# On Fundamental Algebraic Characterizations ofμ-Fuzzy Normal Subgroups

Authors: Ibtisam Masmali; Umer Shuaib; Abdul Razaq; Areeba Fatima; Ghaliah Alhamzi
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we present the study ofμ-fuzzy subgroups and prove numerous fundamental algebraic attributes of this newly defined notion. We also define the concept of μ-fuzzy normal subgroup and investigate many vital algebraic characteristics of these phenomena. In addition, we characterize the quotient group induced by this particular fuzzy normal subgroup and establish a group isomorphism between the quotient groups G/κμ and G/κ∗μ. Furthermore, we initiate the study of level subgroup, open level subgroup, and tangible subgroup of a μ-fuzzy subgroup and emphasize the significance of μ-fuzzy normal subgroups by establishing a relationship between these newly defined notions and μ-fuzzy normal subgroup.


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## Body

## 1. Introduction

The theory of fuzzy logic is based on the concept of relative graded membership as inspired by the processes of human perception. This logic deals with information that is uncertain, imprecise, vague, partly true, or without clear boundaries. Moreover, this theory provides a mathematical framework within which ambiguous conceptual phenomena can be studied with precision. New computing methods based on fuzzy logic can be used in the development of intelligent system for decision-making, identification, pattern recognition, optimization, and control system. This particular logic is currently being used in the industrial practice of advanced information technology. One of the most important applications of group theory is its key role in geometry and cryptography. Geometry is the study of properties of a space that are invariant under a group of transformations of that space. The theory of groups is used to classify the symmetries of molecules, crystal structures, and regular polyhedral. It is also used to solve the old issues of algebra.Fuzzy subsets (FSs) have a central position in modern mathematics. In 1965, Zadeh [1] proposed the idea of FSs. The idea of fuzzy subgroup (FSG) was presented by Rosenfeld [2] in the framework of FSs in 1971. Das [3] defined the level subgroups (LSGs) of a FSG in 1981. Mukherjee and Bhattacharya [4] characterized the notions of fuzzy normal subgroup (FNSG) and fuzzy coset in 1984. Mashour et al. [5] described many important properties of FNSGs in 1990. The characterizations of fuzzy conjugate subgroups and fuzzy characteristic subgroups by their level subgroups were presented in [6]. To see more on the development of theory of FSGs, we refer to [7–11]. The fuzzy logic was effectively used in industrial management [12], coding theory [13] and forecasting systems [14]. Rowlands and Wang [15] applied fuzzy logic to present a useful technique to design fuzzy-SPC evaluation and control method in 2000. Liu [16] developed a new method of constructing quotient groups induced by FNSGs and proved the corresponding isomorphism theorem. In 2005, Mordeson et al. [17] discussed the zero power of a FSG along with various important analytical characteristics of FSGs. Li et al. [18] gave the efficient solutions of linear programming problems based on fuzzy logic. Faraz and Shapiro [19] designed a fuzzy statistical control chart that explains the existence of fuzziness in data in 2010. The authors proposed a new type of FNSGs and fuzzy cosets [20]. Celik et al. [21] introduced the concept of fuzzy soft rings and studied some of their structural properties. Borah and Hazarika [22] innovated the study of mixed fuzzy soft topology with separation axioms along with applications of fuzzy soft sets in chemistry. In addition, useful characteristics of FSGs in various algebraic structures can be viewed in [23–27]. Ullah et al. [28] utilized the idea of complex Pythagorean FS in pattern recognition problems. Alghazzawi et al. [29] introduced ρ anti-intuitionistic fuzzy subgroups and presented a comprehensive study of this concept. In [30] Bejines et al. investigated various properties of aggregation of fuzzy subgroups.We recommend the readers to study [31–39] to obtain more information on this topic.The fuzzy sets have ability to play an effective role to solve many physical problems. These sets provide us the meaningful representations of measuring uncertainty. Despite all of these advantages, we still face vast complications to counter various physical situations. This motivates us to define the notion ofμ-fuzzy set through which one can have multiple options to investigate a specific real-world situation in much efficient way by choosing appropriate value of the parameter μ.Theμ-fuzzy sets are capable to deal with the uncertainty and vagueness of a physical problem much more effectively than with the theory of the classic fuzzy set, especially in the area of logic programming and decision-making, financial services, psychological examinations, medical diagnosis, career determination, and artificial intelligence. For instance, μ-fuzzy sets are used to improve the resolution of a certain picture. A photograph of a person describes his many biological features like tall or short, heavy or thin, old or young, and male or female. Sometimes this picture becomes corrupted due to many distortions in the lenses. Such distortions are removed under the framework of these particular fuzzy sets by choosing the appropriate value of the parameter μ.In this article, we initiate the study ofμ-FSs and their level sets along with many set theoretical properties of these phenomena. We also propose the concept of μ-FSG based over a μ-FS and investigate some of their various algebraic properties. We extend this ideology by defining the notions of μ-FNSG and μ-fuzzy coset and establish a group isomorphism between the quotient groups of these newly defined ideas. In addition, we characterize the concept of level subgroup (LSG), open level subgroup (OLSG), and tangible subgroup (TSG) of a μ-FSG and establish a relationship between a μ-FNSG and the above stated notions.After a brief discussion about the historical background and significance of FSG, the rest of the article is organized as follows: the second section contains a brief review of some fundamental definitions of basic notions which are quite useful to understand the novelty of this study. Section3 is dedicated to introduce the notion of μ-FSG defined over a μ-FS, and many fundamental algebraic characteristics of this notion are investigated. In Section 4, we extend the study of μ-FSGs by defining the notions of μ-FNSG and quotient group induced by this particular FNSG. Furthermore, we establish a group isomorphism between the quotient groups G/κμ and G/κ∗μ. In addition, we define the concepts of LSG, OLSG, and TSG of a μ-FSG and shed light the importance of μ-FNSG by developing an important relationship between these concepts and μ-FNSG.


