A Conference in Honor of
Adalbert Bovdi's
70th Birthday
November 18-23, 2005
Abstracts
Debrecen, Hungary
About coreflective subcategories of the
category of topological modules
Alina Alb
University of Oradea,
Oradea, Romania
[email protected]
We give in this paper some examples of coreective subcategories of the
category of topological modules. Denote by R-TopMod the category of all
topologically left R-modules over a xed topological ring R with identity.
We give conditions on a ring R under which the subcategory of R-TopMod
whose underlying space is a P-space is coreective.
Radical and local rings with Engel conditions
B. Amberg
University of Mainz,
Mainz, Germany
[email protected]
In this talk, the relations between a radical ring and its adjoint group and
between a local ring and its multiplicative group will be discussed.
The set of all elements of an associative ring R forms a semi-group with
neutral element 0 under the operation r ◦ s = r + s + rs for all r, s ∈ R. The
group of all invertible elements of this semi-group is called the adjoint group
of R and is denoted by R◦ . Following Jacobson R is said to be radical if
R = R◦ , which means that R coincides with its Jacobson radical. Obviously
such a ring does not contain an identity element for multiplication. If a ring
R is embedded in any way in a ring R1 with identity, then R◦ is isomorphic
with the subgroup 1 + R of the group of units of R1 .
2
There are many relations between the (Lie) ring-theoretical properties of
the radical ring R and the group-theoretical properties of its adjoint group
R◦ . It can for instance be shown that for a radical ring R, the adjoint group
R◦ satises an n-Engel condition for some positive integer n, if and only if
R is m-Engel as a Lie ring for some positive integer m depending only on n.
Similar connections hold for the Lie structure of a local ring and its multiplicative group. Some of these can even be extended to semi-local rings.
Topological rings with additive linearly
compact groups
Loriana Andrei
University of Oradea,
Oradea, Romania
[email protected]
We will expose some properties of topological rings whose additive group
is linearly compact. It will be presented the structure of simple topological
rings with this property, their decomposition in products of p-rings, examples.
3
Metaideals in commutative rings
Ryszard R. Andruszkiewicz
University of Bialystok,
Bialystok, Poland
[email protected]
A subring A of a ring R is said to be a metaideal in R if there exists a
family M S
of subrings of R including A such that for every non-empty chain
X ⊆ M, X∈X X ∈ M, and for every X ∈ M with X 6= R, there exists
Y ∈ M such that X is an ideal of Y and Y 6= X .
The notion of a metaideal was introduced by Baer in [2], where fundamental properties and examples of metaideals were given. Later, it was
considered by Friedman in [4]. In [5] Krempa and Stankiewicz analysed radical properties of metaideals. Metaideals of nite index (also called accessible
subrings) have found many applications in the radical theory (cf.[3]) and
especially in solving the problem of stabilization of the Kurosh's chains (cf.
[1]). They are also interesting because of on analogy to subnormal subgroups
of groups.
In this talk new examples of metaideals in associative, commutative rings
are constructed. It is proved that: metaideals of a commutative ring form a
sublattice of the lattice of all subrings and for any subring A of a commutative
ring P there exists the largest subring M idP (A) (called metaidealizer) in
which A is a metaideal. Metaidealizers in several cases are described.
References
[1] R. R. Andruszkiewicz and E. R. Puczyªowski, Accessible subrings and
Kurosh's chains of associative rings, Algebra Colloquium 4:1 (1997),
79-88.
[2] R. Baer, Metaideals, Publication 502, Linear Algebras, 33-52, National
Academy of Sciences-National Reserch Council (1957).
[3] K. I. Beidar, On essential extensions, maximal essential extensions and
iterated maximal essential extensions in radical theory, Colloq. Math.
Soc. János Bolyai 61 (North-Holland, 1993), 17-26.
4
[4] P. A. Friedman, Rings with an idealizer condition, Izv. Vyst. Uchebnykh
Zaved.,15 (1960), 213-222 (in Russian).
[5] J. Krempa and E. Stankiewicz, Radicals of metaideals,
Acad.Polon.Sci.,Vol.XXII, no. 4 (1974), 359-365.
Bull.
On the Automorphism Group of Fullerene
Graphs
A. R. Ashra
University of Kashan,
Kashan, Iran
[email protected]
It is well known to associate an Euclidean graph to a molecule. Balasubramanian computed the Euclidean graphs and their automorphism groups for
benzene, eclipsed and staggered forms of ethane and eclipsed and staggered
forms of ferrocene, see [1].
In this talk, we present an algorithm, which is useful for computing symmetry of fullerenes. Using this algorithm, a new simple method is described,
by means of which it is possible to calculate the automorphism group of
Euclidean graph of fullerene graphs. We apply this method to compute the
symmetry of some big fullerenes, as C500 and C720 .
References
[1] K. Balasubramanian, Graph-Theoretical Perception of Molecular Symmetry, Chem. Phys. Letters, 232(1995), 415-423.
[2] A.R. Ashra, On symmetry properties of molecules, Chem. Phys. Letters, 406(2005), 75-80.
5
GROUPS ASSOCIATED WITH NEAR-RINGS
O.D. Artemovych and I.I. Kravets
Crakow University of Technology,
Cracow, Poland
[email protected]
Let N be a left near-ring with the identity element 1 with two operations
+" and ·" and 0 be the zero of the additive group N + . It is well known that
U (R) = {a ∈ N |a is invertible in N } is a group under the multiplication
·". Dene on the set of pairs
B(I, T ) = { (x, y) | x ∈ I, y ∈ T }
the algebraic operation by the rule
(x, y)(u, v) = (yu + x, y · v).
(*)
Then B(I, T ) is a group with the identity element (0, 1) under the operation given by the rule (∗) and, moreover, B(I, T ) = A o B , where A =
{(x, 1)|x ∈ I} is isomorphic to additive group of I and B = {(0, y)|y ∈ T } is
isomorphic to T .
We study the properties of B(I, T ) and their relations with N .
6
On sumsets of geometric progressions
A. Bérczes
University of Debrecen,
Debrecen, Hungary
[email protected]
We consider geometric progressions of complex numbers where the common ratio is not a root of unity. If the set of elements of the progression
consists of n elements, then in general, its sumset (that is the set of the
sums of two distinct elements from this progression) has cardinality n(n−1)
.
2
We describe all cases when the cardinality of the sumset is strictly less than
n(n−1)
.
2
An example of not Engel group generated by
Engel elements
Vasily Bludov
Irkutsk State Teacher Training University,
Irkutsk, Russia
[email protected]
A problem "Does a set of all Engel elements consist a subgroup" is very
old. We do not know all the history of this problem. One can nd a mentioning of the problem in Plotkin's book [3]. Here we consider unbounded
left Engel elements (or nil-elements in terms of [3]). In some partial cases a
set of all Engel elements forms the locally nilpotent radical (see R. Baer [5],
B.I. Plotkin [4], and also [3]). In general, an Engel group may not be locally
nilpotent (E.S. Golod [1]).
In our report we show an example of a not Engel group generated by
Engel elements. In particular, a couple of Engel elements which product is
not Engel will be presented. To construct the example we use the wreath
7
product G = H o D, where H = H(t, a, b, c) is 2-group of R.I. Grigorchuk [2]
and D is dihedral group of order 8, D = hd1 , d2 | d21 = d22 = (d1 d2 )4 = ei.
Thus G is 2-group generated by involutions t, a, b, d1 , d2 It is well known
that every involution in any 2-group is Engel. So G is generated by Engel
elements. Let d = d1 d2 . Consider a subgroup H hdi = He × Hd × Hd2 × Hd3
and an element h = (e, ta, bt, c) ∈ H hdi . Using properties of group H we
show that [h, d, . . . , d] 6= e nevertheless the length of the commutator. So G
is not an Engel group.
These researches were supported by RFBR, grant No 03-01-00320
References
[1] E. S. Golod, Some problems of Burnside type, Proc. International
Congress Math. Moscow 1968, p. 284289 (Russian).
[2] R.I. Grigorchuk, On Bernside's problem on periodical groups, Functional
analysis and applications, 1980, 14, 1, p. 5354 (Russian).
[3] B.I. Plotkin, Groups of automorphisms of algebraic systems, translated from the Russian by K. A. Hirsh, Wolters-NoordHo, Groningen,
1972.
[4] B.I. Plotkin, A radical and nil elements in groups, Izvestia vyshykh uchebnykh zavedenii. Matematika, 1 (1958), p. 130135 (Russian).
