Bipolar Complex Fuzzy Subgroups

Abstract

In this study, firstly, we interpret the level set, support, kernel for bipolar complex fuzzy (BCF) set, bipolar complex characteristic function, and BCF point. Then, we interpret the BCF subgroup, BCF normal subgroup, BCF conjugate, normalizer for BCF subgroup, cosets, BCF abelian subgroup, and BCF factor group. Furthermore, we present the associated examples and theorems, and prove these associated theorems. After that, we interpret the image and pre-image of BCF subgroups under homomorphism and prove the related theorems.

1. Introduction

In mathematics, one of the significant parts of algebra is group theory (GT). Because of its rich and efficient structure, GT examines objects that exist in symmetric shapes. GT has a curial role in categorizing the symmetries of crystals, atoms, molecules, and polyhedral structures. GT is raised as a great technique to analyze codon sequence behavior and genetic code. GT is one of the significant branches of mathematics that has been employed in various areas such as cryptography, algebraic geometry, physics, etc. To solve the genuine life issues which involve uncertainty and ambiguity, one needs proper structure, which was given by Zadeh [1], who called it fuzzy set (FS). The FS theory helps us to achieve an improved way to solve genuine life issues by the appropriate technique of decision-making (DM). The FS generalized the conception of the crisp set by transforming the set $\left\{0,1\right\}$ to the interval $\left[0,1\right]$. The structure of FS is a wonderful mathematical structure to signify the family of elements whose values are ambiguous. The FS holds a supportive degree, which is covered in $\left[0,1\right]$. The conception of FS is employed in numerous fields such as graph theory, engineering, and social and medical sciences by various scholars. Rosenfield [2] described the relationship between FS and groups and called it fuzzy groups in 1971. The fuzzy relations, fuzzy groups, level subgroups, fuzzy functions, and fuzzy equivalence relations are explored by Bhattacharya and Mukherjee [3], Das [4], and Demirci and Recasens [5]. Foster [6] introduced fuzzy topological groups. The fuzzy homomorphism and isomorphism were interpreted by Jin-Xuan [7]. Various researchers generalized this basic conception of FS such as intuitionistic FS [8], interval-valued FS [9], soft sets [10], etc. However, these modifications of FS cannot signify the negative pole or negative opinion of human beings. To overwhelmed Zhang [11] modified the FS to the conception of bipolar FS (BFS) by transforming $\left[0,1\right]$ to $(\left[0,1\right],\left[-1,0\right])$. The structure of the bipolar fuzzy (BF) set holds positive supportive and negative supportive degrees to express both opinions or sides of the human being. Lee [12] presented the conception of the BF valued set. Mahmood and Munir [13] interpreted BF-valued subgroups. Alolaivan et al. [14] developed a certain structure of BF subrings. The homomorphism and anti-homomorphism of BF sub-HX groups were initiated by Muthuraj and Sridharan [15]. The BF K-algebras and BF graphs were initiated by Akram et al. [16], and Akram and Dudek [17]. Manemaran and Chellappa [18] presented the BF groups and BF D-ideals.

In various circumstances, the second dimension is involved in the information which cannot be handled by FS and its generalization. Therefore, Ramot et al. [19] investigated complex FS (CFS) which modified FS by transforming its range from $\left[0,1\right]$ to the unit disc of the complex plane. Then, Tamir et al. [20] improved the CFS by transforming its range from unit disc to unit square. Alolaiyan et al. [21] initiated $(\alpha ,\beta )$- complex fuzzy (CF) subgroups. (CFSGs). Abuhijleh et al. [22] explored CFSGs by relying on Rosenfeld’s approach. The CF subfield was established by Gulzar et al. [23]. The CF group initiated on CF space was given by Al-Husban and Salleh [24]. Mahmood and Ur Rehman [25] modified the conception of the BF set to the BCF set by transformed $(\left[0,1\right],\left[-1,0\right])$ to $(\left[0,1\right]+\iota \left[0,1\right],\left[-1,0\right]+\iota \left[-1,0\right])$ to overcome various kinds of information involving two dimensions with both the positive and negative human aspect or opinion. The BCF set holds a positive supportive degree contained in $(\left[0,1\right]+\iota \left[0,1\right])$ and negative supportive degree contained in $(\left[-1,0\right]+\iota \left[-1,0\right])$ to express two-dimensional data with both opinions or sides of the human being. The structure of the BCF set is a wonderful mathematical structure to signify the family of elements whose values are two-dimensional with both positive and negative opinions. There is a significant role of group symmetry in the examination of molecule structures. The fuzzy intellect appears in it because the isotope molecules decay with a specific ratio. If this specific ratio of decays follows the conditions of the BCF set, then we cannot employ the BF-valued subgroups presented by Mahmood and Munir [13] or any of the existing subgroups to examine the form of the isotope at a specific time. Thus, for such circumstances, where one cannot use the BF-valued subgroups or any other subgroups, we are going to introduce BCF subgroups which will close this gap in the literature. To date, no one has introduced the algebraic structures of the BCF set. Thus, in this analysis, we interpreted the algebraic structures of the BCF set to enhance the conception of the BCF set in the algebraic structures and cover the existing gap in the literature. We know that the BCF set generalized FS, BFS, and CFS, and the main difference between these notions and the BCF set is that the BCF set can handle the information displayed in two dimensions with two-sided opinions of human beings. Thus, our developed BCF subgroup and its related result in this analysis is differentiated from the fuzzy subgroups, BF-valued subgroups, and CF subgroups in a similar manner. One can obtain a fuzzy subgroup, BF-valued subgroup, and CF subgroup from our developed BCF subgroup. Thus, the BCF subgroup generalized these prevailing notions.

