Another Look on the D-sets Generated by A Subset of a Group

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Another Look on the 𝓓-sets Generated by A Subset of a Group

Cristopher John S. Rosero Cebu Normal University

Date Submitted: September 30, 2015 Originality: 84%

Date Revised: December 22, 2015 Plagiarism Detection: Passed

ABSTRACT

Let 𝐺 be a group. A subset 𝐷 of 𝐺 is called a 𝒟-set if every element 𝑥 ∈ 𝐺\𝐷, 𝑥−1 _{∈ 𝐷}_{. If}_𝐴_{is a nonempty subset of} _𝐺,_{then the smallest} _𝐷_{-set that} contains 𝐴 is called the 𝒟-set generated by 𝐴 and is denoted by 〈𝐴〉. This paper re-investigates more properties of the 𝒟-sets generated by a nonempty subset 𝐴 of 𝐺 and shows proofs of some identities using the concept of 𝒟-sets.

Keywords:group, 𝐷-sets, index minimum, 𝐷-set of a group 𝐺 generated by a non-empty subset 𝐴

INTRODUCTION

A new type of substructure called

𝒟-sets of a group was introduced by

Buloron et al in 2014. The research

focusing on the 𝒟-sets of a finite group

emphasizes on the minimum 𝒟-sets and

showed that 𝒟-sets generate the

corresponding semigroup 𝑇 where 𝑇 is the

collection of all the 𝒟-sets in the group 𝐺.

The number of 𝒟-sets and the number of

minimum 𝒟-sets in finite groups were

given by Ontolan et al (2014). The concept

of 𝒟-sets in a group was extended to

another mathematical structure. Hence, the

notion of 𝛾 −sets in a ring was introduced

by Rosero and Baldado (2014).

Furthermore, Rosero and Baldado (2014)

investigated the smallest 𝒟-set that

contains the nonempty subset 𝐴 of a

group 𝐺, denoted by 〈𝐴〉. Rosero and

Baldado characterized sets 𝐴 with unique

〈𝐴〉 and sets whose number of generated 𝒟

-sets is equal to the index minimum.

This paper provides more results

on the 𝒟-sets generated by a non-empty

subset 𝐴 of a group 𝐺. The proofs here

make use of the concepts of 𝒟-sets.

Preliminary Concepts and Results

Throughout the study we refer 𝐺 as

a group. We also note that the groups considered here are finite.

We start with the general definition

of 𝒟-sets of a group. By this, we also give

definition of a minimum 𝒟-set of a

finite 𝐺 and index minimum of 𝐺.

Definition 2.1 (Buloron et al, 2014) Let

𝑒 be the indentity of 𝐺. A subset 𝐷 of 𝐺 is

called a 𝒟-set if for every 𝑥 in

𝐺\𝐷, there exists 𝑦 ∈ 𝐷 such that 𝑥𝑦 =

𝑒 = 𝑦𝑥.

Example 2.2 Let 𝑆3 be the symmetric group of degree three, i.e.

𝑆3 =

{𝑒, (12), (13), (23), (123), (132)}.

𝐷 𝑆 𝐷 𝑆

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𝐷2= {𝑒, (12), (13), (23), (123)}, and 𝐷3 =

{𝑒, (12), (13), (23), (132)}.

Definition 2.3 (Rosero et al, 2014) A 𝒟-set

of 𝐺 with the least number of elements is

called a 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝒟 − 𝑠𝑒𝑡, denoted by

𝒟𝑚𝑖𝑛. The number of all minimum 𝒟-sets

of 𝐺 is called the 𝑖𝑛𝑑𝑒𝑥 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 of

𝐺 and is denoted by 𝑖(𝐺).

It is

known in [4] that if 𝑥 ∈ 𝐺 such that 𝑥2₌

𝑒, then 𝑥 is an element of any

𝒟-set. Hence if 𝑆𝐺 = {𝑠 ∈ 𝐺 ∶ 𝑠2= 𝑒},

then 𝑆𝐺⊆ D for any 𝒟-set 𝐷.

