# Algebraic Properties ofω,θ-Complex Fuzzy Subgroups

Authors: Ahad Abdullah Al-harshni; Dilshad Alghazzawi
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper defined the notion ofω,θ-complex fuzzy sets, ω,θ-complex fuzzy subgroupoid, and ω,θ-complex fuzzy subgroups and described important examples under ω,θ-complex fuzzy sets. Additionally, we discussed the conjugacy class of group with respect to ω,θ-complex fuzzy normal subgroups. We define ω,θ-complex fuzzy cosets and elaborate the certain operation of this analog to group theoretic operation. We prove that factor group with regard to ω,θ-complex fuzzy normal subgroup forms a group and establishes an ordinary homomorphism from group to its factor group with regard to ω,θ-complex fuzzy normal subgroup. Moreover, we create the ω,θ-complex fuzzy subgroup of factor group.


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## Body

## 1. Introduction

The study of group theory has a crucial orbit in mathematics owing to its utilitarian applications in different scientific domains including cryptography, mathematical biology, and chaos theory. The extensive understanding of the group allows us to evaluate the dynamic behavior of the real-world physical problems that have been effectively employed in mathematical problem modeling to tackle many complicated problems in the field of science and technology. On the other hand, the theory of fuzzy set plays a significant role for dealing uncertainty and vagueness. Zadeh [1] first proposed the concept of fuzzy sets and their operations. Subsequently, many researchers have analyzed various aspects of this theory and its applications. This theory also plays a vital role for studying differential equations. Many differential equations exhibit uncertainty and vagueness in their solutions [2]. Fuzzy set theory becomes more important to deal this type of information. Imai et al. [3] proposed the study of BCK/BCI-algebras in 1966 as a generalized conviction of set-theoretic difference. Rosenfeld [4] built up the structure of fuzzy subgroups on fuzzy sets in 1971. In 1979, fuzzy groups were redefined by Anthony and Sherwood [5]. The notions of inclusion, intersection, union, convexity, relation, etc., are extended to such sets, and different properties of these convictions in the context of fuzzy sets are established in [6]. Liu [7] introduced the concept of fuzzy invariant subgroups. In 1988, Choudhury [8] introduced the notion of fuzzy homomorphism between two groups and studied its effect on fuzzy subgroups. Mashour et. al. depicted many different key properties of fuzzy subgroups in [9]. Filep [10] extended the structure and construction of fuzzy subgroups of groups in 1992. Many researchers actively engaged the development of fuzzy set in group theory [11]. Gupta and Qi [12] presented the some typical T-operators and reviewed T-norm, T-conorm, and negation function under T-operators. Malik and Mordeson [13] defined the concept fuzzy subgroups of Abelian groups and discussed their various algebraic properties. Mishref [14] depicted the fuzzy normal series of finite groups. The idea of complex fuzzy set was introduced by Ramot et al. [15, 16] in 2002. In 2009, Zhang [17] et al. developed the various algebraic setups of complex fuzzy sets. In 2009, a new structure and construction of Q-fuzzy groups were described by Solairaju [18]. Al-Husban and Salleh [19]elaborated the notion of complex fuzzy hypergroups based on complex fuzzy spaces. The new applications of complex fuzzy sets in metric spaces, graph theory, group theory, and ring theory were introduced in [20–23]. Ma et al. [24] presented some new mathematical operations and laws of complex fuzzy set such as absorption law, involution law, symmetrical difference formula, simple difference, and disjunctive sum. Moreover, they developed a new algorithm using complex fuzzy sets for applications signals. Trevijano and Elorza [25] initiated the new concept of annihilator of fuzzy subgroups and discussed their various algebraic properties. The recent applications of complex fuzzy sets in ring theory and BCk/BCI algebras may be viewed in [26, 27]. Imtiaz [28] et al explored the new structure of ξ-complex fuzzy sets and ξ-complex fuzzy subgroups. Many authors [29–34] proposed several interesting techniques and approaches to solve complicated systems in fuzzy group theory, fuzzy ring theory, and fuzzy fractional calculus whereas the computational effects are very vague and straightforward. Gulzar [35] et al. presented the novel concept of complex fuzzy subfields. Verma [36] et al. discussed a systematic review on the advancement in study of fuzzy variational problems. Alolaiyan et al. [38] developed the structure of (α, β)-complex fuzzy subgroups.In this article, we present a new conceptω,θ-complex fuzzy set ω,θ-CFS. First, we introduce some basic notions and some important properties which are needed in our paper. Second, in section three, we define ω,θ-complex fuzzy subgroups ω,θ-CFSGs and ω,θ-complex fuzzy normal subgroups and study their properties. Finally, we define the ω,θ-complex fuzzy cosets, and we obtain the quotient group regarding to ω,θ-complex normal fuzzy group ω,θ-CFNSG.


