.
214
Example 2.6. Let G be a Frobenius group with abelian kernel H. Then CentG (H) =
{G, H} and CentH (G) = {H, 1}.
Example 2.7. Let G = H ◦K be the central product of two groups H and K, where
H/Z(G) ∼
= Z2 × Z2 . Then centG (H) = centH (G) = 4 and G, H satisfy Theorem
2.4(i.ii.i and ii.ii.i).
Example 2.8. Let n ≥ 4 and
n
G = ⟨x, y, z : x2 = y 2 = z 2 = 1, xy = x2
n−1
If H = ⟨x2
n−2 +1
, xz = x−1 y, y z = x2
n−1
y⟩.
∼ Z2n−1 × Z2 , then
, yz⟩ ∼
= Q8 and K = ⟨x2 , y⟩ =
{
}
n−1
n−1
CentG (H) = G, ⟨x, y⟩, ⟨x2 y, yz⟩, ⟨x2 yz, y⟩ ,
{
}
n−2
n−2
n−1
CentH (G) = H, ⟨x2 ⟩, ⟨yz⟩, ⟨x2 yz⟩, ⟨x2 ⟩ ,
{
}
CentG (K) = G, ⟨x, y⟩, ⟨x2 , y, z⟩, ⟨x2 , y, xz⟩, ⟨x2 , y⟩ ,
{
}
n−2
n−1
CentK (G) = K, ⟨x2 ⟩, ⟨x2 y⟩, ⟨x2 , y⟩ .
In particular, centG (H) = 4, centH (G) = 5, centG (K) = 5, centK (G) = 4. Hence,
G, H satisfy Theorem 2.4(i.ii.ii) and G, K satisfy Theorem 2.4(ii.ii.ii).
3
Further results and problems
In this section, we shall obtain some more results and show that the difference between two numbers centG (H) and centH (G) can take any integer value, where G is
a finite group and H is a subgroup of G.
Definition 3.1. Let G be a finite group and H be a subgroup of G. Then G and
H are said to be equicentralizers if centG (H) = centH (G). The eccentralizerity of G
and H is defined by ∆(G, H) = centH (G) − centG (H).
Theorem 3.2. Let R denotes the set of all eccentralizerity ∆(G, H), where G ranges
over all finite groups and H ranges over all subgroups of G. Then R = Z.
We conclude this section with some open questions.
Conjecture 3.3. Let G be a finite group and H be a subgroup of G. Then
centG (H) ≤ 2centH (G)
and
215
centH (G) ≤ 2centG (H).
Conjecture 3.4. Let Q be the set of all quotients centG (H)/centH (G), where G
ranges over all finite groups and H ranges over all subgroups of G. Then Q is a
dense subset of [ 12 , 2].
Conjecture 3.5. For all integers m, n with m ≥ 1 and 0 ≤ n ≤ max{0, m−5}, there
exist finite groups Gi with Hi (i = 1, 2), respectively, such that centG1 (H1 ) = m and
centH1 (G1 ) = m + n, centG2 (H2 ) = m + n and centH2 (G2 ) = m.
Question 3.6. Which finite groups are equicentralizer with all of their subgroups?
In particular, is it true that all finite groups equicentralizer with all of their subgroups
are solvable?
Definition 3.7. Let RelCent(G) be the set of all ordered pairs (centG
(H), centH (G)) for all group G, where H ranges over all subgroups of G. A capable group G = H/Z(H) is said to be characterizable by its relative centralizers if
K/Z(K) ∼
= G whenever K is a group such that RelCent(K) = RelCent(H).
By Theorem 2.4, one can easily see that the group Z2 × Z2 is characterizable by
its relative centralizers. This fact suggest us to pose the following conjectures.
Conjecture 3.8. The alternating group A4 is characterizable by its relative centralizers.
Question 3.9. Are there infinitely many finite groups characterizable by their relative centralizers?
References
[1] A. Abdollahi, S. M. Jafarian Amiri and A. Mohammadi Hassanabadi, Groups
with specific number of centralizers, Houston J. Math. 33(1) (2007), 43–57.
[2] A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra
Colloquium 7(2) (2000), 139–146.
[3] A. R. Ashrafi, Counting the centralizers of some finite groups, Korean J. Comput. & Appl. Math. 7(1) (2000), 115–124.
[4] S. M. Belcastro and G. J. Sherman, Counting centralizers in finite groups, Math.
Mag. 5 (1994), 366–374.
216
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Cellular Automata and its application in group theory
Hesam Safa
Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord,
Iran
[email protected]
Abstract
In this paper, we show how applying linear cellular automata
on elements of finite permutation p-groups creates interesting patterns when the group elements are sorted by their lexicographic ordering. In particular, we investigate the relation
between these patterns and group structures.
1
Introduction
Cellular automata (CA) are well known for their applications in the natural (e.g.,
biology, physics) and social sciences (e.g., population dynamics). They have also
been used as processing machines able to carry on abstract tasks, such as learning
and image rendering. In this paper, we propose a new use for CA in the context
of group theory as a mean to explore the properties of permutation p-groups. We
find that, through simple rules, a cellular automaton is able to reveal numerous
structural features of the permutation group on which it operates. As simple rules
already display important links between the cellular automaton and the structure
(e.g., number of cycles, size of the largest cycle), there is a strong potential to apply
other rules and carry on a similar analysis of patterns in order to reveal additional
properties.
A cellular automaton (CA) describes a discrete space in which each element is
referred to as a cell. A state is assigned to each cell. As the CA evolves over
time, the state of each cell changes according to three fundamental features: all cells
are updated simultaneously (synchronicity) by the same rules (uniformity), and these
2010 Mathematics Subject Classification. Primary 20B40; Secondary 68Q80, 20B05.
Key words and phrases. Cellular Automata, Group theory, Lexicographic ordering.
217
rules can take into account the states of neighbor cells (locality). In this paper we use
a 1-dimensional cellular automaton (CA), also called a linear cellular automaton.
Intuitively, this means that the discrete space can be seen as a 1-dimensional array.
Cells are then commonly seen as squares. We consider periodic CAs, which are
intuitively thought of as a torus or a wrap because the previous neighbor of the first
cell is the last cell. From here on, all definitions as well as the short notation CA
refer specifically to the 1-dimensional case. A CA is formally defined below.
Definition 1.1. A cellular automaton A is a triple (S, N, δ) where:
a) S is a finite set, the elements of which are the states of A,
b) each cell is identified by its coordinate in a set N ,
c) δ is the local transition function or local rule of A.
We denote by ci (t) the state of a cell i ∈ N at time t. The computation of the
new state ci (t + 1) for a given cell i uses the local rule, which can be based on the
cell’s previous state ci (t) as well as states of neighbor cells. Thus, in the most general
sense, the local rule is formally defined by ci (t+1) = δ(cj (t), . . . , ci (t), . . . , ck (t)) ∈ S,
j < i < k ∈ N , where j denotes the leftmost cell and k the rightmost cell. Note that
the state obtained at time t + 1 can only use information available at time t and not
earlier. This is a first order automaton, as it goes back one step in time. In this
paper, we use a symmetric local rule. Intuitively, the rule uses the state of as many
cells before as after i. This is formalized in the following definition.
Definition 1.2. δ(cj (t), . . . , ci (t), . . . , ck (t)) is symmetric if and only if |i − j| =
|i − k|. This difference represents the number of cells used on each side, and is
known as the radius of δ.
We are interested in using a CA to investigate the behaviour of a group G. By
AG , we denote the CA in which the states are the permutations of G, and |N | ≤ |G|.
Permutations can be ordered using a ranking which assigns a unique number to
each permutation. To determine the initial state of the CA, the permutations of
G are ordered by lexicographic order [6]. Then, the first cell of the CA contains
the permutation whose ranking by lexicographic order is 1, and so on. Applying
lexicographic ordering is a key part of our method: if the permutations are ordered
differently, then we do not observe the patterns analyzed in this paper.
Also, a row or a state is said to be a fixed point, if applying the rule δ again leaves
the states unchanged. In this paper, the types of rules that we apply and the fact
that the elements are initially ordered using lexicographic ordering always results in a
space-time diagram reaching a fixed point. The next Section explores the conditions
such that a fixed point is obtained and how obtaining this fixed point depends on the
group structure. The fixed point that we use to find structural properties consists of
all cells having 1 as their state, which we denote by 1.
218
2
Main results
Theorem 2.1. Let p be a prime and G be an abelian p-group. Suppose that the
update function of the CA, AG is given by
ci (t + 1) = δ[ci−r (t), . . . , ci−1 (t), ci (t), ci+1 (t), . . . , ci+r (t)]
= ci−r (t) ◦ · · · ◦ ci−1 (t) ◦ ci (t) ◦ ci+1 (t) ◦ · · · ◦ ci+r (t).
for some radius r. If p = 2r + 1 is odd, then 1 is a fixed point for any initial
configuration.
Conjecture. We strongly believe that the above theorem is true for every pgroup.
Acknowledgement
I would like to thank Prof. Vahid Dabbaghian and the IRMACS centre, Simon Fraser
University, Canada, for the support received during my sabbatical stay in 2012.
References
[1] V. Dabbaghian and P. J. Giabbanelli, Investigating p-groups Using Cellular Automata, Discrete Mathematics, Algorithms and Applications, (to appear).
