International Journal of Algebra, Vol. 8, 2014, no. 13, 623 - 628
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ija.2014.4776
D-Sets of Finite Groups1
Cristopher John S. Rosero, Joris N. Buloron,
Jay M. Ontolan
Mathematics and ICT Department
Cebu Normal University
Cebu City, Philippines
Michael P. Baldado Jr.2
Mathematics Department
Negros Oriental State University
Dumaguete City, Philippines
c 2014 Cristopher John S. Rosero, Joris N. Buloron, Jay M. Ontolan and
Copyright Michael P. Baldado Jr. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Abstract
A subset D of a group G is called a D-set if every element of G
which is not in D has its inverse in D. A D-set of minimum cardinality
is called a minimum D-set. This study gave the number of D-sets and
the number of minimum D-sets of a finite group.
Mathematics Subject Classification: 20Dxx, 05Axx
Keyword: D-sets, finite group, index minimum
1
Introduction
Rosero et al. [1] introduced a subset of a group that has some distinctive
properties. This set is called a D-set. More formally, a subset D of a group G
is called a D-set if every element of G which is not in D has its inverse in D.
1
2
This research is funded by Cebu Normal University, Cebu City, Philippines.
Corresponding author
624
Cristopher John S. Rosero et al.
In this study, we count the number D-sets and the number minimum D-sets of
a finite group.
The following concepts are taken from [3]. A binary operation on a nonempty
set G is a function G × G → G. A semigroup is a nonempty set G together with
a binary operation which is associative, that is, a(bc) = (ab)c for all a, b, c ∈ G.
A group is a semigroup G that satisfies the following properties: (1) There is
an element e ∈ G with the property that ge = g = eg for all g ∈ G; (2) For
each g ∈ G, there exists h ∈ G such that gh = e = hg.
The index minimum of a finite group G is the number of minimum D-sets
of G and is denoted by ind(G).
As found in [2], the Addition Principle states that if an event E can be
partition into r events E1 , E2 , . . . , Er , and that there are n1 ways for the event
E1 to occur, n2 ways for the event E2 to occur, . . ., nr ways for the event Er to
occur. Then the total number of ways for event E to occur is n1 + n2 + · · · + nr .
As found in [2], the Multiplication Principle states that if an event E can be
decomposed into r ordered events E1 , E2 , . . . , Er , and that there are n1 ways
for the event E1 to occur, n2 ways for the event E2 to occur, . . ., nr ways for
the event Er to occur. Then the total number of ways for event E to occur is
n1 n2 · · · nr .
2
Number Minimum D-sets
Theorem 2.1 characterizes groups with index minimum equal to 1.
Theorem 2.1 Let G be a finite group and S = {s ∈ G : s2 = e}. Then G = S
if and only if ind(G) = 1
Proof : Suppose G = S. Then G is the only D-set in G. This implies that G is
the only minimum D-set in G. Therefore, ind(G) = 1.
Conversely, suppose that ind(G) = 1 and G 6= S. If G =
6 S then there
exists x ∈ G such that x =
6 x−1 . Let E be a minimum D-set in G. Then
(E\{x}) ∪ {x−1 } and (E\{x−1 }) ∪ {x} are distinct minimum D-sets in G. This
is a contradiction. Therefore G = S.
The elements of S in Theorem 2.1 are called involutions. We note that if
x ∈ G\S, then x 6= x−1 . Since x and x−1 must be in G\S for all x ∈ G\S,
|G\S| is necessarily even. The following remark is clear.
Remark 2.2 Let G be a finite group. Then the following statements are
equivalent:
1. x2 = e, for all x ∈ G
D-sets of finite groups
625
2. ind(G) = 1
3. |T | = 1.
The next result gives the cardinality of a minimum D-set of group G.
Theorem 2.3 Let G be a finite group and E be a minimum D-set of G. Then
|E| = |S| + |G\S| /2.
Proof : Define a relation ∼ on G\S as follows: x ∼ y if and only if x = y or
y = x−1 . Then ∼ is an equivalence relation, that is, ∼ partitions G\S into
equivalence classes Ex = {x, x−1 } with x ∈ G\S. By the Axiom of Choice,
there exists a set ∆ such that ∆ ∩ Ex is a singleton set for all x ∈ G\S. It is
easy to see that ∆∪S is a minimum D-set and |∆ ∪ S| = |S|+|G\S| /2.
Let G be a finite group and S = {s ∈ G : s2 = e}. We denote by c the
number |G\S| /2. The next result gives the number of minimum D-set of a
finite group.
Theorem 2.4 Let G be a finite group. Then ind(G) = 2c .
Proof : Let E be a minimum D-set and S = {s ∈ G : s2 = e}. The equivalence
relation ∼ on G\S that is given by x ∼ y if and only if x = y or y = x−1
−1
−1
partitions G\S into equivalence classes {{x1 , x−1
1 }, {x2 , x2 }, . . . , {xc , xc }}.