## 2. Preliminaries

In this section, we study some fundamental concepts of FSGs which are quite essential to obtain the basic group theoretic outcomes in terms of their respective fuzzy counterparts.Definition 1 (see [1]).
Consider an elementu1 of a universe U. Any function κ from U to the closed unit interval is called a fuzzy set.Remark 2 (see [17]).
Any two FSsκ and ω of a universe U admit the following characteristics. For any u1∈U.(1)
κ⊆ω iff κu1≤ωu1(2)
κ=ω iff κ⊆ω and ω⊆κ(3)
The complement of the FSκ is κ′ and is determined as κ′u1=1−κu1(4)
κ∩ωu1=minκu1,ωu1(5)
κ∪ωu1=maxκu1,ωu1Definition 3 (see [17]).
Theα-cut and strong α-cut of a FS κ are denoted by κα and κα+, respectively, and are defined in the following ways:
(1)
κα=u1∈U:κu1≥α,0≤α≤1(2)
κα+=u1∈U:κu1>α,0≤α≤1Definition 4 (see [17]).
The support of an FSκ is denoted by κ∗ and is described as κ∗=u1∈U:κu1>0.Definition 5 (see [17]).
An FSκ of a group G is called FSG of G if it satisfies the following conditions:
(i)
κu1u2≥minκu1,κu2(ii)
κu1−1=κu1, for all u1,u2∈GDefinition 6 (see [4]).
The FSGκ of a group G is said to be fuzzy normal subgroup of G if κu1u2=κu2u1, for all u1,u2∈G.Definition 7 (see [4]).
Letκ be FSG and u1 be a fixed element of G. Then, fuzzy left coset of κ in G is denoted by u1κ and is interpreted in the following way: u1κu2=κu1−1u2, for all u1,u2∈G. The fuzzy right coset is determined in the same manner.Definition 8 (see [8]).
Letκ and ω be two FSs of groups G1 and G2, respectively, and η:G1⟶G2 be a mapping. Then, ηκ and η−1ω are defined as κu2=supκu1:u1∈η−1u2;ifη−1u2≠∅0;Otherwise, and η−1ωu1=ωηu1 for every u1,u2∈G1.


## 3. Algebraic Properties ofμ-Fuzzy Subgroups

This section is committed to initiate the conception ofμ-FSG defined over μ-FS. We study several algebraic aspects of these phenomena.Definition 9.
Letκ be a FS of universe U and μ∈0,1. Then, FS κμ is called the μ-fuzzy set of universe U w.r.t. κ and is defined as κμu1=spκu1,μ, for all u1∈U, where the algebraic sum sp is described as spκu1,μ=κu1+μ−κu1∙μ.Remark 10.
(1)
Evidently, we obtain the classical FS fromμ-FS for μ=0 whereas the case becomes crisp set for the value of μ=1(2)
Letκμ and ωμ be any two μ-FSs of U and V, respectively, and η:U⟶V be a mapping. Then(3)
ηκμ=ηκμ(4)
η−1ωμ=η−1ωμDefinition 11.
For anyμ-FS κμ and 0≤α≤1, we denoted α-cut of κμ by κμα such that κμα=u1∈U:κμu1≥α.Definition 12.
Letκμ be a μ-FS and 0≤α≤1. The strong α-cut of κμ is denoted by κμα+ and is described as κμα+=u1∈U:κμu1>α.Example 1.
(1)
An important application ofμ-FS in the medical decision-making is to determine the amount of the drug dose for patients by setting an appropriate value of the parameterμ related to age and weight of the patients. The radiation therapy can be improved in the framework of μ-FSs by selecting an appropriate value of μ(2)
Another important significance ofμ-FSs can be viewed in the formulation of default timings for the traffic lights as it effectively supports the fuzzy controller to adjust the timings depending upon the value of the parameter μDefinition 13.
(1)
The union ofμ-FSs κμ and ωμ is denoted by κμ∪ωμ and is defined as follows:
(1)κμ∪ωμu1=maxκμu1,ωμu1.(2)
The intersection ofμ-FSs κμ and ωμ is denoted by κμ∩ωμ and is interpreted as follows:
(2)κμ∩ωμu1=minκμu1,ωμu1.(3)
The complement ofμ-FS κμ is denoted by κμ, and is describe as follows:
(3)κμ,u1=1−κμu1,forallu1∈U.Definition 14.
Letκ be an FS of a group G and μ∈0,1. Then, κ is called a μ-fuzzy subgroup of G if κμ satisfies the following conditions:
(1)
κμu1u2≥minκμu1,κμu2(2)
κμu1−1=κμu1, for all u1,u2∈G
The above conditions can also be defined in the following way.(4)κμu1u2−1≥minκμu1,κμu2,forallu1,u2∈G.Remark 15.
Everyμ-FSG κμ of a group G admits the following characteristics
(a)
κμu1≤κμe, for all u1∈G(b)
κμu1u2−1=κμe which implies that κμu1=κμu2κµNκµ,u1,u2∈G(2)
Every FSG ofG is a μ-FSG of G but the converse is not true, in general(3)
The intersection of a family ofμ-FSGs of G is also a μ-FSG of G(4)
The union of any twoμ-FSGs of G need not be a μ-FSG of G(5)
In the following result, we establish some important relationships betweenα-cut and strong α-cut of μ-FSTheorem 16.
Letκμ and ωμ be any two μ-FS; then, the following attributes hold for any α,β∈0,1.(i)
κα+μ⊆καμα≤β implies that κβμ⊆καμ and κβ+μ⊆κα+μ(ii)
καμ,=κ1−α+μ,(iii)
κμ∩ωμ=καμ∩αωαμ and κμ∪ωμ=καμ∪αωαμ(iv)
κμ∩ωμ=κα+μ∩α+ωα+μ and κμ∪ωμ=κα+μ∪α+ωα+μProof.
(i)
In view of Definition12, for any element u1∈U, κμu1>α. It means that κμu1≥α. Thus, κα+μ⊆καμ(ii)
Letu1∈κβμ, then by using Definition 11, we have κμu1≥β>α. This implies that u1∈καμ. Consequently, κβμ⊆καμ
Similarly, the assertion can be proved for strongα−cut of κμ.
(iii)
Letu1∈καμ,, then 1−κμu1≥α. It follows that κμu1≤1−α. This means that u1∉κ1−α+μ. Therefore, u1∈κ1−α+μ,. Hence
(5)καμ,⊆κ1−α+μ,.
Now supposeu1∈κ1−α+μ,, then u1∉κ1−α+μ. This implies that 1−α≥κμu1; therefore, α≤1−κμu1. It follows that u1∈καμ,. Hence
(6)κ1−α+μ,⊆καμ,.
From (5) and (6), we obtain καμ,=κ1−α+μ,.
(iv)
Letu1∈κμ∩ωμ,α then κμ∩ωμu1≥α. This implies that minκμu1,ωμu1≥α, so κμu1≥α and ωμu1≥α. Therefore, u1∈κμ∩ωαμα, and consequently,
(7)κμ∩ωμα⊆κμ∩ωαμα.
The application of efinition11 on u1∈κμ∩ωαμα yields that κμu1≥α and ωμu1≥α. This shows that minκμu1,ωμu1≥α. Consequently,
(8)κμ∩ωαμα⊆κμ∩ωμα.
From (7) and (8), we obtain the required equality.
The second part of (iv) can be proved in the same manner.(v)
For any elementu1∈κμ∪ωμα+, we obtain κμ∪ωμu1>α. Therefore, maxκμu1,ωμu1>α. This means that κμu1>α or ωμu1>α. Therefore, u1∈κμ∪ωα+μα+, and consequently,
(9)κμ∪ωμα+⊆κμ∪ωα+μα+.
The application of Definition12 on u1∈κμ∪ωα+μα+ yields that κμu1>α or ωμu1>α. This shows that maxκμu1,ωμu1>α. Consequently,
(10)κμ∪ωα+μ⊆α+κμ∪ωμα+.
By comparing the relations (9) and (10), we get the required result.
The other part of (v) can be proved in the same manner.
The following theorem presents a necessary and sufficient condition for aμ-FS to be a μ-FSG.Theorem 17.
Aμ-FS κμ is a μ-FSG of a group G if and only if κμα is a subgroup of G for each 0≤α≤κμe, where e is the identity element of G.Proof.
Supposeκμ is a μ-FSG and u1,u2∈κμα, then in the view of Definition 11, we have κμu1≥α and κμu2≥α. This implies that κμu1u2≥minκμu1,κμu2≥minα,α≥α. Consequently, u1u2∈κμα. Moreover, for u1∈κμα, we have κμu1≥α, implying that κμu1−1=κμu1≥α for all u1∈G. This means that u1−1∈κμα, and hence, κμα is a subgroup of G.
Conversely, assume thatκμα is a subgroup of G. For any u1,u2∈G, let κμu1=α and κμu1=β. Then, clearly, u1∈κμα and u2∈κμβ. Suppose α<β, then κμ⊆βκμα. Since u2∈κμα and κμα is a subgroup of G, therefore, u1u2∈κμα. Then, κμu1u2≥α implying that κμu1u2≥minκμu1,κμu2. For any u1∈G, let κμu1=α. Then, u1∈κμα and u1−1∈κμ.α This implies that κμu1−1=κμu1, and hence, κμ is a μ-FSG of G.