[5] R. Baer, Engelsche Elemente Noetherscher Gruppen, Mat. Ann. 1957,
133, p. 256270.
8
Some unexpected new results on direct
decompositions of generalized lattices
M.C.R. Butler
University of Liverpool,
Liverpool, United Kingdom
[email protected]
In this talk I will describe some (to me) unexpected new results about
direct sum decompositions of generalized lattices over a separable R-order,
Λ, where R denotes a Dedekind domain; a motivating example is the integer
group ring ZG of a nite group G over the ring of integers. A generalized lattice is dened to be an R-projective Λ-module, and the results I will describe
concern orders such that every generalized lattice is F(ully) D(ecomposable)
into a direct sum of (nitely generated) lattices - such an order is said to
have the property (FD). Various examples of orders with (FD) were given by
Campbell, Kovacs and myself in a paper in Archiv der Mathematik last year;
these had only nitely many dierent isoclasses of indecomposable lattices,
and included the group rings ZG with G of prime order, any maximal order,
and any such R-order with R a complete discrete valuation domain. Now,
in the July issue of the Proc. of the London Math. Soc., Wolfgang Rump
has given a deep and fascinating nite combinatorial procedure for deciding
whether an order has the property (FD). I will describe this procedure, and
illustrate it by showing that a group ring ZG with |G| = 6 has the property
(FD) if and only if G is non-abelian!
9
On Thompson's Conjecture and Related Topics
Guiyun Chen
Guiyun Chen
Southwest University,
China
[email protected]
This report is about progress of rersearch on Thompson's Conjecture and
related topics made by the team led by the author.
A decomposition of finite indexed languages for
multiplicative Kleene algebras
Tibor Csáki, Benedek Nagy
1
University of Debrecen,
Debrecen, Hungary
[email protected]
Let L be a subset of the free monoid over the nite set Σ. Moreover,
let Rc denote the syntactic monoid of L. We show that the following two
statements are equivalent.
1) The index of L with respect to Rc is nite.
2) L is a union of nitely many sets of multiplicative Kleene algebras
(Σ, ·, ∗, 0, 1).
1
The research is supported by the grant OTKA T049409.
10
INNER MAPPING GROUPS AND NILPOTENCY
CLASSES
Piroska Csörg®
Eötvös Loránd University,
Budapest, Hungary
[email protected]
By T. Kepka and M. Niemenmaa if the inner mapping group of a nite
loop Q is abelian, then the loop Q is nilpotent. For a long time there was no
example of a nilpotency degree greater than two. In the nineties T. Kepka
raised the following problem: whether every nite loop with abelian inner
mapping group is centrally nilpotent of class at most two? For many years
the prevailing opinion has been that all such loops have to be of nilpotency
degree two. The converse is always true by Bruck, i.e. the nilpotency class
two of a loop Q implies the inner mapping group I(Q) is abelian. After
describing the problem in terms of transversals I tried to characterize by
means of group theory the counterexample of minimal order. I expected to
nd enough properties of the counterexample that would refute its existence.
By using these results, supposing special properties I choose some parameters
and nally I constructed a counterexample loop Q of order 27 , such that the
multiplication group M (Q) is of order 213 , the inner mapping group I(Q) is
elementary abelian of order 26 , for the normal closure L of I(Q) in M (Q), L
is of order 210 and the factor group M (Q)/L is elementary abelian of order
23 , furthermore the nilpotency class of this loop Q is greater than two.
11
On spectra of abelian Group Rings
M. Dokuchaev, A. Gimenez Bueno
University of Sao Paulo,
Sao Paulo, Brazil
[email protected]
We study spectra of integral group rings of nitely generated abelian
groups from the scheme-theoretic viewpoint. We describe the decomposition
of Spec Z[G] into irreducible components, their intersections and singular
points. We also determine the formal completion of Spec Z[G] at a singular
point.
Some results and problems on primitive words
Pál Dömösi
University of Debrecen,
Debrecen, Hungary
[email protected]
A word is primitive if it is not a power of another word. A well-known
unsolved problem of theoretical computer science is whether the language of
all primitive words over a nontrivial alphabet is context-free or not. Among
others, this (in)famous problem motivates the study of the combinatorial
properties of primitive words. In addition, they have special importance in
studies of automatic sequences. The Lyndon-Schützenberger and the ShyrYu Theorems are well-known classical results in this direction. The known
proofs of these famous results are more or less involved. The aim of this
talk is to show new simple proofs of these well-known theorems. Some open
problems are also discussed.
12
Constructing Holonomy Groups for
Krohn-Rhodes Theory
Attila Egri-Nagy2 and Chrystopher L. Nehaniv
University of Hertfordshire,
Hertfordshire, United Kingdom
[email protected]
The holonomy decomposition method for the Krohn-Rhodes Theory works
by the detailed study of how the characteristic semigroup S of an automaton
(A, X, δ) acts on certain subsets of the state set A. It looks for groups induced
by S 1 permuting some set of these subsets of A. These groups are called the
holonomy groups and they are the building blocks for the components of the
Krohn-Rhodes holonomy decomposition.
Finding the holonomy groups is a computationally challenging problem
and exhaustive search is not feasible due to the potentially huge number
of transformations in the characteristic semigroup of the automaton. We
present here a method based on the hierarchical dependency functions of the
wreath product. Our algorithm gives such a generator set for each holonomy
group that is in practice comparable to the generator set of the original
transformation semigroup in terms of size. We also briey show how and to
what extent the holonomy group structure is determined by the set of subsets
it acts on.
2 This
work was supported by a Hungarian National Foundation for Scientic Research
grant (OTKA T049409) and the University of Hertfordshire Algorithms Research Group.
13
Deformations of Lie algebras
Fialowski Alice
Eötvös Loránd University,
Budapest, Hungary
[email protected]
Deformation theory is a useful tool in considering invariants of a given
object. Namely, it describes the local neighbourhood in the variety of the
considered algebraic or analytic objects. In my talk I will show how deformation theory can be used to describe the moduli space of low dimensional
Lie algebras.
Loops which are semidirect products of groups
Ágota Figula
University of Erlangen,
Erlangen, Germany
[email protected]
This contribution is a report on joint work with K. Strambach, Erlangen.
In [1], [2] constructions of proper loops are discussed which are semidirect
products of groups. Whereas in [1] there are few constructions of such loops
in [2] a general theory for loops which are semidirect products of groups is
developed.
In this talk we show that a wide class of proper loops L can be represented
within the group of anities of an ane space A of dimension 2n over a
commutative eld K. They are semidirect products of groups of translations
of A by suitable subgroups of GL(2n, K). For many of them we may take as
elements ane n-dimensional transversal subspaces of A.
14
To realize our examples it is important to know the eigenvalues for certain
products of matrices in GL(n, K).
If the eld K is a topological eld then we obtain topological loops, for
real or complex numbers the constructed loops are smooth. The groups
topologically generated by the left translations of these smooth proper loops
are Lie groups, whereas the groups generated by right translations are smooth
groups of innite dimension. We determine also the Akivis algebras of these
smooth loops and we show that they are semidirect products of Lie algebras.
References
[1] G. F. Birkenmeier, C. B. Davis, K. J. Reeves, S. Xiao, Is a semidirect
product of groups necessarily a group?, Proc. Amer. Math. Soc. 118(3)
(1993), 689-692.
[2] G. F. Birkenmeier, S. Xiao, Loops which are semidirect products of
groups, Communications in Algebra, 23(1) (1995), 81-95.
The free product of cyclic groups of order two
and Bol loops of exponent two
Alexander Grishkov
University of Sao Paulo,
Sao Paulo, Brazil
[email protected]
The most interesting classes of loops are the classes of Moufang loops,
Bol loops and di-associative loops. Recall that a loop is a set L with one
binary operation (.) such that for any a, b ∈ L there exist unique x, y ∈ L
such that a.x = b and y.a = b, moreover, there exists e ∈ L such that
e.c = c.e = c for any c ∈ L. A loop L is a Moufang loop if it satises the
following identities ((z.x).y).x = z.((x.y).x) and x.(y.(x.z)) = (x.(y.x)).z. If
L satises only the rst identity it calls the (right) Bol loop. Analogously
15
we dene a (right) Bol semiloops as a set with one binary operation and
identity ((z.x).y).x = z.((x.y).x). In this paper we construct a free Bol loop
of exponent n and describe the Bol loops of exponent 2 and nilpotent class 2.