−1
Thus, E = S ∪ {a1 , a2 , . . . , ac } for some a1 ∈ {x1 , x−1
1 }, a2 ∈ {x2 , x2 }, . . . ,
−1
of ways
ac ∈ {xc , xc }. Therefore, the number
to choose
a minimum D
−1 −1 set, by Multiplication Principle, is {x1 , x1 } · {x2 , x2 } · · · |{xc }, x−1
c }| =
2 · 2 · 2 · · · 2 = 2c .
3
Number of Finite D-sets
In this section, an expression for the number of all D-sets of a finite group
G is given. We note here that if E is a minimum D-set, then it is of the
−1
form E = S ∪ {a1 , a2 , . . . , ac } for some a1 ∈ {x1 , x−1
1 }, a2 ∈ {x2 , x2 }, . . . ,
−1
−1
−1
−1
ac ∈ {xc , xc } where {{x1 , x1 }, {x2 , x2 }, . . . , {xc , xc }} is a partition of G\S
in the sense of the proof of Theorem 2.4. Moreover, if E is a D-set, then
E ∪ {v} for all v ∈ V (G) is also a D-set. Thus, if D is a D-set, then either
|G\D| = c, or |G\D| = c − 1, or |G\D| = c − 2, and so on.
Theorem 3.1 Let G be a finite group and T be the set of all D-sets of G.
Then |T | = 3c .
626
Cristopher John S. Rosero et al.
Proof : Let G be a finite group and T be the set of all D-sets of G. We would
like to find the cardinality of T . Consider the following disjoint cases:
Case
Case
Case
Case
0.
1.
2.
3.
|G\D| = 0
|G\D| = 1
|G\D| = 2
|G\D| = 3
No.
No.
No.
No.
..
.
of
of
of
of
D-sets
D-sets
D-sets
D-sets
with
with
with
with
|G\D| = 0
|G\D| = 1
|G\D| = 2
|G\D| = 3
is
is
is
is
1.
2c/1!.
2c(2c − 2)/2!.
2c(2c − 2)(2c − 4)/3!.
Case c. |G\D| = c No. of D-sets with |G\D| = c is 2c(2c − 2) · · · 1/c!.
Hence, by Addition Principle,
|T | = 1 +
c
X
2c(2c − 2)(2c − 4) · · · [2c − (i − 1)] /i!
i=1
i
Y
= 1+
c
X
j=1
[2c − 2(j − 1)]
i!
i=1
i
Y
= 1+
c
X
j=1
[2c − f (j)]
i!
i=1
(1)
.
Let
A=
i
Y
[2c − f (j)]
(2)
j=1
and
B = (2c − 1)(2c − 3) · · · 5 · 3 · 1
!
!
!
2c
2c − 2
2c − 4
···
2
2
2
=
c!
(2c)!
=
c!2c
4
2
!
so that
(2c)! = AB(2c − 2i) [2c − 2(i + 1)] · · · (4)(2)
= AB
c−1
Y
[2c − 2(j)]
j=i
= 2c−i AB
c−1
Y
j=i
[c − (j)] .
2
2
!
D-sets of finite groups
627
Hence,
(2c)!
A =
2c−i B
c−1
Y
[c − (j)]
j=i
(2c)!
=
2c−i
c−1
Y
·
1
B
·
c!2c
(2c)!
[c − (j)]
j=i
(2c)!
=
2c−i
c−1
Y
[c − (j)]
j=i
i
=
c!2
c−1
Y
[c − (j)]
j=i
=
c!2i
.
(c − i)!
(3)
From (2) and (3) we obtain
i
Y
[2c − f (j)] =
j=1
c!2i
(c − i)!
(4)
thus by (1) and (4),
i
Y
|T | = 1 +
= 1+
= 1+
c
X
j=1
[2c − f (j)]
i!
i=1
c
X
c!2i
i=1 i!(c − i)!
c
X
i=1
c
i
!
2i .
(5)
628
Cristopher John S. Rosero et al.
In view of the Binomial Theorem found in [2],
3c = (1 + 2)c
!
c
X
c
=
2i
i
i=0
=
c
0
= 1+
!
0
2 +
c
X
i=1
c
X
i=1
c
i
c
i
!
2i
!
2i
= |T |.
Remark 3.2 Let G1 and G2 be finite groups and T1 and T2 be the set of all
D-sets of G1 and G2 , respectively. If ind(G1 ) = ind(G2 ) then |T1 | = |T2 |.
References
[1] J. N. Buloron, C. S. Rosero, J. M. Ontolan and M. P. Baldado Jr., Some
Properties of D-sets of a Group, International Mathematical Forum, 9(2014),
1035-1040.
[2] C. C. Chong and K. K. Meng, Principles and Techniques in Combinatorics,
World Scientific Publishing Co. Pte. Ltd. , Singapore, 1992.
[3] T. W. Hungerford, Algebra, Springer-Verlag, New York, 1974.
Received: July 7, 2014