It is discussed in [1] that the

relation ~ defined on 𝐺\𝑆𝐺given 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,\𝑆𝐺 =

{𝑎1, 𝑎1−1} ∪ {𝑎2, 𝑎2−1} ∪ … ∪ {𝑎𝑐,𝑎𝑐−1}.

If 𝑎𝑖 ≠ 𝑎𝑗 for 𝑖 ≠ 𝑗, then we call the given

partition a 𝑐𝑎𝑛𝑜𝑛𝑖𝑐𝑎𝑙 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛 of 𝐺\𝑆𝐺.

Remark 2.4[3] Let 𝐺 be a finite group and

𝐷 be a 𝒟-set of 𝐺. Then

𝐷 = 𝑆𝐺 ∪ {𝑥1, 𝑥2, … , 𝑥𝑐},

where 𝑥𝑖𝜖 {𝑎𝑖, 𝑎𝑖−1} for 𝑖 = 1,2, … , 𝑐 and

𝐺\𝑆𝐺 ={𝑎1, 𝑎1−1} ∪ {𝑎2, 𝑎2−1} ∪ … ∪

{𝑎𝑐,𝑎𝑐−1} is a canonical partition, if and

only if 𝐷is a minimum 𝒟-set.

Definition 2.5 Let𝑃𝐺 = {𝑃1, 𝑃2, … , 𝑃𝑐} =

{{𝑎1, 𝑎1−1},{𝑎2, 𝑎2−1},..., {𝑎𝑐,𝑎𝑐−1}}, with 𝑃𝑖 ≠ 𝑃𝑗for 𝑖 ≠ 𝑗 and 𝑃𝑖 {𝑎𝑖, 𝑎𝑖−1}

,

be the set of all paired non- involuntary

elements of a group 𝐺. The cardinality or

the number of elements of the set 𝑃𝐺 is the

𝒞 −number of 𝐺 and written as 𝑐𝐺.

It is mentioned in (Rosero et al.,

2014) that the 𝒞 −number of 𝐺 is half the

total number of all its non-involuntary elements.

Definition 2.6 Let 𝑀1, 𝑀2 ⊆ 𝐺 and 𝑀1=

{𝑎₁, 𝑎2, … , 𝑎𝑐} and 𝑀2=

{𝑎₁−1_{, 𝑎}

2−1, … , 𝑎𝑐−1} be the set of all non-

involuntary elements of 𝐺 such that 𝑎𝑖 ≠

𝑎𝑗and 𝑎𝑖−1≠ 𝑎𝑗−1, for 𝑖 ≠ 𝑗, for all 𝑖, 𝑗, ∈

{1,2, … , 𝑐} and 𝑀1∩ 𝑀2= ∅. Then 𝑀1

and 𝑀2are said to be a separating set of of

non-involutions. Let 𝑆𝐺 = {𝑠 ∈ 𝐺: 𝑠2= 𝑒}

, be the set of all involutions of 𝐺 including

the identity element. Then 𝐺 = 𝑆𝐺 ∪

𝑀1 ∪ 𝑀2and |𝐺| = |𝑆𝐺| + 2𝑐𝐺.

Lemma 2.7 If |𝐺| = 𝑛 is odd, then 𝑐𝐺 = 𝑛−1

2 .

𝑃𝑟𝑜𝑜𝑓 ∶ Any group of odd order, the

identity is the only involution. Hence 𝑆𝐺 =

{𝑒}. If 𝑛 is the order of the group, then|𝐺| =

𝑛 = 𝑆𝐺+ 2𝑐𝐺, which implies that 𝑐𝐺 = 𝑛−1

2 .

The following results are found in (Rosero et al., 2014) and they will be used in the succeeding discussions.

Theorem 2.8 Let 𝐺 be a finite

group. If 𝑐𝐺is the 𝒞 −

𝑛𝑢𝑚𝑏𝑒𝑟 of 𝐺, then 𝑖 (𝐺) = 2𝑐𝐺_.

Theorem 2.9 Let 𝐺 be a finite

groupand 𝑇𝐺be the collection of all of its

𝒟 − 𝑠𝑒𝑡𝑠. If 𝑐𝐺is the 𝒞 −

𝑛𝑢𝑚𝑏𝑒𝑟 of 𝐺, then |𝑇𝐺| = 3𝑐𝐺.