## 4. Fundamental Algebraic Characterizations ofμ-Fuzzy Normal Subgroups

In this section, we categorize the ideas ofμ-fuzzy cosets and μ-FNSGs and prove fundamental algebraic characteristics of these phenomena. We also define the quotient group induced by μ-FNSG and obtain a group isomorphism between G/κμ and G/κ∗μ. Furthermore, we innovate the study of the concepts of LSG, OLSG, and TSG of a μ-FSG and develop a relationship of these notions with μ-FNSG.


### 4.1. Definition

Letκμ be a μ-FSG and u1 be any fixed element of a group G.Then, μ-fuzzy left coset of κμ in G is represented by u1κμ and is defined as follows:
(11)u1κμu2=κμu1−1u2+μ−κμu1−1u2.μforallu1,u2∈G.Likewise, the notion ofμ-fuzzy right coset κμu1 can be established.


### 4.2. Definition

Aμ-FSG κμ is called μ-fuzzy normal subgroup of ωμ if u1κμ=κμu1 for all u1∈G.In other words, aμ-FNSG can also be defined as κμu1u2=κμu2u1, for all u1,u2∈G.Theorem 18.
The following statements are equivalent for anyμ-FSG κμ of a group G.(i)
κμ is a μ-FNSG of G(ii)
κμu1u2u1−1=κμu2(iii)
κμu1u2u1−1≥κμu2(iv)
κμu1u2u1−1≤κμu2,for all u1,u2∈GProof.
i⇒ ii: For any two elements u1,u2∈G, we have κμu1u2u1−1=κμu1−1∙u1u2=κμu2.
ii⇒ iii: The required inequality is a straightforward implication of ii.
iii⇒ iv: Consider κμu2=κμu1−1∙u1u2u1−1∙u1−1−1≥κμu1u2u1−1.
In other words,κμu1u2u1−1≤κμu2.
iv⇒ i: Consider
(12)κμu1u2=κμu1u2∙u1u1−1=κμu1∙u2u1∙u1−1,(13)κμu1u2≤κμu2u1.
Moreover,(14)κμu2u1=κμu2u1∙u2u2−1=κμu2∙u1u2∙u2−1.
The required result can be acquired by using (12) and (13).
In the following theorem, we find a necessary and sufficient condition for aμ-FSG to be μ-FNSG.Theorem 19.
Aμ-FSG κμ is a μ-FNSG of a group G if and only if καμ is a normal subgroup of G for each 0≤α≤κμe, where e is the identity element of G.Proof.
Supposeκμ is a μ-FNSG. The application of Definition 11 on u1,u2∈καμ, we have κμu1≥α and κμu2≥α. Consider κμu1u2u1−1=κμu2≥α. Consequently, u1u2u1−1∈καμ, and hence, καμ is a normal subgroup of G.
Conversely, assume thatκαμ is a normal subgroup of G. Let u1,u2∈G and κμu2=α, then clearly, u2∈καμ. Since καμ is a normal subgroup of G, therefore, u1u2u1−1∈καμ. This means that κμu1u2u1−1≥α=κμu2. It follows that κμ satisfies fact iii of Theorem 18 and according to Theorem 20κμ is a μ-FNSG of G.