[2] M. Mares and M. Straka, Linear-time ranking of permutations, in Proceedings
of the 15th Annual European Symposium on Algorithms, eds. M. Hoffmann and
E. Welszl (Springer-Verlag, 2007), 187–193.
219
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
On t-extensions of abelian groups
Hossein Sahleh1 and Ali Akbar Alijani2∗
1
Department of Mathematics, Faculty of Sciences, Guilan University, Rasht, Iran
[email protected]
2
Department of Mathematics, Faculty of Sciences, Guilan University, Rasht, Iran
[email protected]
Abstract
Let A,B and C be abelian groups and tA, tB and tC the
maximal torsion subgroups of A,B and C respectively. An
extension 0 → A → B → C → 0 is called a t-extension
if 0 → tA → tB → tC → 0 is an exact sequence. In this
paper, we show that the set of all t-extensions of A by C
is a subgroup of Ext(C, A) which contains P ext(C, A), the
subgroup of all pure extensions of A by C.
1
Introduction
Let ℜ denote the category of abelian groups with homomorphisms as morphisms. For
the groups A and C in ℜ, we let Ext(C, A) denote the group of extensions of A by
C [1]. The elements represented by pure extensions of A by C form a subgroup of
Ext(C, A) which is denoted by P ext(C, A). For group A, tA denote the maximal
ϕ
ψ
torsion subgroup of A. An extension 0 → A −→ B −→ C → 0 will be called tϕ
ψ
extension if 0 → tA −→ tB −→ tC → 0 is an extension. Let Extt (C, A) denote the
set of elements in Ext(C, A) represented by t-extensions. In this paper, we show that
Extt (C, A) is a subgroup of Ext(C, A) containing P ext(C, A). We now lay down
some notation and terminology to be used throughout the paper.
∗
Speaker
2010 Mathematics Subject Classification. Primary 20K35; Secondary 20K27.
Key words and phrases. Abelian groups, extensions, pure extensions.
220
Definition 1.1. A sequence of groups A,B and C and homomorphisms ϕ and ψ
ϕ
ψ
0 → A −→ B −→ C → 0
is called exact if ϕ is monic, ψ epic and Imϕ = Kerψ.
ϕ1
ψ1
Definition 1.2. Let A and C be two groups and E1 : 0 → A −→ B1 −→ C → 0
ϕ2
ψ2
and⊕
E2 : 0 → A −→ B2 −→ C → 0 two t-extensions of A by C. Then the extension
E1 E2 is defined as follows:
⊕
⊕ (ϕ1 ⊕ ϕ2 )
⊕
⊕
(ψ1
ψ2 )
0→A
A −→ B1
B2 −→ C
C→0
ϕ1
ψ1
ϕ2
ψ2
Definition 1.3. Let E : 0 → A −→ B −→ C → 0 and E ′ : 0 → A −→ X −→ C → 0
be two extensions of A by C. E and E ′ are said to be equivalent if there is an
isomorphism β : B → X such that the following diagram
0
0
/A
ϕ1
1A
/A
ϕ2
/B
ψ1
β
/X
ψ2
/C
/0
1C
/C
/0
is commutative.
ϕ
ψ
ϕ
ψ
ϕ′
ψ′
Definition 1.4. Let E : 0 → A −→′ B −→ ′ C → 0 be an extension and α : A → A′
⊕
ϕ
ψ
a morphism. Then αE : 0 → A′ −→ B ′ −→ C → 0 where B ′ = (A′ B)/N ,N =
{(−α(a), ϕ(a)) : a ∈ A},ϕ′ (a′ ) = (a′ , 0)+N and ψ ′ ((a′ , b)+N ) = ψ(b). The extension
αE is called a pushout of E.
Definition 1.5. Let E : 0 → A −→ B −→ C → 0 be an extension and γ : C ′ → C
a morphism. Then Eγ : 0 → A −→ B ′ −→ C ′ → 0 where B ′ = {(b, c′ ) : b ∈ B, c′ ∈
C ′ , ψ(b) = γ(c′ )},ϕ′ (a) = (ϕ(a), 0) and ψ ′ ((b, c′ )) = c′ . The extension Eγ is called a
pullback of E.
For each extension E of A by C, let [E] denote the class of all extensions E ′ of
A by C such that E ′ is equivalent to E.
Theorem 1.6. Let A and C be two groups. Then, the class Ext(C, A) of all
equivalence classes of extensions of A by C is an abelian group with respect to the
operation defined by
⊕
[E1 ] + [E2 ] = [∇A (E1
E2 )△C ]
where E1 and E2 are extensions
⊕ of A by C and ∇A : A × A → A, ∇A (a1 , a2 ) =
a1 + a2 and △C : C → C
C, △C (c) = (c, c) are the diagonal and codiagonal
homomorphism, respectively.
221
Proof. See [1].
Definition 1.7. A subgroup A of a group B is said to be a pure subgroup of B if
and only if nA = A ∩ nB for each positive integer n.
ϕ
ψ
Definition 1.8. An extension 0 → A −→ B −→ C → 0 is called pure extension if
ϕ(A) is pure in B.
Remark 1.9. The set of all pure extensions of A by C denote by P ext(C, A) which
is a subgroup of Ext(C, A).
Definition 1.10. An extension
⊕ of Aπ2 by C is called split if it is equivalent to the
ι1
trivial extension 0 → A −→ A C −→ C → 0 where ι1 (a) = (a, 0) and π2 (a, c) = c
for all a ∈ A and c ∈ C.
Remark 1.11. Let A and C be two groups. Then Ext(C, A) = 0 if and only if
every extension of A by C splits.
Definition 1.12. A group A is called cotorsion if Ext(Q, A) = 0 where Q is the
group of rationales.
2
Main results
ϕ
ψ
Definition 2.1. Let A,B and C be abelian groups. An extension 0 → A −→ B −→
ϕ
ψ
C → 0 is called a t-extension if 0 → tA −→ tB −→ tC → 0 is an extension.
Lemma 2.2. A pushout of a t-extension is t-extension. A pullback of a t-extension
is t-extension.
Lemma 2.3. An extension equivalent to a t-extension is t-extension.
Theorem 2.4. Let A and C be two groups. Then, the class Extt (C, A) of all
equivalence classes of t-extensions of A by C is an subgroup of Ext(C, A) with
respect to the operation defined by
⊕
[E1 ] + [E2 ] = [∇A (E1
E2 )△C ]
where E1 and E2 are t-extensions of A by C and ∇A and △C are the diagonal and
codiagonal homomorphisms,respectively.
Theorem 2.5. Let A and C be two groups. Then, P ext(C, A) ⊆ Extt (C, A).
Lemma 2.6. Let A be a torsion-free group and C a torsion group. Then Extt (C, A) =
0.
222
Corollary 2.7. Let A be a torsion-free group and C a torsion group.
P ext(C, A) = 0.
Then
Theorem 2.8. Let⊕A be a group. Then, Extt (C, A) = 0 for every group C if
and only if A ∼
= B D where B is a torsion divisible group and D a torsion-free
cotorsion group.
Theorem 2.9. Let C be a group. Then, Extt (C, A) = 0 for every group A if and
only if C is a free group.
References
[1] L. Fuchs, Infinite Abelian Groups, Vol. I, Academic Press, New York, 1970.
223
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
OD-characterization of almost simple groups related to
L2 (p2 )
Masoumeh Sajjadi1∗ , Masoumeh Bibak2 and Gholamreza Rezaeezadeh3
1,2
3
Department of Mathematics, Payame Noor University, Iran
[email protected], [email protected]
Department of Mathematics, Faculty of Basic Science, Shahrekord University,
Shahrekord, Iran
[email protected]
Abstract
A group G is an almost simple group, if S E G . Aut(S),
for some non-abelian group S. In many articles it has
been shown that many finite almost simple groups are ODcharacterizable or k-fold OD-characterizable for certain k ≥
2. In this paper we denote L2 (p2 ) by Lp , and we characterize
almost simple groups related to Lp by the order and degree
pattern. In fact by Theorem 2.1 we prove that Lp , Lp : 21 ,
Lp : 22 and Lp : 23 are OD-characterizable, and Lp : 22
is 9-fold OD-characterizable, where p is a prime belongs to
{5, 11, 13, 17}.
1
Introduction
Throughout this article, all groups under consideration are finite. For any group
G, we denote by π(G) the set of all prime devisors of |G| and the set of orders of
the elements of G is denoted by πe (G). The prime graph Γ(G) of a group G is a
simple graph whose vertex set is π(G) and two distinct primes p and q are joined
by an edge if and only if G contains an element of order pq. For p ∈ π(G), we put
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D05; Secondary 20D06, 20D60.
Key words and phrases. Almost simple group, Prime graph, Degree pattern.
224
deg(p) := |{q ∈ π(G)|p ∼ q}|, which is called the degree of p. If |G| = pα1 1 pα2 2 ...pαk k ,
p,i s different primes, we define D(G) := (deg(p1 ), deg(p2 ), ..., deg(pk )), where p1 <
p2 < ... < pk , which is called the degree pattern of G.
Definition 1.1. The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |G| = |H| and D(G) = D(H).
In particular, a 1-fold OD-characterizable group is simply called OD-characterizable.
The interest in characterizing finite groups by degree pattern started in 2005 by
M.R. Darafsheh,et.al., that the authors proved that if G is a finite group such that
|G| = |M | and D(G) = D(M ), where M is one of the following simple groups: (1)
sporadic simple groups, (2) alternating Ap with p and p − 2 primes, (3) some simple
groups of Lie type, then G ∼
= M.