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3 𝓓 −sets Generated by a Subset

Definition 3.1 (Rosero and Baldado) Let

𝐴 be a non-empty subset of 𝐺. The smallest

𝒟 −set of 𝐺 that contains 𝐴 is called the 𝒟

-set generated by 𝐴, denoted by 〈𝐴〉.

Example 3.2 Consider the group 𝑈9 =

{1, 2, 4, 5, 7, 8} under multiplication

𝑚𝑜𝑑𝑢𝑙𝑜 9. We have 𝑆𝑈9 = {1, 8} and

{{2, 5}, {4, 7}} is a canonical partition

of 𝑈9which implies that the 𝒞 −

𝑛𝑢𝑚𝑏𝑒𝑟 of 𝑈9 𝑖𝑠 2. Let 𝐴 = {2, 4}. To

determine the 〈𝐴〉, we have

|𝑇𝑈9| = 3

𝑐_𝐺9 _{= 3}2_{= 9}

and the elements of 𝑇𝑈₉are as follows:

𝐷1 =

{1, 8, 2, 5,4,7}

𝐷2 = {1, 8, 2, 5,4}

𝐷3 = {1, 8, 2, 5,7}

𝐷4 = {1, 8, 4, 7,2}

𝐷5 = {1, 8, 4,7,5}

𝐷6 = {1, 8, 5,7}

𝐷7 = {1, 8, 5,4}

𝐷8 = {1, 8, 2,7}

𝐷9 = {1, 8, 2, 4}

Observe that 𝐴 = {2,4} is the subset of

𝐷1 , 𝐷2, 𝐷4 𝑎𝑛𝑑 𝐷9. The smallest set of

which is 𝐷9. Thus 〈𝐴〉 = 〈{2,4}〉 =

𝐷9= {1,8,2,4}.

Definition 3.3 (Rosero and Baldado) Let

𝐺 be a group. The number of distinct 𝒟 −

𝑠𝑒𝑡𝑠 of 𝐺 generated by 𝐴 is denoted by 𝑖(𝐴).

Corollary 3.4 (Rosero and Baldado)

𝐼𝑓 𝑆𝐺= {𝑠 ∈ 𝐺 ∶ 𝑠2= 𝑒} 𝑎𝑛𝑑 𝐴 ⊆

𝑆𝐺, 𝑡ℎ𝑒𝑛 𝑖(𝐴)=2𝑐𝐺.

Proposition 3.5 Let 𝐺 be a group with 𝑐𝐺 ≠ 0. If 𝐴 ⊆ 𝐺 𝑎𝑛𝑑 𝐴 =

𝑎 ∉ 𝑆𝐺then 〈𝐴〉 = 𝒟𝑚𝑖𝑛and 𝑖(𝐴) =

2𝑐𝐺−1_.

𝑃𝑟𝑜𝑜𝑓 : Suppose 𝐴 is a singleton subset of

𝐺 and 𝐴 ⊈ 𝑆𝐺. Let 𝐷𝑟 = 𝑆𝐺 ∪

{𝑎, 𝑏₁, 𝑏2, 𝑏3, … } = 𝑆𝐺 ∪ 𝑀 be the sets of

𝒟 −sets of 𝐺 that contain 𝐴 where

𝑀 ={ 𝑎, 𝑏1, 𝑏2, 𝑏3, …}. To get the smallest

𝒟-set that contains 𝐴, 𝑎−1 therefore should

not be an element of 𝑀 and

𝑏1−1, 𝑏2−1, 𝑏3−1,... are not elements of the

set 𝑀, that is, 𝑀 should contain no inverse

to each of its elements. Thus, 𝑀 is a

separating set of 𝐺. The smallest 𝒟-set then

should be a 𝒟𝑚𝑖𝑛. Therefore, 〈𝐴〉 = 𝒟𝑚𝑖𝑛.