### 4.3. Definition

The support of aμ-FS κμ is denoted by κ∗μ and is described as κ∗μ=u1∈U:κμu1>0.In the following result, we show that support ofκμ is a normal subgroup of G.Theorem 20.
Letκμ be a μ-FNSG of a group G. Then, κ∗μ is normal subgroup of G.Proof.
The application of Definitions (9) and (15) on u1,u2∈κ∗μ, we have κμu1u2−1>0. Consequently, u1u2−1∈κ∗μ; therefore, κ∗μ is a subgroup of G. Moreover, by applying the normality of κμ, we have κμu1u2u1−1=κμu2>0, for any u1∈G and u2∈κ∗μ. Consequently, u1u2u1−1∈κ∗μ; thus, κ∗μ is a normal subgroup of G.
In the following result, we show thatκ∗μ is normal subgroup of G.Theorem 21.
Letκμ be a μ-FNSG of a group G; then, the set κ∗μ=u1∈G:κμu1=κμe is a normal subgroup of G.Proof.
The application of Definition (9) on u1,u2∈κ∗μ, we have
(15)κμu1u2−1≥κμe.
We also know that(16)κμe≥κμu1u2−1.
By the comparison of (15) and (16), we get κμu1u2−1=κμe.
Hence,κ∗μ is a subgroup of G.
Moreover, by applying the normality ofκμ, we have κμu1u2u1−1=κμu2=κμe, for any u1∈G and u2∈κ∗μ. Consequently, u1u2u1−1∈κ∗μ; this shows that κ∗μ is a normal subgroup of G.The following example describes the fact that leads to note that the converses of the Theorems20 and 21 are not true.Example 2.
Consider a subgroupH=e,ab of the dihedral group D3=e,a,a2,b,ab,a2b. Clearly, H is not normal in D3. The FS κ of D3 is as follows:
(17)κu1=1.0,ifu1=e,0.5,ifu1=ab,0.4,otherwise.
Theμ-FS κμ of D3 correspond to the value μ=0.6 is given by
(18)κ0.6u1=1.00,ifu1=e,0.80,ifu1=ab,0.76,otherwise.
Then,κ∗0.6=e and κ∗0.6=D3. Obviously, κ∗0.6 and κ∗0.6 are normal subgroups of D3. But κ0.6 is not a 0.6−FNSG of D3 because κ0.80.6=H.The following result describes another approach to obtain aμ-FNSG.Theorem 22.
Aμ-FSG κμ is a μ-FNSG of a group G if and only if κμ is constant on the conjugate classes of G.Proof.
Supposeκμ is a μ-FNSG of G. Then
(19)κμu2−1u1u2=κμu1u2u2−1=κμu1,forallu1,u2∈G.
This means thatκμ is a constant on the conjugate classes of G.
Conversely, suppose thatκμ is constant on each conjugate class of G. Then, κμu1u2=κμu1u2u1u1−1=κμu1u2u1u1−1=κμu2u1, for all u1,u2∈G. Consequently, κμ is a μ-FNSG of G.Theorem 23.
Everyμ-FNSG κμ of G admits the following properties for all u1,u2∈G:
(i)
u1κμ=u2κμ⇔u1κ∗μ=u2κ∗μ(ii)
κμu1=κμu2⇔κ∗μu1=κ∗μu2Proof.
(i) Suppose thatu1κμ=u2κμ, where u1,u2∈G. By using Definition (12) in the above relation, we obtain
(20)κμu1−1u2=u1κμu2=u2κμu2=κμu2−1u2=κμe.
So,u1−1u2∈κ∗μ. This means that u2∈u1κ∗μ, and ultimately, we have
(21)u2κ∗μ⊆u1κ∗μ.
Similarly(22)u1κ∗μ⊆u2κ∗μ.
By comparing the relations (21) and (22), we get the required equality.
Conversely, letu1κ∗μ=u2κ∗μ. Then, u1−1u2∈κ∗μ and u2−1u1∈κ∗μ. By applying Definition (12) on u1κμ, we have
(23)u1κμg1=κμu1−1g1=κμu1−1u2u2−1g1≥minκμu1−1u2,κμu2−1g1=minκμe,κμu2−1g1≥κμu2−1g1,
which implies that(24)u1κμg1=u2κμg1.
Similarly(25)u2κμg1=u1κμg1.
The required equality can be obtained by using (24) and (25).
(ii) This part proved in the same way.Theorem 24.
Letκμ be a μ-FNSG of a group G and κμu1=κμu2; then, κμu1=κμu2, for all u1,u2∈G.Proof.
Letu1,u2∈G and κμu1=κμu2; then, κμu1u2=κμu2u2. This implies that κμu2u1−1=κμe. Consider
(26)κμu1=κμu1u2−1u2≥minκμu1u2−1,κμu2=minκμe,κμu2.
This means that(27)κμu1≥κμu2.
Likewise,(28)κμu1≥κμu2.
In view of (27) and (28), we obtain κμu1=κμu2.Theorem 25.
Letκμ be a μ-FNSG of G; then, for all u1,u2,v1,v2∈G, u1κμ=v1κμ and u2κμ=v2κμ implies u1u2κμ=v1v2κμ.Proof.
The application of Theorem23 and using the given condition on κμ, we have u1−1v1,u2−1v2∈κ∗μ. Consider
(29)u1u2−1v1v2=u2−1u1−1v1v2=u2−1u1−1v1v2=u2−1u1−1v1u2−1u2v2=u2−1u1−1v1u2u2−1v2,
which implies thatu1u2−1v1v2∈κ∗μ. It follows that v1v2∈u1u2κ∗μ; therefore
(30)v1v2κ∗μ⊆u1u2κ∗μ.
Similarly(31)u1u2κ∗μ⊆v1v2κ∗μ.
By comparing the relations (30) and (31), we get the required equality.Theorem 26.
Suppose thatκμ is a μ-FNSG of G and G/κμ is the set of all μ-fuzzy cosets of κμ in G. Define a binary operation ∗ on G/κμ in the following way: κμu1∗κμu2=κμu1u2, where u1,u2∈G. Then, G/κμ,∗ forms a group.Proof.
Letu0,v0,u1,u2∈G and κμu0,κμv0,κμu1,κμu2∈G/κμ such that κμu0=κμu1 and κμv0=κμu2. Then, for eachg1∈G, we have κμu1u2g1=κμg1u2−1u1−1 and κμu0v0g1=κμg1v0−1u0−1. Moreover,
(32)κμg1u2−1u1−1=κμg1v0−1v0u2−1u1−1=κμg1v0−1u0−1u0v0u2−1u1−1≥minκμg1v0−1u0−1,κμu0v0u2−1u1−1.
Sinceκμu0=κμu1 and κμv0=κμu2, therefore κμg1u1−1=κμg1u0−1 and κμg1u2−1=κμg1v0−1. Particularly, we have
(33)κμu0v0u2−1u1−1=κμu0v0u2−1u0−1=κμv0u2−1=κμe.
Thus,κμu0v0u2−1u1−1=κμe.
Also, we know thatκμe≥κμg1v0−1u0−1. This implies that