As corollary we proved that any Bol loop of exponent 2 and nilpotent class 2
may be embedded in a right alternative algebra over a eld of characteristic
2. In the last section we proved that the free Bol loop of exponent 2 with
two generators has the automorphism group isomorphic to free product of
three cyclic groups of order two.
We note that the study of free Bol loop of exponent two with two generators is important since one of the most interesting open questions in the
loop theory is the question of existence of simple nite Bol loops which is not
Moufang loops. It is easy to see that any simple non-abelian nite Bol loop
of exponent 2 (if it exists!) is not Moufang loop. G.Nagy [1] proved that
if there exists simple non-abelian nite Bol loop of exponent 2 than there
exists one with two generators. We note that this question is not trivial (see
[2]).
References
[1] G. Nagy, On the tangent algebra of algebraic commutative Moufang
loops. Mathematica 45(68) (2003), no. 2, p.147160
[2] Ashbaher, On Bol loops of exponent 2. J. Algebra 288 (2005), no. 1,
p. 99136
16
ON REPRESENTATION OF FINITE p-GROUPS
OVER
INTEGRAL DOMAINS
P. M. Gudivok
University of Uzhgorod,
Uzhgorod, Ukraine
[email protected]
Let K be an integral domain of characteristic zero, which is not eld.
A nite group G is called wild over the ring K , if the description of nonequivalent matrix K -representations of the group G includes the problem of
the classication up to similarity of pairs of n × n-matrices over some eld
for an arbitrary natural n.
The problem of the wildness of a nite p-group over the ring K have
been solved [15], if K is a complete discrete valuation ring or K is the ring
of formal power series in m indeterminate with coecients from complete
discrete valuation ring.
We have obtain the next results.
Theorem. Let G be a nite p-group of order |G| and K be a noetherian
integral domain of characteristic zero, K ∗ be the multiplicative group of the
ring K and p ∈
/ K ∗ . The group G is wild over the ring K if one of the
following condition holds:
1) G is a non-cyclic p-group and p 6= 2;
2) G is the cyclic p-group of order |G| = pr (r > 2, p 6= 2);
3) G is a non-cyclic 2-group of order |G| > 4;
4) G is the cyclic 2-group of order |G| > 8;
5) G is a non-cyclic 2-group or a cyclic 2-group of order |G| > 4, K
is local ring with residue class eld of characteristic 2, Rad K 6= 2K
(Rad K is the Jacobson radical of the ring K);
6) G is the cyclic p-group of order p2 , K is local factorial ring, which is
not discrete valuation ring.
17
References
[1] P. M. Gudivok, On representations of nite groups over discrete valuation
ring, Trudy Mat. Ins. Akad. Nauk SSSR 148 (1978), 96105.
[2] P. M. Gudivok, Integral representations of nite groups and the matrix
pair problem, Materials Twenty-ninth Sci. Conf. Professors and Instructors, Math. Section, Uzhgorod. Gos. Univ., Uzhgorod, 1975. Manuscript
no. 705-76, deposited at VINITI, (1976), 231240.
[3] E. Dieterich, Group rings of wild representations type, Math. Ann. 266
(1983), 122.
[4] P. M. Gudivok, V. M. Orosz, A. V. Roiter, On representations of nite pgroups over the ring of formal power series with integer p-adic coecients,
Ukr. Mat. Z. 44 (1992), 753765.
[5] V. M. Bondarenko, P. M. Gudivok, On representations of nite p-groups
over the ring of formal power series with integer P -adic coecients, Innite groups and concerning algebraic structure, Ins. Mat. Akad. Nauk
Ukraini, Kyiv, 1993, 514.
Sylow Objects in Finite Groups and Formations
Wenbin Guo
Jiangxi Normal University,
Nancheong, China
[email protected]
It is well known that the classic Sylow theorem is the most important
result of groups and has numerous applications. In particular, we should
mention that Sylow objects such as p-subgroups and their normalizers have
played an important role in the problems of classication of nite simple
groups.
Within the framework of the theory of formations, Sylow objects played
an important role, too. Remenber that if a nite group G belongs to a
18
saturated formation F and G has a composition factor of order p, then the
class Np of all nite p-groups is contained in F. Analogy to Sylow subgroups
is seen here, therefore the formations of the Np type can be called Sylow
objects in the theory of formations.
In this report, we give a brief introduction on some of the new research
along the two directions.
Computational aspects of the first Zassenhaus
conjecture
Christian Höfert
Universität Stuttgart,
Stuttgart, Germany
[email protected]
The so-called rst Zassenhaus conjecture states, that every normalized
torsion unit of the integral group ring ZG of a nite group G is conjugated
to an element of G. This conjugation takes place within QG.
Especially computational aspects of the conjecture will be discussed and
applied on small groups.
Solving equations and checking identities over
finite groups
Gábor Horváth
Eötvös Loránd University,
Budapest, Hungary
[email protected]
One of the oldest problems of classical algebra is to determine whether
an equation can be satised or not. An other interesting problem is to check
whether two expressions are identically equivalent, i.e. they are equal over
all substitutions. We investigate the complexity of these problems over nite
groups.
19
Invariants of Lie superalgebras
Malgorzata E. Hryniewicka
University of Bialystok,
Bialystok, Poland
[email protected]
It is well know Bergman's and Isaacs' fundamental result about the existence of xed elements, which states that if G is a nite group of automorphisms of a non-nilpotent ring R with no |G|-torsion then RG is no-nilpotent.
The next important result is due to V. K. Kharchenko. He proved that if
R has no nilpotent elements, and G is any nite group acting as automorphisms on R, then RG is non-trivial. The problem of the existence of nontrivial invariants for actions of Lie algebras was solved by K. I. Beidar and P.
Grzeszczuk. More precisely, they proved that if R is an algebra over a eld
F such that R contains no non-zero nilpotent elements and if L is a nite
dimensional Lie algebra of F -algebraic derivations acting on R, then the subalgebra of invariants RL is non-zero. The purpose of this talk is the extension
of Beidar's and Grzeszczuk's result to the actions of Lie superalgebras. This
is a joint work with J. Bergen and P. Grzeszczuk.
Irreducible representations of non-abelian
groups wich have an abelian normal subgroup of
index prime
A. Iranmanesh, N. Ahanjideh and M. Ferodi
Tarbiat Modarres University,
Tehran, Iran
[email protected], [email protected]
Let G be a non-abelian p-group which contains an abelian normal subgroup H of index p in G . James and Liebeck in [1] showed that G has a
20
normal subgroup K such that |K| = p, K ≤ H ∩ G0 ∩ Z(G). Moreover, every
irreducible character of G was given by either the lift of an irreducible character of G/K or ψ G for some linear character ψ of H which satises K ≤ Kerψ .
In this paper ,at rst we nd irreducible characters and representations of G
with simpler method,then we generalize for every non-abelian group with an
abelian subgroup of index prime.
[1] G. James & M. Liebeck, Representations and Characters of Groups,
cambridge university press, 1993.
On Simple Kn -Groups for n = 5, 6
A. Jafarzadeh and A. Iranmanesh
Tarbiat Modarres University,
Tehran, Iran
[email protected], [email protected]
A nite non-abelian simple group is called to be a simple Kn -group, if
the order of G has exactly n distinct prime factors. M. Herzog and W. J. Shi
gave a characterization of simple Kn -groups for n = 3, 4. Here π(n) refers
to the number of all prime factors of a natural number n, and q is a prime
power.
In this talk, we characterize all simple Kn -groups for n = 5, 6 and prove
theorems below:
Theorem. Each simple K5 -group is isomorphic to one of L2 (q) where q
satises π(q 2 − 1) = 4 or L3 (q) where π((q 2 − 1)(q 3 − 1)) = 4 or U3 (q) where
π((q 2 − 1)(q 3 + 1)) = 4 or O5 (q) where π(q 4 − 1) = 4 or Sz(22m+1 ) where
π((22m+1 − 1)(24m+2 + 1)) = 4 or R(q) where q is an odd power of 3 and
π(q 2 − 1) = 3 and π(q 2 − q + 1) = 1 or one of the 30 other simple groups
A11 , A12 , M22 , J3 , HS, He, M cL, L4 (4), L4 (5), L4 (7), L5 (2), L5 (3), L6 (2),
O7 (3), O9 (2), P Sp6 (3), P Sp8 (2), U4 (4), U4 (5), U4 (7), U4 (9), U5 (3), U6 (2),
O8+ (3), O8− (2),3 D4 (3), G2 (4), G2 (5), G2 (7), G2 (9).