Now, since 𝑀 is a separating set and

|𝑀| = 𝑐𝐺, then 𝑀 =

{𝑎, 𝑏1, 𝑏2, 𝑏3, … 𝑏𝑐𝐺−1}. Let 𝑘 = 𝑐𝐺 − 1.

All the minimum 𝒟-sets that contain 𝐴 are

of the form 𝑆𝐺 ∪ {𝑎} ∪ 𝐵 where 𝐵 =

{𝑏𝑖|𝑏𝑖 ∈ (𝑏1, 𝑏2, … , 𝑏𝑘) ∈ ∏𝑘𝑖=1𝑃𝑖} with

𝑃 = {𝑃1, 𝑃2, … , 𝑃𝑘}. Thus, all we need to

do is to count the total number of set 𝐵’s

produced. Note that the total number of 𝐵’s

is just the cardinality of ∏𝑘_𝑘=1𝑃𝑖. By the Multiplication Property and because each

𝑃𝑖 (for all 𝑖 ∈ {1,2, … , 𝑘}) is of order 2, we have 𝑖(𝐴) = | ∏𝑘𝑖=1𝑃𝑖 |= |𝑃1| ∙ |𝑃2| ∙ … ∙

|𝑃𝑘| = 2 ∙ 2 ∙ … ∙ 2 = 2𝑘= 2𝑐𝐺−1.

The next results determine the

number of smallest 𝐷-sets of 𝐺 generated

by a non-empty subset 𝐴 of 𝐺. Let 𝑐𝐴be the

𝐶-number of 𝐴 and let 𝑆𝐴= {𝑎 ∈ 𝐴 ∶ 𝑎2=

𝑒} be the set of involutions of 𝐴.

Proposition 3.6 〈𝐴〉 = 𝒟𝑚𝑖𝑛 and

𝑖(𝐴) = 1 if and only if 𝑐𝐴 = 0 and 𝑐𝐴 =

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𝑃𝑟𝑜𝑜𝑓 : Suppose 𝑐𝐺 = |𝐴| − |𝑆𝐴|. Then

|𝐴| = |𝑆𝐴| + 𝑐𝐺. This implies that 𝐴 is of the form 𝑆𝐴 ∪ {𝑎1, 𝑎2, … , 𝑎_𝑐𝐺} where 𝑎𝑖 ≠

𝑎𝑗−1 for all 1 ≤ 𝑖 ≤ 𝑐𝐺 and 1 ≤ 𝑗 ≤ 𝑐𝐺.

For all 𝑠 ∈ 𝑆𝐺, this implies that 𝑠 ∈ 𝐷 for

any 𝒟-set 𝐷 of 𝐺. Let 𝑀 =

{𝑎1, 𝑎2, … , 𝑎𝑐𝐺}. Since 𝑐𝐴 = 0, then 𝑀

therefore is a separating set of non-involutions. Thus, the smallest set such that

𝐷𝑖⋂3𝑖=1𝑐𝐺𝑀 is a 𝒟𝑚𝑖𝑛, for all 𝒟-sets 𝐷𝑖 of 𝐺,

implying 〈𝐴〉 =𝒟𝑚𝑖𝑛. By the property of a

𝒟-set, either 𝑎𝑖 ∈ 𝐷 or 𝑎1−1 ∈ 𝐷 for all

𝑖 ∈ {1,2, … , 𝑐}. Hence, 𝑖(𝐴) = 1.

Conversely, suppose 〈𝐴〉 = 𝒟_𝑚𝑖𝑛 and

𝑖(𝐴) = 1. Then 𝐴 contains a separating set

of non-involutions of 𝐺. Hence 𝑐𝐴 = 0.

Thus, 𝐴 = 𝑆𝐴 ∪ {𝑎1, 𝑎2, … , 𝑎𝑐} implying

|𝐴| = |𝑆𝐴| + 𝑐𝐺. Therefore, 𝑐𝐺= |𝐴| −

|𝑆𝐴|.

Corollary 3.7 Let 𝐺 be a group and 𝐴 ⊆ 𝐺. If 𝑐𝐴 = 0 and 𝑐𝐺= |𝐴| − |𝑆𝐴|, then

|〈𝐴〉 | = |𝑆𝐺 | + 𝑐𝐺.