The short review of the composing article is as follows: In Section 2, the basic notion of FS, its related theory, BF set, its related theory, and BCF set are reviewed. In Section 3, we diagnosed the level set, kernel, bipolar complex characteristic function (BCCF), BCF point, BCF subgroup, BCF normal subgroup, BCF normalizer, BCF conjugate, BCF cosets, BCF abelian subgroups, and their related examples and theorems. Additionally, we interpreted the image and pre-image of BCF subgroups under homomorphism. The concluding remarks are explained in Section 4.

2. Preliminaries

Here, the basic notion of FS, its related theory, BF set, its related theory, and BCF set is reviewed. Let ${\rm X}$ be a non-empty set and $Ğ$ be a group in this article.

Definition 1 ([1]).

A FS over ${\rm X}$ is identified as

$$Ɓ=\left\{(ӿ,{\mu }_{Ɓ}(ӿ))\vee ӿ\in {\rm X}\right\}$$

(1)

apparently, ${\mu }_{Ɓ}(ӿ):{\rm X}\to \left[0,1\right]$ named as the supportive degree.

Definition 2 ([2]).

A fuzzy subset (FSS) $Ɓ={\mu }_{Ɓ}(ӿ)$ of Ğ is known as a fuzzy subgroup (FSG) of Ğ if $\forall ӿ,ƴ\in Ğ$, we have

• ${\mu }_{Ɓ}(ӿƴ)\ge min\left\{{\mu }_{Ɓ}(ӿ),{\mu }_{Ɓ}(ƴ)\right\}$
• ${\mu }_{Ɓ}({ӿ}^{-1})\ge {\mu }_{Ɓ}(ӿ)$

Or equivalently ${\mu }_{Ɓ}({ӿ}^{-1}ƴ)\ge {\mu }_{Ɓ}(ӿƴ)$

Definition 3 ([2]).

A FSG $Ɓ={\mu }_{Ɓ}(ӿ)$ of Ğ is known as a fuzzy normal subgroup (FNSG) of Ğ if $\forall ӿ,ƴ\in Ğ$, we have

$${\mu }_{Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{Ɓ}(ӿ)$$

(2)

Definition 4 ([11]).

A BF set over ${\rm X}$ is identified as

$$Ɓ=\left\{(ӿ,{\mu }_{PƁ}(ӿ),{\mu }_{N-Ɓ}(ӿ))\vee ӿ\in {\rm X}\right\}$$

(3)

apparently, ${\mu }_{P-Ɓ}(ӿ):{\rm X}\to \left[0,1\right]$ and ${\mu }_{N-Ɓ}(ӿ):{\rm X}\to \left[0,1\right]$, named as the positive supportive degree and negative supportive degree.

Definition 5 ([13]).

A bipolar FSS $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})$ of Ğ is called bipolar FSG (BFSG) of Ğ if $\forall ӿ,ƴ\in Ğ$, we have

• ${\mu }_{P-Ɓ}(ӿƴ)\ge min\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ƴ)\right\}$
• ${\mu }_{P-Ɓ}({ӿ}^{-1})\ge {\mu }_{P-Ɓ}(ӿ)$
• ${\mu }_{N-Ɓ}(ӿƴ)\le max\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ƴ)\right\}$
• ${\mu }_{N-Ɓ}({ӿ}^{-1})\le {\mu }_{N-Ɓ}(ӿ)$

Definition 6 ([13]).

A BFSG of Ğ $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})$ is called bipolar FNSG of Ğ if$\forall ӿ,ƴ\in Ğ$ we have

• ${\mu }_{P-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{P-Ɓ}(ӿ)$
• ${\mu }_{N-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{N-Ɓ}(ӿ)$

Definition 7 ([25]).

A BCF set over ${\rm X}$ is identified as

$$Ɓ=\left\{(ӿ,{\mu }_{P-Ɓ}(ӿ),{\mu }_{N-Ɓ}(ӿ))\vee ӿ\in {\rm X}\right\}$$

(4)

apparently, ${\mu }_{P-Ɓ}(ӿ)={\mu }_{RP-Ɓ}(ӿ)+\iota {\mu }_{IP-Ɓ}(ӿ)$ and ${\mu }_{N-Ɓ}(ӿ)={\mu }_{RN-Ɓ}(ӿ)+\iota {\mu }_{-Ɓ}(ӿ)$, named as the positive supportive degree and negative supportive degree with ${\mu }_{RP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}(ӿ)\in \left[0,1\right]$ and ${\mu }_{RN-Ɓ}(ӿ),{\mu }_{-Ɓ}(ӿ)\in \left[-1,0\right]$. For ease, we will consider the structure of the BCF set as $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ in this article.

3. BCF Subgroups and Related Concepts

Here, we are going to diagnose the level set, kernel, bipolar complex characteristic function (BCCF), BCF point, BCF subgroup, BCF normal subgroup, BCF normalizer, BCF conjugate, BCF cosets, BCF abelian subgroups, and their related examples and theorems. Moreover, we interpret the image and pre-image of BCF subgroups under homomorphism. Throughout this analysis, for the two BCF sets ${Ɓ}_1=({\mu }_{P-{Ɓ}_1},{\mu }_{N-{Ɓ}_1})=({\mu }_{RP-{Ɓ}_1}+\iota {\mu }_{IP-{Ɓ}_1},{\mu }_{RN-{Ɓ}_1}+\iota {\mu }_{-{Ɓ}_1})$ and ${Ɓ}_2=({\mu }_{P-{Ɓ}_2},{\mu }_{N-{Ɓ}_2})=({\mu }_{RP-{Ɓ}_2}+\iota {\mu }_{IP-{Ɓ}_2},{\mu }_{RN-{Ɓ}_2}+\iota {\mu }_{-{Ɓ}_2})$, ${Ɓ}_1\le {Ɓ}_2$ if ${\mu }_{P-{Ɓ}_1}\le {\mu }_{P-{Ɓ}_2}$ and ${\mu }_{N-{Ɓ}_1}\ge {\mu }_{N-{Ɓ}_2}$ that is, ${\mu }_{RP-{Ɓ}_1}\le {\mu }_{RP-{Ɓ}_2}$, ${\mu }_{IP-{Ɓ}_1}\le {\mu }_{IP-{Ɓ}_2}$ and ${\mu }_{RN-{Ɓ}_1}\ge {\mu }_{RN-{Ɓ}_2}$, ${\mu }_{-{Ɓ}_1}\ge {\mu }_{-{Ɓ}_2}$.