21
Theorem. Each simple K6 -group is isomorphic to one of L2 (q) where π(q 2 −
1) = 5 or L3 (q) where π((q 2 − 1)(q 3 − 1)) = 5 or L4 (q) where π((q 2 − 1)(q 3 −
1)(q 4 − 1)) = 5 or U3 (q) where π((q 2 − 1)(q 3 + 1)) = 5 or U4 (q) where
π((q 2 − 1)(q 3 + 1)(q 4 − 1)) = 5 or O5 (q) where π(q 4 − 1) = 5 or G2 (q) where
π(q 6 − 1) = 5 or Sz(22m+1 ) where π((22m+1 − 1)(24m+2 + 1)) = 5 or R(32m+1 )
where π((32m+1 − 1)(36m+3 + 1)) = 5 or one of the 38 other simple groups
A13 , A14 , A15 , A16 , M23 , M24 , J1 , Suz, Ru, Co2 , Co3 , F i22 , HN, L5 (7), L6 (3),
L7 (2), O7 (4), O7 (5), O7 (7), O9 (3), P Sp6 (4), P Sp6 (5), P Sp6 (7), P Sp8 (3),
U5 (4), U5 (5), U5 (9), U6 (3), U7 (2), F4 (2), O8+ (4), O8+ (5), O8+ (7),
+
−
O10
(2), O8− (3), O10
(2),3 D4 (4),3 D4 (5).
On torsion units of integral group rings
W. Kimmerle
Universität Stuttgart,
Stuttgart, Germany
[email protected]
Let R be a ring. The basic question which properties of a nite group G
are reected by its R - representations leads in a natural way to the study of
the torsion subgroups of the unit group of the group ring RG. The question
may be rephrased to the problem to determine the inuence of G on the
structure of the torsion subgroups of RG.
The talk deals mainly with the case R = Z when G is a nite group with
at least two prime graph components.
In particular the following problems are considered.
1) The rst Zassenhaus conjecture Z1 for integral group rings, i.e. the
question whether a torsion unit is conjugate to a trivial unit within the
rational group algebra, and a conjecture of A.A.Bovdi closely related
to Z1.
2) The prime graph of the normalized unit group of ZG.
3) The isomorphism problem for integral group rings of nite groups with
decomposable prime graph.
22
On weakly prime Frobenius rings
V.V.Kirichenko, M.A.Khibina and I.V.Dudchenko
Natonal Univesity of "Kyiv-Mohyla Akademy",
Kyiv, Ukraine
[email protected]
Let A be a nonzero associative ring with the Jacobson radical R and
L ⊆ A, N ⊆ A are two-sided ideals in A. A ring A is called weakly prime
if every product LN , where L 6⊂ R and N 6⊂ R, is nonzero.
Let A be a semiperfect ring, 1 = e1 + . . . + en be a decomposition of 1 ∈ A
into a sum of pairwise orthogonal idempotents, Aij = ei Aej (i, j = 1, . . . , n).
Proposition. A semiperfect ring A is weakly prime if and only if all Peirce
components Aij are nonzero, i.e., Aij 6= 0 for i, j = 1, . . . , n.
Theorem. A quiver Q(A) of a right Noetherian semiperfect weakly prime
ring A is strongly connected.
Denote by Mn (Z) a ring of all square n × n-matrices over the ring of
integers Z. Let E ∈ Mn (Z). A matrix E = (αij ) is called an exponent
matrix if αij + αjk ≥ αik for i, j, k = 1, . . . , n and αii = 0 for i = 1, . . . , n.
An exponent matrix E is called a reduced exponent matrix αij + αji > 0 for
i 6= j .
Let E = (αij ) be a reduced exponent matrix. Set E(1) = (βij ), where
βij = αij for i 6= j and βii = 1 for i = 1, . . . , n, and E(2) = (γij ), where
γij = min (βik + βkj ). Obviously, [Q] = E(2) − E(1) is a (0, 1)-matrix.
1≤k≤n
Theorem. (see [1, Ÿ14.7]) The matrix [Q] = E(2) − E(1) is the adjacency
matrix of the strongly connected simply laced quiver Q = Q(E).
The quiver Q(E) is called the quiver of a reduced exponent matrix
E. A strongly connected simply laced quiver is called admissible if it is a
quiver of a reduced exponent matrix.
For more detail information see [1, Ch.14].
Theorem. For any admissible quiver Q there is a countable set of weakly
prime semidistributive Frobenius rings An such that Q(An ) = Q.
23
Theorem. For any permutation σ ∈ Sn , such that i 6= σ(i) for i = 1, . . . , n,
there is a weakly prime semidistributive Frobenius ring A with the Nakayama
permutation σ .
Let σ ∈ Sn be as above and k be a eld.
Theorem. There is a weakly prime semidistributive and n2 -dimensional
Frobenius k -algebra A with the Nakayama permutation σ .
References
[1] M.Hazewinkel, N.Gubareni and V.V.Kirichenko, Algebras, Rings and
Modules, V.I, Mathematics and Its Applications, V.575, Kluwer Academic Publishers, 2004.
On finite subgroups of the group GL(q, Rp )
A. Kirilyuk
University of Uzhgorod
Uzhgorod, Ukraine
[email protected]
D. A. Suprunenko [1] and V. P. Yuferev [2] have described up to conjugation all minimal irreducible solvable subgroups of the general linear group
GL(q, F ) where q is a prime and F is an arbitrary eld. The nonconjugate
nonabelian minimal irreducible solvable subgroups of the group GL(q, Rp )
where Rp is a ring of integers of a nite extension Fp of the rational padic eld Qp was founded for q = 2, 3 in [3, 4]. The nonconjugate minimal
nilpotent subgroups of the group GL(q, Rp ) have been described in [5] for
(Fp : Qp ) ≤ 2.
Sylow p-subgroups of the general linear group over dicrete valuation rings
have been studied in [6, 7]. The nonconjugate p-subgroups of the group
GL(n, Zp ) (Zp is a ring of the rational p-adic integers) was described in [6, 8]
for n ≤ 3(p − 1). The problem of the isomorphism as well as conjugation of
24
the Sylow p-subgroups of the general linear group over a principal integral
domains R of the characteristic zero in the case p is an invertible element of
R have been studied in [9, 10]. Nonconjugate irreducible Sylow p-subgroup
of the group GL(3, Rp ) have been classicate in [11].
We describe all nonconjugated irreducible subgroups of the group GL(q, Rp )
in the case (Fp : Qp ) ≤ 2.
References
[1] Suprunenko D. A., The minimal irreducible solvable subgroups of the
prime degree, Trudy Mosk. Matem. Ob., 29, (1973), pp. 223334.
[2] Yuferev V. P., The classikation of the minimal irreducible linear groups
of the prime degree, Izv. AN BSSR. Ser. Phis.-Mat. Nauk, 2, (1974),
pp. 510.
[3] Kirilyuk A.A., The minimal irreducible solvable subgroups of the group
GL(2, Rp ), Materialy XXXII Naych. Conf. Uzhgorod Univ., Uzhgorod,
Dep., N. 207979, (1978), pp. 166198.
[4] Kirilyuk A. A., The irreducible solvable p-subgroups of the group
GL(3, Rp ), Nauk. Visnik Uzhgorod. Univ. Ser. Mat. i Inform.,v. 7, (2002),
pp. 4957.
[5] Gudivok P. M. and Kirilyuk A. A., On minimal irreducible subgroups of
the general linear group over the ring of P -adic integer, Nauk. Visnik
Uzhgorod. Univ. Ser. Mat. i Inform.,v. 7,(2002), pp. 37437.
[6] Gudivok P. M. and Kirilyuk A. A., Sylow p-subgroups of the general linear
group over discrete valuation rings, Dokl. AN Ukr. SSR, ser. A, N. 5,
(1979), pp. 326329.
[7] Gudivok P. M., On Sylow p-subgroups of the general linear group over
complete discrete valuation rings, Ukr. math. j., 43, N. 78, (1991),
pp. 918924.
[8] Vaschuk F. G. and Gudivok P. M., On integral p-adic representation of the
nite abelian p-groups, Dop. AN Ukr. SSR, ser. A, N. 1, (1986), pp. 36.
[9] Gudivok P. M. and Rud'ko V. P., On Sylow subgroups of the general linear
group over integral domains, Dop. NAN Ukraine, N. 8, (1995), pp. 57.
25
[10] Gudivok P. M., Rud'ko V. P., Yurthcenko N. V., On Sylow p-subgroups
of the general linear groups over principal integral domains, Nauk. Visnik
Uzhgorod. un. Ser. math. and inform., v. 6, (2001), pp. 3146.
