CENTRALIZER AND NORMALIZER OF A MULTIGROUP

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International Journal of Recent Research in Mathematics Computer Science and Information Technology

Vol. 9, Issue 2, pp: (42-54), Month: October 2022 – March 2023, Available at: www.paperpublications.org

CENTRALIZER AND NORMALIZER OF A MULTIGROUP

1.2 Department of Mathematics, Faculty of Natural and Applied Sciences, Nassarawa State University, Keffi.

DOI: https://doi.org/10.5281/zenodo.7496529

Published Date: 01-January-2023

Abstract: The concept of multiset generalises the classical set, and so multigroup is an algebraic structure of multiset. In thispaper we study the concept of centralizerand normalizer of a group under thecontext of multiset. The closure of these concept was studied under union, intersection, arithemetic addition, compposition and arithmetic multiplication among others. We have also shown that a centralizer and normalizer is not empty and are sub multi group of a multi group alongside defining the normal sub multi group of a multi group.

Keywords: Multiset, Multigroup, Commutative multigroup, Centralizer and Normalizer of mulrigroup.

1. INTRODUCTION

Multisets seems to generalized the Cantorian sets ([1],[7],[8],[9],[10],[13],[14]). It was first suggested by N.G, de Bruijn ([3]) in a private communication which opposes the basic principle that an element can belong to a set onlyonce. This study generated the study of group theory under multiset perspective which we called multi group ([5],[11],[12]).

Research is ongoing concerning multigroup as it began by a good number of researchers up untill Tella and Daniel, (2013), and was improve by Nazmul et al, (2013), Ejegwa and Ibrahim, (2017) worked on much of the properties of the classical group theory under multiset contaxt Some of the research work done is ranging from multiset operation of multi group, homomorphic nature of multi group, sub multi group and normal sub multi group, Abelian multi group, Centre of a multi group among others. ([15],[16],[17],[18])

This paper seeks to extend and to reconsider in a different perspective the study of centralizer, and normalizer of a multi group with an intension to study more results on it. That is, the operations on, the sub and normal sub multi group of these concepts and their properties of. In addition to this section we present some preliminary definitions, notations and the introductions of some of the definitions of our terms in chapter two. In chapter three we present some basic results on the concepts. We summarise our work in chapter four and then conclude.

2. PRELIMINARY DEFINITIONS AND NOTATIONS

Definition 2.1[1]. An mset �� over the set �� can be defined as a function ����:�� →ℕ={0,1,2, } where the value ����(��) denotethe number of times or multiplicity or count function of �� ������ For example, Let �� = [��,��,��,��,��,��,��,��], then ����(��)=3,����(��) =3,����(��) = 2 ����(��)=0⇒�� ∉��

The mset �� over the set �� is said to be empty if ����(��)=0�������������� ∈�� We denote the empty mset by ∅. Then ��∅(��)= 0,∀�� ∈�� ��������(��)>0,��ℎ������ ∈��

Definition2.2[1]:The cardinality of a mset �� denoted |��| or ��������(��) is the sum of all the multiplicities of its elements given by the expression ��������(��) =∑ ����(��) ��∈��

Definition 2.3[2]: Let �� be an mset drawn from a set ��. The support set of �� denoted by ��∗ is a subset of �� and ��∗ = {������:����(��)>0} that is ��∗ is an ordinary set. ��∗ is also called root set.

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International Journal of Recent Research in Mathematics Computer Science and Information Technology

Vol. 9, Issue 2, pp: (42-54), Month: October 2022 – March 2023, Available at: www.paperpublications.org

Definition 2.4[1]: Equal msets. Two msets �� and �� are said to be equal denoted ��=�� if and only if for any objects �� ∈

��,����(��)=����(��) This is to say that �� =�� if the multiplicity of every element in �� is equal to its multiplicities in B and conversely. Clearly, ��=��⟹��∗ =��∗ , though the converse need not hold.

For example: let �� =[��,��,��,��,��]�������� =[��,��,��

��,��,��,��]��ℎ��������

∗ =��∗

={��,��,��}�������� ≠��

Definition 2.5[1] Submsets: Let X be a set and let A and B be msets over X A is a submset of B, denoted by �� ⊆��������⊇

��, if ����(��)≤����(��)�������������� ∈��. Also if ��⊆����������≠��, then �� is called proper submset of �� denoted by �� ⊂��. In other words ��⊂�� if�� ⊆��andthereexistatleastone�� ∈ �� suchthat ����(��)<����(��).Weassertthatanmset �� iscalled the parent mset in relation to the mset ��

Definition 2.6[1]: Regular or Constant mset: A mset �� is called regular or constant if all its elements are of the same multiplicities, i.e for any ��,�� ∈ ��suchthat�� ≠��,����(��)=����(��)

Definition 2.7[1]: The notations ⋀ and ⋁:[6]. The notations ⋀ and ⋁ denote the minimum and maximum operator respectively for instance ����(��)⋀����(��) =������{����(��),����(��)}����������(��)⋁����(��) =������{����(��),����(��)}.