Proposition 3.8 〈𝐴〉 = 𝐺 and 𝑖(𝐴) = 1 if and only if 𝑐𝐴 = 𝑐𝐺 .

𝑃𝑟𝑜𝑜𝑓 : Suppose 𝑐𝐴 = 𝑐𝐺. Then

𝐴 contains all the non-involuntary

elements of 𝐺. Hence, 𝐴 can be written as

𝐴 = 𝑆𝐴 ∪ 𝑀1 ∪ 𝑀2 where 𝑀1 𝑎𝑛𝑑 𝑀2 are the two separating sets of

non-involutions of the group 𝐺. Let 𝑀 =

𝑀1 ∪ 𝑀2. Since for all 𝑠 ∈ 𝑆𝐺 is also an

element of any 𝒟-set 𝐷 of 𝐺, then the only

𝒟-set that contains 𝐴 is the group 𝐺, i.e.

𝐷𝑖⋂3𝑖=1𝑐𝐺𝑀 = 𝐺 for all 𝒟-sets 𝐷𝑖of 𝐺. The converse is trivial.

Corollary 3.9 If 𝑐𝐴 = 𝑐𝐺, then |〈𝐴〉| =

|𝐺|.

From the previous discussions and

results, any element 𝑎 of 𝐴 such that 𝑎 ∈

𝑆𝐺, belongs to every 𝒟-set of 𝐺. Thus, the

elements that can affect and determine the

smallest 𝒟-set that contains 𝐴 are the

non-involution elements of 𝐴. The next result

will determine in general the smallest 𝒟-set

that contains a non-empty subset 𝐴 of 𝐺.

Proposition 3.10 Let 𝐺 be a group and 𝐴 ⊆ 𝐺. Let 𝑘 = 𝑐𝐺+ 𝑐𝐴+ |𝑆𝐴| −

|𝐴| where 1 ≤ 𝑐𝐴< 𝑐𝐺. Then 𝑖(𝐴) =

2𝑘and |〈𝐴〉| = |𝑆𝐺| + 𝑐𝐺 + 𝑐𝐴.

𝑃𝑟𝑜𝑜𝑓 : Without loss of generality, 𝐴 can

be written as 𝑆𝐴 ∪

{𝑎1, 𝑎1−1, … , 𝑎_𝑐𝐴, 𝑎_𝑐𝐴−1, 𝑏1, … , 𝑏𝑟} where

𝑟 = |𝐴| − |𝑆𝐴| − 2𝑐𝐴 and 𝑏𝑖 are the

non-involuntary elements of 𝐴 where 𝑖 ∈

{1,2, … , 𝑟} such that their corresponding

inverses are not in 𝐴. Then = 𝑆𝐴∪

{𝑎1, 𝑎1−1, … , 𝑎_𝑐𝐴 , 𝑎_𝑐𝐴−1} ∪ {𝑏1, … , 𝑏𝑟}.

The smallest 𝒟-sets that contain 𝐴 are of

the form 𝒟𝑚 = 𝑆𝐺 ∪ 𝐴 ∪ {𝑐1, … , 𝑐𝑘} =

𝑆𝐺 ∪ 𝐴 ∪ 𝐶 where

𝑘 = 𝑐𝐺− 𝑐𝐴− 𝑟 and the elements of 𝐶 are

of the non-involuntary elements of 𝐺 such

that their corresponding inverses are not in

𝒟𝑚. Hence 𝐶 = {𝑐𝑖|𝑐𝑖 ∈ (𝑐1, 𝑐2, … , 𝑐𝑘) ∈

∏𝑘_𝑖=1𝑃𝑖} with 𝒫 = {𝑃1, 𝑃2, … , 𝑃𝑘}. To

arrive at the total number of the smallest 𝒟

-set generated by 𝐴 is to count the total

number of 𝐶 produced. Hence,

𝑖(𝐴) = |∏𝑘_𝑖=1𝑃𝑖| = |𝑃1| ∙ |𝑃2| ∙ … ∙ |𝑃𝑘|

= 2 ∙ 2 ∙ … ∙ 2 = 2𝑘_.