## 2. Preliminaries

In this section, we recall important concepts of complex fuzzy set and complex fuzzy subgroup.Definition 1.
(see [1]). A fuzzy subset is just like a function from a universal set to a unit interval [0, 1].Definition 2.
(see[15]). A CFS M of the universe of discourse X is mapping from nonempty set to unit disk and is described by the rule αM:X⟶x∈Ç:z≤1, where Ç is set of complex numbers. The αMx=ζMxeiφMx is complex membership function of CFS M, where i=−1.Definition 3.
(see [22]). Let R be a underlying universe and δ=x,δx:x∈R be fuzzy subset. Then, π-fuzzy subset of R is described as(1)δπ=x,δπx:δπx=2πδx,x∈R.Definition 4.
(see [22]). Let G be a group. A π-fuzzy set δπ of G is a π-fuzzy subgroup of G if(1)
δπxy≥minδπx,δπy for all x,y∈G(2)
δπx−1≥δπx−1 for all x∈GDefinition 5.
(see [22]). Let G be a underlying universe where δ=x,ζδxeiφδx:x∈R and γ=x,ζγxeiφγx:x∈R be CFSs of G. Then,(1)
A CFSδ is homogeneous CFS if ∀x,y∈R, ζδx≤ζδy if and only if φδx≤φδy(2)
A CFSδ is homogeneous CFS with γ if, ∀x,y∈R such that ζδx≤ζγy if and only if φδx≤φγyDefinition 6.
(see [17]). Let δ=x,ζδxeiφδx:x∈G and γ=x,ζγxeiφγx:x∈G be a CFSs of set X. Then, the following set-theoretic operation is defined as(1)
δ∩γx=ζδ∩γxeiφδ∩γx=minζδxeiφδx,ζγxeiφγx,∀x∈G(2)
δ∪γx=ζδ∪γxeiφδ∪γx=maxζδxeiφδx,ζγxeiφγx,∀x∈GDefinition 7.
(see [22]). Let G be a group and A=x,μAxeiφAx:x∈G be a homogeneous CFS of G. Then, A is said to be CFSG of G if the given below axioms are satisfied.(1)
ζδxyeiφδxy≥minζδxeiφδx,ζδyeiφδy(2)
ζδx−1eiφδx−1≥ζδxeiφδx , for all x,y∈GDefinition 8.
(see [22]). Let G be group. A CFS δ=x,ζδxeiφδx:x∈G of G is called a CFNSG of G if: ζδxyeiφδxy=ζδyxeiφδyx for all x,y∈G.


## 3.ω,θ-Complex Fuzzy Subgroups

In this section, we defineω,θ-CFSG and study their properties. We investigate the properties of ω,θ-CFNSGs and erect the quotient group with respect to ω,θ-CFNSGs.Definition 9.
Letδ=x,ζδxeiφδx;x∈G be complex fuzzy set of group G. For any ω∈0,1 and θ∈0,2π, then the complex fuzzy set δω,θ is an ω,θ-complex fuzzy set of G with regard to complex fuzzy set δ and is defined as ζδωxeiφδθx=ζδxeiφδx∗ωeiθ=ζδx.ωeiφδθx.θ/2π.Example 1.
Consider a universal setZ=a,b,c,d,e. The complex fuzzy set δ of Z is described as δ=a,0.2eπ/3i,b,0.1eπ/4i,c,0.4eπ/6i,d,0.7eπ/3i,e,0.3eπ/5i. In the view Definition 9, we get ω,θ-complex fuzzy set of Z with respect to complex fuzzy set δ, for the value of ω=0.2, and θ=eπi, as follows: δω,θ=a,0.04eπ/6i,b,0.02eπ/8i,c,0.08eπ/12i,d,0.14eπ/6i,e,0.06eπ/10i.Definition 10.