Definition 2.8[9]: Union (∪) of msets.Let �� and �� be two msets over a given domain set ��. The union of ��and�� denoted by ��∪�� is the mset defined by ����∪��(��)=������{����(��),����(��)},

That is object �� occurring �� times in �� and �� times in �� occur maximum {��,��} times in ��∪ ��, if such maximum exist. For example: Suppose �� =[��,��,��,��,��,��],�� =[��,��,��,��],��ℎ������∪�� =[��,��,��,��,��,��]

Definition 2.9[9]:Intersection (∩) of msets. Let �� and �� be two mset over a given domain set �� The intersection of two mset �� and �� denoted by ��∩ ��, is the mset for which ����∩��(��)=������{����(��),����(��)}for all �� ∈��.

In other words, ��∩�� is the smallest mset which is contained in both �� and ��. That is an objects x occuring �� times in �� and �� in ��, occurs minimum (��,��) times in ��∩ ��

From the above example, ��∩�� =[��,��,��]

Definition 2.10[9]: Addition or sum of Mset. Let �� and �� be two msets over a given domain set ��. The direct sum or arithmetic addition of �� and �� denoted by �� �� or ��⊎�� is the mset defined by ����+��(��)=����(��)+����(��) for all �� ∈��

That is, an object �� occurring a times in �� and b times in ��, occurs ��+�� times in ��⊎��

Using the above example we conclude that, ��⊎�� =[��,��,��,��,��,��,��,��,��] clearly∣A⊎B∣= ∣��∪��∣ + ∣��∩��∣.

Definition 2.11[9]: Difference of msets.Let A and B be two mset over a given domain set X. then the difference of �� from ��, denoted by �� �� is the mset such that ���� ��(��)=������{����(��) ����(��),0}for all �� ∈��. If �� ⊆��, then ���� ��(��)= ����(��) ����(��).

It is sometimes called the arithmetic difference of �� from �� If �� ⊈�� this definition still holds. It follows that the deletion of an element �� from an mset �� give rise to a new mset ��′ =�� �� such that ����′(��)={����(��) 1,0}.

Definition 2.12[8]: Symmetric Difference. Let �� be a set and ��, �� a mset over ��. Then the symmetric difference, denoted ��∆��, is defined by ����∆��(��)=|����(��) ����(��)|.

Definition 2.13[8]:Compliment in msets:Let �� ={��1,��2, } be a family of msets generated from the set ��. Then, the maximum mset �� is defined by ����(��)=��������∈������(��) for all �� ∈�� Complement of an mset ��, denoted by ��, is defined by �� =�� �������� ����(��)={����(��) ����(��),�������������� ∈��} where �� is a universal mset.

Theorem 2.14[8] Let �� and �� be any mset then the following holds:

(i) (��∪��)∗ =��∗ ∪��∗

(ii) (��∩��)∗ =��∗ ∩��∗

Theorem 2.15 [11]. For any �� ∈��(��), ��∗ =(����)∗ =(����)∗ �������������� ∈��������ℎ��ℎ������ ≥1

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Theorem 2.16[11]: Let ��,�� ∈��(��), �� ⊆�� →��∗ ⊆��∗

Definition 2.17[6]: Let �� be a group. An mset �� over �� is said to be a multigroup (mgroup for short) over �� if the count function ����(��) satisfied the following conditions:

(i) ����(����) ≥����(��)∧����(��)∀��,�� ∈��.

(ii) ����(�� 1)≥����(��)∀�� ∈��

It follows immediately that:

����(�� 1)=����(��)∀�� ∈��

We denote the set of all mgroups over �� by ����(��)

Theorem 2.18[5]: Let �� be a group. If �� is a mgroup over ��, then

(i) ����(��)≥����(��),∀�� ∈�� and �� identity in ��

(ii) ����(����)≥����(��),∀��,∀�� ∈ℕ

(iii) ����(�� 1)=����(��),∀�� ∈��

(iv) �� =�� 1

Definition 2.19[5]; Composition and Inverse. Let ��,�� ∈����(��), then we call

(i) ��∘�� as the composition between two mgroups defined as

����∘��(��) =⋁{����(��)∧����(��):��,�� ∈�� ∋���� =��} and

(ii) �� 1 is called the inverse of mgroup �� and defined as ���� 1(��) =����(�� 1) for all ��∈��

Theorem 2.20[5]. Let ��,�� ∈����(��), then the following assertions holds:

(i) [�� 1] 1 =��

(ii) �� ⊆�� ⟹�� 1 ⊆�� 1

(iii) (��∘��) 1 =�� 1 ∘�� 1

(iv) [∩��∈�� ����] 1 =∩��∈�� [���� 1]

(v) [∪��∈�� ����] 1 =∪��∈�� [���� 1]

(vi) (��∘��) ∘�� =��∘(��∘��)

Theorem 2.21[5]. Let ��,�� ∈ ����(��) Then ��∩�� ∈ ����(��)

Theorem 2.21[5]. Let ��,�� ∈ ����(��) Then ��∪�� ∉ ����(��)

Theorem 2.22[17]: If ��,�� ∈ ����(��), then the sum of �� and �� is a multigroup of ��

Theorem 2.23[17]: Let �� ∈ ����(��) and if ��,�� ∈�� with ����(��)≠����(��), then ����(����)=����(����)=����(��)⋀����(��)

Theorem 2.24[17]: Let �� ∈ ����(��) and �� be a nonempty submultiset of ��. Then the following statements are equivalent.