A group G is an almost simple group, if S E G . Aut(S), for some non-abelian
group S. In many articles it has been shown that many finite almost simple groups
are OD-characterizable or k-fold OD-characterizable for certain k ≥ 2, see [1, 2, 3,
4, 5].
In [1], for U := U3 (17), it is shown that finite almost simple groups U and
U : 2 are OD-characterizable, U : 3 is 3-fold OD-characterizable, and U : S3 is
5-fold OD-characterizable and in [3], for L := L2 (49), it is shown that finite almost
simple groups L, L : 21 , L : 22 and L : 23 are OD-characterizable; L : 22 is 9-fold
OD-characterizable( 22 is the Klein, s four group).
In this article our main aim is to show the recognizability of the almost simple
groups related to Lp := L2 (p2 ) by the degree pattern in the prime graph and the order
of the groups. By using Theorem 3.1 in [5], Lp is OD-characterizable, so in this
paper we investigate the remainder cases.
2
Main results
Theorem 2.1. Let M be an almost simple group related to Lp = L2 (p2 ). If G is a
finite group such that D(G) = D(M ) and |G| = |M |, then the following assertions
hold:
(a) If M = Lp , then G ∼
= Lp ;
(b) If M = Lp : 21 , then G ∼
= Lp : 21 ;
∼
(c) If M = Lp : 22 , then G = Lp : 22 ;
(d) If M = Lp : 23 , then G ∼
= Lp : 23
2
(e) If M = Lp : 2 , then G ∼
= Lp : 22 , Z2 × (Lp : 21 ), Z2 × (Lp : 22 ), Z2 × (Lp :
23 ), Z2 · (Lp : 21 ), Z2 · (Lp : 22 ), Z2 · (Lp : 23 ), Z4 × Lp or (Z2 × Z2 ) × Lp .
Conjecture Let M be an almost simple group related to L = L2 (p2 ), where
p > 5 is a prime. If G is a finite group such that D(G) = D(M ) and |G| = |M |,
225
then L, L : 21 , L : 22 and L : 23 are OD-characterizable and L : 22 is 9-fold ODcharacterizable.
References
[1] M.R. Darafsheh, G.R. Rezaeezadeh, M. Sajjadi and M. Bibak, ODCharacterization of almost simple groups related to U3 (17), Quasigroups and
related Systems. 21 (2013), 49–58.
[2] G.R. Rezaeezadeh, M.R. Darafsheh, M. Sajjadi and M. Bibak, ODCharacterization of almost simple groups related to L3 (25), Accepted in Bull.
Iranian Math. Soc.
[3] L.C. Zhang and W.J. Shi, OD-Characterization of almost simple groups related
to L2 (49), Archivum Mathematicum. Masaryk Univ., Brno. 44 (2008), 191–199.
[4] L.C. Zhang and W.J. Shi, OD-Characterization of almost simple groups related
to U3 (5), Acta Mathematica Sinica, English Series. 26 (2010), no. 1, 161–168 .
[5] L.C. Zhang and W.J. Shi, OD-Characterization of Projective Special Linear
Groups L2 (q), Algebra Colloquium. 19 (2012), no 3, 509–524.
226
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
A new characterization of Ap where p and p − 2 are
twin primes
Seyed Sadegh Salehi Amiri
Department of Mathematics, Babol Branch, Islamic Azad University, Babol, Iran
[email protected]
Abstract
Let G be a finite group and πe (G) be the set of elements of
order G. Let k ∈ πe (G) and mk be the number of elements
of order k in G. Set nse(G):={mk |k ∈ πe (G)}. Assume p
and p − 2 are twin primes. We prove that if G is a group
such that nse(G)=nse(Ap ) and p ∈ π(G), then G ∼
= Ap . As
a consequence of our results we prove that, Ap is uniquely
determined by its nse and order.
1
Introduction
If n is an integer, then we denote by π(n) the set of all prime divisors of n. Let G be
a finite group. Denote by π(G) the set of primes p such that G contains an element
of order p. Also the set of element orders of G is denoted by πe (G).
Set mi = mi (G)=|{g ∈ G| the order of g is i}|. In fact mi is the number of
elements of order i in G, and nse(G):={mi | i ∈ πe (G)}, the set of sizes of elements
with the same order.
For the set nse(G), the most important problem is related to Thompson’s problem. In 1987, J. G. Thompson posed the following problem: ([3, Problem 12.37])
Problem 1: For each finite group G and each integer d ≥ 1, let G(d) = {x ∈ G|
2010 Mathematics Subject Classification. Primary 20D06, ; Secondary 20D20, 20D60.
Key words and phrases. Element order, set of the numbers of elements of the same order,
alternating group.
227
xd = 1}. G1 and G2 are called the same order type if and only if, |G1 (d)| = |G2 (d)|,
d = 1, 2, 3, . . . . Suppose G1 and G2 are finite group of the same order type. If G1
is solvable, is G2 necessarily solvable?
Unfortunately, as so far, no one can prove it completely, or even give a counterexample. However, if groups G1 and G2 are of the same order type, we see clearly
that |G1 | = |G2 | and nse(G1 ) = nse(G2 ). So it is natural to investigate the Thompson’s Problem by |G| and nse(G).
In [2, 5], it is proved that the groups A5 , A6 , A7 and A8 are uniquely determined
only by nse(G). Also, In [1], it has been proved that the sporadic groups are characterizable by their nse and order. In [5], the authors gave the following problem:
Problem 2: Is a group G isomorphic to An (n ≥ 4) if and only if nse(G) =
nse(An )?
In this paper, we give a positive answer to this problem for some type of the
alternating groups and show that the alternating groups Ap with p and p − 2 prime
are characterizable by nse(Ap ), when p ∈ π(G). In fact the main theorem of our
paper is as follows:
Main Theorem: Let G be a group such that nse(G)=nse(Ap ), where p and p − 2
are twin primes. If p ∈ π(G), then G ∼
= Ap .
We note that there are finite groups which are not characterizable by nse(G) and
|G|. In 1987, Prof. G. J. Thompson gave an example as follows:
Let G1 = (C2 × C2 × C2 × C2 ) ⋊ A7 and G2 = L3 (4) ⋊ C2 be the maximal subgroups
of M23 . Then nse(G1 ) = nse(G2 )= {1, 435, 2240, 5040, 5760, 6300, 6720, 8064} and
|G1 | = |G2 | = 40320, but G1 ≇ G2 . Also there is a another example as follow: Let
H1 = C4 × C4 and H2 = C2 × Q8 , where C2 and C4 are cyclic groups of orders 2
and 4, respectively and Q8 is a quaternion group of order 8. It is easy to see that
nse(H1 ) = nse(H2 )= {1, 3, 12} and |H1 |=|H2 | = 16, but H1 is an abelian group and
H2 is a non-abelian group. Therefore H1 ≇ H2 .
2
Preliminary Results
We first quote some comments and lemmas that are used in deducing the main
theorem of this paper.
Let α ∈ Sn be a permutation and let α have ti cycles of length i, i = 1, 2, ..., l,
in its cycle decomposition. The cycle structure of α is denote by 1t1 2t2 ...ltl where
1t1 + 2t2 ... + ltl = n. One can easily show that two permutations in Sn are conjugate
if and only if they have the same cycle structure. Let α ∈ Sn and assume that the
cycle decomposition of α contains t1 cycles of length 1, t2 cycles of length 2, ..., tl
cycles of length l. Then |clSn (α)| = n!/1t1 2t2 · · · ltl t1 !t2 ! · · · tl !.
228
Lemma 2.1. [5] Let G be a finite group and m be a positive integer dividing |G|.
If Lm (G) = {g ∈ G|g m = 1}, then m | |Lm (G)|.
Let mn be the number of elements of order n. We note that mn = kϕ(n), where k
is the number of cyclic subgroups of order n in G and ϕ is the Euler totient function.
Also we note that if n > 2, then ϕ(n) is even. If n ∈ πe (G), then by 2.1 and the
above notation we have:

ϕ(n) | mn



∑
n | d|n md



(∗)
Lemma 2.2. [5] Let G be a group containing more than two elements. Let k ∈ πe (G)
and mk be the number of elements of order k in G. If s = sup{mk |k ∈ πe (G)} is
finite, then G is finite and |G| ≤ s(s2 − 1).
Lemma 2.3. [4] Let G be a finite group and p ∈ π(G) be odd. Suppose that P is
a Sylow p−subgroup of G and n = ps m, where (p, m) = 1. If P is not cyclic and
s > 1, then the number of elements of order n is always a multiple of ps .
References
[1] A. K. Asboei, S. S. salehi Amiri, A. Iranmanesh, A. Tehranian, A new characterization of sporadic simple groups by NSE and order, J. Algebra Appl, 12 (2),
(2013) 1-3.
[2] A. K. Asboei, S. S. salehi Amiri, A. Iranmanesh, A. Tehranian, A new characterization of A7 and A8 , An. Stint. Univ. Ovidius Constanta, 21 (3), (2013),
43-50.
[3] V. D. Mazurov and E. I. Khukhro, Unsolved Problems in group theory: the
Kourovka Notebook, 16 ed. (Novosibirsk, Inst. Mat. Sibirsk. Otdel. Akad, 2006).
[4] G. Miller, Addition to a theorem due to Frobenius, Bull. Am. Math. Soc. 11,
(1904), 6-7.
[5] R. Shen, C. G. Shao, Q. Jiang, W. Shi, V. Mazuro, A New Characterization of
A5 , Monatsh Math, (2010), 337-341.