(34)κμg1u2−1u1−1≥κμg1v0−1u0−1.
Similarly, we have(35)κμg1v0−1u0−1≥κμg1u2−1u1−1.
By comparing (33) and (35), we have
(36)κμg1v0−1u0−1=κμg1u2−1u1−1.
Thus,κμu0v0g1=κμu1u2g1. Hence, ∗ is well defined.
(1)
Furthermore,∗ is associative(2)
κμu1−1 is inverse of κμu1(3)
κμe=κμ is identity element in G/κμ
Hence,G/κμ,∗ is a group.
Note thatG/κμ is known as the μ-fuzzy quotient group induced by κμ.
In the following result, we establish a group isomorphism betweenG/κμ and G/κ∗μ.Theorem 27.
Letκμ∈μ-FNSG. Then, G/κμ≅G/κ∗μ.Proof.
The application of Theorem19 and using the fact that κμ∈μ−FNSG, we have κ∗μ that is a normal subgroup of G. This ensures the existence of G/κ∗μ. Define a map η:G/κμ⟶G/κ∗μ by the rule ηu1κμ=u1κ∗μ, u1∈G. Consider
(37)u1κμ=u2κμ.
In view of Theorem23, the above relation becomes
(38)u1κ∗μ=u2κ∗μ.
This implies that(39)ηu1κμ=ηu2κμ.
This shows thatη is a well defined mapping.
The functionη is injective because
(40)ηu1κμ=ηu2κμ,
implies that(41)u1κ∗μ=u2κ∗μ
By applying heorem23, we have
(42)u1κμ=u2κμ.
Clearly,η is surjective because for each u1κ∗μ∈G/κ∗μ, there exists u1κμ∈G/κμ such that ηu1κμ=u1κ∗μ.
Moreover,η is homomorphism as for each u1κμ,u2κμ∈G/κμ,
(43)ηu1κμ·u2κμ=ηu1u2κμ=u1u2κ∗μ=u1κ∗μ·u2κ∗μ=ηu1κμ·ηu2κμ.
Consequently,η is a group isomorphism between G/κμ and G/κ∗μ.
In the upcoming theorem, we establish a natural homomorphism between the groupG and its quotient group by μ-FNSG κμ.Theorem 28.
Letκμ∈μ-FNSG and u1∈G. Then, the map η:G⟶G/κμ defined by ηu1=κμu1 is a natural homomorphism with its kernel κ∗μ.Proof.
Letu1,u2∈G, then
(44)ηu1u2=κμu1u2,=κμu1·κμu2=ηu1.ηu2.
It means thatη is a homomorphism.
Also(45)kerη=u1∈G:ηu1=κμe=u1∈G:κμu1=κμe=u1∈G:κμu1u1=κμeu1=u1∈G:κμe=κμu1kerη=κ∗μ.Definition 29.
Letκμ and ωμ be any two μ-FSGs and u1 be a fixed element of a group G. Then, κμ is μ-fuzzy conjugate to ωμ if there exists an element g1∈G such that κμu1=ωμg1u1g1−1. If such is the case, then we write κμ=ωμg1, where ωμg1u1=ωμg1u1g1−1.Theorem 30.
For anyμ-FSG κμ of G, then, ⋂g1∈Gκμg1 is a μ-FNSG of G and is the largest μ-FNSG of G that is contained in κμ.Proof.
Given thatκμg1 is a μ-FSG and g1 be any element of G, in view of Remark 15, we have ⋂g1∈Gκμg1 that is a μ-FSG of G. Note that, for all u1∈G,
(46)κμg1∣g1∈G=κμg1u1∣g1∈G.
Thus,(47)ming1∈Gκμg1u1u2u1−1=ming1∈Gκμg1u1u2u1−1g1−1=ming1∈Gκμg1u1u2g1u1−1=ming1∈Gκμg1u1u2=ming1∈Gκμg1u2,
for allu1,u2∈G. Hence, by Theorem 18, ⋂g1∈Gκμg1 is a μ-FNSG of G.
Now letωμ be a μ-FNSG of G with ωμ⊆κμ. Then, ωμ=ωμg1⊆κμg1 for all g1∈G. So ωμ⊆⋂g1∈Gκμg1. Therefore, ⋂g1∈Gκμg1 is the largest μ-FNSG of G that is contained in κμ.Definition 31.
Letκμ be a μ-FSG of G and α=κμe be the tip of κμ. Then
(i)
The level subgroup ofκμ is denoted by κτμτ,α and is defined as follows:
(48)κτμτ,α=u1∈G:τ≤κμu1≤α,0≤τ≤α.(ii)
The open level subgroup ofκμ is denoted by κτμτ,α and is defined as follows:
(49)κτμτ,α=u1∈G:τ<κμu1≤α,0≤τ<α.(iii)
The tangible subgroup ofκμ is denoted by κτ∈Imκμμτ,α and is defined as follows:
(50)κτ∈Imκμμτ,α=u1∈G:τ≤κμu1≤α,τ∈Imκμ,0<τ≤α.Example 3.
Consider the quaternion groupQ8=±1,±i,±j,±k of order 8. The FS κ of Q8 is defined as follows:
(51)κu1=0.90,ifu1=1,0.72,ifu1∈−1,±j,0.50,otherwise.
Theμ-FS κμ of Q8 correspond to the value μ=0.40 is given by
(52)κ0.4u1=0.94,ifu1=1,0.83,ifu1∈−1,±j,0.70,otherwise.
Note that the tip ofκμ is α=0.94 and Imκ0.40=0.70,0.83,0.94.(i)
The LSG corresponding to the valueτ=0.83 is given by
(53)κ0.830.400.83,0.94=±1,±j.(ii)
The OLSG corresponding to the valueτ=0.90 is given by κ0.900.400.90,0.94=1(iii)
The TSG corresponding to the valueτ=0.70 is given by κ0.70∈Imκμ0.400.70,0.94=Q8
Clearly,κ0.830.400.83,0.94, κ0.900.400.90,0.94, and κ0.70∈Imκμ0.400.70,0.94 are the subgroups of Q8.The following result describes another approach to obtain aμ-FNSG.Theorem 32.
For anyμ-FSG κμ of a group G, the following statements are equivalent:
(i)
κμ is a μ-FNSG in G(ii)
Every LSG ofκμ is normal in G(iii)
Every TSG ofκμ is normal in G(iv)
Every OLSG ofκμ is normal in GProof.
i⇒ ii: The application of Theorem 19 ensures the validity of i ⇒ ii.
ii⇒ iii: Since every TSG of κμ is a LSG of κμ, which is normal in G. Thus, every TSG of κμ is normal in G.
iii⇒ iv: Let 0≤τ≤α, where α is the tip of κμ. Then,
(54)κτμτ,α=⋃τ<λλ∈Imκμκλ∈Imκμμλ,α,
which is normal inG, being the union of an increasing chain of normal subgroups of G.
iv⇒ i: If possible, let there exist two elements u1,u2∈G such that κμu1u2>κμu2u1=τ. Then, u1u2∈κτμτ,α, since κτμτ,α is normal in G. Therefore, u2u1∈κτμτ,α. This implies that τ<κμu1u2, which is not true.