(i) �� is a submultigroup of ��

(ii) ����(����)=����(��)⋀����(��) and ����(�� 1)=����(��)∀��,�� ∈��

(iii) ����(���� 1)=����(��)⋀����(��)∀��,�� ∈��

Definition 2.25[5]: Let �� ∈ ����(��) and if ∀��,�� ∈��, we defined a commutative multigroup or an Abelian multigroup as ����(����) =����(����)

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For example; If �� ={��,��,��,��} be a commutative group then let �� ={��,��,��,��}4,3,3,2. Hence �� is an Abelian or commutative multigroup.

Theorem 2.26[5]: Let �� be a commutative multigroup of a group ��. Then,

(i) ����([��,��])=����(��)

(ii) ����([��,��])=����(��)

3. CENTRALIZER AND NORMALIZER OF A MULTIGROUP

Definition 3.1: Let �� be a multigroup over a group ��. Then the centralizer of ��, denoted ��(��) and defined ��(��) ={�� ∈��/����(����)=����(����)∀�� ∈��}

Example: Suppose the group �� = {0,1,2}=��3 and let �� ∈����(��), that is �� ={0,1,2}3,2,1. Clearly ��(��) is canbe shown

Thus ��(��) is a the centre of the multigroup ��

We denote ��(��) as the class of all centralizers of a multigroup.

Proposition 3.2: Let �� ∈����(��). Then the centre of ��, ��(��) is a subgroup of ��.

Proof: Let ����(��)≠∅. Then at least �� ∈��(��). Now let ��,�� ∈��(��).

Since ����([��,��])=����(��) and for all ��,�� ∈��

Consequently, ����([��,��])=��

if �� ∈��(��) then �� 1 ∈��(��) Now

Again,

This implies that ���� ∈��(��)

Showing �� 1 ∈��(��). Hence ��(��) is a subgroup of �� Proposition 3.3: Let ��,�� ∈����(��). If ��∩��

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��(��) =����(������ 1�� 1) ≥����(����)∧����(�� 1�� 1) =����(���� 1)∧����(���� 1) =����(��)∧����(��) =����(��)
���� ∈��(��) and
����([��,��])=����(������ 1�� 1) ≥����(����)∧����(�� 1�� 1) =����(����)∧����(����) 1 =����(����)⋀����(����) =����(����)
����([��,�� 1])=����(������ 1�� 1) ≥����(����)∧����(�� 1�� 1) =����(���� 1) 1 ∧����(���� 1) 1 =����(�� 1��)∧����(�� 1��) =����(��)∧����(��) =����(��)
let ��,�� ∈��(��) we want to show that
and
∈����(��)
��(��)∩��(��)∈ ��(��)
then

Therefore ��(��)∩��(��)∈ ��(��)

Proposition 3.4: Let �� ∈����(��) then ��(��) ∈����(��)

Proof: Since �� ∈����(��), then ��∗ is a group and ∀��,�� ∈��.

Now ����(����) ≥����(��)∧����(��) and ����(��) 1 =����(��)∀��,�� ∈��. Also ��(��)∈����(��) shows that ��(��)∗ is the centralizer of the group �� then ����(��)(����)≥����(��)(��)⋀����(��)(��) and ����(��)(��) 1 =����(��)(��)∀��,�� ∈��

Hence ��(��)∈����(��)

Proposition 3.5: Let ��,�� ∈����(��). If ��∪�� ∈����(��) then ��(��)∪��(��)∈ ��(��)

Proof: Let �� ∈����(��). Then ����(����) ≥����(��)∧����(��) and ����(��) 1 =����(��)∀��,�� ∈��

�� ∈����(��). Then ����(����) ≥����(��)∧����(��) and ����(��) 1 =����(��)∀��,�� ∈��

Now, ����∪��(����)=⋁{����∪��(��),����∪��(��)}

Therefore ��∪�� ∉����(��)

For ��(��)∪��(��) ∉ ��(��),

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Vol. 9,
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at: www.paperpublications.org Page | 46 Paper Publications Proof: Let �� ∈����(��). Then ����(����) ≥����(��)∧����(��) and ����(��) 1 =����(��)∀��,�� ∈�� �� ∈����(��). Then ����(����) ≥����(��)∧����(��) and ����(��) 1 =����(��)∀��,�� ∈��. Now, ����∩��(����)=⋀{����∩��(��),����∩��(��)} ≥⋀{[����(��)∧����(��)],[����(��)∧����(��)]} ≥����(��)∧����(��)⋀����(��)∧����(��) =[����(��)⋀����(��)]∧[����(��)⋀����(��)] =����∩��(��)⋀����∩��(��) And ����∩��(��) 1 =����∩��(��) Therefore ��∩�� ∈����(��). For ��(��)∩��(��) ∈ ��(��), Then ����(��)∩��(��)(����) ={����(��)∩��(��)(����)⋀����(��)∩��(��)(����)}∀��,�� ∈�� ≥⋀{[����(��)(��)∧����(��)(��)],[����(��)(��)∧����(��)(��)]} =����(��)(��)∧����(��)(��)⋀����(��)(��)∧����(��)(��) =[����(��)(��)⋀����(��)(��)]∧[����(��)(��)⋀����(��)(��)] =[����(��)(��)⋀����(��)(��)]⋀[����(��)(��)⋀����(��)(��)] =����(��)∩��(��)(��)⋀����(��)∩��(��)(��) =����(��)∩��(��)(����)
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Issue
pp:
Month: October 2022 – March
Available
≥⋁{[����
��
��
��
≥����
��(��) =[����(��)⋀����(��)]⋁[����(��)⋀����(��)] ≠����∪��(��)⋁����∪��(��)
(��)∧��
(��)],[��
(��)∧��
(��)]}
(��)∧����(��)⋁����(��)∧��

International Journal of Recent Research in Mathematics Computer Science and Information Technology

Vol. 9, Issue 2, pp: (42-54), Month: October 2022 – March 2023, Available at: www.paperpublications.org

Therefore ��(��)∩��(��)∉ ��(��).

Remark: If ��∗ is an abelian group and �� ∈����(��) Then ��(��)=��

Proposition 3.6: Let ��,�� ∈����(��). If ��+�� ∈����(��) then ��(��)+��(��)∈ ��(��)

Proof:

Now, ����+��(����) ={����(����)+����(����)}

��(��)∧����(��)+����(��)∧����(��)

And ����+��(��) 1 =����+��(��)

Therefore ��+�� ∈����(��)

For ��(��)+��(��)∈ ��(��),

��+��(��)

then ����(��)+��(��)(����) ={����(��)+��(��)(��)⋀����(��)+��(��)(��)}∀��,�� ∈ ��.

≥{[����(��)(��)+����(��)(��)]⋀[����(��)(��)+����(��)(��)]}

=����(��)(��)∧����(��)(��)+����(��)(��)∧����(��)(��)

=[����(��)(��)⋀����(��)(��)]+[����(��)(��)⋀����(��)(��)]

=[����(��)(��)⋀����(��)(��)]+[����(��)(��)⋀����(��)(��)] =����(��)+��(��)(��)⋀����(��)+��(��)(��)

=����(��)+��(��)(����)

Therefore ��(��)+��(��) ∈ ��(��)

Proposition 3.7: Let ��,�� ∈����(��). If �� �� ∈����(��) then ��(��) ��(��)∈ ��(��)

Proof: Let �� ∈����(��). Then ����(����) ≥����(��)∧����(��) and ����(��) 1 =����(��)∀��,�� ∈��

�� ∈����(��). Then ����(����) ≥����(��)∧����(��) and ����(��) 1 =����(��)∀��,�� ∈��.

Now, ������(����) ={����(����).����(����)}

≥{[����(��)∧����(��)] [����(��)∧����(��)]}

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Page | 47 Paper Publications then ����(��)∪��(��)(����) ={����(��)∪��(��)(����)⋀����(��)∪��(��)(����)}∀��,�� ∈�� ≥⋁{[����(��)(��)∧����(��)(��)],[����(��)(��)∧����(��)(��)]} =����(��)(��)∧����(��)(��)⋁����(��)(��)∧����(��)(��) =[����(��)(��)⋀����(��)(��)]⋁[����(��)(��)⋀����(��)(��)] =[����(��)(��)⋀����(��)(��)]⋁[����(��)(��)⋀����(��)(��)] =����(��)∪��(��)(��)⋁����(��)∪��(��)(��) ≠����(��)∪��(��)(����)
����(����) ≥����(��)∧����(��) and ����(��) 1 =����(��)∀��,�� ∈�� �� ∈����(��)
��(��) and ����(��) 1 =����(��)∀��,�� ∈��
Let �� ∈����(��). Then
. Then ����(����) ≥����(��)∧��
=��
=[����(��)⋀����(��)]+[����(��)⋀����(��)] =����+��
≥{[����(��)∧����(��)]+[����(��)∧����(��)]}
(��)⋀��

�� ∈����(��). Then ����(����) ≥����(��)∧����(��) and ����(��) 1 =����(��)∀��,�� ∈��.

Now, ��������(����) ={����(����)������(����)}

≥{[����(��)∧����(��)]��[����(��)∧����(��)]}

=����(��)∧��

And ��������(��) 1 =��������(��)

Therefore ������ ∈����(��)

For ��(��)����(��)∈ ��(��), then ����(��)����(��)(����) ={����(��)����(��)(��)⋀����(��)����(��)(��)}∀��,�� ∈�� ≥{[����(��)(��)������(��)(��)]⋀[����(��)(��)������(��)(��)]} =����(��)(��)∧����(��)(��)������(��)(��)∧����(��)(��) =[����(��)(��)⋀����(��)(��)]��[����(��)(��)⋀����(��)(��)] =[����(��)(��)⋀����(��)(��)]��[����(��)(��)⋀����(��)(��)]

=����(��)����(��)(��)⋀����(��)����(��)(��)

=����(��)����(��)(����)