229
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Commuting graphs on conjugacy classes of finite
groups
Maryam Shadab1 and Amin Saeidi2∗
1
Islamic Azadi University, Shahr-e Rey branch, Tehran, Iran
[email protected]
2
Islamic Azadi University, Central branch, Tehran, Iran
[email protected]
Abstract
Let G be a finite group. We define the commuting conjugacy
class graph of G as follows: the vertex set is the set of noncentral conjugacy classes of G and two distinct conjugacy
classes [x] and [y] in G are joint by an edge if x commutes
with an element of [y]. In this talk, we study the structure of
groups in which the commuting conjugacy class graph contains an isolated vertex.
1
Introduction
Let G be a finite group. A usual way to study the structure of G is to attach certain graphs to it. By studying the structure of these graphs, we may obtain useful
information about G. In these talk, we fix a finite group G and investigate a graph
defined on the set of the conjugacy classes of G. There are different ways to define
the vertex set of a graph with conjugacy classes as edges (see an overview in [3]).
The graphs that we consider here have been introduced and studied in [2]: the vertex set is the set of non-identity conjugacy classes of G and two distinct conjugacy
classes [x] and [y] in G are joint by an edge if x commutes with an element of [y].
In other words, if there exists an element g ∈ G such that [x, y g ] = 1. Since every
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D10; Secondary 20F65.
Key words and phrases. Conjugacy class graphs, isolated vertex, special p-groups.
230
central element of G is a singleton set of conjugacy class of G, one observes that
central elements as vertices are joint to all other vertices of the graph. So for the
groups with a nontrivial center, the graph is connected and the diameter may not
exceed two. One may ask what can we say about a subgraph obtained by omitting the
conjugacy classes of central elements. Here we concentrate to these subgraph that
we denote by Γcc (G). We are mainly interested in the case when Γcc (G) cintains an
isolated vertex. All groups we consider are finite and non-abelian. We refer to [1]
for notations.
2
preliminaries
Lemma 2.1. [2, Theorem 19] If Γcc (G) is an empty graph, then G is one of the
groups D8 , Q8 or S3 .
Lemma 2.2. If Γcc (G) has an isolated vertex, then |G| is even.
Lemma 2.3. Let G be a nilpotent group. If Γcc (G) has an isolated vertex, then
(i) G is a 2-group;
(ii) Z(G) is elementary abelian;
(iii) |Z2 (G)| ≤ 2|Z(G)|2 .
Lemma 2.4. Let [x] be an isolated vertex. Then x is a p-element, i.e |x| = pα for
a prime p and a positive integr α.
3
Main results
We start with the following result which is a a generalization of Lemma 2.1.
Theorem 3.1. Let G be a p-group in which |G : Z(G)| = p2 . Then it contains
exactly p + 1 connected components. Moreover, each component is complete of type
Km where m = p−1
|G|.
p3
Example 3.2. Let G be a non-abelian p-group of order p3 . Then Γcc (G) =
where Ki are complete graphs of order p − 1.
p+1
∪
Ki ,
i=1
Remark 3.3. The previous example implies that two non-isomorphic p-groups may
have isomorphic graphs.
231
Proposition 3.4. Let G be a 2-group. If [x] is an isolated vertex of Γcc (G), then
|CG (x) : Z(G)| = 2
Proposition 3.5. Let G be a nilpotent group of class 2, and let Γcc (G) has an
isolated vertex. Then G is a special 2-group.
Corollary 3.6. Let [x] be an isolated vertex. Then CG (x) is a p-group.
Acknowledgement
The authors gratefully acknowledge the financial and other support of this research,
provided by the Islamic Azad University, Shahr-e Rey Branch, Tehran, Iran
References
[1] M. Aschbacher, Finite Group Theory, Cambridge Univ. Press, 1986.
[2] M. Herzog, P. Longobardi and M. Maj, On a commuting graph on conjugacy
classes of groups, Comm.Algebra, 37 (2009), 3369–3387.
[3] M.L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mountain J. Math. 38 (2008), 175-212.
232
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
A quotient oF topological fundamental groups
Hamid Torabi1∗ Ali Pakdaman2 and Behrooz Mashayekhy3
1
2
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic
Structures, Ferdowsi University of Mashhad, Mashhad, Iran
hamid [email protected]
Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan, Iran
[email protected]
1
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic
Structures, Ferdowsi University of Mashhad, Mashhad, Iran
[email protected]
Abstract
In this talk, we discuss on the topological properties of a
quotient of topological fundamental groups via a new subgroups of fundamental group, namely small generated subgroup, constructed by small loops which presence of them is
equivalent to absence of homotopically Hausdorffness properties.
1
Introduction
In 2002, a work of Biss initiated the development of a theory in which the familiar
fundamental group π1 (X, x) of a topological space X becomes a topological space
denoted by π1top (X, x) by endowing it with the quotient topology inherited from the
path components of based loops in X with the compact-open topology. Among other
things, Biss claimed that π1top (X, x) is a topological group. However, there is a gap
in his proof. Brazas discovered some interesting counterexamples for continuity of
multiplication in π1top (X, x) (for more details, see [1]).
∗
Speaker
2010 Mathematics Subject Classification. Primary 20F38; Secondary 20k45 , 20F34.
Key words and phrases. Topological fundamental group, SG subgroup, Semi-locally small
generated space.
233
In fact, π1top (X, x) was a quasitopological group, that is, a group with a topology
such that inversion and all translations are continuous. Althogh, Brazas by removing
some open subsets of π1qtop (X, x) make it a topological group, but it is an interesting
question that when these two topologies are equivalent. In the sequel, by introducing
some spaces, we give a partial answer to this question.
If a space X is not homotopically Hausdorff, then there exist x ∈ X and a
nontrivial loop in X based at x which is homotopic to a loop in every neighborhood
U of x. Z. Virk [4] called these loops as small loops and showed that for every x ∈ X
they form a subgroup of π1 (X, x) which is named small loop group and denoted by
π1s (X, x). In general, various points of X have different small loop groups and hence
in order to have a subgroup independent of the base point, Virk [4] introduced the
SG (small generated) subgroup, denoted by π1sg (X, x), as the subgroup generated by
the following set
{[α ∗ β ∗ α−1 ] | [β] ∈ π1s (X, α(1)), α ∈ P (X, x)},
where P (X, x) is the space of all paths from I into X with initial point x ((see [3]
for further details)).
Throughout this article, all the homotopies between two paths are relative to end
points, X is a topological space with the base point x ∈ X.
2
Main results
Definition 2.1. ([4]) The small loop group π1s (X, x) of (X, x) is the subgroup of the
fundamental group π1 (X, x) consisting of all homotopy classes of small loops. The
SG subgroup of π1 (X, x), denoted by π1sg (X, x), is the subgroup generated by the
following set
{[α ∗ β ∗ α−1 ] | [β] ∈ π1s (X, α(1)), α ∈ P (X, x)},
where P (X, x) is the space of all paths in X with initial point x.
Definition 2.2. We call a space X semi-locally small generated if and only if for each
x ∈ X there exists an open neighborhood U of x such that i∗ π1 (U, x) ≤ π1sg (X, x),
where i : U ↩→ X is the inclusion map.
Theorem 2.3. If (X, x) is a pointed topological space and U is an open neighborhood of the identity element [ex ] ∈ π1top (X, x), then π1sg (X, x) ⊆ U.
Corollary 2.4. Every nonempty open or closed subset of π1qtop (X, x) is a disjoint
union of some cosets of π1sg (X, x).
234
Proof. Since π1qtop (X, x) is the disjoint union of all cosets of π1sg (X, x), it suffices to
prove the theorem for open subsets of π1qtop (X, x). For this, let V be a nonempty
open subset of π1qtop (X, x) and g ∈ V . Then g −1 V is an open subset of π1qtop (X, x)
containing [ex ] and hence by∪Theorem 2.3 , π1sg (X, x) ⊆ g −1 V which implies that
gπ1sg (X, x) ⊆ V . Hence V = g∈V gπ1sg (X, x).
The natural quotient map p : π1qtop (X, x) −→
π1 (X,x)
π1sg (X,x)
on the algebraic quotient group
previous corollary we can prove that:
π1 (X,x)
π1sg (X,x)
induce the quotient topology
which we denote it by ( ππsg1 (X,x)
)top . By the
(X,x)
1
Theorem 2.5. For a topological space X, π1qtop (X, x) is a topological group if an
only if ( ππsg1 (X,x)
)top is topological group.
(X,x)
1
Theorem 2.6. For a topological space X, π1qtop (X, x) is a indiscrete topological
group if an only if ( ππsg1 (X,x)
)top is indiscrete topological group.
(X,x)
1
)top is discrete topological group
Theorem 2.7. For a topological space X, ( ππsg1 (X,x)
1 (X,x)
if and only if X is semi-locally small generated.
Corollary 2.8. If X is semi-locally small generated, then π1qtop (X, x) is topological
group.
By the following example, we use Theorem 2.5 to find a non semi-locally small
generated space with topological fundamental group as a topological group.
Example 2.9. Let HA be the Harmonic Archipelago space and let X = [0, 1] ∪n
({1/n} × HAn+1 ), where HAn is scaled Harminic Archipelago by the scaler 1/n.
)top is topological group.
π1qtop (X, x) is topological group since ( ππsg1 (X,x)
(X,x)
1
References
[1] J. Brazas, The fundamental group as a topological group, Topology Appl. 160
(2013), no. 1, 170–188.
