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EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 9, No. 1, 2016, 34-38

ISSN 1307-5543 – www.ejpam.com

D-Sets Generated by a Subset of a Group

Cristopher John S. Rosero1, Michael P. Baldado Jr. 2,∗

1 Mathematics and ICT Department, Cebu Normal University, Cebu City, Philippines
2 Math Department, Negros Oriental State University, Dumaguete City, Philippines

Abstract. A subset D of a group G is a D -set if every element of G, not in D, has its inverse in D. Let

A be a non-empty subset of G. A smallest D -set of G that contains A is called a D -set generated by A,

denoted by 〈A〉. Note that 〈A〉 may not be unique. This paper characterized sets A with unique 〈A〉 and

sets whose number of generated D -sets is equal to the index minimum.

2010 Mathematics Subject Classifications: 20D99

Key Words and Phrases: Groups, D -sets, Index minimum

1. Introduction

Let G be a group. A subset D of G is called a D -set if for every x ∈ G\D, x−1 ∈ D. A

smallest D -set in G is called a minimum D -set. The number of minimum D -sets of G is called

the index minimum of G. Please refer to [3] for the concepts that are not defined in this paper.

In [1], we proved that if x2 = e, then x is an element of any D -set. Thus, if

S =
�

s ∈ G : s2 = e
	

, then S ⊆ D for all D -set D.

It is mention in [2] that the relation ∼ defined on G\S given by x ∼ y if and only if x = y

or x−1 = y is an equivalence relation, and the equivalence class containing x is {x , x−1}. Thus,

G\S = {a1, a−1
1
}∪{a2, a−1

2
}∪· · ·∪{ac , a−1

c }. If ai 6= a j for i 6= j, then we call the given partition

a canonical partition of G\S, and c is called the C -number of G. Clearly, c = |G\S|/2.

Remark 1. Let G be a finite group and D be a D -set of G. Then D = S ∪ {x1, x2, . . . , xc}, where

x i ∈ {ai , a−1
i
} for i = 1,2, . . . , c and G\S = {a1, a−1

1
} ∪ {a2, a−1

2
} ∪ · · · ∪ {ac , a−1

c } is a canonical

partition, if and only if D is a minimum D -set.

To see this, assume that D = S∪{x1, x2, . . . , xc}, where x i ∈ {ai , a−1
i
} for i = 1,2, . . . , c and

G\S = {a1, a−1
1
} ∪ {a2, a−1

2
} ∪ · · · ∪ {ac , a−1

c } is a canonical partition, and D is not a minimum

D -set. Let D′ be a minimum D -set. Then
�

�D′
�

� < |D|. Let x ∈ D\D′. Since the elements of

∗Corresponding author.

Email addresses: crisrose_18@yahoo.com (C. Rosero), mbaldadojr@yahoo.com (M. Baldado)

http://www.ejpam.com 34 c© 2016 EJPAM All rights reserved.



C. Rosero, M. Baldado / Eur. J. Pure Appl. Math, 9 (2016), 34-38 35

S must be in D′, x = x i for some i ∈ {1,2, . . . , c}. Since D′ is a D -set, x−1
i
∈ D′. Hence,

x i , x−1
i
∈ D. This is a contradiction.

Conversely, assume that D is a minimum D -set and D 6= S ∪ {x1, x2, . . . , xc}, where

x i ∈ {ai , a−1
i
} for i = 1,2, . . . , c and G\S = {a1, a−1

1
} ∪ {a2, a−1

2
} ∪ · · · ∪ {ac , a−1

c } is a canonical

partition. Since the elements of S must be in D and D 6= S ∪ {x1, x2, . . . , xc}, where

x i ∈ {ai , a−1
i
} for i = 1,2, . . . , c and G\S = {a1, a−1

1
} ∪ {a2, a−1

2
} ∪ · · · ∪ {ac , a−1

c } is a canonical

partition, there exist x ∈ G\S such that {x , x−1} ∈ D (since one of x and x−1 must be in D).

If {x , x−1} ∈ D, then D\{x} is a D -set smaller than D. This is a contradiction.

The following results are found in [2]. They will be used in the succeeding sections.

Theorem 1. Let G be a finite group. If c is the C -number of G, then i (G) = 2c .

Theorem 2. Let G be a finite group and T be the family of all of its D -sets. If c is the C -number

of G, then |T |= 3c .