Since 𝑟 = |𝐴| − |𝑆𝐴| − 2𝑐𝑎 and 𝑘 = 𝑐𝐺−

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𝑘 = 𝑐𝐺− 𝑐𝐴− 𝑟

= 𝑐𝐺− 𝑐𝐴− (|𝐴| − |𝑆𝐴| − 2𝑐𝐴) = 𝑐𝐺− 𝑐𝐴− |𝐴| + |𝑆𝐴| + 2𝑐𝐴 = 𝑐𝐺+ 𝑐𝐴+ |𝑆𝐴| − |𝐴|

Hence,

𝑖(𝐴) = 2𝑐𝐺+𝑐𝐴+|𝑆𝐴|−|𝐴|_{= 2}𝑘_.

To get the cardinality of the 𝒟𝑚

generated by 𝐴,

𝐷𝑚= 𝑆𝐺∪ 𝐴 ∪ 𝐶

= 𝑆𝐺∪ 𝑆𝐴∪ {𝑎1,𝑎1,−1, … , 𝑎_𝑐𝐴, 𝑎_𝑐𝐴−1}

∪ {𝑏1, … , 𝑏𝑟} ∪ {𝑐1, … , 𝑐𝑘} = 𝑆𝐺∪ {𝑎1,𝑎1,−1, … , 𝑎_𝑐1, 𝑎_𝑐1−1}

∪ {𝑏1, … , 𝑏𝑟} ∪ {𝑐1, … , 𝑐𝑘}

Thus

|〈𝐴〉| = |𝐷𝑚|

= |𝑆𝐺| ∪ |{𝑎1,𝑎1,−1, … , 𝑎_𝑐𝐴, 𝑎_𝑐𝐴−1}|

∪ |{𝑏1, … , 𝑏𝑟} ∪ {𝑐1, … , 𝑐𝑘}|

= |𝑆𝐺| + 2𝑐𝐴+ 𝑟 + 𝑘

= |𝑆𝐺| + 2𝑐𝐴+ (|𝐴| − |𝑆𝐴| − 2𝑐𝐴)

+( 𝑐𝐺+ 𝑐𝐴+ |𝑆𝐴| − |𝐴|)

= |𝑆𝐺| + 2𝑐𝐴+ |𝐴| − |𝑆𝐴| − 2𝑐𝐴+ 𝑐𝐺

+𝑐𝐴+ |𝑆𝐴| − |𝐴|

= |𝑆𝐺|+ 𝑐𝐺+ 𝑐𝐴+ 2𝑐𝐴− 2𝑐𝐴+ |𝐴|

−|𝐴| − |𝑆𝐴| − |𝑆𝐴| = |𝑆𝐺|+ 𝑐𝐺+ 𝑐𝐴

4 𝒟-SETS GENERATED BY A

SUBGROUP

It is obvious that 𝐻 = {𝑒} then

〈𝐻〉 = 𝒟𝑚𝑖𝑛. Furthermore, if 𝐻 = {ℎ ∈

𝐺: ℎ2= 𝑒} is a subgroup of 𝐺, then 〈𝐻〉 =

𝒟𝑚𝑖𝑛 and 𝑖(𝐻) = 2 𝑐𝐺.

Remark 4.1 Let 𝐺 be a group of odd order say 𝑝. Then |𝑇𝐺| = 3𝑘and

𝑖(𝐺) = 2𝑘where 𝑘 =𝑝−1

2 .

𝑃𝑟𝑜𝑜𝑓: Since |𝐺| = 𝑝 is odd, then by

Lemma 2, 𝑐𝐺 = 𝑝−1

2 . Hence, by

Theorem 2.8, we have |𝑇_𝐺| = 3𝑝−12 _and

𝑖(𝐺) = 2𝑝−12 _.

Proposition 4.2 Let 𝐺 be a group of

order 𝑝𝑞 where 𝑝 and 𝑞 are distinct odd

primes. If 𝐻 and 𝐾 are two non-trivial

subgroups of 𝐺, then

𝑖(𝐻 ∪ 𝐾) = √2(𝑝−1)(𝑞−1) and

|〈𝐻 ∪ 𝐾〉| =𝑝𝑞+1

2 .