[2] E. H. Spanier, Algebraic Topology, McGraw-Hill Book Co., New York-Toronto,
Ont.-London 1966
[3] H. Torabi, A. Pakdaman, B. Mashayekhy, Topological fundamental groups and
small generated coverings, to appear in Mathematica Slovaca.
[4] Z. Virk, Small loop spaces, Topology and its Applications. 157 (2010) 451-455.
235
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
On 11− decomposable finite groups
M. Yousefi∗ and A. R. Ashrafi
Department of Pure Mathematics, Faculty of Mathematical Sciences,University of Kashan,
Kashan, I. R. Iran
s− [email protected]
Abstract
Let G be a finite group and NG denote the set of non-trivial
proper normal subgroups of G. An element K of NG is said
to be n-decomposable if K is a union of n distinct conjugacy
classes of G. G is called n−decomposable, if NG ̸= ∅ and
every element of NG is n-decomposable.
In this paper, we investigate the structure of non-solvable
non-perfect finite group G, when G is 11−decomposable.
We prove that such a group is isomorphic to P SL(2, 16).2,
Aut(P SL(2, 19)), P SL(2, 25) : 22 , Aut(Sz(32)) or U3 (5).2.
Here, P SL(2, 16).2 and U3 (5).2 are extensions of the groups
P SL(2, 16) and U3 (5) and P SL(2, 25) : 22 denotes one of the
third split extension of P SL(2, 25) in the small group library
of GAP [M. Schonert et al., GAP, Groups, Algorithms and
Programming, Lehrstuhl für Mathematik, RWTH, Aachen,
1992].
1
Introduction
Let G be a finite group and let NG be the set of non-trivial proper normal subgroups
of G. An element K of NG is said to be n−decomposable if K is a union of n distinct
conjugacy classes of G. If NG ̸= ∅ and every element of NG is n−decomposable,
then we say that G is n−decomposable.
∗
Speaker
2010 Mathematics Subject Classification. Primary 20E34; Secondary 20D10.
Key words and phrases. Conjugacy class, n-decomposable group.
236
Asharfi and Sahraee [1] settled the problem of classifying n-decomposable finite
groups. They characterized the solvable n−decomposable finite groups under certain
conditions and also the structure of 2−, 3− and 4−decomposable finite groups. If G
is solvable, then the problem of classifying such groups is so difficult. In [2, 3, 4], the
authors continue this problem by characterization of n− decomposable finite groups,
when 5 ≤ n ≤ 10. They applied some deep results of Wu Jie Shi in the field of the
quantitative structure of finite groups [6]. It is merit to state here that such type of
problems in group theory is started by publishing some pioneering works of Wu Jie
Shi.
Throughout this paper, as usual, G′ denotes the derived subgroup of G, Z(G) is
the center of G, xG , x ∈ G, denotes the conjugacy class of G with the representative
x and G is called non-perfect, if G′ ̸= G. Also, SmallGroup(n, i) denotes the
ith group of order n in the small group library of GAP. All groups considered are
assumed to be finite. Our notation is standard and can be taken from the standard
books of group theory.
2
Results and Discussions
Shahryari and Shahabi [5], investigated the structure of finite groups which contains a
2−decomposable subgroup H. In this case, H ≤ G′ , |H|(|H|−1) divides |G| and H is
an elementary abelian normal subgroup of G. Moreover, they proved that, under certain conditions, G is a Frobenius group with kernel H. Ashrafi and Sahraei [1], used
the mentioned paper to characterize the structure of 2−, 3− and 4−decomposable
finite groups. Also, they obtained the structure of solvable n−decomposable finite
groups.
In this section, we first report on recent results on our problem. We first assume that G is a non-perfect 5− or 6−decomposable finite group. By [2, Theorems 5 and 6], G is 5− or 6−decomposable if and only if G ∈ {Z5 × A5 , A6 ·
23 , Aut(P SL(2, 7)), Aut(P SL(2, 7))} or G ∈ {S6 , A6 · 22 }, respectively. In [3, Theorem 2.5 and 2.6], the problem of classifying non-perfect 7− and 8−decomposable
finite groups are considered into account. We assume that G is such a group. If
G is a non-perfect 7−decomposable finite group then G is isomorphic to an abelian
group of order 49, Aut(P SL(2, 11)), Z7 xA6 , Aut(Sz(8)) or a Frobemus group of
order 61 pr (pr 1), p > 5 is prime, and r is a positive integer, such that the kernel of
G is elementary abelian of order pr and its complement is cyclic. Let G be a nonperfect 8−decomposable finite group. Then G is isomorphic to Aut(P SL(2, 13)),
P SL(2, 27) : 3, P SL(3, 4) : 2 (including P SL(3, 4).21; P SL(3, 4).22 and P SL(3, 4).23);
P SL(3, 4) : 3, S7 or a Frobenius group of order 17 2r (2r 1), r is a positive integer, such
that the kernel of G is elementary abelian of order 2r and its complement is cyclic.
237
In [4, Theorems 2.6 and 2.7], the authors proved that if then G is isomorphic to
Aut(P SL(3, 3)), Aut(P SL(2, 32)), Z3 ⋉ (Z5 × Z5 ), or a non-abelian group of order pq, where p and q are primes and p − 1 = 8q. Moreover, it is proved that
a non-perfect 10−decomposable finite group G is isomorphic to Aut(P SL(2, 17)),
Aut(U 3(3)), P SL(2, 25).23 or D38 .
In this paper we continue the study of this problem and classify the non-solvable
non-perfect 11−decomposable finite groups. We prove that:
Mail Theorem: Suppose G is a non−solvable non−perfect 11−decomposable finite
group, then G is isomorphic to P SL(2, 16).2, Aut(P SL(2, 19)), P SL(2, 25) : 22 ,
Aut(Sz(32)) or U3 (5).2.
References
[1] A. R. Ashrafi and H. Sahraei, On Finite Groups Whose Every Normal Subgroup
is a Union of the Same Number of Conjugacy Classes, Vietnam J. Math., 30
(3) (2002) 289–294.
[2] A. R. Ashrafi and Y. Q. Zhao, On 5− and 6−decomposable finite groups, Math.
Slovaca 53 (4) (2003)373–383.
[3] A. R. Ashrafi and W. J. Shi, On 7− and 8−decomposable finite groups, Math.
Slovaca 55 (3) (2005) 253–262.
[4] A. R. Ashrafi and W. J. Shi, On 9− and 10−decomposable finite groups, J. Appl.
Math. Comput. (2008) 26 169–182.
[5] M. Shahryari and M. A. Shahabi, Subgroups which are the union of two conjugacy classes, Bull. Iranian Math. Soc. Vol. 25, No. 1 (1999), 59-71.
[6] W. J. Shi, The Quantitative Structure of Groups and Related Topics, Group
Theory in China, Zhe-Xian Wan and Sheng-Ming Shi(Eds.), 163-181, Science
Press New York, Ltd. and Kluwer Academic Publishers, 1996.
238
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
The nonabelian tensor square of some finite groups
Rosita Zainal1∗ , Nor Muhainiah Mohd Ali2 , Nor Haniza Sarmin3 ,
Samad Rashid4 and Adnin Afifi Nawi5
1,2,3,5
4
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi
Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
[email protected] , [email protected] , [email protected] ,
adnin− [email protected]
Department of Mathematics, , Faculty of Science , Shahr-e-Rey Branch, Islamic Azad
University, Tehran, Iran
[email protected]
Abstract
The nonabelian tensor square, G ⊗ G, is a special case of the
nonabelian tensor product which has its origin in homotopy
theory. In this research, the nonabelian tensor square of
groups of order 8p, where p is and odd prime are determined.
1
Introduction
The nonabelian tensor square of a group G, denoted by G⊗G, is the group generated
by the symbols g ⊗ h and defined by the relations gg ′ ⊗ h=(g g ′ ⊗ g h)(g ⊗ h) and
g ⊗ hh′ = (g ⊗ h)(h g ⊗ h h′ ) for all g, g ′ , h, h′ ∈ G, where G acts on itself by
conjugation, i.e. g g ′ =gg ′ g −1 .
Many researches on the nonabelian tensor square of various groups have been
conducted over the years. In 2011, Rashid et al. [1] determined the nonabelian
tensor square of groups of order p2 q where p and q are primes. Recently, Basri
et al. in [2] computed the nonabelian tensor square of metacyclic p-groups.
Next, the classification of groups of order 8p, where p is an odd prime is stated
in the following theorem:
∗
Speaker
2010 Mathematics Subject Classification. Primary 20D15; Secondary 20F99 , 20J05, 20J99.
Key words and phrases. Nonabelian tensor square, finite groups.
239
Theorem 1.1. [4] Let G be a group of order 8p, where p is an odd prime. Then
exactly one of the following holds:
(1.1) G ∼
= D4 × Zp .
∼
(1.2) G = Q2 × Zp .
(1.3) G ∼
= D2p × Z2 .
(1.4) G ∼
= Qp × Z2 .
(1.5) G ∼
= ⟨Dp × Z4 .
⟩
(1.6) G ∼
= a, b|a8 = bp = 1, a−1 ba = b−1 .
(1.7) G ∼
=D
⟨ 4p .
⟩
(1.8) G ∼
= a, b, c|a4 = b2 = cp = 1, b−1 ab = a−1 , a−1 ca = c−1 , bc = cb .
(1.9) G ∼
=Q
⟨ 2p .
⟩
(1.10) G ∼
= a, b|a8 = bp = 1, a−1 ba = bα ,
root of α4 ≡ 1(mod p), 4 divides
p − 1.