## 4.1. Definition

Letκμ be a μ-FSG and u1 be any fixed element of a group G.Then, μ-fuzzy left coset of κμ in G is represented by u1κμ and is defined as follows:
(11)u1κμu2=κμu1−1u2+μ−κμu1−1u2.μforallu1,u2∈G.Likewise, the notion ofμ-fuzzy right coset κμu1 can be established.


## 4.2. Definition

Aμ-FSG κμ is called μ-fuzzy normal subgroup of ωμ if u1κμ=κμu1 for all u1∈G.In other words, aμ-FNSG can also be defined as κμu1u2=κμu2u1, for all u1,u2∈G.Theorem 18.
The following statements are equivalent for anyμ-FSG κμ of a group G.(i)
κμ is a μ-FNSG of G(ii)
κμu1u2u1−1=κμu2(iii)
κμu1u2u1−1≥κμu2(iv)
κμu1u2u1−1≤κμu2,for all u1,u2∈GProof.
i⇒ ii: For any two elements u1,u2∈G, we have κμu1u2u1−1=κμu1−1∙u1u2=κμu2.
ii⇒ iii: The required inequality is a straightforward implication of ii.
iii⇒ iv: Consider κμu2=κμu1−1∙u1u2u1−1∙u1−1−1≥κμu1u2u1−1.
In other words,κμu1u2u1−1≤κμu2.
iv⇒ i: Consider
(12)κμu1u2=κμu1u2∙u1u1−1=κμu1∙u2u1∙u1−1,(13)κμu1u2≤κμu2u1.
Moreover,(14)κμu2u1=κμu2u1∙u2u2−1=κμu2∙u1u2∙u2−1.
The required result can be acquired by using (12) and (13).
In the following theorem, we find a necessary and sufficient condition for aμ-FSG to be μ-FNSG.Theorem 19.
Aμ-FSG κμ is a μ-FNSG of a group G if and only if καμ is a normal subgroup of G for each 0≤α≤κμe, where e is the identity element of G.Proof.
Supposeκμ is a μ-FNSG. The application of Definition 11 on u1,u2∈καμ, we have κμu1≥α and κμu2≥α. Consider κμu1u2u1−1=κμu2≥α. Consequently, u1u2u1−1∈καμ, and hence, καμ is a normal subgroup of G.
Conversely, assume thatκαμ is a normal subgroup of G. Let u1,u2∈G and κμu2=α, then clearly, u2∈καμ. Since καμ is a normal subgroup of G, therefore, u1u2u1−1∈καμ. This means that κμu1u2u1−1≥α=κμu2. It follows that κμ satisfies fact iii of Theorem 18 and according to Theorem 20κμ is a μ-FNSG of G.