Letδω,θ and γω,θ be two ω,θ-CFSs of G. Then, A ω,θ−CFS δω,θ is homogeneous ω,θ-CFS if for all x,y∈G, we have ζδωx≤ζδωy if and only if φδθx≤φδθy. A ω,θ-CFS δω,θ is homogeneous ω,θ-CFS with γω,θ if for all x,y∈G, we have ζδωx≤ζδωy if and only if φδθx≤φδθy.Remark 1.
Ifω=1 and θ=2π in the above definition, we obtain a classical complex fuzzy set.Remark 2.
Letδω,θ and γω,θ be two ω,θ-CFSs of G. Then, δ∩γω,θ=δω,θ∩γω,θ.Definition 11.
Letδω,θ be an ω,θ-CFS of group G, for ω∈0,1 and θ∈0,2π. A ω,θ-complex fuzzy subgroupoid of G is a fuzzy algebraic structure which satisfies the following condition:(2)ζδωxyeiφδθxy≥minζδωxeiφδθx,ζδωyeiφδθy.Example 2.
Consider the groupG of fourth roots of unity under usual multiplication. Define(3)δx=0.4eπ/3i,ifx∈1,i0.2eπ/6i,ifx∈−1,−i.
Takeδω,θ with ω=0.1 and θ=π. Then, ζδ0.1xyeiφδπxy≥minζδ0.1xeiφδπx,ζδ0.1yeiφδπy for each x,y∈1,−1,i,−i. Thus, δω,θ is ω.θ-complex fuzzy subgroupoid of G.Definition 12.
Letδω,θ be an ω,θ− CFS of group G for ω∈0,1 and θ∈0,2π. A ω,θ-CFSG of the group G is a fuzzy algebraic structure that satisfies the following axioms:(1)
ζδωxyeiφδθxy≥minζδωxeiφδθx,ζδωyeiφδθy(2)
ζδωx−1eiφδθx−1≥ζδωxeiφδθx for all x,y∈GExample 3.
Consider the dihedral groupD3=j,s:j3=s2=e,sj=j2s. Let δ be CFS of G such that(4)δ=0.8eπ/3iifx∈s0.7eπ/6iotherwise.
Defineδω,θ for the value ω=0.5 and θ=π/2 as follows:(5)δ0.5,π/2x=0.4eπ/12iifx∈s0.35eπ/24iotherwise.
Takex=sj, y=s, and xy=j2, we have(6)ζδ0.5xyeiφδπ/2xy=0.35eπ/24i.(7)minζδ0.5xeiφδπ/2x,ζδ0.5yeiφδπ/2y,=min0.35eπ/24i,0.4eπ/12i=0.35eπ/24i.
Moreover,(8)δ0.5,π/2x−1≥δ0.5,π/2x.
Similarly, both conditions of Definition12 are satisfied for all elements of D3. Hence, δ0.5,π/2x is CFSG of D3.Remark 3.
Ifδω,θ be an ω,θ−CFS of the group G for ω∈0,1. Then,(9)ζδωx−1yeiφδθx−1y≥minζδωxeiφδθx,ζδωyeiφδθy.Theorem 1.
Letδω,θ be an ω,θ−CFSG of the group G,∀x,y∈G. Then,(1)
ζδωyeiφδθy≤ζδωeeiφδθe(2)
ζδωxy−1eiφδθxy−1=ζδωeeiφδθe which implies that ζδωxeiφδθx=ζδωyeiφδθyProof.
The proof of this theorem is obvious.Theorem 2.
LetG be finite group. If δω,θ is an ω,θ−complex fuzzy subgroupoid of G, then δω,θ is ω,θ-CFSG of G.Proof.
Assume thatx is a member of G, thus, the order of x is a finite number n such that xn=e, where e is the identity element of the group G. We know that x−1=xn−1 by the application of Definition 11. Then, we have(10)ζδωx−1eiφδθx−1=ζδωxn−1eiφδθxn−1=ζδωxn−2xeiφδθxn−2x≥ζδωxeiφδθx.Theorem 3.