Definition 8.

Suppose a BCF subset $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ of $Ğ$ , then

• ${\mu }_{P-{Ɓ}_{\tau ,\xi }}=\left\{ӿ\in Ğ:{\mu }_{RP-Ɓ}(ӿ)\ge \tau \wedge {\mu }_{IP-Ɓ}(ӿ)\ge \xi ,\tau ,\xi \in \left[0,1\right]\right\}$ and ${\mu }_{N-{Ɓ}_{\tau ,\xi }}=\left\{ӿ\in Ğ:{\mu }_{NP-Ɓ}(ӿ)\le \tau \wedge {\mu }_{-Ɓ}(ӿ)\le \xi ,\tau ,\xi \in \left[-1,0\right]\right\}$ is called a level set.
• ${Ɓ}^{″}=\left\{ӿ\in Ğ:{\mu }_{P-Ɓ}(ӿ)\ge 0\wedge {\mu }_{N-Ɓ}\le 0\right\}=\left\{\begin{array}{c}ӿ\in Ğ:{\mu }_{RP-Ɓ}(ӿ)\ge 0,{\mu }_{IP-Ɓ}(ӿ)\ge 0\wedge {\mu }_{RN-Ɓ}\le 0,{\mu }_{-Ɓ}(ӿ)\le 0\end{array}\right\}$ is said to be the support of $Ɓ$
• ${Ɓ}^{}=\left\{ӿ\in Ğ:{\mu }_{P-Ɓ}(ӿ)=1+\iota 1\wedge {\mu }_{N-Ɓ}=-1-\iota 1\right\}$ is said to be the kernel of $Ɓ$.

Definition 9.

For any subset $H$ of a group $Ğ$, the BCCF is signified by $({\mu }_{P-H},{\mu }_{N-H})$ and described as

$${\mu }_{P-H}(ӿ)=\{\begin{array}{c}1+\iota 1 if ӿ\in H\\ 0+\iota 0,otheriwse\end{array}$$

$${\mu }_{N-H}(ӿ)=\{\begin{array}{c}-1-\iota 1 if ӿ\in H\\ 0+\iota 0,otheriwse\end{array}$$

Remark 1.

Evidently, BCCF is a BCF subset of Ğ.

Definition 10.

For $ƴ\in Ğ$, and $p,q\in (0,1]$ the BCF subset $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ of Ğ is of the structure

$${\mu }_{P-Ɓ}(ӿ)=\{\begin{array}{c}p+\iota q if ӿ=ƴ\\ 0+\iota 0, if ӿ\ne ƴ\end{array}$$

$${\mu }_{N-Ɓ}(ӿ)=\{\begin{array}{c}-p-\iota q if ӿ=ƴ\\ 0+\iota 0, if ӿ\ne ƴ\end{array}$$

is said to be BCF point with values $p$ and $q$ and support ӿ or said to be BCF singleton subset of Ğ.

Definition 11.

For two BCF subsets of Ğ ${Ɓ}_1=({\mu }_{P-{Ɓ}_1},{\mu }_{N-{Ɓ}_1})=({\mu }_{RP-{Ɓ}_1}+\iota {\mu }_{IP-{Ɓ}_1},{\mu }_{RN-{Ɓ}_1}+\iota {\mu }_{-{Ɓ}_1})$ and ${Ɓ}_2=({\mu }_{P-{Ɓ}_2},{\mu }_{N-{Ɓ}_2})=({\mu }_{RP-{Ɓ}_2}+\iota {\mu }_{IP-{Ɓ}_2},{\mu }_{RN-{Ɓ}_2}+\iota {\mu }_{-{Ɓ}_2})$, their product is interpreted as

$${Ɓ}_1\circ {Ɓ}_2=({\mu }_{P-{Ɓ}_1}\circ {\mu }_{P-{Ɓ}_2},{\mu }_{N-{Ɓ}_1}\circ {\mu }_{N-{Ɓ}_2})$$

$$({\mu }_{RP-{Ɓ}_1}\circ {\mu }_{RP-{Ɓ}_2}+\iota {\mu }_{IP-{Ɓ}_1}\circ {\mu }_{IP-{Ɓ}_2},{\mu }_{RN-{Ɓ}_1}\circ {\mu }_{RN-{Ɓ}_2}+\iota {\mu }_{-{Ɓ}_1}\circ {\mu }_{-{Ɓ}_2})$$

(5)

where,

$$\begin{gathered} ({\mu }_{RP-{Ɓ}_1}\circ {\mu }_{RP-{Ɓ}_2})(ӿ)=\underset{ӿ=ƴȥ}{\sup }\left\{\min ({\mu }_{RP-{Ɓ}_1}(ƴ), {\mu }_{RP-{Ɓ}_2}(ȥ))\right\} \\ ({\mu }_{IP-{Ɓ}_1}\circ {\mu }_{IP-{Ɓ}_2})(ӿ)=\underset{ӿ=ƴȥ}{\inf }\left\{\max ({\mu }_{IP-{Ɓ}_1}(ƴ), {\mu }_{IP-{Ɓ}_2}(ȥ))\right\} \\ ({\mu }_{RN-{Ɓ}_1}\circ {\mu }_{RN-{Ɓ}_2})(ӿ)=\underset{ӿ=ƴȥ}{\sup }\left\{\min ({\mu }_{RN-{Ɓ}_1}(ƴ), {\mu }_{RN-{Ɓ}_2}(ȥ))\right\} \\ ({\mu }_{IN-{Ɓ}_1}\circ {\mu }_{IN-{Ɓ}_2})(ӿ)=\underset{ӿ=ƴȥ}{\inf }\left\{\max ({\mu }_{IN-{Ɓ}_1}(ƴ), {\mu }_{IN-{Ɓ}_2}(ȥ))\right\} \end{gathered}$$

Definition 12.