## 4.3. Definition

The support of aμ-FS κμ is denoted by κ∗μ and is described as κ∗μ=u1∈U:κμu1>0.In the following result, we show that support ofκμ is a normal subgroup of G.Theorem 20.
Letκμ be a μ-FNSG of a group G. Then, κ∗μ is normal subgroup of G.Proof.
The application of Definitions (9) and (15) on u1,u2∈κ∗μ, we have κμu1u2−1>0. Consequently, u1u2−1∈κ∗μ; therefore, κ∗μ is a subgroup of G. Moreover, by applying the normality of κμ, we have κμu1u2u1−1=κμu2>0, for any u1∈G and u2∈κ∗μ. Consequently, u1u2u1−1∈κ∗μ; thus, κ∗μ is a normal subgroup of G.
In the following result, we show thatκ∗μ is normal subgroup of G.Theorem 21.
Letκμ be a μ-FNSG of a group G; then, the set κ∗μ=u1∈G:κμu1=κμe is a normal subgroup of G.Proof.
The application of Definition (9) on u1,u2∈κ∗μ, we have
(15)κμu1u2−1≥κμe.
We also know that(16)κμe≥κμu1u2−1.
By the comparison of (15) and (16), we get κμu1u2−1=κμe.
Hence,κ∗μ is a subgroup of G.
Moreover, by applying the normality ofκμ, we have κμu1u2u1−1=κμu2=κμe, for any u1∈G and u2∈κ∗μ. Consequently, u1u2u1−1∈κ∗μ; this shows that κ∗μ is a normal subgroup of G.The following example describes the fact that leads to note that the converses of the Theorems20 and 21 are not true.Example 2.
Consider a subgroupH=e,ab of the dihedral group D3=e,a,a2,b,ab,a2b. Clearly, H is not normal in D3. The FS κ of D3 is as follows:
(17)κu1=1.0,ifu1=e,0.5,ifu1=ab,0.4,otherwise.
Theμ-FS κμ of D3 correspond to the value μ=0.6 is given by
(18)κ0.6u1=1.00,ifu1=e,0.80,ifu1=ab,0.76,otherwise.
Then,κ∗0.6=e and κ∗0.6=D3. Obviously, κ∗0.6 and κ∗0.6 are normal subgroups of D3. But κ0.6 is not a 0.6−FNSG of D3 because κ0.80.6=H.The following result describes another approach to obtain aμ-FNSG.Theorem 22.
Aμ-FSG κμ is a μ-FNSG of a group G if and only if κμ is constant on the conjugate classes of G.Proof.
Supposeκμ is a μ-FNSG of G. Then
(19)κμu2−1u1u2=κμu1u2u2−1=κμu1,forallu1,u2∈G.
This means thatκμ is a constant on the conjugate classes of G.
Conversely, suppose thatκμ is constant on each conjugate class of G. Then, κμu1u2=κμu1u2u1u1−1=κμu1u2u1u1−1=κμu2u1, for all u1,u2∈G. Consequently, κμ is a μ-FNSG of G.Theorem 23.
Everyμ-FNSG κμ of G admits the following properties for all u1,u2∈G:
(i)
u1κμ=u2κμ⇔u1κ∗μ=u2κ∗μ(ii)
κμu1=κμu2⇔κ∗μu1=κ∗μu2Proof.
(i) Suppose thatu1κμ=u2κμ, where u1,u2∈G. By using Definition (12) in the above relation, we obtain
(20)κμu1−1u2=u1κμu2=u2κμu2=κμu2−1u2=κμe.
So,u1−1u2∈κ∗μ. This means that u2∈u1κ∗μ, and ultimately, we have
(21)u2κ∗μ⊆u1κ∗μ.
Similarly(22)u1κ∗μ⊆u2κ∗μ.
By comparing the relations (21) and (22), we get the required equality.
Conversely, letu1κ∗μ=u2κ∗μ. Then, u1−1u2∈κ∗μ and u2−1u1∈κ∗μ. By applying Definition (12) on u1κμ, we have
(23)u1κμg1=κμu1−1g1=κμu1−1u2u2−1g1≥minκμu1−1u2,κμu2−1g1=minκμe,κμu2−1g1≥κμu2−1g1,
which implies that(24)u1κμg1=u2κμg1.
Similarly(25)u2κμg1=u1κμg1.
The required equality can be obtained by using (24) and (25).
(ii) This part proved in the same way.Theorem 24.
Letκμ be a μ-FNSG of a group G and κμu1=κμu2; then, κμu1=κμu2, for all u1,u2∈G.Proof.
Letu1,u2∈G and κμu1=κμu2; then, κμu1u2=κμu2u2. This implies that κμu2u1−1=κμe. Consider
(26)κμu1=κμu1u2−1u2≥minκμu1u2−1,κμu2=minκμe,κμu2.
This means that(27)κμu1≥κμu2.
Likewise,(28)κμu1≥κμu2.
In view of (27) and (28), we obtain κμu1=κμu2.Theorem 25.
Letκμ be a μ-FNSG of G; then, for all u1,u2,v1,v2∈G, u1κμ=v1κμ and u2κμ=v2κμ implies u1u2κμ=v1v2κμ.Proof.
The application of Theorem23 and using the given condition on κμ, we have u1−1v1,u2−1v2∈κ∗μ. Consider
(29)u1u2−1v1v2=u2−1u1−1v1v2=u2−1u1−1v1v2=u2−1u1−1v1u2−1u2v2=u2−1u1−1v1u2u2−1v2,
which implies thatu1u2−1v1v2∈κ∗μ. It follows that v1v2∈u1u2κ∗μ; therefore
(30)v1v2κ∗μ⊆u1u2κ∗μ.
Similarly(31)u1u2κ∗μ⊆v1v2κ∗μ.
By comparing the relations (30) and (31), we get the required equality.Theorem 26.
Suppose thatκμ is a μ-FNSG of G and G/κμ is the set of all μ-fuzzy cosets of κμ in G. Define a binary operation ∗ on G/κμ in the following way: κμu1∗κμu2=κμu1u2, where u1,u2∈G. Then, G/κμ,∗ forms a group.Proof.
Letu0,v0,u1,u2∈G and κμu0,κμv0,κμu1,κμu2∈G/κμ such that κμu0=κμu1 and κμv0=κμu2. Then, for eachg1∈G, we have κμu1u2g1=κμg1u2−1u1−1 and κμu0v0g1=κμg1v0−1u0−1. Moreover,
(32)κμg1u2−1u1−1=κμg1v0−1v0u2−1u1−1=κμg1v0−1u0−1u0v0u2−1u1−1≥minκμg1v0−1u0−1,κμu0v0u2−1u1−1.
Sinceκμu0=κμu1 and κμv0=κμu2, therefore κμg1u1−1=κμg1u0−1 and κμg1u2−1=κμg1v0−1. Particularly, we have
(33)κμu0v0u2−1u1−1=κμu0v0u2−1u0−1=κμv0u2−1=κμe.
Thus,κμu0v0u2−1u1−1=κμe.
Also, we know thatκμe≥κμg1v0−1u0−1. This implies that
(34)κμg1u2−1u1−1≥κμg1v0−1u0−1.
Similarly, we have(35)κμg1v0−1u0−1≥κμg1u2−1u1−1.
By comparing (33) and (35), we have
(36)κμg1v0−1u0−1=κμg1u2−1u1−1.
Thus,κμu0v0g1=κμu1u2g1. Hence, ∗ is well defined.
(1)
Furthermore,∗ is associative(2)
κμu1−1 is inverse of κμu1(3)
κμe=κμ is identity element in G/κμ
Hence,G/κμ,∗ is a group.
Note thatG/κμ is known as the μ-fuzzy quotient group induced by κμ.