LetG be a group. If δω,θ be an ω,θ-CFSG of G and for some x∈G, ζδωxeiφδθx=ζδωeeiφδθe, then ζδωxyeiφδθxy=ζδωyeiφδθy for all y∈G.Proof.
Given thatζδωxeiφδθx=ζδωeeiφδθe. Then, from Theorem 2, we have(11)ζδωyeiφδθy≤ζδωxeiφδθx,∀y∈G.
Consider(12)ζδωxyeiφδθxy≥minζδωxeiφδθx,ζδωyeiφδθy,ζδωxyeiφδθxy≥ζδωyeiφδθy.From Theorem3.9.
Now, assume that(13)ζδωyeiφδθy=ζδx−1xyiφδx−1xy≥minζδωxeiφδθx,ζδωxyeiφδθxy.
Again, from Theorem2, we have(14)minζδωxeiφδθx,ζδωxyeiφδθxy=ζδωxyeiφδθxy.
Therefore, we obtain(15)ζδωyeiφδθy≥ζδωxyeiφδθxy,∀y∈G.
From (12) and (15), we have(16)ζδωyeiφδθy=ζδωxyeiφδθxy,∀y∈G.Theorem 4.
.Ifδ is CFSG of the group G, then δ is an ω,θ-CFSG of G.Proof.
Assume thatδ is a CFSG of G, for all x,y∈G. Consider(17)ζδωxyeiφδθxy=ζδxyeiφδxy∗ωeiθ≥minζδxeiφδx,ζδyeiφδy∗ωeiθ=minζδxeiφδx∗ωeiθ,ζδyeiφδy∗ωeiθ=minζδωxeiφδθx,ζδωyeiφδθy.
Further, we assume that(18)ζδωx−1eiφδθx−1=ζδx−1iφδx−1∗ωeiθ≥ζδxeiφδx∗ωeiθ=ζδωxeiφδθx.Theorem 5.
LetPω,θ and Qω,θ be two ω,θ-CFSGs of G, then Pω,θ∩Qω,θ is also ω,θ-CFSGs of G.Proof.
Given thatPω,θ and Qω,θ be two ω,θ-CFSGs of G, for any x,y∈G. Consider(19)ζP∩QωxyeφP∩Qθxy=ζPω∩QωxyeiφPθ∩Qθxy=minζPωxyeiφPxθxy,ζQωxyeiφQxθxy≥minminζPωxeiφPθx,ζPωyeiφPθy,minζQωxeiφQθx,ζQωyeiφQθy.=minminζPωxeiφPθx,ζQωxeiφQθx,minζPωyeiφPθy,ζQωyeiφQθy.=minζPω∩QωxeiφQθ∩Pθx,μQω∩PωyeiφPθ∩Qθy,=minζP∩QωxeφP∩Qθx,ζP∩QωyeφP∩Qθy.
Furthermore,(20)ζP∩Qωx−1eφP∩Qθx−1=ζPω∩Qωx−1eiφQθ∩Pθx−1=minζPωx−1eiφPθx−1,ζQωx−1eiφQθx−1≥minζPωxeiφPθx,ζQωxeiφQθx=ζP∩QωxeφP∩Qθx.
Consequently,Pω,θ∩Qω,θ is ω,θ of G.Remark 4.
The converse of Theorem10 may not be true.Example 4.
Consider a permutation groupS4. Consider two ω,θ-CFSGs δ0.7,2π and γ0.7,2π of S4, we take ωeiθ=0.7e2π are described as(21)δ0.7,2πx=0.6eπ/3ifx∈130.5eπ/5otherwise.(22)γ0.7,2πx=0.7e2πifx∈13240.4eπ/6otherwise.
This implies that(23)δ0.7,2π∪γ0.7,2πx=0.7e2πifx∈13240.6eπ/3ifx∈13−13240.5eπ/5otherwise.
Takex=1234,y=13, and xy=1432. Therefore,
δ0.7,2π∪γ0.7,2πx=0.7e2π,δ0.7,2π∪γ0.7,2πy=0.6eπ/3 and δ0.7,2π∪γ0.7,2πxy=0.5eπ/5δ0.7,2π∪γ0.7,2πxy≱minδ0.7,2π∪γ0.7,2πx,δ0.7,2π∪γ0.7,2πy.
Hence, this proves the claim.Definition 13.