∈��

Therefore ��(��)����(��)∈ ��(��)

Proposition 3.9: Let ��,�� ∈����(��), then ��(��)����(��)⊆��(������)

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2023,
at: www.paperpublications.org Page | 48 Paper Publications =����(��)∧����(��) ����(��)∧����(��) =[����(��)⋀����(��)] [����(��)⋀����(��)] =������(��)⋀������(��) And ������(��) 1 =������(��) Therefore �� �� ∈ ����(��) For ��(��) ��(��) ∈ ��(��), then ����(��)��(��)(����) ={����(��)��(��)(��)⋀����(��)��(��)(��)}∀��,�� ∈�� ≥{[����(��)(��) ����(��)(��)]⋀[����(��)(��) ����(��)(��)]} =����(��)(��)∧����(��)(��) ����(��)(��)∧����(��)(��) =[����(��)(��)⋀����(��)(��)].[����(��)(��)⋀����(��)(��)] =[����(��)(��)⋀����(��)(��)] [����(��)(��)⋀����(��)(��)] =����(��)��(��)(��)⋀����(��)��(��)(��) =����(��)��(��)(����)
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Issue
pp:
Month: October 2022
March
Available
Therefore ��(��) ��(��) ∈ ��(��) Proposition 3.8: Let ��,�� ∈����(��). If ������ ∈����(��) then ��(��)����(��)∈ ��(��)
��
≥����(��)∧����(��) and ����(��) 1 =����(��)∀��,��
Proof: Let �� ∈����(��). Then ��
(����)
(��) =[����(��)⋀����(��)]��[����(��)⋀����(��)] =��������(��)⋀��������(��)
��(��)������(��)∧����

International Journal of Recent Research in Mathematics Computer Science and Information Technology

Vol. 9, Issue 2, pp: (42-54), Month: October 2022 – March 2023, Available at: www.paperpublications.org

Proof: Let �� ∈ ��(��) and �� ∈ ��(��), for all �� ∈��

Similarly, ��������(��(����))=��������((����)��).

Hence ���� ∈��(������)

Thus ��(��)����(��)⊆��(������)

Proposition 3.10: Let �� ∈����(��), then ��(��)⊆��

Proof: It is obvious that

Thus ���� ∈��(��)

Also,

Thus �� 1 ∈��(��)

Therefore ��(��)⊆��

Remark: Let �� ∈����(��), then ��(��) =��∗. If A is commutative or regular multigroup. Otherwise ��(��) ⊆��

Proposition 3.11: Let �� ∈������(��) be abelian, then

(i) ����([��,��])=����(��)

(ii) ����([��,��])≥����(��)

Where (��) is the identity element of �� and [��,��] is the commutator of ����������������

Proof: (i) Let ��,�� ∈�� such that ���������� commutes with each other. Now

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Now ��������((����)��)=⋁{����(��)⋀����(��)∀��,�� ∈��,������ =����} =⋁{����(�������� 1)⋀����(��)∀�� ∈��,������ =����} =⋁{����(�������� 1)⋀����(��)∀�� ∈��,������ =����} =⋁{����(��)⋀����(��)∀��,�� ∈��,������ =����} =⋁{����(��)⋀����(�� 1������)∀�� ∈��,������ =����} =⋁{����(��)⋀����(�� 1������)∀�� ∈��,������ =����} =⋁{����(��)⋀����(��)∀��,�� ∈��,������ =����} =��������(��(����))
≠∅
�� ∈��(��). Let ��,�� ∈��(��). We
to
���� ∈��(��)
����(����) =����(����)≥����(��)⋀����(��) =����(��)⋀����(��) ⟺����(���� 1���� 1)=����(��) ⟺����([��,��])=����(��)
��(��)
and since at least
want
show that
Then
����([��,��])=����(��)∀�� ∈�� . Hence ����([�� 1,��])=����(���� 1�� 1��)=����(���� 1�� 1������ 1) =����(�� 1�� 1������ 1��) =����([��,��]) =����([��,��] 1)=����([��,��])=����(��)
let �� ∈ ��(��). Then
����([��,��])=����(�� 1�� 1����) =����(�� 1���� 1��) ≥����(�� 1��)∧����(�� 1��) =����(��)∧����(��) =����(��) →����
����(��)=����(������ 1�� 1)≥����((������ 1�� 1)��)
([��,��])≥����(��) and

International Journal of Recent Research in Mathematics Computer Science and Information Technology

Vol. 9, Issue 2, pp: (42-54), Month: October 2022 – March 2023, Available at: www.paperpublications.org

����(���� ������������[��,��])≥����(���� ������������[��,��])=����((����)��).

⇒����(��������)=����((����)��)

Hence ����((����)��)=����(��������)

Definition 3.13: Let ��,�� ∈����(��), such that �� ⊆��, then we defined the Normalizer of �� in �� as ��(��)={�� ∈��/����(�� 1����)=����(��)∀�� ∈��}

Example: Let �� ={��,��,��,��} be a group under the multiplicative operation such that ���� =��,���� =��,���� =�� and ��2 =

��2 =��2 =��. Also let �� ={��,��,��,��}

4,3,2,2 be a multigroup and �� ={��,��,��,��}3,2,1,1 be a submultigroup ��. Then the normalizer of ��,��(��) is given by;

����(�� 1����)=����(������)=����((����)��)=����(����) =����(��)=1 by definition.