A BCF subset $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ of Ğ is called BCF subgroup of Ğ (BCFSG) if $\forall ӿ,ƴ\in Ğ$, we have

• ${\mu }_{P-Ɓ}(ӿƴ)\ge min\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ƴ)\right\} \Rightarrow {\mu }_{RP-Ɓ}(ӿƴ)\ge min\left\{{\mu }_{RP-Ɓ}(ӿ),{\mu }_{RP-Ɓ}(ƴ)\right\}$ and ${\mu }_{IP-Ɓ}(ӿƴ)\ge min\left\{{\mu }_{IP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}(ƴ)\right\}$
• ${\mu }_{P-Ɓ}({ӿ}^{-1})\ge {\mu }_{P-Ɓ}(ӿ)\Rightarrow {\mu }_{RP-Ɓ}({ӿ}^{-1})\ge {\mu }_{RP-Ɓ}(ӿ)$ and ${\mu }_{IP-Ɓ}({ӿ}^{-1})\ge {\mu }_{IP-Ɓ}(ӿ)$
• ${\mu }_{N-Ɓ}(ӿƴ)\le max\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ƴ)\right\} \Rightarrow {\mu }_{RN-Ɓ}(ӿƴ)\le max\left\{{\mu }_{RN-Ɓ}(ӿ),{\mu }_{RN-Ɓ}(ƴ)\right\}$ and ${\mu }_{-Ɓ}(ӿƴ)\le max\left\{{\mu }_{-Ɓ}(ӿ),{\mu }_{-Ɓ}(ƴ)\right\}$
• ${\mu }_{N-Ɓ}({ӿ}^{-1})\le {\mu }_{N-Ɓ}(ӿ)\Rightarrow {\mu }_{RN-Ɓ}({ӿ}^{-1})\le {\mu }_{RN-Ɓ}(ӿ)$ and ${\mu }_{-Ɓ}({ӿ}^{-1})\le {\mu }_{-Ɓ}(ӿ)$

Or equivalently

• ${\mu }_{P-Ɓ}({ӿ}^{-1}ƴ)\ge min\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ƴ)\right\} \Rightarrow {\mu }_{RP-Ɓ}({ӿ}^{-1}ƴ)\ge min\left\{{\mu }_{RP-Ɓ}(ӿ),{\mu }_{RP-Ɓ}(ƴ)\right\}$ and ${\mu }_{IP-Ɓ}({ӿ}^{-1}ƴ)\ge min\left\{{\mu }_{IP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}(ƴ)\right\}$
• ${\mu }_{N-Ɓ}({ӿ}^{-1}ƴ)\le max\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ƴ)\right\} \Rightarrow {\mu }_{RN-Ɓ}({ӿ}^{-1}ƴ)\le max\left\{{\mu }_{RN-Ɓ}(ӿ),{\mu }_{RN-Ɓ}(ƴ)\right\}$ and ${\mu }_{-Ɓ}({ӿ}^{-1}ƴ)\le max\left\{{\mu }_{-Ɓ}(ӿ),{\mu }_{-Ɓ}(ƴ)\right\}$

Example 1.

Consider $Ğ=K_4=\left\{1,ӿ,ƴ,ӿƴ\right\}$ is a group with a Cayley table (Table 1).

Table 1. The Cayley table of $K_4$.

	$1$	$ӿ$	$ƴ$	$ӿƴ$
$1$	$1$	$ӿ$	$ƴ$	$ӿƴ$
$ӿ$	$ӿ$	$1$	$ӿƴ$	$Ƴ$
$ƴ$	$ƴ$	$ӿƴ$	$1$	$ӿ$
$ӿƴ$	$ӿƴ$	$ƴ$	$ӿ$	$1$

Table 1. The Cayley table of $K_4$.

	$1$	$ӿ$	$ƴ$	$ӿƴ$
$1$	$1$	$ӿ$	$ƴ$	$ӿƴ$
$ӿ$	$ӿ$	$1$	$ӿƴ$	$Ƴ$
$ƴ$	$ƴ$	$ӿƴ$	$1$	$ӿ$
$ӿƴ$	$ӿƴ$	$ƴ$	$ӿ$	$1$

and properties that ${ӿ}^2={ƴ}^2={(ӿƴ)}^2=1$ and $ӿƴ=ƴӿ$. Now take any BCF subset of Ğ, $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})$ such that ${\mu }_{P-Ɓ}(1)=(0.8+\iota 0.73,-0.15-\iota 0.1)$, ${\mu }_{P-Ɓ}(ӿ)={\mu }_{P-Ɓ}(ƴ)=(0.5+\iota 0.48,-0.37-\iota 0.61)$, ${\mu }_{P-Ɓ}(ӿƴ)=(0.6+\iota 0.75,-0.37-\iota 0.7)$, then we have to show that

• ${\mu }_{P-Ɓ}(ӿƴ)\ge min\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ƴ)\right\} \Rightarrow {\mu }_{RP-Ɓ}(ӿƴ)\ge min\left\{{\mu }_{RP-Ɓ}(ӿ),{\mu }_{RP-Ɓ}(ƴ)\right\}$ and ${\mu }_{IP-Ɓ}(ӿƴ)\ge min\left\{{\mu }_{IP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}(ƴ)\right\}\forall ӿ,ƴ\in K_4$.