[11] Kirilyuk A. A., Irreducible Sylow p-subgroups of GL(3, Rp ), Nauk. Visnik
Uzhgorod. un. Ser. math. and inform., v. 7, (2002), pp. 4957.
Wreath products in unit groups of modular
group algebras of some finite 2-groups
Alexander Konovalov, Victor Bovdi
Zaporozhye State University,
Zaporozhye, Ukraine
[email protected]
Let p be a prime number, G be a nite p-group and K be a eld of
characteristic p. Denote by I(KG) the augmentation ideal of the modular
group algebra KG. The group of normalized units V (KG) consists of all
elements of the form 1 + x, where x belongs to I(KG).
In [1] Shalev stated the question whether V (KG) possesses a section
isomorphic to the wreath product of a cyclic group of order p and the commutator subgroup of G. In [2] he proved that this is true for the case of an
odd p and a cyclic commutator subgroup of G.
When p = 2 and G is a 2-group of almost maximal class, in [3] and [4]
the second author constructed a section of V (KG) isomorphic to the wreath
product of a group of order 2 and the commutator subgroup of G.
This was the only known result that conrms the conjecture by Shalev
for 2-groups until we proved the following theorem:
Theorem. Let K be eld of characteristic 2 and let G be a nite nonabelian 2-group with the cyclic commutator subgroup G0 , and there exists a
central element z of order 2 in Z(G)\G0 . Then the wreath product of the
cyclic group of order 2 and the derived subgroup of G is involved into the
V (KG).
26
Note that the family of groups from the conditions of the theorem extends
the result of [4] signicantly. For example, using the Small Groups Library
of the GAP system [5], we can nd that the number of such groups of order
2n for n = 4, 5, 6, 7, 8, 9 is is accordingly 4, 20, 72, 231, 662 and 1750.
References
[1] A. Shalev, On the conjectures concerning units in p-group algebras,
Rend. Circ. Mat. Palermo (2) Suppl. No. 23 (1990), 279288
[2] A. Shalev, The nilpotency class of the unit group of a modular group
algebra. I, Israel J. Math. 70 (1990), no. 3, 257266
[3] A. B. Konovalov, Ukraïn. Mat. Zh. 47 (1995), no. 1, 3945; translation
in Ukrainian Math. J. 47 (1995), no. 1, 4249 (1996)
[4] A. B. Konovalov, Wreath products in the unit group of modular group
algebras of 2-groups of maximal class, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 17 (2001), no. 3, 141149
[5] The GAP Group, GAP Groups, Algorithms, and Programming, Version 4.4, 2004; http://www.gap-system.org
On infinite crossed products
Jan Krempa
Warsaw University,
Warsaw, Poland
[email protected]
In this talk I'm going to discuss some properties of crossed products of
rings and semigroups, investigated earlier by Professor A.A. Bovdi.
27
Functions on classical groups
and some unsolved questions
V.M. Levchuk
Krasnoyarsk State University,
Krasnoyarsk, Russia
[email protected]
The generalize Eulerian functions (see [1]) on nite Lie-type groups
of a small rank are investigated. We consider some unsolved questions. By
analogously with J. Whiston, J. Saxl, P. Cameron and P. Cara we investigate
the maximal size of independent generating sets of some classical groups.
The research is supported by Russian Fond of Basic Investigations, code
of grant 03-01-00905.
[1] Ph. Hall, The Eulerian functions of a group // Qurt. J. Math., 1936,
V.7, P.134-151.
ON GALOIS STABLE SUBGROUPS OF GLn IN
RELATIVE EXTENSIONS OF NUMBER FIELDS
Dmitry Malinin
University of the South Pacic,
Suva, Fiji islands
dmalininmail.ru
In order to generalize our joint with H.-J. Bartels results of [1] to the
case of certain relative number eld extensions (that are composites of two
elds of coprime discriminants over one of them), we consider the following
situation: K is a nite Galois extension of Q unramied outside the rational
primes p1 , p2 , ...pk . Let F/Q be unramied in p1 , p2 , ...pk and L := KF . So we
suppose that (d(F/Q), pi ) = 1 for all indices i and the discriminant d(F/Q)
of F/Q, and we consider nite subgroups G of GLn (L ) that are stable under
the natural action of the Galois group Gal(L/F ).
28
Theorem. There exist G as above such that G * GLn (F Q(ζpm1 pm2 ...pmk ) ) if
1
2
k
and only if there exists an unramied over F intermediate extension between
L and F .
Theorem. Let K be a nite Galois extension of some number eld F , and
let K/F have no unramied subextensions K1 /F , where K1 ⊂ K . Suppose
that G is a nite subgroup of GLn (K ) which is stable under the natural action
of the Galois group Gamma of the eld extension K/F . Suppose also that
the order of G is coprime to the discriminant d(F/Q). Then G ⊂ GLn (F Kab )
holds, where Kab is the maximal abelian subextension of K over Q.
Theorem. Let K be a nite totally ramied Galois extension of Qp the eld
of p-adic rational numbers, let K contain a primitive p-root ζp of 1, and let
G be a nite p-subgroup of GLn (K ) that is stable under the natural action of
the Galois group Gamma of the eld K . Then G ⊂ GLn (Kab ) holds, where
Kab is the maximal abelian subextension of K over Qp .
References
[1] H.J. Bartels, D.A. Malinin, Finite Galois stable subgroups of GLn , Noncommutative Algebra and Geometry, Chapter I, (2005), 122.
Broué's abelian defect group conjecture for
alternating groups
Andrei Marcus
"Babes-Bolyai" University,
Cluj-Napoca, Romania
[email protected], [email protected]
We establish Broué's abelian defect group conjecture for the alternating
groups, using the Chuang-Rouquier theorem proving this for the symmetric
groups and a descent result.
29
On a Subalgebra of the Centre of a Group Ring
Harald Meyer
University of Bayreuth,
Bayreuth, Germany
[email protected]
Let p be a prime, G a nite group with p | |G| and F a eld of characteristic p. Let C1 , ..., Cr be the p-regular conjugacy classes in G, C1+ , ..., Cr+
the class sums in F G and ZpG0 = hC1+ , ..., Cr+ i the F -subspace of the centre
Z(F G) of F G generated by the p-regular class sums. In this talk we discuss
the following question:
For which groups G is ZpG0 an algebra?
We show that ZpG0 is an algebra if the Sylow-p-subgroups of G are abelian and
we develop a method to decide whether ZpG0 is an algebra for a given group G
or not. Using our method we can give examples for groups G in both cases.
Nilpotency and dimension series for loops
Jacob Mostovoy
Universidad Nacional Autonoma de Mexico,
Morelos, MEXICO
[email protected]
In this talk I will explain how to set up a nilpotency theory for loops that
preserves many features of the associative case. In particular, I will discuss
the relation between the lower central series and the dimension series, and the
algebraic structure on the graded abelian groups associated to these series.
30
Isomorphism classes of 1-dimensional
differentiable loops
Peter Nagy
University of Debrecen,
Debrecen, Hungary
[email protected]
The normal form with respect to a distinguished parametrization of a C r dierentiable loop is investigated if the loop is dened on the real line and
the group topologically generated by the left translations is locally compact.
We give a classication of isomorphism classes of such loops by pairs of C r dierentiable real functions satisfying a dierential inequality.
Some algebraic problems concerning the
neighborhood of cellular automata
Hidenosuke Nishio
Univesity of Kyoto,
Kyoto, Japan
[email protected]
A cellular automaton CA is a discrete (symbolic) dynamical system on
a Cayley graph of a nitely generated group G. Its global map is dened
by a local rule simultaneously applied to every vertex (cell). A local rule is
dened on the neighborhood. Usually, like von Neumann neighborhood, the
neighborhood is assumed to be the same as the set of generators of G and
their inverses, however, we consider a dierent setting, i.e. a neighborhood
N is an arbitrary nite subset of the vertices and the neighbors of a vertex
x are dened by a consecutive application of N to x. In other words the set
of the neighbors constitutes a semigroup generated by N , which is generally
a subalgebra of G.
31
This talk addresses some algebraic problems arising from this setting like
(1) Does N generate G? This kind of problems is called the Horse Power
Problem after Hamiltonian cycle of a knight(horse) on a chess board.
(2) How does the neighborhood aect the global behavior of a CA?