Letδω,θ be an ω,θ-CFSG of the group G. Then, ω,θ-CFS xδω,θp=p,ζxδωpeiφxδθp of G is said to be a ω,θ-complex fuzzy left coset of G established by ω,θ and x. It is defined as(24)ζxδωpeiφxδθp=ζδωx−1peiφδθx−1p=ζδx−1peiφδx−1p∗ωeiθ,∀p,x∈G.
On the same way, we describe theω,θ-complex fuzzy right coset δω,θxp=p,ζδωxpeiφδθxp,x∈G is described as(25)ζδωxpeiφδθxp=ζδωpx−1eiφδθpx−1=ζδpx−1eiφδpx−1∗ωeiθ∀p,x∈G.Example 5.
TakeG=1,1234,1432,1324,1423,1234,24,13 a group of order 8. Describe ω,θ-CFSG of G, ω=1 and θ=π/2, such that(26)δ1,π/2p=0.6eπ/2ifx∈1,13240.5eπ/4ifx∈1423,12340.4eπ/6otherwise.
From Definition13, we have ζxδ1,π/2peiφxδ1,π/2p=ζδ1,π/2x−1peiφδ1,π/2x−1p. Hence, the 1,π/2-complex fuzzy left coset of δ1,π/2p in G for x=24 is defined as(27)xδ1,π/2p=0.6eπ/2ifx∈13,240.5eπ/4ifx∈1432,12340.4eπ/6otherwise.
In the same way, we can define1,π/2-complex fuzzy right coset of δ1,π/2p, for all x∈G.Definition 14.
LetG be group and δω,θ be an ω,θ-CFSG of G. Then, δω,θ is said to be a complex fuzzy normal subgroup (invariant) ω,θ-CFNSG if δω,θxy=δω,θyx. Equivalently, ω,θ-CFSG δω,θ is ω,θ-CFNSG of the group G if δω,θxy=xδω,θy, for all x,y∈G.Remark 5.
Letδω,θ be an ω,θ-CFNSG of the group G. Then, δω,θy−1xy=δω,θx for all x,y∈G.Theorem 6.
Every CFNSG of the groupG is an ω,θ-CFNSG of G.Proof.
Suppose thatz,x are elements of G. Then,(28)ζδx−1zeiφδx−1z=ζδz−1xeiφδz−1x.
This implies that(29)ζδx−1zeiφδx−1z∗ωeiθ=minζδz−1xeiφδz−1x∗ωeiθ,
which implies that(30)ζxδωzeiφxδθz=ζδωxzeiφxδθz.
This implies thatxδω,θz=δω,θxz. Consequently, δω,θ is ω,θ-CFNSG of G.Theorem 7.
Letδω,θ be an ω,θ-CFSG of the group G. Then, δω,θ remains the same in the conjugacy class of G if and only if δω,θ is an ω,θ-CFNSG.Proof.
Suppose thatδω,θ is an ω,θ-CFNSG of the group G. Then,(31)ζδωy−1xyeiφδθy−1xy=ζδωxyy−1eiφδθxyy−1=ζδωxeiφδθx,∀x,y∈G.
Conversely, assume thatδω,θ remains the same in every conjugate class of G. Then,(32)ζδωxyeiφδθxy=ζδωxyxx−1eiφδθxyxx−1=ζδωxyxx−1eiφδθxyxx−1=ζδωyxeiφδθyx,∀x,y∈G.Theorem 8.
LetG be a group and δω,θ be an ω,θ-CFSG of G. Then, δω,θ is an ω,θ-CFNSG if and only if ζδωx,yeiφδθx,y≥ζδωxeiφδθx,∀x,y∈G.Proof.
Takeδω,θ is an ω,θ-CFNSG of G. Let x,y be two members of G. Assume that(33)ζδωx−1y−1xyeiφδθx−1y−1xy≥minζδωy−1xyeiφδθy−1xy,ζδωx−1eiφδθx−1=minζδωxeiφδθx,ζδωxeiφδθxζδωx,yeiφδθx,y≥ζδxiφδθx.