Let $ӿ=1,$ and $ƴ=ӿ$, then

${\mu }_{RP-Ɓ}(1ӿ)={\mu }_{RP-Ɓ}(ӿ)=0.5$ and $min\left\{{\mu }_{RP-Ɓ}(1),{\mu }_{RP-Ɓ}(x)\right\}=min\left\{0.8,0.5\right\}=0.5\Rightarrow {\mu }_{RP-Ɓ}(1ӿ)\ge min\left\{{\mu }_{RP-Ɓ}(1),{\mu }_{RP-Ɓ}(ӿ)\right\}$, and ${\mu }_{IP-Ɓ}(1ӿ)={\mu }_{IP-Ɓ}(ӿ)=0.48$ and $min\left\{{\mu }_{IP-Ɓ}(1),{\mu }_{IP-Ɓ}(x)\right\}=min\left\{0.73,0.5\right\}=0.5\Rightarrow {\mu }_{IP-Ɓ}(1ӿ)\ge min\left\{{\mu }_{IP-Ɓ}(1),{\mu }_{IP-Ɓ}(ӿ)\right\}$. This shows that ${\mu }_{P-Ɓ}(1ӿ)\ge min\left\{{\mu }_{P-Ɓ}(1),{\mu }_{P-Ɓ}(ӿ)\right\}$

Now let $ӿ=1,$ and $ƴ=ƴ$, then

${\mu }_{RP-Ɓ}(1ƴ)={\mu }_{RP-Ɓ}(ƴ)=0.5$ and $min\left\{{\mu }_{RP-Ɓ}(1),{\mu }_{RP-Ɓ}(ƴ)\right\}=min\left\{0.8,0.5\right\}=0.5\Rightarrow {\mu }_{RP-Ɓ}(1ƴ)\ge min\left\{{\mu }_{RP-Ɓ}(1),{\mu }_{RP-Ɓ}(ƴ)\right\}$, and ${\mu }_{IP-Ɓ}(1ƴ)={\mu }_{IP-Ɓ}(ƴ)=0.48$ and $min\left\{{\mu }_{IP-Ɓ}(1),{\mu }_{IP-Ɓ}(ƴ)\right\}=min\left\{0.73,0.48\right\}=0.48\Rightarrow {\mu }_{IP-Ɓ}(1ƴ)\ge min\left\{{\mu }_{IP-Ɓ}(1),{\mu }_{IP-Ɓ}(ƴ)\right\}$. This shows that ${\mu }_{P-Ɓ}(1ƴ)\ge min\left\{{\mu }_{P-Ɓ}(1),{\mu }_{P-Ɓ}(ƴ)\right\}$

Next, let $ӿ=1,$ and $ƴ=ӿƴ$, then

${\mu }_{RP-Ɓ}(1(ӿƴ))={\mu }_{RP-Ɓ}((ӿƴ))=0.6$ and $min\left\{{\mu }_{RP-Ɓ}(1),{\mu }_{RP-Ɓ}((ӿƴ))\right\}=min\left\{0.8,0.6\right\}=0.6\Rightarrow {\mu }_{RP-Ɓ}(1(ӿƴ))\ge min\left\{{\mu }_{RP-Ɓ}(1),{\mu }_{RP-Ɓ}((ӿƴ))\right\}$, and ${\mu }_{IP-Ɓ}(1(ӿƴ))={\mu }_{IP-Ɓ}((ӿƴ))=0.75$ and $min\left\{{\mu }_{IP-Ɓ}(1),{\mu }_{IP-Ɓ}((ӿƴ))\right\}=min\left\{0.73,0.75\right\}=0.48\Rightarrow {\mu }_{IP-Ɓ}(1(ӿƴ))\ge min\left\{{\mu }_{IP-Ɓ}(1),{\mu }_{IP-Ɓ}((ӿƴ))\right\}$. This shows that ${\mu }_{P-Ɓ}(1(ӿƴ))\ge min\left\{{\mu }_{P-Ɓ}(1),{\mu }_{P-Ɓ}(ӿƴ)\right\}$.

Now let $ӿ=ӿ,$ and $ƴ=ƴ$, then

${\mu }_{RP-Ɓ}(ӿƴ)=0.6$ and $min\left\{{\mu }_{RP-Ɓ}(ӿ),{\mu }_{RP-Ɓ}(ƴ)\right\}=min\left\{0.5,0.5\right\}=0.5\Rightarrow {\mu }_{RP-Ɓ}(ӿƴ)\ge min\left\{{\mu }_{RP-Ɓ}(ӿ),{\mu }_{RP-Ɓ}(ƴ)\right\}$, and ${\mu }_{IP-Ɓ}(ӿƴ)=0.75$ and $min\left\{{\mu }_{IP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}(ƴ)\right\}=min\left\{0.48,0.48\right\}=0.48\Rightarrow {\mu }_{IP-Ɓ}(ӿƴ)\ge min\left\{{\mu }_{IP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}(ƴ)\right\}$. This shows that ${\mu }_{P-Ɓ}(ӿƴ)\ge min\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ƴ)\right\}$.