In the following result, we establish a group isomorphism betweenG/κμ and G/κ∗μ.Theorem 27.
Letκμ∈μ-FNSG. Then, G/κμ≅G/κ∗μ.Proof.
The application of Theorem19 and using the fact that κμ∈μ−FNSG, we have κ∗μ that is a normal subgroup of G. This ensures the existence of G/κ∗μ. Define a map η:G/κμ⟶G/κ∗μ by the rule ηu1κμ=u1κ∗μ, u1∈G. Consider
(37)u1κμ=u2κμ.
In view of Theorem23, the above relation becomes
(38)u1κ∗μ=u2κ∗μ.
This implies that(39)ηu1κμ=ηu2κμ.
This shows thatη is a well defined mapping.
The functionη is injective because
(40)ηu1κμ=ηu2κμ,
implies that(41)u1κ∗μ=u2κ∗μ
By applying heorem23, we have
(42)u1κμ=u2κμ.
Clearly,η is surjective because for each u1κ∗μ∈G/κ∗μ, there exists u1κμ∈G/κμ such that ηu1κμ=u1κ∗μ.
Moreover,η is homomorphism as for each u1κμ,u2κμ∈G/κμ,
(43)ηu1κμ·u2κμ=ηu1u2κμ=u1u2κ∗μ=u1κ∗μ·u2κ∗μ=ηu1κμ·ηu2κμ.
Consequently,η is a group isomorphism between G/κμ and G/κ∗μ.
In the upcoming theorem, we establish a natural homomorphism between the groupG and its quotient group by μ-FNSG κμ.Theorem 28.
Letκμ∈μ-FNSG and u1∈G. Then, the map η:G⟶G/κμ defined by ηu1=κμu1 is a natural homomorphism with its kernel κ∗μ.Proof.
Letu1,u2∈G, then
(44)ηu1u2=κμu1u2,=κμu1·κμu2=ηu1.ηu2.
It means thatη is a homomorphism.
Also(45)kerη=u1∈G:ηu1=κμe=u1∈G:κμu1=κμe=u1∈G:κμu1u1=κμeu1=u1∈G:κμe=κμu1kerη=κ∗μ.Definition 29.
Letκμ and ωμ be any two μ-FSGs and u1 be a fixed element of a group G. Then, κμ is μ-fuzzy conjugate to ωμ if there exists an element g1∈G such that κμu1=ωμg1u1g1−1. If such is the case, then we write κμ=ωμg1, where ωμg1u1=ωμg1u1g1−1.Theorem 30.
For anyμ-FSG κμ of G, then, ⋂g1∈Gκμg1 is a μ-FNSG of G and is the largest μ-FNSG of G that is contained in κμ.Proof.
Given thatκμg1 is a μ-FSG and g1 be any element of G, in view of Remark 15, we have ⋂g1∈Gκμg1 that is a μ-FSG of G. Note that, for all u1∈G,
(46)κμg1∣g1∈G=κμg1u1∣g1∈G.
Thus,(47)ming1∈Gκμg1u1u2u1−1=ming1∈Gκμg1u1u2u1−1g1−1=ming1∈Gκμg1u1u2g1u1−1=ming1∈Gκμg1u1u2=ming1∈Gκμg1u2,
for allu1,u2∈G. Hence, by Theorem 18, ⋂g1∈Gκμg1 is a μ-FNSG of G.
Now letωμ be a μ-FNSG of G with ωμ⊆κμ. Then, ωμ=ωμg1⊆κμg1 for all g1∈G. So ωμ⊆⋂g1∈Gκμg1. Therefore, ⋂g1∈Gκμg1 is the largest μ-FNSG of G that is contained in κμ.Definition 31.
Letκμ be a μ-FSG of G and α=κμe be the tip of κμ. Then
(i)
The level subgroup ofκμ is denoted by κτμτ,α and is defined as follows:
(48)κτμτ,α=u1∈G:τ≤κμu1≤α,0≤τ≤α.(ii)
The open level subgroup ofκμ is denoted by κτμτ,α and is defined as follows:
(49)κτμτ,α=u1∈G:τ<κμu1≤α,0≤τ<α.(iii)
The tangible subgroup ofκμ is denoted by κτ∈Imκμμτ,α and is defined as follows:
(50)κτ∈Imκμμτ,α=u1∈G:τ≤κμu1≤α,τ∈Imκμ,0<τ≤α.Example 3.
Consider the quaternion groupQ8=±1,±i,±j,±k of order 8. The FS κ of Q8 is defined as follows:
(51)κu1=0.90,ifu1=1,0.72,ifu1∈−1,±j,0.50,otherwise.
Theμ-FS κμ of Q8 correspond to the value μ=0.40 is given by
(52)κ0.4u1=0.94,ifu1=1,0.83,ifu1∈−1,±j,0.70,otherwise.
Note that the tip ofκμ is α=0.94 and Imκ0.40=0.70,0.83,0.94.(i)
The LSG corresponding to the valueτ=0.83 is given by
(53)κ0.830.400.83,0.94=±1,±j.(ii)
The OLSG corresponding to the valueτ=0.90 is given by κ0.900.400.90,0.94=1(iii)
The TSG corresponding to the valueτ=0.70 is given by κ0.70∈Imκμ0.400.70,0.94=Q8
Clearly,κ0.830.400.83,0.94, κ0.900.400.90,0.94, and κ0.70∈Imκμ0.400.70,0.94 are the subgroups of Q8.The following result describes another approach to obtain aμ-FNSG.Theorem 32.
For anyμ-FSG κμ of a group G, the following statements are equivalent:
(i)
κμ is a μ-FNSG in G(ii)
Every LSG ofκμ is normal in G(iii)
Every TSG ofκμ is normal in G(iv)
Every OLSG ofκμ is normal in GProof.
i⇒ ii: The application of Theorem 19 ensures the validity of i ⇒ ii.
ii⇒ iii: Since every TSG of κμ is a LSG of κμ, which is normal in G. Thus, every TSG of κμ is normal in G.
iii⇒ iv: Let 0≤τ≤α, where α is the tip of κμ. Then,
(54)κτμτ,α=⋃τ<λλ∈Imκμκλ∈Imκμμλ,α,
which is normal inG, being the union of an increasing chain of normal subgroups of G.
iv⇒ i: If possible, let there exist two elements u1,u2∈G such that κμu1u2>κμu2u1=τ. Then, u1u2∈κτμτ,α, since κτμτ,α is normal in G. Therefore, u2u1∈κτμτ,α. This implies that τ<κμu1u2, which is not true.


## 5. Conclusions

In this work, we introduced the notion ofμ-FSG defined over μ-FS and studies various important properties. We also presented the concept of μ-FNSGs and proved some important related results. Furthermore, we have defined the quotient group G/κμ and proved the existence of isomorphism between G/κμ and G/κ∗μ. In addition, we have defined the notions of LSG, OLSG, and TSG of a μ-FSG and have developed a relationship between these notions and μ-FNSG. We are hopeful this article provides a base to study many more important group theoretic topics like isomorphism theorems, Cayley’s theorem, Lagrange’s theorem, Sylow’s theorem, and many others inμ-fuzzy environment.



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