Theorem 2 says that if A⊆ G, then i (A)≤ 3c .

2. D -Sets Generated by a Subset

All groups considered here are finite groups. Let G be a group and A be a non-empty subset

of G. A smallest D -set of G that contains A is called a D -set generated by A, denoted by 〈A〉.
For example, consider the additive group Z6 = {0,1,2,3,4,5}. Note that SZ6

= {0,3},
and {{1,5}, {2,4}} is a canonical partition of Z6. Thus, the C -number of Z6 is 2. Hence by

Theorem 2, |T |= 32 = 9. The elements of T would be

D1 = {0,3,1,5,2,4}

D2 = {0,3,1,2,5}

D3 = {0,3,1,4,5}

D4 = {0,3,1,2,4}

D5 = {0,3,2,4,5}

D6 = {0,3,1,2}

D7 = {0,3,1,4}

D8 = {0,3,2,5}

D9 = {0,3,4,5} .

Observe that A= {1,4} is a subset of D1, D3, D4, and D7 and the smallest set among these

is D7. Thus, 〈1,4〉= D7 = {0,3,1,4}.
Note that 〈A〉 may not be unique. To see this, let B = {0,3}. Then D6 and D9 are the

smallest D -sets of G containing {0,3}. Hence, 〈0,3〉 = D6 or D9. We denote by i (A) the

number of distinct D -sets of G generated by A.

Remark 2. For any nonempty subset A of a finite group G, 〈A〉 always exist since G is itself a

D -set. Hence, i (A)> 0.



C. Rosero, M. Baldado / Eur. J. Pure Appl. Math, 9 (2016), 34-38 36

Remark 3. Let G be a group and D be a minimum D -set of G. If x ∈ D\S, then x−1 /∈ D.

To see this, suppose that x ∈ D\S and x−1 ∈ D. Then D\{x} is a D -set smaller than D.

This is a contradiction.

Theorem 3. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

and A⊆ G. Then A⊆ S if and only if

A⊆ D for all minimum D -set D of G.

Proof. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

and A⊆ G. In [2], S ⊆ D for all D -set D

of G. So, if A⊆ S, then A⊆ D for all D -set D of G. In particular, A⊆ D for all minimum D -set

D of G.

Conversely, assume that A⊆ D for all minimum D -set D of G and A 6⊆ S. Let x ∈ A\S and

D be a minimum D -set containing A. Since A⊆ D and x ∈ A\S, x ∈ D\S. Hence by Remark 3,

x−1 /∈ D. Note that D1 = D\{x} ∪ {x−1} is a minimum D -set that do not contain A. This is a

contradiction.

Corollary 1. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

. If A⊆ S, then i (A) = 2c .

Proof. This follows from Theorem 3 and Theorem 1.

The next result characterizes sets A with unique generated D -set.

Theorem 4. Let G be a finite group and A⊆ G. Then, i (A) = 1 if and only if A⊆ D and A⊇ (D\S)
for some D -set D of G.

Proof. Let G be a finite group and A ⊆ G. Suppose that i (A) = 1, and, A 6⊆ D or A (D\S)
for all D -set D of G. If A 6⊆ D for all D -set D of G, then i (A) = 0. This is a contradiction (by

Remark 2). So we assume that A ⊆ D. Let D1 be a smallest D -set containing A. If A (D\S)
for all D -set D of G, then A∪ S is not a D -set, that A∪ S is properly contained in D1. Let

x ∈ D1\(A ∪ S). Then D1\{x} ∪ {x
−1} is a D -set containing A with

�

�D1

�

� ≤ |D|. This is a

contradiction.

Conversely, assume that A ⊆ D, A ⊇ (D\S) for some D -set D of G, and i (A) > 1. If A ⊆ D

and A ⊇ (D\S) for some D -set D of G, then D = A ∪ S is a smallest D -set containing A.

Since i (A) > 1, let D1 be another smallest D -set containing A. Since A ⊇ (D\S), D1 ⊇ (D\S).
If D 6= D1 and D1 ⊇ (D\S), D is a proper subset of D1 (since D1 must contain S). Hence

|D| 6=
�

�D1

�

�. This is a contradiction.

Theorem 5. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

, and A ⊆ G. If x i 6= x−1
j

for all

x i , x j ∈ A\S, then A is a subset of a minimum D -set. Hence, i (A)≤ 2c .