⟨where α4 is a 2primitive
⟩
p
−1
α
∼
(1.11) G = a, b, c|a = b = c = 1, ab = ba, a ca = c , bc = cb ,
root ⟩of α4 ≡ 1(mod p), 4 divides p − 1.
⟨where8 α ispa primitive
(1.12) G ∼
= a, b|a = b = 1, a−1 ba = bα ,
where α is a primitive root of α8 ≡ 1(mod p), 8 divides p − 1.
(1.13) G ∼
= Z4 × A4 .
(1.14) G ∼
= SL(2, 3).
(1.15) G ∼
= S4 .
(1.11) G ∼
= ⟨a, b, c, d|a4 = b2 = c2 = dp = 1, ab = ba, ac = ca, bc = cb,
d−1 ad = b, d−1 bd = c, d−1 cd = ab⟩.
Some basic results that are used to compute the nonabelian tensor square of finite
groups are stated in the following theorems.
Theorem 1.2. [3] Let G and H be groups. Then
(G × H) ⊗ (G × H) ∼
= (G ⊗ G) × (G ⊗ H) × (H ⊗ G) × (H ⊗ H).
Theorem 1.3. [3] Let G be the metacyclic group, G = ⟨a, b|bn = am = e, aba−1 =
bl ⟩, where lm ≡ 1 (mod n) and n is an odd number. Then G ⊗ G = Zm × Zm1 ×
Zm2 × Zm3 where m1 = (n, l − 1), m2 = (n, l − 1, 1 + l + l2 + ... + lm−1 ), m3 =
(n, 1 + l + l2 + ... + lm−1 ).
Theorem 1.4. [5] If G is a finite group such that the derived subgroup G′ is cyclic
and (|G′ |, |Gab |) = 1, then G⊗G ∼
= G′ ×(Gab ⊗Z Gab ), where Gab is the abelianization
ab
′
of a group G, G = G/G .
Theorem 1.5. [6] Let G be a finite solvable group of derived length 2. Then
|G ⊗ G| divides |Gab ⊗Z Gab ||G′ ∧ G′ ||G′ ⊗Z[Gab ] I(Gab )|, where I(Gab ) is the kernel
of Z[Gab ] → Z.
240
Theorem 1.6. [6] Let G be a finite group and i > 0. Then there is an exact
sequence 1 → [G′ , Gφ ] → τ (G, G) → τ (Gab , Gab ) → 1 where φ : G → Gφ is
an isomorphism, τ (G, G) is the subgroup [G, Gφ ] of η(G, G) = ⟨G, Gφ |[g, hφ ]g1 =
[g g1 , (hg1 )φ ], [g, hφ ]h1φ = [g h1 , (hh1 )φ ]⟩ for all g1 , g, h, h1 ∈ G and [G′ , Gφ ] 6 τ [G, G].
2
Main results
The nonabelian tensor square for all groups of order 8p are stated in the next theorem.
Theorem 2.1. Let G be a group of order 8p, where p is an odd prime. Then exactly
one of the following holds:


Z4p × Z32
; if G is of type (1.1), (1.4),





(1.5), (1.7) or (1.11),




2

Z4p × Z4 × Z2 ; if G is of type (1.2) or (1.9),




Z8p
; if G is of type (1.6), (1.10) or (1.12),



Z × Z8
; if G is of type (1.3),
2p
2
G⊗G∼
=

Z4p × M
; if G is of type (1.8),





Z6 × Q2
; if G is of type (1.13),





Z3 × Q2
; if G is of type (1.14),





Z3 × Â4
; if G is of type (1.15),



Z × Z2
; if G is of type (1.16),
14
2
where M is an abelian group of order 8.
Proof. Let G ∼
= D4 × Zp . By Theorem 1.2,
G⊗G∼
= (D4 × Zp ) ⊗ (D4 × Zp ) ∼
= Z4p × Z32 .
By using the same proof as group of type (1.1), we prove for a group of type (1.2),
then G ⊗ G ∼
= Z4p × Z4 × Z22 .
For the group of type (1.3), we consider the following cases:
Case 1: If p = 3, then G ⊗ G ∼
= (S3 × Z22 ) ⊗ (S × Z22 ) ∼
= Z6 × Z82 .
2
Case 2: If p ̸= 3, then G ⊗ G ∼
= (D2p × Z2 ) ⊗ (D2p × Z22 ) ∼
= Z2p × Z82 .
Next, for the group of type (1.4), G = H × Z2 , where ⟨a, b|a4 = bp = 1, a−1 ba =
−1
b ⟩. Then G ⊗ G = Z4p × Z32 . For the group of type (1.5), we have G′ ∼
= Zp and
∼
∼
1.2,
G
⊗
G
Z
×
((Z
×
Z
)
⊗
(Z
Gab ∼
Z
×
Z
and
by
using
Theorem
= p
= 4
4
2
4 × Z2 )) =
2
Z
3
Z4p × Z2 . The proof for the group of type (1.6) is similar with the group of type
(1.5). Now, Let G ∼
= D2p , then G ⊗ G ∼
= Z4p × Z32 . The proof can be found in [3].
241
Since the proof for the groups of type (1.8) and (1.9) are similar, we will only show
the proof for the group of type (1.8). For this group, G′ ∼
= Z2p and Gab ∼
= Z2 × Z2 ,
then G is a finite solvable group. By Theorem 1.5, |G ⊗ G| divides 24 · 2p, where
Gab ∼
= Z2 ×Z2 , Gab ⊗Gab ∼
= Z42 , G′ ∧G′ = 1, G′ ⊗Z[Gab ] I(Gab ) ∼
= Z2p and I(Gab ) is the
kernel of Z[Gab ] → Z. The exact sequence 1 → [G′ , Gφ ] → τ (G, G) → τ (Gab , Gab ) →
1 in Theorem 1.6 shows that |[G′ , Gφ ]| divides 2p. The commutative diagram shows
that (G ∧ G)/M (G) ∼
= G′ , that is, G ∧ G ∼
= Z4p and the epimorphism G ⊗ G → G ∧ G
shows that G ⊗ G has an element of order 4p. Since G ⊗ G is abelian, |G ⊗ G| divides
24 · 2p, |[G′ , Gφ ]| divides 2p and G ⊗ G has an element of order 4p. Then, we obtain
|G ⊗ G| = 24 · 2p and G ⊗ G ∼
= Z4p × M , where M is an abelian group of order 8.
For the group of type (1.10), by choosing n = p, m = 8 and l = 1, thus G is a
metacyclic group. Then by using Theorem 1.3, G ⊗ G ∼
= Z8p . The computation of
the nonabelian tensor square for the group of type (1.12) is similar to that of the
group of type (1.10).
Next, for the group of type (1.11), G′ ∼
= Zp and Gab ∼
= Z4 × Z2 and by using
3
∼
Theorem 1.4, we have G ⊗ G = Z4p × Z2 . The proof of G ⊗ G for the groups of types
(1.13) until (1.15) have been computed in [3].
Lastly, for the group of type (1.6), Gab ∼
= Z7 . We have |G⊗G| = |G||M (G)| = 56.
The epimorphism G ⊗ G → Gab ⊗ Gab ∼
= Z7 and exp(G ⊗ G) = 14, thus G ⊗ G ∼
=
2.
Z7 × Z32 ∼
Z
×
Z
= 14
2
Acknowledgement
The authors would like to acknowledge Ministry of Education (MOE) Malaysia and
Universiti Teknologi Malaysia (UTM) for the financial funding through the Research
University Grant (RUG) Vote No 04H13 and UTM Mobility Program. The first
author is also indebted to MOE Malaysia for MyPhD Scholarship.
References
[1] S. Rashid, N.H. Sarmin, A. Erfanian and N.M. Mohd Ali, On The Nonabelian
Tensor Square and Capability of Groups of Order p2 q, Arch. Math. 97(2011).
[2] A.M. Basri, N.H. Sarmin, N.M. Mohd Ali, J.R. Beuerle, On some metacyclic
p-groups and their nonabelian tensor square, Indian Journal of Science and Technology. 6 (2013), no.2, 67–70.
242
[3] R. Brown, D.L. Johnson, and E.F Robertson, Some computations of nonabelian
tensor products of groups, J. Algebra. 111 (1987), 177–202.
[4] S.H. Miah, On the isomorphism of group algebras of groups of order 8q, J. Lond.
Math. Soc. 2 (1975) no.9, 549–556.
[5] I.N. Nakaoka, Nonabelian tensor square of solvable groups, J. Group Theory. 3
(2007), 157–167.
[6] I.N. Nakaoka, N.R. Rocco, Nilpotent actions on nonabelian tensor products of
groups, Mat. Contemp. 21 (2001), 223–238.
243
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
Epicenter of Lie rings and the Lazard correspondence
Seiran Zandi
Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran
[email protected]
Abstract
Let G be a finite p-group of nilpotency class less than p − 1
and let L be the Lie ring corresponding to G via the Lazard
correspondence. We show that the epicenters of G and L are
isomorphic as abelian groups. Thus the group G is capable
if and only if the Lie ring L is capable.
1
Introduction
The Lazard correspondence establishes a natural equivalence between the category of
p-groups of nilpotency class less than p, and the category of Lie rings of prime power
order and nilpotency class less than p. For more details we refer the reader to [2]
and [4]. This means amongst other things that subgroups correspond to Lie subrings,
normal subgroups to ideals, and in general many properties transfer from the groups
to the Lie rings and vice versa.