Hence ��(��) is a normalizer of �� in ��

We denote the class of all normalizers of �� in �� as Ɲ(��)

Proposition 3.14: Let �� be a multigroup and if ��1 and ��2are normal sub multi group of ��. Then ��(��1)∩��(��2)∈

Proof: Since ��(��1),��(��2)∈Ɲ(��), we have

ISSN 2350-1022
Page | 50 Paper Publications =����((������ 1�� 1)(������ 1�� 1)) ≥����(������ 1�� 1)∧����(������ 1�� 1) =����(�� 1�� 1����)
����([��,��])=����(�� 1�� 1����) =����(�� 1�� 1����) ≥����(�� 1)∧����(�� 1����) ≥����(��)∧����(��)=����(��)
����([��,��])≥����(��)
⇒����(��)≥����([��,��]) Hence ����([��,��])=����(��) (i)
Thus
�� ∈����(��) be abelian and �� ∈��. Then ����((����)��)=����(��������) for all ��,�� ∈��.
Let ��,�� ∈��. We
����((����)��)=����(���� ������������)=����(���� ��������2��[��,��]) ≥����(���� ��������2��)∧����([��,��])=����(��2�� ��������2)=����(��2�� ����3��)=����(��2�� ����3��[��,��]) ≥����(��2�� ����3��)≥⋯≥����(���� 1�������� 1)=����(���� 1������[��,���� 1])≥����(���� 1������)=����(��������) ⇒����((����)��)=����(��������). Also, ����(��������)=����(���� 1������)=����(���� 1��������[���� 1,��])≥����(���� 1�������� 1)≥⋯≥����(���� ��������2��)=
Proposition 3.12: Let
Proof:
have
Ɲ(��)
����(��1)(�� 1����)=����(��1)(��)∀��,�� ∈��∗ ����(��2)(�� 1����)=����(��2)(��)∀��,�� ∈��∗ Then ����(��1)∩��(��2)(�� 1����)=⋀{����(��1)(�� 1����),����(��2)(�� 1����)} =⋀{����(��1)(��),����(��2)(��)} =����(��1)(��)⋀����(��2)(��)∀�� ∈ ��1 ∗ ,�� ∈��1 ∗ ∩��2 ∗ =����(��1)∩��(��2)(��) ∀�� ∈��1 ∗ ,�� ∈��1 ∗ ∩��2 ∗

International Journal of Recent Research in Mathematics Computer Science and Information Technology Vol. 9, Issue 2, pp: (42-54), Month: October 2022 – March 2023, Available at: www.paperpublications.org

Hence ��(��1)∩��(��2)∈Ɲ(��)

Proposition 3.15: Let �� be a multigroup and if

Proof: Since ��(��1),��(��2)∈Ɲ(��), we have

Proof: Since ��(��1),��(��2)∈Ɲ(��), we have

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��1 and ��2
normal submultigroup
��
��(��1)∪��(��2)∉ Ɲ(��)
are
of
. Then
����(��1)(�� 1����)=����(��1)(��)∀��,�� ∈��∗ ����(��2)(�� 1����)=����(��2)(��)∀��,�� ∈��∗ Then ����(��1)∪��(��2)(�� 1����)=⋁{����(��1)(�� 1����),����(��2)(�� 1����)} =⋁{����(��1)(��),����(��2)(��)} ≠����(��1)(��)⋁����(��2)(��)∀�� ∈ ��1 ∗ ,�� ∈��1 ∗ ∪��2 ∗ ≠����(��1)∪��(��2)(��) ∀�� ∈��1 ∗ ,�� ∈��1 ∗ ∪��2 ∗ Hence ��(��1)∪��(��2)∉Ɲ(��) Proposition 3.16: Let �� be a multigroup and if ��1 and ��2 are normal submultigroup of ��. Then ��(��1)+��(��2)∈ Ɲ(��) Proof: Since ��(��1),��(��2)∈Ɲ(��), we have ����(��1)(�� 1����)=����(��1)(��)∀��,�� ∈��∗ ����(��2)(�� 1����)=����(��2)(��)∀��,�� ∈��∗ Then ����(��1)+��(��2)(�� 1����) ={����(��1)(�� 1����)+����(��2)(�� 1����)} ={����(��1)(��)+����(��2)(��)} =����(��1)(��)+����(��2)(��)∀�� ∈��1 ∗ ,�� ∈��1 ∗ +��2 ∗ =����(��1)+��(��2)(��) ∀�� ∈��1 ∗ ,�� ∈��1 ∗ +��2 ∗
and if ��1 and ��2are normal submultigroup of ��. Then ��(��1) ��(��2)∈Ɲ(��)
1),��(��2)∈Ɲ(��), we have ����(��1)(�� 1����)=����(��1)(��)∀��,�� ∈��∗ ����(��2)(�� 1����)=����(��2)(��)∀��,�� ∈��∗ Then ����(��1)��(��2)(�� 1����)={����(��1)(�� 1����) ����(��2)(�� 1����)} ={����(��1)(��).����(��2)(��)} =����(��1)(��) ����(��2)(��)∀�� ∈��1 ∗ ,�� ∈��1 ∗ ��2 ∗ =����(��1)��(��2)(��) ∀�� ∈��1 ∗ ,�� ∈��1 ∗ ��2 ∗ Hence
multigroup
��1 and ��2 are normal sub multigroup of ��.
��(��1)����(��2)∈Ɲ(��)
Hence ��(��1)+��(��2)∈Ɲ(��) Proposition 3.17: Let �� be a multigroup
Proof: Since ��(��
��(��1).��(��2)∈Ɲ(��) Proposition 3.18: Let �� be a
and if
Then
����(��1)(�� 1����)=����(��1)(��)∀��,�� ∈��∗