Emulation of Synchronous Automata Networks
with Dynamically Changing Topologies by
Asynchronous Automata Networks
Chrystopher L. Nehaniv
University of Hertfordshire,
Hertfordshire, United Kingdom
[email protected]
A synchronous automata network is a locally nite network of automata
whose behaviour depends only on local interactions and an external input
sequence (see e.g. [1] or [2]).3 Up to now, the work on automata networks
has assumed a xed unchanging interconnection topology. In [3, or Ch. 7 of
2], for a given synchronous automata network, it is shown how to construct
an asynchronous network emulating the behaviour of the synchronous network for every possible input sequence (Asynchronous Emulation Theorem),
and the issue of generalizing this to networks whose topology changes was
formulated as an open problem.
In this research, we rst generalize the notion of (synchronous) automata
network to dynamic automata network whose interconnection digraph Γt =
(Vt , Et ) with vertices Vt and edges Et ⊆ Vt × Vt may change as a function
of discrete time t. Each automaton Av,t at a given node v ∈ Vt present
in the network at time t is updated according to a feedback function ϕv,t
depending only on the state of the local neighborhood at that time and an
external global input. A behaviour of the dynamic automata network is then
a sequence, (qv,t )v∈Vt ,t∈N , of congurations of states of the local automata
3 These
are equivalent to general (or Glu²kov) products of automata, and are natural
generalizations of cascade products, of automata constructions with feedback and parallel
components, and also of cellular automata allowing directed graphs and external input.
32
over time, where each qv,t is a state of Av,t . Each external input sequence
and initial state of the network uniquely determines a possible behaviour of
the network.
We then generalize the Asynchronous Emulation Theorem from (static)
automata networks to a class of dynamic automata networks whose growth
and change satisfy certain local constraints. Thus any such synchronous
dynamic automata network can be emulated by an asynchronous one in a
strict sense that allows one to completely recover the behaviour of the former.
Interestingly, these same constraints that allow us to prove the theorem
are generally satised in the growth and development of many dierentiated
multicellular organisms.
References
[1] F. Gécseg, Products of Automata, EATCS Monographs in Theoretical Computer
Science 7, Springer Verlag, 1986.
[2] P. Dömösi and C. L. Nehaniv, Algebraic Theory of Finite Automat a Networks:
An Introduction, SIAM Monographs on Discrete Mathematics and Applications 11,
Society for Industrial and Applied Mathematics, Philadelphia, 2005.
[3] C. L. Nehaniv, Asynchronous Automata Networks Can Emulate Any Synchronous Automata Network, International Journal of Algebra & Computation,
14(5-6):719-739, 2004.
Abstract groups from the viewpoint of
algorithmic complexity
Péter P. Pálfy
Eötvös Loránd University,
Budapest, Hungary
[email protected]
Group theoretic algorithms have been developed for various classes of
groups: permutation groups, matrix groups, nitely presented groups, etc.
33
The computational model for abstract groups is a concept called black-box
group introduced by L. Babai. Surprisingly, the computational diculties
in black-box groups are often related to Abelian sections of the the group,
since notoriously hard problems as factoring integers or nding discrete logarithms may arise in this context. In contrast, for groups with nonabelian
composition factors polynomial-time randomized algorithms have been constructed, based on statistics of element orders in simple groups. This is a
joint work with L. Babai, W.M. Kantor, Á. Seress, and J. Saxl.
Unipotent Linear Groups over Skew Fields
V.M. Petechuk
University of Uzhgorod,
Uzhgorod, Ukraine
Recall that under the type of the element g of the full linear group
GL(n, V ) over a skew eld we understand the dimension of (g − 1)V . The
unipotent elements of the type 1 are called transvections.
Triangulation of the unipotent linear groups over elds has been proven,
using Burnside's theorem, by Kolchin. Kolchin's proof remains true even
when the unipotent linear groups are considered over skew elds that are
nite-dimensional over their centers. Using O.I. Kostrykin's theorem, Heineken
showed that in the case of the skew eld with characteristic exceeding (n−1)!
unipotent linear group can be triangulated. Brook and Mochizuki lowered
this bound to (n−1)(n−[n/2]). D.O. Suprunenko and O.E. Zaleski (independetly) showed that the irreducible unipotent linear groups over skew elds
do not contain transvections. V.M. Petechuk showed that the irreducible
unipotent linear groups over skew elds do not contain elements of types 2,
3, 4 and that for n = 5 the unipotent linear groups can be triangulated over
an arbitrary skew eld. Earlier V.N. Serezhkin proved that for n = 4 the
unipotent linear groups can be triangulated over an arbitrary skew eld.
The author obtained the following results:
Theorem 1 For n = 6 the unipotent linear group can be triangulated
over an arbitrary skew eld.
34
Theorem 2 Irreducible unipotent linear groups over skew elds do not
contain elements of type 5 and 6.
Complements of Connected Subgroups in
Algebraic Groups
P. Plaumann
University of Erlangen,
Erlangen, Germany
[email protected]
This contribution is a report on joint work with K. Strambach, Erlangen and
G. Zacher, Padova.
Groups with a complemented subgroup lattice have been treated in various
classes of groups, like nite groups, topological groups or Lie groups. Using detailed information on the maximal subgroups of nite simple groups
M. Costantini and G. Zacher have recently shown in [1] that all these groups
have a complemented lattice of subgroups.
If G is a connected algebraic group over an algebraically closed eld, it is
adequate to consider the lattice ΛG of closed connected subgroups of G. We
classify the connected algebraic groups G having a complemented lattice ΛG.
In order not to remain restricted to ane groups we make use of Rosenlicht's
theory of algebraic groups in the sense of A. Weil (see [2]). With this tool our
problem can be reduced to ane algebraic groups. The case of a simple ane
algebraic group can be treated using elementary arguments from algebraic
geometry and needs no detailed information about the classication of simple
ane algebraic groups.
Our main result is the complete description of algebraic groups G having a
complemented lattice ΛG. For nite groups such a theorem is not known.
Theorem. The lattice of closed connected subgroups of a connected algebraic
group G over an algebraically closed eld is complemented if and only if G is
an almost direct product of an abelian variety with an ane group L, such
that L is a semidirect product of a vector group with a reductive group.
Two useful tools for our work are the notion of the Frattini subgroup in ΛG
and the notion of distributive pairs of closed subgroups of an algebraic group.
For this reason we discuss properties of these concepts in some detail.
35
References
[1] M. Constantini, G. Zacher, The nite simple groups have complemented subgroup lattices, Pac. J. Math. 213, 245-251 (2004).
[2] M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J.
Math. 78, 401-443 (1956).
Boundedly generated groups
Pyber László
Alfréd Rényi Institute of Mathematics,
Budapest, Hungary
[email protected]
A group G is called boundedly generated (BG) if it is the set-theoretic
product of nitely many cyclic subgroups.Following the discovery that the
groups SL(n, Z) (n ≥ 3) have bounded generation,it has been shown that
many other S-arithmetic groups over number elds have this property. Recently Muranov has constructed boundedly generated innite simple groups.His
examples suggest that in general there are "too many" BG groups. On the
other hand it is conjectured by Tavgen that if a BG group is residually nite then it is linear.In joint work with D. Segal we conrm this if G is also
just-innite or residually nite-soluble.
36
The p-groups with some conditions on cyclic
subgroups
O.S. Pylyavska, Ju.V. Shatohina
Natonal Univesity of "Kyiv-Mohyla Akademy" ,
Kyiv, Ukraine
[email protected]
Z.Janko proposed the next problem. Suppose that a p-group G satises
the following condition: If Z is a cyclic subgroup of G, that either Z ≤ Z(G)
or Z ∩ Z(G) = 1. We obtain the full classication of groups with this
condition.
Also we give a full list of such groups with additional condition: each
subgroup H of G is abelian or it has a commutator subgroup of order p.
References
[1] V., Pylyavska, Determination of groups, in which every proper subgroup
is abelian or has derived subgroup of order (p ≥ 3). Naukovi zapysky
NaUKMA.Phyz.-Math.Sci.23 (2005) (in printing).
Zeta Function to Algebraic Minimal Curves
Jaime Edmundo Apaza Rodriguez
Universidade Estadual Paulista UNESP- Brazil,
Sao Paulo, Brazil
[email protected]
Let C/Fq be a non-singular algebraic curve with genus g and let ZC/Fq (t)
be its zeta function. A curve C/Fq is called maximal (minimal) if attains
the upper (lower) bound in Hasse-Weil theorem to the number of rational
37
points. In this work we are studying some properties the maximal curves and
the minimal curves by using the zeta function ZC/Fq (t) (polinomial LC/Fq (t),
normalized polinomial ΛC/Fq (u) and auxiliar polinomial AC/Fq (v)). Many
examples are displayed.