𝑃𝑟𝑜𝑜𝑓: Suppose 𝐺 is a group of order 𝑝𝑞

where 𝑝 and 𝑞 are distinct odd primes.

Then by Sylow Theory, 𝐺 has a unique

subgroup 𝐻 of order 𝑝 and a subgroup 𝐾 of

order .

Now, since 𝐻, 𝐾 ≤ 𝐾, this implies

𝐻 ∩ 𝐾 ≤ 𝐺. Let |𝐻 ∩ 𝐾| = 𝑠. Since 𝐻 ∩ 𝐾 ≤ 𝐻 and 𝐻 ∩ 𝐾 ≤ 𝐾, then by Lagrange

Theory, 𝑠 divides |𝐻| and 𝑠 divides |𝐾|.

Since gcd(|𝐻|, |𝐾|) = gcd(𝑝, 𝑞) = 1, we

must have 𝑠 = 1. Hence, 𝐻 ∩ 𝐾 = {𝑒},

implying that |𝐻 ∪ 𝐾| = |𝐻| + |𝐾| − |𝐻 ∩

𝐾| = 𝑝 + 𝑞 − 1. Since 𝐻, 𝐾 and 𝐺 are of

odd order, then 𝑆𝐺 = 𝑆𝐻 = 𝑆𝐾= {𝑒}.

Thus, 𝑐𝐺=

𝑝𝑞−1 2 , 𝑐𝐻 =

𝑝−1

2 , 𝑎𝑛𝑑 𝑐𝐾 = 𝑞−1

(6)

Now, the smallest 𝐷-sets that

contains 𝐻 ∪ 𝐾are of the form,

𝒟

_𝑚

= 𝑆

_𝐺

∪ (𝑐

_𝐻

∪ 𝑐

_𝐾

) ∪ 𝐶

=

{𝑆𝐺, 𝑎1, 𝑎1−1, … , 𝑎𝑝−1, 𝑎𝑝−1−1, 𝑏1, 𝑏1−1, … , 𝑏𝑞−1,𝑏𝑞−1,−1, 𝑐1, … , 𝑐𝑟}

where

𝑟 = 𝑐

𝐺

− 𝑐

𝐻

− 𝑐

𝐾

=

𝑝𝑞−1

2

−

𝑝−1 2

−

𝑞−1 2

=

𝑝𝑞−1−𝑝+1−𝑞+1 2

=

𝑝𝑞−𝑝−𝑞+1

2

=

(𝑝−1)(𝑞−1)

2

and the elements of 𝐶 are the

non-involuntary elements of 𝐺 such that their

corresponding inverses are not in 𝒟𝑚.

Hence, 𝐶 = {𝑐𝑖|𝑐𝑖 ∈ (𝑐1, 𝑐2, … , 𝑐𝑟) ∈

∏𝑟𝑖=1𝑃𝑖} with 𝒫 = {𝑃1,𝑃2, … , 𝑃𝑟}.

Obtaining the total number of the

smallest 𝒟-set generated by 𝐻 ∪ 𝐾 is to

count the total number of 𝐶 produced. In

this case, the total number of 𝐶 is just the

cardinality of ∏𝑟𝑖=1𝑃𝑖 . Thus,

𝑖(𝐻 ∪ 𝐾) = |∏ 𝑃

_𝑖

𝑟

𝑖=1

|

= |𝑃

₁

| ∙ |𝑃

₂

| ∙. . .∙ |𝑃

_𝑟

|

= 2 ∙ 2 ∙ … ∙ 2 = 2

𝑟

= 2

(𝑝−1)(𝑞−1)2

= √2

(𝑝−1)(𝑞−1)

_.