A group G is capable if there exists a group H with H/Z(H) ∼
= G.Similarly, a
Lie ring L is capable if there exists a Lie ring K with K/Z(K) ∼
= L. Given a p-group
G of nilpotency class less than p and its associated Lie ring Lie(G), we show that
the epicenters of G and Lie(G) are isomorphic as abelian groups. Furthermore, the
group G is capable if and only if the Lie ring Lie(G) is capable.
2
Preliminary
The aim of this section is to collect some facts and results that will be applied in the
next section of the paper.
2010 Mathematics Subject Classification. Primary 20D15; Secondary 17B30, 20F40.
Key words and phrases. Lie rings, Epicenter of Lie rings, Lazard correspondence, p-groups.
244
A Lie ring L is an additive abelian group with a (not necessarily associative)
multiplication denoted by [·, ·] that satisfies the following properties:
• [x, x] = 0 for all x ∈ L. (Anti-commutativity)
• [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0 for all x, y, z ∈ L. (Jacobi identity)
• [x + y, z] = [x, z] + [y, z] and [x, y + z] = [x, y] + [x, z] for all x, y, z ∈ L.
(Bilinearity)
The product [x, y] is also called the commutator of x and y.
Given a Lie ring L and two subrings U and V , we define [U, V ] as the subring of
L generated by all commutators [u, v] with u ∈ U and v ∈ V . This allows to define
the lower central series L = L1 ≥ L2 ≥ L3 ≥ . . . via Li = [Li−1 , L]. The Lie ring
L is nilpotent if this series terminates at {0}. In this case, the class c(L) is the
length of the lower central series of L. The center of L is Z(L) = {x ∈ L | [x, y] =
0 for all y ∈ L}.
In a group G we denote the multiplication with g · h (or just gh for short) and
the commutator with Jg, hK = g −1 h−1 gh. Generalized commutators are right-normed
throughout our paper; that is
[x, x, y] = [x, [x, y]] and Jx, x, yK = Jx, Jx, yKK.
Definition 2.1. Let G be a group. We define the epicenter Z ∗ (G) of G as the
smallest normal subgroup of G such that G/Z ∗ (G) is capable. Hence a group G is
capable if and only if Z ∗ (G) is trivial.
It is clear that Z ∗ (G) ≤ Z(G). So Z ∗ (G) is abelian. Similarly, we define the
epicenter of Lie rings.
Definition 2.2. Let L be a Lie ring. We define the epicenter Z ∗ (L) of L as the
smallest ideal of L such that L/Z ∗ (L) is capable.
Proposition 2.3. Let G be a finite p-group of class less than p and let L be its
Lazard correspondent. Let X be a subset of G and hence of L.
1. The subring of L generated by X and the subgroup of G generated by X
coincide as sets and are in Lazard correspondence.
2. The ideal of L generated by X and the normal subgroup of G generated by X
coincide as sets and are in Lazard correspondence.
Proposition 2.4. Let G be a finite p-group of class less than p and let L be its
Lazard correspondent.
245
1. Z(G) and Z(L) coincide as sets and are in Lazard correspondence. Hence
Z(G) and Z(L) are isomorphic as abelian groups.
2. G′ and L2 coincide as sets and are in Lazard correspondence.
Finally, we also get a correspondence between the quotients.
Proposition 2.5. Let G be a finite p-group of class c < p and let L be its Lazard
correspondent. Let G0 be a normal subgroup in G and L0 the corresponding ideal
in L. Then ψ : G/G0 → L/L0 : xG0 7→ x + L0 is a well-defined bijection and it
induces the Lazard correspondence between G/G0 and L/L0 .
3
Main results
In [1] it is proved that the epicenter Z ∗ (G) of a group G is the intersection of all
normal subgroups N of G so that G/N is capable. Thus Z ∗ (G) ≤ Z(G) holds. This
translates directly to Lie rings. We first prove the following preliminary statement.
Lemma 3.1.
1. Let G be a finite p-group which is capable. Then there exists a finite p-group
H with H/Z(H) ∼
= G.
2. Let L be a finite capable Lie ring with pn elements. Then there exists a finite
Lie ring K with K/Z(K) ∼
= L and K has pm elements for some m ≥ n.
Note that a group G is capable if and only if Z ∗ (G) = {1} and similar for a Lie
ring.
Theorem 3.2. Let G be a finite p-group of class less than p − 1 and let L = Lie(G).
Then Z ∗ (G) and Z ∗ (L) are isomorphic as abelian groups.
Proof. By Lemma 3.1 there exists a finite Lie ring K of p-power order so that
K/Z(K) ∼
= L/Z ∗ (L). By Proposition 2.3 the group G has a normal subgroup N so
that G/N corresponds to L/Z ∗ (L) via the Lazard correspondence. As c(K) < p, we
can form H = Grp(K). This group H satisfies Z(H) = Z(K). Further, we obtain
that H/Z(H) and K/Z(K) are Lazard correspondents. Thus H/Z(H) ∼
= G/N .
∗
∗
∗
Hence Z (G) ≤ N and |Z (G)| ≤ |Z (L)| = |N |. Similarly, one can show that
|Z ∗ (L)| ≤ |Z ∗ (G)|. Thus Z ∗ (G) = N and Z ∗ (G) ∼
= Z ∗ (L) as abelian groups.
246
References
[1] F. R. Beyl, U. Felgner and P. Schmid, On groups occurring as center factor
groups, J. Algebra. (1) 61 (1979), 161–177.
[2] S. Cicalò, W. Graaf and M. R. Vaughan-Lee, An effective version of the Lazard
correspondence, J. Algebra, (1) 352 (2012), 430–450.
[3] C. C. Sims, Computation with finitely presented groups, Encyclopedia of Mathematics and its Applications, Cambridge. Press, 1994.
[4] E. I. Khukhro, p-automorphisms of finite p-groups, London Mathematical Society
Lecture Note Series 246, Cambridge University Press, Cambridge, 1998.
247
The Extended Abstract of
The 6th International Group Theory Conference
12–13 March 2014, Golestan University, Gorgan, Iran.
A generalization of Mohres’s Theorem on groups with
all subnormal subgroups
Mohammad Zarrin
Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj, Iran
[email protected], [email protected]
Abstract
Möhres’s Theorem says that an arbitrary group whose proper
subgroups are all subnormal is solvable. Here we generalize
Möhres’s Theorem, by proving that every group with at most
56 non-subnormal subgroups is solvable. Also we show that
the derived length of a solvable group with a finite number k
of non-n-subnormal subgroups is bounded in terms of n and
k.
1
Introduction and results
Let G be a group. A subgroup H of G is said to be subnormal in G, if there exists
a finite series of subgroups of G, such that
H = H0 ▹ H1 ▹ . . . ▹ Hk = G.
If H is subnormal in G, then the def ect of H in G is the shortest length of such a
series. We shall say that a subgroup H of G is n-subnormal if H is subnormal of
defect at most n.
In a famous paper of 1965, J.E. Roseblade [8] proved that a group with all subgroups n-subnormal is nilpotent of class µ(n), where µ(n) is a function that depends
only on n (we can take µ(n) to be best possible for all n), and so it is solvable of
class at most
[log2 (µ(n))] + 1.
(⋆)
2010 Mathematics Subject Classification. 20E15.
Key words and phrases. Norm; Subnormal subgroup; Solvable group.
248
However, µ(n) is not explicitly given in [8] and the exact values are only known for
n ∈ {1, 2}. In fact, results of Heineken [2] and Mahdavianary [6] state that a group
with all cyclic subgroups 2-subnormal is nilpotent of class not exceeding 3. As a
corollary of this result, it follows that µ(2) ≤ 3. Moreover, it follows from [10] that
this bound is sharp. (In case n = 1, G is a Dedekind group and so µ(1) ≤ 2.)
In 1990 Möhres [7] proved that a group with all subgroups subnormal (or a
group without non-subnormal subgroups) is solvable. In this paper we study groups
with finitely many non-subnormal subgroups. Let k be a non-negative integer. We
say that a group G is an Rn (k)-group (M(k)-group, resp.) if G has exactly k nonn-subnormal (non-subnormal, resp.) subgroups. We note that (for fixed k and n)
Rn (k) is contained in M(l) for some l less than or equal to k. To prove this, let
n, k be fixed and G be in Rn (k). Then G has exactly k non-n-subnormal subgroups,
so certainly G has at most k non-n-subnormal subgroups, so G has at most k nonsubnormal subgroups and so G has exactly l non-subnormal subgroups for some l
less than or equal to k and so G lies in M(l) for some l less than or equal to k.
Clearly the Rn (k)-groups G with k = 0 are those in which all subgroups are nsubnormal, and the M(k)-groups G with k = 0 are those with all subgroups subnormal. We note that if G is an Rn (k)-group (or M(k)-group), then k ̸= 1. This is
because every conjugate of a non-n-subnormal subgroup is also a non-n-subnormal
subgroup. Therefore in considering Rn (k)-groups (or M(k)-groups) we may assume
that k ≥ 2.
According to (⋆), the derived length of a solvable Rn (0)-group is ≤ [log2 (µ(n))]+
1. Here (see also [12]) we obtain a result which is of independent interest, namely,
the derived length of solvable Rn (k)-groups is bounded in terms of n and k (2 ≤ k).
Theorem 1.1. Let G be an arbitrary solvable Rn (k)-group and d be the derived
length of G . Then d ≤ [log2 (µ(n))] + k, where µ(n) is the function of Roseblade’s
theorem.