Hearing the platycosms
Juan Pablo Rossetti, John H. Conway
Univesity of Cordoba,
Cordoba, Argentina
[email protected]
We call a closed locally Euclidean 3-manifold `platycosm' (`at universe'),
since they are the simplest alternative universes for us to think living in.
There are ten types of platycosms. We name, describe and give parameters to them in a unied way, so they can be easily remembered.
We will sketch the proof of the following theorem: There is, up to scale, a
unique pair of isospectral (non-isometric) platycosms. They are a tetracosm
and a didicosm of certain sizes.
Group algebras satisfying a certain Lie identity
Meena Sahai
Lucknow University,
Lucknow, India
[email protected]
Let K be a eld of characteristic p 6= 2 and let G be any group. In the
proposed talk we give a characterization of group algebras KG satisfying the
Lie identity [[x, y], [u, v], [z, t]] = 0 for all x, y, u, v .
38
On unreducible tottaly sutureted formations
with nilpotent defect 3
Vasiliy G. Safonov
Gomel State University of F.Skorina,
Gomel, Belarus
[email protected]
All groups considered are nite. The following terminology can nd in
[1, 2].
A formation is a class of groups closed under taking homomorphic images
and subdirect products. Consider a function f : {primes} → {f ormations}
with the following rule. A chief factor H/K of a group G is called f -central
if G/CG (H/K) ∈ f (p) for any p ∈ π(H/K). A chief series of G is called f central if every its factor is f -central. Then the class of groups with f -central
chief series is a formation, and f is called a local satellite of that formation.
It is well known that a non-empty formation F has a local satellite if and
only if it is saturated, i.e. G/Φ(G) ∈ F always implies G ∈ F.
We consider any formation as a 0-multiply saturated formation. When
n ≥ 1, a formation F 6= ∅ is called n-multiply saturated if it has a local
satellite such that all its non-empty values are (n − 1)-multiply saturated
formations. A totally saturated formation is a non-empty formation which
is n-multiply saturated for any non-negative integer n.
Let F and H be some totally saturated formations. A length of the lattice
F/∞ F ∩ H of totally saturated formations X with F ∩ H ⊆ X ⊆ F is called
H-defect of a totally saturated formation F (or H∞ -defect of formation F).
If H is a formation of all nilpotent groups N, then H-defect of the totally
saturated formation F is called nilpotent defect of F.
Let F be a totally saturated formation, {Fi |i ∈ I} be a set of all proper
totally saturated subformations of F, and X = l∞ form(∪i∈I Fi ). Then F is
called an irreducible totally saturated formation if F 6= X.
Theorem. Let F be an irreducible totally saturated formation. Then a nilpotent defect of F equals 3 if and only if F = Np Nq Np , where p, q are primes
and p 6= q .
References
1. Shemetkov, L.A., Skiba, A.N. Formations of Algebraic Systems; Nauka:
Moscow, 1989; 256 pp.
39
2. Skiba, A.H. Algebra of Formations; Belarus Science: Minsk, 1997:
240 pp.
Lie properties of restricted enveloping algebras
Salvatore Siciliano
University of Lecce,
Lecce, Italy
[email protected]
We examine the Lie structure of a restricted enveloping algebra u(L),
where L is a restricted Lie algebra over a eld of characteristic p > 0. In
particular, we present some recent results about the Lie derived length of
u(L). Moreover, Lie nilpotent restricted enveloping algebras are considered
and their Lie nilpotency indices studied.
Weakly s-quasinormal subgroups of finite groups
Alexander N. Skiba
Gomel State University of F.Skorina,
Gomel, Belarus
[email protected]
All considered groups are nite. A subgroup H of a group G is called
s-quasinormal in G if H permutes with all Sylow subgroups of G.
Denition. Let G be a group, H ≤ G. Then we say that H is weakly squasinormal in G if G has a s-quasinormal subgroup T such that HT = G
and H ∩ T ⊆ HG .
40
Theorem. Let F be a saturated formation contaning all supersoluble groups,
G be a group with a normal subgroup N such that G/N ∈ F. Suppose that all
maximal subgroups of all Sylow subgroups of F ∗ (N ) are weakly s-quasinormal
in G. Then G ∈ F.
On some finite hypergroups
M. Stefanescu, G. Pinotsis
Ovidius University of Constanta,
Constanta, Romania
[email protected]
A hypergroup is a pair (H,.), where H is a nonempty set and "." is a
multioperation, i.e. a map from H 2 to the set of nonempty subsets of H,
with some property. We consider some nite hypergroups which have special
properties. We give all such hypergroups with less than 8 elements and the
algebras which are attached to them , if possible. Also we consider such
hypergroups in connection with fuzzy sets.
41
About the ϕ-unitary subgroup of the group of
units in a finite commutative group algebra
A. Szakács
Tessedik Sámuel College,
Békéscsaba, Hungary
[email protected]
Let G be a nite abelian p-group, K the eld GF (pm ) of pm elements and
V (KG) the P
group of normalized units in the group algebraPKG.
For x = g∈G αg g ∈ KG, we denote by x∗ the element g∈G αg g −1 . The
mapping x → x∗ (x ∈ KG) is an automorphism of order 2 (involution) of
the algebra KG. An element u ∈ V (KG) is called unitary if u−1 = u∗ . The
set of all unitary elements of the group V (KG) is obviously a subgroup; we
call it the unitary subgroup of V (KG), and we denote it by V∗ (KG).
S. P. Novikov had raised the problem of determining the invariants and the
basis (the minimal set of generators) of V∗ (KG). It was solved by A. A. Bovdi
and the author in [1], [2].
It is interesting to consider an arbitrary automorphism φ : KG → KG
of second order and to describe the φ-unitary subgroup Vφ (KG) = {x ∈
V (KG)|x−1 = xφ } of the group V (KG). This problem was proposed by
A. Bovdi in [3].
The rst step in this way may be the study of the ϕ-unitary subgroup
for such automorphism ϕ : KG → KG which can be obtained from an
automorphism
ϕ
e : G → G of P
the group G in the following way: if x =
P
ϕ
e
Here we describe suchlike ϕα
g
∈
KG
then
x
=
g∈G g
g∈G αg ϕ(g).
unitary subgroups for a special automorphism ϕ
e : G → G of the group G.
Throughout we shall use the following notations:
K = GF (pm ) the eld of pm elements;
G = H × C a nite abelian p-group;
ϕ
e : G → G, ϕ(h)
e
= h−1
e = c (c ∈P
C);
P(h ∈ H), ϕ(c)
ϕ
ϕ : KG → KG, x = g∈G αg ϕ(g)
e
where x = g∈G αg g ∈ KG;
¡
¢
p
G[p] = {g ∈ G|g = 1 } the lower layer of G;
|G| the¡ order of G¢;
i
i
Gp = {g p |g ∈ G };
fi (G) the number of components of order pi in the decomposition of the
group G into a direct product of cyclic groups;
<(G) = f1 (G) + f2 (G) + · · · the p-rank of G;
42
Vϕ (KG) the ϕ-unitary subgroup of V (KG);
H = ha1 i × · · · × has i;
C = has+1 i × · · · × han i;
qi the order of the element ai (i = 1, . . . , n);
m−1
ε, εp , . . . , εp
a GF (p)-basis of the eld K ;
pj
xj,α = 1 + ε (a1 − 1)α1 · · · (as − 1)αs (as+1 − 1)αs+1 · · · (an − 1)αn .
It was proved in [2] that the set b(G) = {xj,α |0 ≤ j < m, α ∈ L(G)} is a
basis for the group V (KG).
Theorem. Let p > 2. Then
fj (Vϕ (KG)) =
¡ j ¢ ¡ j+1 ¢
¡ ¡ j−1 ¢
m ¡ ¡ pj−1 ¢
[ |G
| − 2 |Gp | + |Gp
| − ( |C p
|
2
¡ j ¢ ¡ j+1 ¢ ¢ ¢
− 2 |C p | + |C p
| ) ]
(j = 1, 2, 3, . . .).
In fact in case p > 2 we have V∗ (KG) ' Vϕ (KG) × V∗ (KC).
Theorem. If p > 2 then the set
¡
¢
b∗ (G) = {(xj,α )ϕ (xj,α )−1 |xj,α ∈ b(G), α1 + · · · + αs is an odd number }
is a basis for Vϕ (KG).