International Journal of Recent Research in Mathematics Computer Science and Information Technology

Vol. 9, Issue 2, pp: (42-54), Month: October 2022 – March 2023, Available at: www.paperpublications.org

Hence ����(�� 1����)=����(��) and so �� ∈��(��)

Therefore ��(��)=��

Conversely, suppose ��(��)=��,∀��,�� ∈��. We want to prove that �� is a normal subgroup of ��. Then �� ∈��, ����(��)=

Hence ����(�� 1����)=����(��)

That is �� is a normal sub multigroup of ��

Remark: Let �� be a normal sub multigroup of ��, then ��(��)=�� =��

If �� ={�� ∈��/����(����(����) 1)=����(��)∀�� ∈��} and �� ={�� ∈��/����(����) =����(����)∀�� ∈��}

Proposition 3.21: Let �� ∈����(��), and ��1 ⊆��, ��2 ⊆��, then ��(��1)∩��(��1)⊆��(��1 ∩��2).

Proof: Let �� ∈��(��1) and �� ∈��(��2) which implies �� ∈ ��(��1) for any �� ∈��(��1)∩��(��

, we get

Thus, �� ∈��(��1 ∩��2)

Hence ��(��1)∩��(��1)⊆��(��1 ∩��2)

ISSN 2350-1022
Page | 52 Paper Publications ����(��2)(�� 1����)=����(��2)(��)∀��,�� ∈��∗ Then, ����(��1)����(��2)(�� 1����) =⋁{����(��1)(�� 1����)⋀����(��2)(�� 1����)∀�� ∈��1 ∗ ,�� ∈��1 ∗����2 ∗ ,�� 1���� =��} =⋀{����(��1)(��)⋀����(��2)(��)∀�� ∈��1 ∗ ,�� ∈��1 ∗����2 ∗ ,�� 1���� =��} =����(��1)����(��2)(��) ∀�� ∈��1 ∗ ,�� ∈��1 ∗����2 ∗ Hence ��(��1)����(��2)∈Ɲ(��) Proposition 3.19: Let �� ∈����(��), and ��(��) a normalizer of �� over ��. Then the normalizer of �� is a subgroup of ��. Proof: ��(��)≠∅. Then at least �� ∈ ��(��). Let ��,�� ∈��(��). Now ����(�� 1����)=����(��) and ����(�� 1����)=����(��)∀�� ∈ �� Consequently, ����((����) 1��(����))=����(�� 1�� 1������) =����(�� 1����)⋀����(�� 1����) =����(��)⋀����(��) =����(��)
���� ∈��(��)
����(�� 1����) 1 =����(���� 1�� 1)=����(������ 1)=����(��) ⇒�� 1 ∈��(��).
3.20: �� is a normal sub multigroup of �� if and only if ��(��)=��
Let ��
a normal sub multigroup of ��
����(�� 1����)=����((�� 1��)��)=����((�� 1��)��)=����(�� ��)=����(��)
Then
Again
Proposition
Proof:
be
, then ∀�� ∈��
����(�� ��)��������
1�� ����((�� 1��)��)=����(�� 1����)∀�� ∈��,�� ∈��(��)
=��
∈��
����1∩��2(������ 1)=����1(������ 1)⋀����2(������ 1) =����1((���� 1)��)⋀����2((���� 1)��) =����1(��.��)⋀����2(��.��) =����1(��)⋀����2(��) =����1∩��2(��)
1) for any �� ��

International Journal of Recent Research in Mathematics Computer Science and Information Technology

Vol. 9, Issue 2, pp: (42-54), Month: October 2022 – March 2023, Available at: www.paperpublications.org

Corollary 3.22: Let �� ∈����(��), and

The

Hence �� ∈∈��(��1����2)

Therefore, ��(��1)∩��(��1)=��(��1����2)

Remark: Suppose �� ∈����(��), and ��1 ⊆��,��2 ⊆��. If ��1 ⊆��2, then N( ��1)⊆��(��2)

4. CONCLUSION

Centralizer and Normalizer of the classical group were studied under the perspective of multiset and an extended results from the existing ones were established. The closure of each of the concept under union, intersection, arithmetic addition, and arithmetic multiplication, composition were also studied. Finally, we have also shown that a centralizer and normalizer of a multi group is not empty and are sub multi group of a multi group. We have also define the normal sub multi group of these concepts and some results were put down. Other aspect of the classical group can also be studied under multiset perspecrtive