Next, let $ӿ=ӿ,$ and $ƴ=ӿƴ$, then

${\mu }_{RP-Ɓ}(ӿ(ӿƴ))={\mu }_{RP-Ɓ}(ƴ)=0.5$ and $min\left\{{\mu }_{RP-Ɓ}(ӿ),{\mu }_{RP-Ɓ}(ӿƴ)\right\}=min\left\{0.5,0.6\right\}=0.5\Rightarrow {\mu }_{RP-Ɓ}(ӿ(ӿƴ))\ge min\left\{{\mu }_{RP-Ɓ}(ӿ),{\mu }_{RP-Ɓ}(ӿƴ)\right\}$, and ${\mu }_{IP-Ɓ}(ӿ(ӿƴ))={\mu }_{IP-Ɓ}(ƴ)=0.48$ and $min\left\{{\mu }_{IP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}(ӿƴ)\right\}=min\left\{0.48,0.75\right\}=0.48\Rightarrow {\mu }_{IP-Ɓ}(ӿ(ӿƴ))\ge min\left\{{\mu }_{IP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}(ӿƴ)\right\}$. This shows that ${\mu }_{P-Ɓ}(ӿ(ӿƴ))\ge min\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ӿƴ)\right\}$

Now let $ӿ=ƴ,$ and $ƴ=ӿƴ$, then

${\mu }_{RP-Ɓ}(ƴ(ӿƴ))={\mu }_{RP-Ɓ}(ӿ)=0.5$ and $min\left\{{\mu }_{RP-Ɓ}(ƴ),{\mu }_{RP-Ɓ}(ӿƴ)\right\}=min\left\{0.5,0.6\right\}=0.5\Rightarrow {\mu }_{RP-Ɓ}(ƴ(ӿƴ))\ge min\left\{{\mu }_{RP-Ɓ}(ƴ),{\mu }_{RP-Ɓ}(ӿƴ)\right\}$, and ${\mu }_{IP-Ɓ}(ƴ(ӿƴ))={\mu }_{IP-Ɓ}(ӿ)=0.48$ and $min\left\{{\mu }_{IP-Ɓ}(ƴ),{\mu }_{IP-Ɓ}(ӿƴ)\right\}=min\left\{0.48,0.75\right\}=0.48\Rightarrow {\mu }_{IP-Ɓ}(ƴ(ӿƴ))\ge min\left\{{\mu }_{IP-Ɓ}(ƴ),{\mu }_{IP-Ɓ}(ӿƴ)\right\}$. This shows that ${\mu }_{P-Ɓ}(ƴ(ӿƴ))\ge min\left\{{\mu }_{P-Ɓ}(ƴ),{\mu }_{P-Ɓ}(ӿƴ)\right\}$

2.

Each element is self inverse so it is evident that $\forall ӿ\in Ğ$ ${\mu }_{P-Ɓ}({ӿ}^{-1})\ge {\mu }_{P-Ɓ}(ӿ)\Rightarrow {\mu }_{RP-Ɓ}({ӿ}^{-1})\ge {\mu }_{RP-Ɓ}(ӿ)$ and ${\mu }_{IP-Ɓ}({ӿ}^{-1})\ge {\mu }_{IP-Ɓ}(ӿ)$.

3.

${\mu }_{N-Ɓ}(ӿƴ)\le max\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ƴ)\right\} \Rightarrow {\mu }_{RN-Ɓ}(ӿƴ)\le max\left\{{\mu }_{RN-Ɓ}(ӿ),{\mu }_{RN-Ɓ}(ƴ)\right\}$ and ${\mu }_{-Ɓ}(ӿƴ)\le max\left\{{\mu }_{-Ɓ}(ӿ),{\mu }_{-Ɓ}(ƴ)\right\}$

Let $ӿ=1,$ and $ƴ=ӿ$, then

${\mu }_{RN-Ɓ}(1ӿ)={\mu }_{RN-Ɓ}(ӿ)=-0.37$ and $max\left\{{\mu }_{RN-Ɓ}(1),{\mu }_{RN-Ɓ}(x)\right\}=max\left\{-0.15,-0.37\right\}=-0.15\Rightarrow {\mu }_{RN-Ɓ}(1ӿ)\le max\left\{{\mu }_{RN-Ɓ}(1),{\mu }_{RN-Ɓ}(ӿ)\right\}$, and ${\mu }_{-Ɓ}(1ӿ)={\mu }_{-Ɓ}(ӿ)=-0.61$ and $max\left\{{\mu }_{-Ɓ}(1),{\mu }_{-Ɓ}(x)\right\}=max\left\{-0.1,-0.61\right\}=-0.1\Rightarrow {\mu }_{-Ɓ}(1ӿ)\le max\left\{{\mu }_{-Ɓ}(1),{\mu }_{-Ɓ}(ӿ)\right\}$. This implies that ${\mu }_{N-Ɓ}(1ӿ)\le max\left\{{\mu }_{P-Ɓ}(1),{\mu }_{P-Ɓ}(ӿ)\right\}$.

Now let $ӿ=1,$ and $ƴ=ƴ$, then

${\mu }_{RN-Ɓ}(1ƴ)={\mu }_{RN-Ɓ}(ƴ)=-0.37$ and $max\left\{{\mu }_{RN-Ɓ}(1),{\mu }_{RN-Ɓ}(ƴ)\right\}=max\left\{-0.15,-0.37\right\}=-0.15\Rightarrow {\mu }_{RN-Ɓ}(1ƴ)\le max\left\{{\mu }_{RN-Ɓ}(1),{\mu }_{RN-Ɓ}(ƴ)\right\}$, and ${\mu }_{-Ɓ}(1ƴ)={\mu }_{-Ɓ}(ƴ)=-0.61$ and $max\left\{{\mu }_{-Ɓ}(1),{\mu }_{-Ɓ}(ƴ)\right\}=max\left\{-0.1,-0.61\right\}=-0.1\Rightarrow {\mu }_{-Ɓ}(1ƴ)\le max\left\{{\mu }_{-Ɓ}(1),{\mu }_{-Ɓ}(ƴ)\right\}$. This implies that ${\mu }_{N-Ɓ}(1ƴ)\le max\left\{{\mu }_{P-Ɓ}(1),{\mu }_{P-Ɓ}(ƴ)\right\}$.