Theorem. Let p = 2. Then there exist such groups T (KG) and B(KG) for
which
Vϕ (KG) = V (KC)[2] × H × T (KG) × B(KG).
The invariants of the groups T = T (KG) and B = B(KG):
f1 (T ) = <(T ) = m|C|(|H[2]| − 1),
fi (B) = di−1 − 2di + di+1 − fi+1 (H)
where
¡ j¢ ¡
¢¢
1 ¡ ¡ 2j ¢
j
( |G | − |C | · |H 2 [2] | ).
2
The order of Vϕ (KG) equals
dj = m
¡
¢
1
2
|Vϕ (KG)| = |H 2 [2] | · |K| 2 (|G|+|C|·|H[2]|)−|C |
and the 2-rank
¡ 1¡
¡ ¢ ¡ ¢ ¡
¢¢
¡ ¢¢
<(Vϕ (KG) = m { (|G| + |C| · |H[2]| − |G2 | + |C 2 | · |H 2 [2] | ) − |C 2 | }.
2
We describe the basis of the group Vϕ (KG) too.
43
References
[1] Bovdi, A. A. and Szakács, A., Unitary subgroup of the multiplicative
group of a modular group algebra of a nite abelian p-group, Mat. Zametki, 45, (1989), No 6, 2329. (see also Math. Notes, 45, (1989), No
56, 445450.)
[2] Bovdi, A. A. and Szakács, A., A basis for the unitary subgroup
of the group of units in a nite commutative group algebra, Publ.
Math.(Debrecen), 46, (1995), No 12, 97120.
[3] Bovdi, A. On the group of units in modular group algebras, An. St. Univ.
Ovidius Constantza. 4, (1996), f.2, 2230.
[4] Sandling, R., Units in the modular group algebra of a nite abelian pgroup, J. Pure Appl. Algebra. 33, (1984), 337346.
On Irreducible Modular Representations of
Given Degree of Finite p-Group over Semiprime
Commutative Local Ring
Alexander Tylyshchak
University of Uzhgorod,
Uzhgorod, Ukraine
[email protected]
All irreducible matrix representations of nite group of order p over commutative local ring R of characteristic ps (s > 0, pR maximal ideal of
ring R) have been described up to equivalence in [1,2]. In [35] it's making
up clear, when the set of all nonequivalent irreducible matrix representations
of nite p-group over commutative Artinian local ring of characteristic ps is
nite.
44
Let G be a nite p-group of order |G| > 1 and R be a commutative
Noetherian local ring of characteristic p which is not an integral domain but
dos not contain nonzero nilpotent elements with innite residue class eld
R/Rad R. It has been shown that if n is even or |G| > 2 then set of all
nonequivalent irreducible matrix representations of given degree n > 1 of
group G over ring R is innite.
It has also been shown that all matrix representations of group of order
2 of odd degrees n > 1 over some commutative Noetherian local ring of
characteristic 2 which is not an integral domain but dos not contain nonzero
nilpotent elements are reducible.
References
[1] V. S. Drobotenko, E. S. Drobotenko, Z. P. Zhilinskaya, and
E. Ya. Pogorilyak, Representations of the cyclic group of prime order p
over residue classes mod ps , Ukrain. Mat. Z. 17 (1965), 12391242.
[2] T. Hannula, The integral representation ring a(Rk G), Trans. Amer. Math.
Soc. 133 (1968), 553559.
[3] P. M. Gudivok, V. S. Drobotenko, A. I. Lichtman, On representation of
nite groups over the ring of residue classes mod ps , Ukrain. Mat. Z. 16
(1964), 8189.
[4] P. M. Gudivok, O. A. Tylyshchak, On irreducible modular representations
of nite p-groups over commutative local rings, Nauk. Visnik Uzhgorod
Univ. Ser. Mat. 3 (1998), 7883.
[5] A. A. Tylyshchak, On irreducible modular representations of given degree
of nite p-group over commutative local ring, Nauk. Visnik Uzhgorod
Univ. Ser. Mat. and Inform. 7 (2002), 108114.
45
GROUPS WITH MANY FINITE-BY-NILPOTENT
SUBGROUPS
L. Yonyk
Ivan Franko National University of Lviv,
Lviv, Ukraine
[email protected]
We say that a group G satises the minimal condition on non-niteby-nilpotent" subgroups if there exists no innite properly descending chain
of non-nite-by-nilpotent" subgroups in G. Every minimal non-nite-bynilpotent" group (i.e. non-nite-by-nilpotent" group with all proper subgroups nite-by-nilpotent) satises the minimal condition on non-nite-bynilpotent" subgroups [1].
We study groups with the minimal condition on non-nite-by-nilpotent"
subgroups and prove that a soluble group G satises the minimal condition
on non-nite-by-nilpotent" subgroups if and only if G is either nite-bynilpotent, or minimal non-nite-by-nilpotent", or ƒernikov.
References
[1] M. Xu, Groups whose proper subgroups are nite-by-nilpotent, Archiv
Math. 66 (1996), 353-359.
46
Asymptotic properties of infinite algebras.
Em Zelmanov
University of California,
San Diego, USA
[email protected]
We will discuss (i) some very old problems concerning nil algebras of slow
(polynomial) growth and their relations with branch groups and algebras, and
(ii) very fast growing algebras and their relations with groups and algebras
with property tau.
Conjugately dense subgroups of free products
of groups and linear groups
S. Zyubin4
Tomsk Polytechnic University,
Tomsk, Russia
[email protected]
A subgroup of any group is called conjugately dense if it has nonempty
intersection with each class of conjugate elements of the group. In the
group GLn (K) over algebraically closed eld, examples of proper conjugately
dense subgroups are triangular subgroup, all it's conjugate, and overgroups of
these subgroups. P. Neumann set the following problem in Kourovka Notebook. Describe all irreducible conjugately dense subgroups H of the group
GLn (K) over arbitrary eld K . In the same place P. Neumann conjectured
that H = GLn (K) with the exception of a case when n = charK = 2,
K is quadratically closed eld and H is conjugated to monomial subgroup
[1, Problem 6.38a]. The author and V.M. Levchuk conrmed Neumann's
4 This
research was supported by the Russian Foundation for Basic Research (Grant
030100905)
47
conjecture for the group GL2 (K) over a locally nite eld K [2]. Further,
the author proved that the conjecture is true for the group GL3 (K) over a
locally nite eld K [3]. Now we establish the following results.
Theorem 1. Let group G be a nontrivial free product G = G1 ∗ G2
A
with amalgamated subgroup A, B be a largest normal subgroup of G that lying
in A, and quotient group G/B contain α conjugacy classes. Suppose (1) for
any conjugacy class there exists element such that any its power either doesn't
belong to A or already belongs to B ; (2) the following inequalities are fullled:
max{|G1 : A|, |G2 : A|} > 2 and max{|G1 : A|, |G2 : A|, ℵ0 } > α. Then
group G has max{|G1 : A|, |G2 : A|, 2α } or greater distinct conjugately dense
subgroups.
Decomposition of groups SL2 (K) and GL2 (K)) over eld K with discrete valuation into free product with amalgamation permits prove the next
theorem.
Theorem 2. Let K be a eld with nontrivial, discrete valuation and
k be it's residue class eld. Suppose |K| = |k| and char k = 0 then
groups SL2 (K) and GL2 (K) have 2|K| pairwise nonconjugated conjugately
dense subgroups.
From theorem 2 it follows that Neumann's conjecture is wrong for the
group GL2 (K) over eld K that satises conditions of this theorem. Also we
have another consequences of Theorem 1.
Corollary 1. Modular group P SL2 (Z) has continuum pairwise nonconjugated conjugately dense subgroups.
Corollary 2. Free group F (X) with rank > 1 has 2max{|X|, ℵ0 } pairwise
nonconjugated conjugately dense subgroups.
References
[1] The Kourovka Notebook , Unsolved Problems in Group Theory, 15th
ed. (2002), Inst. Mat. (Novosibirsk).
[2] Zyubin S.A. and Levchuk V.M., Conjugately dense subgroups of group
GL2 (K) over locally nite eld K (in Russian), Sbornik trudov konf.
"Symm. and Di. Equations", Krasnoyarsk (2000), 110-112.
[3] Zyubin S.A., Conjugately dense subgroups of 3-dimensional linear
groups over locally nite eld, Int. J. of Algebra and Computation: Proc.
of Conf., Gaeta, 2003, (to be published).
48