Conversely, suppose thatζδωx,yeiφδθx,y≥ζδxeiφδθx. Let x,r∈G(34)Considerζδωx−1rxeiφδθx−1rx=ζδωrr−1x−1rxeiφδθrr−1x−1rx≥minζδωreiφδθr,ζδωr,xeiφδθr,x=ζδωreiφδθr,(35)Thus,ζδωx−1rxeiφδθx−1rx≥ζδωreiφδθr∀r,x∈G.(36)Now,ζδωreiφδθr=ζδωxx−1rxx−1eiφδθxx−1rxx−1≥minζδωxeiφδθx,ζδωx−1rxeiφδθx−1rx.
Now, we present two potential cases:Case 1.
Ifminζδωxeiφδθx,ζδωx−1rxeiφδθx−1rx=ζδωxeiφδθx
Then, we obtainζδωreiφδθr≥ζδωxeiφδθx,∀r,x∈G, which means that δω,θ remains a constant, and the result is valid trivially.Case 2.
Ifminζδωxeiφδθx,ζδωx−1rxeiφδθx−1rx=ζδωx−1rxeiφδθx−1rx.
Then, from (36), we have(37)ζδωreiφδθr≥ζδωx−1rxeiφδθx−1rx.
From Equations (35) and (37), we have(38)ζδωreiφδθr=ζδωx−1rxeiφδθx−1rx.
Hence,δω,θ is a constantTheorem 9.
Ifδω,θ is an ω,θ-complex fuzzy normal subgroup of G, Then, the set δω,θe=p∈G:δω,θp−1=δω,θe is an invariant subgroup of G.Proof.
We haveδω,θe≠ϕ because e∈δω,θe. Let p,q∈δω,θe be any members. Consider ζδωpqeiφδθpq≥minζδωpeiφδθp,ζδωqeiφδθq=minζδωeeiφδθe,ζδωeeiφδθe which means that ζδωpqeiφδθpq≥ζδωeeiφδθe. However, ζδωpqeiφδθpq≥ζδωeeiφδθe. Therefore, ζδωpqeiφδθpq≥ζδωeeiφδθe. It means that δω,θp−1=δω,θe; hence, pq∈δω,θe. Furthermore, ζδωq−1eiφδθq−1≥ζδωqeiφδθq=ζδωeeiφδθe. However, ζδωqeiφδθq≤ζδωeeiφδθe. Thus, δω,θe is subgroup of group G. Let p∈δω,θe and q∈G. We know ζδωq−1pqeiφδθq−1pq=ζδωeeiφδθe. This implies that q−1pq∈δω,θe. Therefore, δω,θe is invariant subgroup.Theorem 10.
Letδω,θ is an ω,θ-CFNSG of G. Then,(i)
xδω,θ=yδω,θ if and only if x−1y∈δω,θe(ii)
δω,θx=δω,θyif and only ifxy−1∈δω,θeProof.
(i)
For anyx,y∈G, we have xδω,θ=yδω,θ. Consider(39)ζδωx−1yeiφδθx−1y=ζδx−1yeiφδx−1y∗ωeiθ=ζxδωyeiφxδωy=ζyδωyeiφyδωy=ζδωy−1yeiφδθy−1y=ζδωeeiφδθe.Therefore,x−1y∈δω,θe. Conversely, let x−1y∈δω,θe implies that ζδωx−1yeiφδθx−1y=ζδeeiφδe∗ωeiθ. Consider(40)ζxδωaeiφxδωa=ζδx−1aeiφδx−1a∗ωeiθ=ζδωx−1aeiφδθx−1a=ζδωx−1yy−1aeiφδθx−1a≥minζδωx−1yeiφδθx−1y,ζδωy−1aeiφδθy−1a=minζδωeeiφδθe,ζδωy−1aeiφδθy−1a=ζδωy−1aeiφδθy−1a=ζyδωaeiφyδωa.Exchange the function ofx and y, and we get ζyδωaeiφyδθa≥ζxδωaeiφxδθa. Therefore, ζxδωaeiφxδθa=ζyδωaeiφyδθa.(ii)
In the same way, we can show that as part (i).Theorem 11.
LetH be a group and δω,θ be an ω,θ-CFNSG of H and let x,y,r and t be any members in H. If xδω,θ=rδω,θ and yδω,θ=tδω,θ. Then, xyδω,θ=rtδω,θ.Proof.