Next, let $ӿ=1,$ and $ƴ=ӿƴ$, then

${\mu }_{RN-Ɓ}(1(ӿƴ))={\mu }_{RN-Ɓ}((ӿƴ))=-0.53$ and $max\left\{{\mu }_{RN-Ɓ}(1),{\mu }_{RN-Ɓ}((ӿƴ))\right\}=min\left\{-0.15,-0.53\right\}=-0.15\Rightarrow {\mu }_{RN-Ɓ}(1(ӿƴ))\le max\left\{{\mu }_{RN-Ɓ}(1),{\mu }_{RN-Ɓ}((ӿƴ))\right\}$, and ${\mu }_{-Ɓ}(1(ӿƴ))={\mu }_{-Ɓ}((ӿƴ))=-0.78$ and $max\left\{{\mu }_{-Ɓ}(1),{\mu }_{-Ɓ}((ӿƴ))\right\}=max\left\{-0.1,-0.78\right\}=-0.1\Rightarrow {\mu }_{-Ɓ}(1(ӿƴ))\le max\left\{{\mu }_{-Ɓ}(1),{\mu }_{-Ɓ}((ӿƴ))\right\}$. This implies that ${\mu }_{N-Ɓ}(1(ӿƴ))\le max\left\{{\mu }_{P-Ɓ}(1),{\mu }_{P-Ɓ}(ӿƴ)\right\}$.

Now let $ӿ=ӿ,$ and $ƴ=ƴ$, then

${\mu }_{RN-Ɓ}(ӿƴ)=-0.53$ and $max\left\{{\mu }_{RN-Ɓ}(ӿ),{\mu }_{RN-Ɓ}(ƴ)\right\}=min\left\{-0.37,-0.37\right\}=-0.37\Rightarrow {\mu }_{RN-Ɓ}(ӿƴ)\le max\left\{{\mu }_{RN-Ɓ}(ӿ),{\mu }_{RN-Ɓ}(ƴ)\right\}$, and ${\mu }_{-Ɓ}(ӿƴ)=-0.78$ and $max\left\{{\mu }_{-Ɓ}(ӿ),{\mu }_{-Ɓ}(ƴ)\right\}=min\left\{-0.61,-0.61\right\}=-0.61\Rightarrow {\mu }_{-Ɓ}(ӿƴ)\le max\left\{{\mu }_{-Ɓ}(ӿ),{\mu }_{-Ɓ}(ƴ)\right\}$. This implies that ${\mu }_{N-Ɓ}(ӿƴ)\le max\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ƴ)\right\}$.

Next, let $ӿ=ӿ,$ and $ƴ=ӿƴ$, then

${\mu }_{RN-Ɓ}(ӿ(ӿƴ))={\mu }_{RN-Ɓ}(ƴ)=-0.37$ and $max\left\{{\mu }_{RN-Ɓ}(ӿ),{\mu }_{RN-Ɓ}(ӿƴ)\right\}=max\left\{-0.37,-0.53\right\}=-0.37\Rightarrow {\mu }_{RN-Ɓ}(ӿ(ӿƴ))\le max\left\{{\mu }_{RN-Ɓ}(ӿ),{\mu }_{RN-Ɓ}(ӿƴ)\right\}$, and ${\mu }_{-Ɓ}(ӿ(ӿƴ))={\mu }_{-Ɓ}(ƴ)=-0.61$ and $max\left\{{\mu }_{-Ɓ}(ӿ),{\mu }_{-Ɓ}(ӿƴ)\right\}=max\left\{-0.61,-0.78\right\}=-0.61\Rightarrow {\mu }_{-Ɓ}(ӿ(ӿƴ))\le max\left\{{\mu }_{-Ɓ}(ӿ),{\mu }_{-Ɓ}(ӿƴ)\right\}$. This implies that ${\mu }_{N-Ɓ}(ӿ(ӿƴ))\le max\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ӿƴ)\right\}$.

Now let $ӿ=ƴ,$ and $ƴ=ӿƴ$, then

${\mu }_{RN-Ɓ}(ƴ(ӿƴ))={\mu }_{RN-Ɓ}(ӿ)=-0.37$ and $max\left\{{\mu }_{RN-Ɓ}(ƴ),{\mu }_{RN-Ɓ}(ӿƴ)\right\}=max\left\{-0.37,-0.53\right\}=-0.37\Rightarrow {\mu }_{RN-Ɓ}(ƴ(ӿƴ))\le max\left\{{\mu }_{RN-Ɓ}(ƴ),{\mu }_{RN-Ɓ}(ӿƴ)\right\},$ and ${\mu }_{-Ɓ}(ƴ(ӿƴ))={\mu }_{-Ɓ}(ӿ)=-0.61$ and $max\left\{{\mu }_{-Ɓ}(ƴ),{\mu }_{-Ɓ}(ӿƴ)\right\}=max\left\{-0.61,-0.78\right\}=-0.61\Rightarrow {\mu }_{-Ɓ}(ƴ(ӿƴ))\le max\left\{{\mu }_{-Ɓ}(ƴ),{\mu }_{-Ɓ}(ӿƴ)\right\}$. This implies that ${\mu }_{N-Ɓ}(ƴ(ӿƴ))\le max\left\{{\mu }_{P-Ɓ}(ƴ),{\mu }_{P-Ɓ}(ӿƴ)\right\}$.

4.

Each element is a self-inverse so it is evident that $\forall ӿ\in Ğ$ ${\mu }_{N-Ɓ}({ӿ}^{-1})\le {\mu }_{N-Ɓ}(ӿ)\Rightarrow {\mu }_{RN-Ɓ}({ӿ}^{-1})\le {\mu }_{RN-Ɓ}(ӿ)$ and ${\mu }_{-Ɓ}({ӿ}^{-1})\le {\mu }_{-Ɓ}(ӿ)$.

Thus, the BCF subset $Ɓ$ of Ğ is a BCFSG.

Theorem 1.