Proof. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

, and A ⊆ G. From Remark 1, if D is a

minimum D -set, then D = S ∪
�

a1, a2, . . . , ac

	

where ai ∈
�

x i , x−1
i

	

for i = 1,2, . . . , c where
�

{x i , x−1
i
} : i = 1,2, . . . , c

	

is a partition of G\S in the sense of Remark 1. Thus, if x i 6= x−1
j

for

all x i , x j ∈ A\S, Then A is a subset of a minimum D -set.

We recall the disjoint union of sets. Let X and Y be sets. The disjoint union of X and Y ,

denoted by X ∪̇ Y , is found by combining the elements of X and Y , treating all elements to be

distinct. Thus,
�

�X ∪̇ Y
�

�= |X |+ |Y |.



C. Rosero, M. Baldado / Eur. J. Pure Appl. Math, 9 (2016), 34-38 37

Theorem 6. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

, Q =
�

x ∈ A\S : x−1 ∈ A
	

, and A⊆ G.

Then i (A) = 2c−n, where n= |A| − |A∩ S| − |Q|2 .

Proof. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

, Q = {x ∈ A\S : x−1 ∈ A} and A ⊆ G.

Consider P = A\ (S ∪Q). Note that if x ∈ P, then x−1 ∈ G\A. Thus,

A=(A∩ S) ∪̇Q ∪̇ P

=(A∩ S) ∪̇
�

x1, x−1
1 , x2, x−1

2 , . . . , xk, x−1
k

	

∪̇
�

xk+1, xk+2, . . . , xn

	

.
(1)

It can be shown that if D is a smallest D -set containing A, then D is of the form

D =S ∪̇Q ∪̇ P ∪̇
�

xn+1, xn+2, . . . , xc

	

=S ∪̇
�

x1, x−1
1 , x2, x−1

2 , . . . , xk, x−1
k

, xk+1, xk+2, . . . , xn

	

∪̇
�

xn+1, xn+2, . . . , xc

	

.
(2)

By this, the number of ways to choose a smallest D -set containing A is 2 · 2 · · · · · · 2 = 2c−n,

where n=
|Q|
2 + (n− k). Since |A|= |A∩ S|+ |Q|+ (n− k), n= |A| − |A∩ S| − |Q|2 .

3. D -Sets Generated by a Subgroup

The following are consequences of Theorem 6.

Corollary 2. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

, and H ≤ G. Then i (A) = 2c−n, where

n= |H\S|/2.

Proof. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

, and H ≤ G. If H ≤ G, then x , x−1 ∈ H

for all x ∈ H. Let Q = {x ∈ A\S : x−1 ∈ A}. Then Q = H\S, that is, |Q| = |H\S|. Thus, by

Theorem 6, i (H) = 2
c−
�

|H|−|H∩S|− |H\S|2

�

= 2
c−
�

|H\S|− |H\S|2

�

= 2
c−
�

|H\S|
2

�

.

Corollary 3. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

, and H ≤ G. If S ⊆ H, then

i (H) = 2
c−
�

|H|−|S|
2

�

.

Proof. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

, and H ≤ G. If S ⊆ H, then

|H\S|= |H| − |S|. Thus, by Corollary 2, i (H) = 2
c−
�

|H|−|S|
2

�

.

Corollary 4. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

, and H ≤ G. If H ∼= Zp where p is an

odd number, then i (H) = 2
c−
�

|H|−1
2

�

.

Proof. Let G be a finite group, S =
�

s ∈ G : s2 = e
	

, and H ≤ G. If H ∼= Zp where p is an

odd number, then S = {e}. Thus, by Corollary 3, i (H) = 2
c−
�

|H|−1
2

�

.

ACKNOWLEDGEMENTS The authors would like to thank Cebu Normal University, Cebu City

and Negros Oriental State University, Dumaguete City for allowing the authors to use some of

their facilities and resources in the conduct of this research.



REFERENCES 38

References

[1] J. N. Buloron, C. S. Rocero, J. M. Ontolan and M. P. Baldado Jr. Some Properties of D -sets

of a Group, International Mathematical Forum, 9, 1035-1040, 2014.

[2] J. N. Buloron, C. S. Rocero, J. M. Ontolan and M. P. Baldado Jr. D -Sets of Finite Group,

International Journal of Algebra, 8, 623-628, 2014.

[3] T. W. Hungerford. Algebra, Springer-Verlag New York, Inc, 1976.