We havexδω,θ=rδω,θ and yδω,θ=tδω,θ. This means that x−1r,y−1t∈δω,θe. Consider xy−1rt=y−1x−1rt=y−1x−1ryy−1t=y−1x−1ryy−1t. As δω,θe is normal subgroup of H. Thus, xy−1rt∈δω,θe. As result, xyδω,θ=rtδω,θ.Theorem 12.
Assume thatH/δω,θ=xδω,θ;x∈H is the assemblage of all ω,θ-complex fuzzy cosets of δω,θ is an ω,θ-CFNSG of the group H. Therefore, we can define a well-defined binary operation ⋆ of H/δω,θ and is described as xδω,θ⋆yδω,θ=xyδω,θ∀x,y∈H.Proof.
We havexδω,θ=yδω,θ and fδω,θ=gδω,θ for any f,g,x,y∈H. Let c∈H by any element. Then, xδω,θ⋆fδω,θc  =  xfδω,θc  =  c,ζxfδωceiφxδωc. Consider(41)ζxfδωceiφxδωc=ζxfδωceiφxδωc⋆ωeiθ=ζδωxf−1ceiφδθxf−1c=ζδωf−1x−1ceiφδθf−1x−1c=ζδωf−1x−1ceiφδθf−1x−1c=ζfδωx−1ceiφxδθx−1c=ζgδωx−1ceiφxδθx−1c=ζδg−1x−1ceiφδg−1x−1c=ζδx−1cg−1eiφδx−1cg−1=ζxδωcg−1eiφxδθcg−1=ζyδωcg−1eiφyδθcg−1=ζδωy−1cg−1eiφδθy−1cg−1=ζδωy−1cg−1eiφδθy−1cg−1=ζδωy−1g−1ceiφδθy−1g−1c=ζδωyg−1ceiφδθyg−1c=ζygδωceiφygδθc.
Therefore,⋆ is well-defined operation and satisfied the associative property on H/δω,θ. Furthermore, ζδωeiφδθ⋆ζxδωeiφxδθ=ζeδωeiφeδθ⋆ζxδωeiφxδθ=ζxδωeiφxδθ, and this means that ζδωeiφδ is an identity element of H/δω,θ. Obviously, the inverse of each member of H/δω,θ exists if ζxδωeiφxδθ∈H/δω,θ, and then, there exists an element ζx−1δωeiφx−1δθ∈H/δω,θ such that ζx−1xδωeiφx−1xδθ=ζδωeiφδθ. Consequently, H/δω,θ is a group.Lemma 1.
LetG be group then we can define L:G⟶G/δω,θ to be a classical homomorphism from G onto G/δω,θ and is described by Lx=xδω,θ with the kernel L=δω,θe.Proof.
Letx,y be an arbitrary member of G, and then,(42)Lxy=xyδω,θ=μxyδωeiφxyδθ=ζxδωeiφxδθ⋆ζyδωeiφyδθ=xδω,θ⋆yδω,θ=Lx⋆Ly.
Therefore,L is a homomorphism. In the same way, f is also a homomorphism(43)Now,KerL=x∈G:Lx=eδω,θ=x∈G:xδω,θ=eδω,θ=x∈G:xe−1∈δω,θe=x∈G:x∈δω,θe=δω,θe.Theorem 13.
Letδω,θe be a normal subgroup of G. If δω,θ=x,ζδωxeiφδθx:x∈G is ω,θ-CFSG, then the ω,θ-CFS δ¯ω,θ=xδω,θe,ζ¯δωxδω,θeeiφ¯δθxδω,θe:x∈G of G/δω,θe is also ω,θ-CFSG of G/δω,θe, where ζ¯δωxδω,θeeiφ¯δθxδω,θe=maxζδωxaeiφδθxa:a∈δω,θe.Remark 6.
Ifδω,θ is an ω,θ-CFSG of G, x∈G, and ζδωxyeiφδθxy=ζδωyeiφδθy, for all y∈G, then ζδωxeiφδθx=ζδωeeiφδθe.


## 4. Conclusion

We have presented a new theory ofω,θ-CFSs and ω,θ-CFSGs as a powerful extension of complex fuzzy sets and complex fuzzy subgroups. Moreover, we have created the ω,θ-CFSG of the factor group. Each concept introduced in this paper is explained with clear examples. This algebraic structure will play an important role to handle many other group theory problems under a new complex fuzzy environment [37, 38].



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