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Page | 53 Paper Publications
��1 ⊆��,��2 ⊆��, such that ����1(��)=����2(��)
��(��1)∩��(��1)=��(��1 ∩��2)
Recall that ��(��1)={�� ∈��/����(����) =����(����)∀�� ∈��} ={�� ∈��/����(������ 1�� 1)=����(��)∀�� ∈��} Let �� ∈��(��1 ∩��2) Then by definition ����1∩��2(������ 1�� 1)=����1(������ 1�� 1)⋀����2(������ 1�� 1) =����1(��)⋀����2(��) =����1∩��2(��) Which implies �� ∈��(��2) and �� ∈��(��1). Thus �� ∈��(��1 ∩��2) since ����1(������ 1�� 1)=����1(��) ⇒����(����)=����(����) and similarly in the case of ��,����1(��) =����2(��) Hence ��(��1)∩��(��1)=��(��1 ∩��2) Corollary 3.23: Let �� ∈����(��), and ��1 ⊆��,��2 ⊆��.
1
1
1
2
. Then
Proof:
Then ��(��
)∩��(��
)=��(��
����
)
∈��(��1 ∩��2
�� ∈��(��2) and �� ∈��(��1). Then for all �� ∈�� ����1����2(��) =⋁{����1(��)⋀����1(��)∀��,�� ∈��,�� =����} =⋁{����1(�� 1����)⋀����2(�� 1����)∀��,�� ∈��,�� =����} =⋁{����1(��)⋀����2(��)∀��,�� ∈��,�� 1���� =����} =����1����2(�� 1����) ⇒����1∩��2(��)=����1����2(�� 1����)
Proof: Let ��
), that is
inequality
�� = ���� ⇒ �� 1������ =���� ⇒���� =(������ 1)(������ 1) and since �� =������ 1 and �� =������ 1 implying ������ 1 =�� and ������ 1 =��. Again ����1����2(�� 1����) ≤����1����2(��(�� 1����)�� 1)=����1����2(��)
����1����2(��) ≥����1����2(�� 1����).
����1����2(��) =����1����2(�� 1����)
holds since
So,
Thus

International Journal of Recent Research in Mathematics Computer Science and Information Technology

Vol. 9, Issue 2, pp: (42-54), Month: October 2022 – March 2023, Available at: www.paperpublications.org

REFERENCES

[1] Blizard, W..(1989); Multiset Theory, Notre Dame, J. Formal Logic 30, 36-66.

[2] K.P. Girish and Sunil Jacob John, (2012); On Multiset Topologies; Theory and Application of Mathematical and Computer Sciences(2)(2012),37-52

[3] Knuth, D.E. (1981); The art of computer programming, Vol. 2 (seminumerical Algorithms), Second Edition, AddisonWesley, Reading, Massachusetts.

[4] M. Raafat and A.M.Abd El-Latif,2010; On Multi Weak Structures; International Journal of Advances in Mathematics; Volume 2017,Number 6, 42-50.

[5] Nazmul, S.K., Majumbar, P. and Samantha, S.K. (2013) – on multiset and multigroups, Annals of Fuzzy Mathematics and Informatics, 6(3), 643-656.

[6] Parker – Rhodes, A.F. (1981); The Theory of Indistinguishables, D. Reidel, Dordrecht.

[7] Petrovsky, A. B, (1997); Structuring Techniques in Multiset Spaces. Springer. 174-184

[8] Wildberger, N.J. (2003); A new look at multiset, School of Mathematics, UNSW, Sydney.

[9] Shravan K. and Tripathy B.C., (2018); Generalised closed Sets in multiset topological spaces, Proyecciones Jour. Math. Vol. 37, 2.pp. 223-237.

[10] Singh, D., Ibrahim, A,M., Yohanna, T. and Singh, J.N. (2011) – Complemention in multiset Theory, International Mathematica Forum, 6(38), 1877.

[11] Tella, Y. (2013) – A Study of group theory in the context of Multiset Theory. International Journal of Science and Technology, 2(8), 609-615.

[12] Tella, Y. and Daniel, S. (2013); Symmetric groups under multiset perspective, Journal of Mathematics, 7(5), 47-52.

[13] Yager, R. R, (1986); On the theory of bags. International Journal of Genaral System, 13; 23-37

[14] W.D. Blizard,The development of multiset theory, Mordern Logic 1, 319-352(1991).

[15] P.A. Ejegwa and A.M. Ibrahim, Some homomorphic properties of multigroup Bul. Acid Stiinte Repub. Mold. Mat. 83 (1), 67-76 (2017)

[16] Anagha Baby, Shinoj T.K and Sunil Jacob John, On Abelian fuzzy multigroup and orders of fuzzy Mulgtigroups, New theory 2, 80-93 (2015).

[17] Ejegwa,P.A.andIbrahim(2017a);Normalsub multigroupsandco multisetofa multigroups.Quassi.Related Systems, 25, 231–244.

[18] P. A. Ejegwa, On Abelian fuzzy multigroup, J. fuzzy math. 26(3) (2018) 655-668.

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