Solving for the cardinality of the 𝒟𝑚

generated by 𝐻 ∪ 𝐾, we have

|𝒟𝑚| = |𝑆𝐺| ∪ (𝑐𝐻∪ 𝑐𝐾) ∪ 𝐶

= |𝑆𝐺∪ {𝑎1, 𝑎1−1, … , 𝑎𝑝−1, 𝑎𝑝−1−1} ∪

{𝑏1, 𝑏1−1, … , 𝑏𝑞−1,𝑏𝑞−1,−1} ∪ {𝑐1, … , 𝑐𝑟}|

=1 +𝑝−1

2 + 𝑞−1

2 + 𝑟

=1 +𝑝−1₂ +𝑞−1₂ +(𝑝−1)(𝑞−1)₂ =1 +𝑝−1₂ +𝑞−1₂ +𝑝𝑞−𝑝−𝑞+1₂ =2

2+ 𝑝−1

2 + 𝑞−1

2 +

𝑝𝑞−𝑝−𝑞+1 2

=

2+𝑝−1+𝑞−1+𝑝𝑞−𝑝−𝑞+1

2

=

𝑝𝑞+1

2

=

|〈𝐻 ∪ 𝐾〉|

Proposition 4.3 If 𝐺 is a group of even

order, then there exists a non-trivial

subgroup 𝐻 of 𝐺 such that |𝒟_{𝑚𝑖𝑛𝐻}| =

1 and |𝑇𝐻| = 1.

𝑃𝑟𝑜𝑜𝑓: It is mentioned in (Rosero et al,

2014) that the order of 𝐺 and the number of

elements of 𝑆𝐺 have the same parity. Hence

if 𝐺 is a group of even order, then 𝑆𝐺 has

also an even number of elements. Thus,

there exists 𝑎 ≠ 𝑒 ∈ 𝐺 such that 𝑎2 = 𝑒.

Let 〈𝑎〉 = {𝑒, 𝑎} = 𝐻. Now, 𝐻 is a

non-trivial subgroup of 𝐺 with |𝐻| = 2. Since

𝑆𝐻 = 𝐻, then 𝑐𝐻 = 0. Therefore,

|𝒟_{𝑚𝑖𝑛𝐻}| = 1 and |𝑇_𝐻| = 1.

Proposition 4.4 Let 𝐺 be a finite group of

order 𝑛 such that 𝑛 is divisible by an odd

prime 𝑝, then there exists a nontrivial

subgroup 𝐻 of 𝐺 such that

𝑖(𝐻) = √2

𝑝−1

and

(7)

𝑃𝑟𝑜𝑜𝑓: Suppose 𝐺 is a group of order 𝑛

where 𝑝 divides 𝑛 for an odd prime 𝑝. Then

by Sylow Theory, 𝐺 contains an element of

order 𝑝 say 𝑔. Let 𝐻 = 〈𝑔〉 =

{𝑒, 𝑔1, 𝑔2, … 𝑔𝑝−1}. Since |𝐻| = 𝑝 is odd,

then 𝑆𝐻= {𝑒}.This implies that 𝑐𝐻 =

𝑝−1 2 .

Thus from Theorem 2.8 and

Theorem 2.9, 𝑖(𝐻) = 2𝑐𝐻 _{= 2} 𝑝−1

2 =

√2𝑝−1_and_|𝑇

𝐻| = 3𝑐𝐻 = 3

𝑝−1

2 = √3𝑝−1_.

REFERENCES

C. S. Rosero, J. N. Buloron, J. M. Ontolan

and M. P. Baldado Jr. (2014), 𝒟 -

sets of Finite Groups, International

Journal of Algebra, 8,

623-628.http://dx.doi.

org/10.12988/ija.2014.4776

C. S. Rosero, and M. P. Baldado Jr. (2014),

Some Properties of 𝛾-sets in a Ring,

International Journal of Algebra, 8, 883-888.

http://dx.doi.org/10. 12988/ija.2014.41098

C. S. Rosero, and M. P. Baldado Jr., On the

𝒟 -sets Generated by a Subset of a

Group, (Submitted)

J. N. Buloron, C. S. Rosero, J. M. Ontolan and M. P. Baldado Jr. (2014), Some

Properties of 𝒟 -sets of a Group,

International Mathematical Forum,

9, 1035-1040. http://dx.doi.org/

10.12988/imf.2014.45104