Also, we generalize Möhres’s Theorem as follows (note that Möhres’s Theorem
says that every M(0)-group is solvable).
Theorem 1.2. Suppose that G is an M(k)-group (Rn (k)-group) for some n with
k ≤ 56. Then G is solvable.
2
Proofs
If G is an arbitrary group, the norm B1 (G) of G is the intersection of all the
normalizers of subgroups of G (in fact, B1 (G) is the intersection of all the normalizers
249
of non-1-subnormal subgroups of G). This concept was introduced by R. Baer, and
it is well-known ([5] and [9]) that Z(G) ≤ B1 (G) ≤ Z2 (G) . Now we define Bn (G)
as the intersection of all the normalizers of non-n-subnormal subgroups of G, i.e.,
∩
Bn (G) =
NG (H),
H∈Rn (G)
{
where Rn (G) = H | H is a non-n-subnormal subgroup of G} (with the stipulation that Bn (G) = G if all subgroups of G are n-subnormal). Clearly Bi (G) ≤
Bi+1 (G). Moreover, in view of the proof of Theorem A, below, we can see that
Bn (G) is a nilpotent normal subgroup of G of class ≤ µ(n),
(⋆⋆)
where µ(n) is the function of Roseblade’s theorem. In fact Bn (G) is a substantial
generalization of the norm B1 (G).
Proof of Theorem 1.1. The group G acts on the set
{
Rn (G) = H | H is a non-n-subnormal subgroup of G}
by conjugation. By assumption |Rn (G)| = k (note that k ≥ 2). Now the subgroup
Bn (G) is the kernel of this action, so Bn (G) is normal in G and G/Bn (G) ↩→ Symk .
But it is surely well known that the derived length of every solvable subgroup of
the symmetric group Sn of degree n (n ≥ 1) is at most n − 1. Hence G/Bn (G) has
derived length ≤ k − 1. Therefore to complete the proof it is enough to show that
Bn (G) is solvable of class at most [log2 (µ(n))] + 1. To see this, according to the
main result in [8], it is enough to show that every subgroup of Bn (G) is n-subnormal.
Suppose on the contrary that there exists a non-n-subnormal, say H of Bn (G). It
follows that H is a non-n-subnormal of G and so, by definition of Bn (G), we obtain
that H ▹ Bn (G), which is impossible. Hence Bn (G) is a nilpotent group of class at
most µ(n) and so it is solvable of class at most [log2 (µ(n))] + 1. This completes the
proof.
Combining the results quoted in the introduction and the above Theorem, we
obtain a nice corollary as follows:
Corollary 2.1. Let G be a solvable R2 (k)-group, d the derived length of G and
k ≥ 2. Then d ≤ k + 1.
Remark 2.2. In view of the proof of Theorem A, we can see that if G is an arbitrary
group with a finite number k of non-n-subnormal subgroups, then the factor group
G
Bn (G) is finite and
G
|
|≤ k!.
Bn (G)
250
This result suggests that the behaviour of non-n-subnormal subgroups has a strong
influence on the structure of the group.
Remark 2.3. The Wielandt subgroup W (G) of a group G is defined to be the
intersection of all the normalizers of subnormal subgroups of G; this concept is
′
naturally analogous to the norm of a group. Also we shall denote by W (G) the
intersection of all the normalizers of non-subnormal subgroups of G. Now by a
similar argument as in the proof of Theorem A, mentioned for Bn (G), we can see
′
that every subgroup of W (G) is subnormal and so, by Möhres’s Theorem,
′
W (G) is a solvable normal subgroup of G.
Moreover, if G is an arbitrary group with finitely many, k of non-subnormal subgroups, then the factor group W ′G(G) is finite and
|
G
|≤ k!.
W ′ (G)
In the sequel, we want to prove Theorem B.
Lemma 2.4. Let G be an M(n)-group and H ≤ G, then H is an M(t)-group for
some t ≤ n.
Lemma 2.5. Let G be an M(t)-group, K a normal subgroup of G,
K ∈ M(m). Then t ≥ m + n.
G
K
∈ M(n) and
For any prime power q, we denote by Ln (q) and Sz(q), respectively, the projective special linear group of degree n over the finite field of size q and the Suzuki
group over the field with q elements. If G is a finite group, then for each prime
divisor p of |G|, we denote by vp (G) the number of Sylow p-subgroups of G.
References
[1] D. J. Garrison and L.-C. Kappe, Metabelian groups with all cyclic subgroups
subnormal of bounded defect, in proccedings of infinite groups 1994 (walter de
gruyter, 1996), PP. 73-85.
[2] H. Heineken, A class of three-Engel groups, J. Algebra 17 (1971), 341-345.
[3] B. Huppert and N. Blackburn, Finite groups, III (Springer-Verlag, New York,
1982).
251
[4] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967.
[5] W. P. Kappe, Die A-Norm einer Gruppe, Illinois J. Math. 5 (1961), 187-197.
[6] S. K. Mahdavianary, A special class of three-Engel groups, Arch. Math. (Basel)
40 (1983), 193-199.
[7] W. Möhres, Auflösbarkeit von Gruppen, deren Untergruppen alle subnormal
sind, Arch. Math. (Basel) 54 (1990), 232-235.
[8] J. E. Roseblade, On groups in which every subgroup is subnormal, J. Algebra
2 (1965), 402-412.
[9] E. Schenkman, On the norm of a group, Illinois J. Math. 4 (1960), 150-152.
[10] M. Stadelmann, Gruppen, deren Untergruppen subnormal vom Defekt zwei
sind, Arch. Math. 30 (1978), 364-371.
[11] J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are
solvable (Part I), Bull. Amer. Math. Soc. (NS) 74 (1968), 383-437.
[12] M. Zarrin, Non-subnormal subgroups of groups, Journal of Pure and Applied
Algebra 217 (2013) 851-853.
252
Index
Abdollahi*, A., 28
Abdul Hamid*, M., 31
Aghabozorgi*, G. H., 35
Aghajari*, Z., 193
Ahanjideh, N., 50, 54, 157
Ahmadidelir*, K., 40
Akbarossadat*, S. N., 207
Alijani*, A. A., 220
Alizadeh Sanati, M., 46
Araskhan, M., 116
Asadian*, B., 50
Asgary*, S., 54
Ashrafi, A. R., 236
Babaei*, E., 58
Badrkhani Asl*, M., 123
Bahlekeh*, A., 63
Bibak*, M., 67
Bibak, M., 224
Davvaz, B., 35
Ercan*, G., 2
Erfanian, A., 31, 141
Faramarzi Salles*, A., 70
Farhami, N., 96
Farrokhi D. G.*, M., 73
Farrokhi D. G., M., 141, 212
Foroudi Ghasemabadi, M., 178
Ghanei, F., 119
Gholamian*, A., 76
Gholamian, A., 161
Ghoraishi*, S. M., 80
Ghorbany*, M., 84
Guloglu*, İ. Ş., 3
Hassanzadeh*, M., 181
Hatamian, R., 105
Hokmabadi, A., 88
Hoseini Ravesh, M., 127
Iranmanesh, A., 178
Jafari*, S. H., 92
Jafarpour, M., 35
Jahandideh*, M., 96
Johari*, F., 101
Kaheni*, A., 105
Kahkeshani, R., 109
Kayvanfar*, S., 9
Kayvanfar, S., 105
Kazemi Esfeh, H., 96
Khoddami*, A. R., 113
Khosravi*, H., 116
Khosravi, H., 70
Kuzucuoğlu*, M., 16
Mahdipour*, Z., 46
Mashayekhi, B., 165, 174
Mashayekhy, B., 233
Mirdamadi*, S. E., 196
Mirebrahimi*, H., 119
Mirebrahimi, H., 165
Moghaddam, M. R. R., 123, 127, 132, 136
Mohammadian*, A., 141
Mohammadzadeh*, E., 88
Mohammadzadeh*, H., 144
Mohammadzadeh, F., 88
Mohd Ali, N. M., 31, 169, 239
Mousavi, H., 149
Najafi*, A., 153
Najafi*, M., 157
Nasrabadi*, M. M., 161
253
Nasrabadi, M. M., 76
Nasri*, T., 165
Nawi*, A. A., 169
Nawi, A. A., 239
Niroomand, P., 101
Zainal*, R., 239
Zainal, R., 169
Zamani, Y., 58
Zandi*, S., 244
Zarrin*, M., 248
Zenkov, V., 149
Pakdaman*, A., 174
Pakdaman, A., 233
Parvizi Mosaed*, H., 178
Parvizi, M., 101
Pourmirzaei, A., 181
Rashid, S., 169, 239
Rasouli, H., 153
Rastgoo*, T., 149
Razzaghmaneshi*, B., 185, 189
Rezaeezadeh, G., 67, 193, 196, 224
Rezaei*, M., 199
Robati*, S. M., 203
Rostamyari*, A., 132
Sadeghifard*, M. J., 136
Saeedi*, F., 212
Saeedi, F., 73, 207
Saeidi*, A., 230
Safa*, H., 217
Saffarnia*, S., 127
Sahleh, H., 220
Sajjadi*, M., 224
Sajjadi, M., 67
Salehi Amiri*, S. S., 227
Sarmin, N. H., 31, 169, 239
Shadab, M., 230
Shahryari*, M., 20
Shum*, K. P., 26
Torabi*, H., 233
Torabi, H., 174
Yazdany Moghaddam*, M., 109
Yousefi*, M., 236
254