A $H\ne \varnothing$ subset ofĞ is a subgroup of Ğ if and only if its BCCF is a BCF subgroup of Ğ.

Theorem 2.

Suppose a class of BCF subgroups of Ğ $\left\{{Ɓ}_j=({\mu }_{P-{Ɓ}_j},{\mu }_{N-{Ɓ}_j})=({\mu }_{RP-{Ɓ}_j}+\iota {\mu }_{IP-{Ɓ}_j},{\mu }_{RN-{Ɓ}_j}+\iota {\mu }_{-{Ɓ}_j})\vee j\in I\right\}$, then $\wedge {Ɓ}_j$ is a BCF subgroup of Ğ.

Proof. 

To $\wedge {Ɓ}_j$ is a BCF subgroup of Ğ, we need to prove that $\forall ӿ,ƴ\in Ğ$, the following holds

• ${\mu }_{P-Ɓ}({ӿ}^{-1}ƴ)\ge min\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ƴ)\right\} \Rightarrow {\mu }_{RP-Ɓ}({ӿ}^{-1}ƴ)\ge min\left\{{\mu }_{RP-Ɓ}(ӿ),{\mu }_{RP-Ɓ}(ƴ)\right\}$ and ${\mu }_{IP-Ɓ}({ӿ}^{-1}ƴ)\ge min\left\{{\mu }_{IP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}(ƴ)\right\}$
• ${\mu }_{N-Ɓ}({ӿ}^{-1}ƴ)\le max\left\{{\mu }_{P-Ɓ}(ӿ),{\mu }_{P-Ɓ}(ƴ)\right\} \Rightarrow {\mu }_{RN-Ɓ}({ӿ}^{-1}ƴ)\le max\left\{{\mu }_{RN-Ɓ}(ӿ),{\mu }_{RN-Ɓ}(ƴ)\right\}$ and ${\mu }_{-Ɓ}({ӿ}^{-1}ƴ)\le max\left\{{\mu }_{-Ɓ}(ӿ),{\mu }_{-Ɓ}(ƴ)\right\}$

• $\forall ӿ,ƴ\in Ğ$,

  $$\begin{array}{cc}\hfill (\wedge {\mu }_{RP-Ɓ})({ӿ}^{-1}ƴ)& =\wedge ({\mu }_{RP-Ɓ}({ӿ}^{-1}ƴ))\ge \wedge ({\mu }_{RP-Ɓ}(ӿ)\wedge {\mu }_{RP-Ɓ}(ƴ))\hfill \\ & =min\left\{\wedge {\mu }_{RP-Ɓ}(ӿ),\wedge {\mu }_{RP-Ɓ}(ƴ)\right\}\hfill \end{array}$$

  Similarly, $\forall ӿ,ƴ\in Ğ$,

  $$\begin{array}{cc}\hfill (\wedge {\mu }_{IP-Ɓ})({ӿ}^{-1}ƴ)& =\wedge ({\mu }_{IP-Ɓ}({ӿ}^{-1}ƴ))\ge \wedge ({\mu }_{IP-Ɓ}(ӿ)\wedge {\mu }_{IP-Ɓ}(ƴ))\hfill \\ & =min\left\{\wedge {\mu }_{IP-Ɓ}(ӿ),\wedge {\mu }_{IP-Ɓ}(ƴ)\right\}.\hfill \end{array}$$
• $\forall ӿ,ƴ\in Ğ$,

  $$\begin{array}{cc}\hfill (\wedge {\mu }_{RN-Ɓ})({ӿ}^{-1}ƴ)& =\wedge ({\mu }_{RN-Ɓ}({ӿ}^{-1}ƴ))\le \wedge ({\mu }_{RN-Ɓ}(ӿ)\vee {\mu }_{RN-Ɓ}(ƴ))\hfill \\ & =max\left\{\wedge {\mu }_{RP-Ɓ}(ӿ),\wedge {\mu }_{RP-Ɓ}(ƴ)\right\}\hfill \end{array}$$

  Similarly, $\forall ӿ,ƴ\in Ğ$,

  $$\begin{array}{cc}\hfill (\wedge {\mu }_{-Ɓ})({ӿ}^{-1}ƴ)& =\wedge ({\mu }_{-Ɓ}({ӿ}^{-1}ƴ))\le \wedge ({\mu }_{-Ɓ}(ӿ)\vee {\mu }_{-Ɓ}(ƴ))\hfill \\ & =max\left\{\wedge {\mu }_{RP-Ɓ}(ӿ),\wedge {\mu }_{RP-Ɓ}(ƴ)\right\}\hfill \end{array}$$

  $\Rightarrow \wedge {Ɓ}_j$ is a BCF subgroup of Ğ.□

Remark 2.

For two BCF subgroups ${Ɓ}_1$ and ${Ɓ}_2$, it is not always possible that ${Ɓ}_1\vee {Ɓ}_2$ is a BCF subgroup of Ğ.

Definition 13.

A BCF subgroup of Ğ $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ is called BCF normal subgroup of Ğ if $\forall ӿ,ƴ\in Ğ$ we have

• ${\mu }_{P-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{P-Ɓ}(ӿ) \Rightarrow {\mu }_{RP-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{RP-Ɓ}(ӿ)$ and ${\mu }_{IP-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{IP-Ɓ}(ӿ)$
• ${\mu }_{N-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{N-Ɓ}(ӿ) \Rightarrow {\mu }_{RN-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{RN-Ɓ}(ӿ)$ and ${\mu }_{-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{-Ɓ}(ӿ)$

Example 2.

Consider $Ğ=C_4=\left\{\pm 1,\pm \iota \right\}$ with a Cayley table (Table 2).

Table 2. The Cayley table of $C_4$.

	$1$	$-1$	$\iota$	$-\iota$
$1$	$1$	$-1$	$\iota$	$-\iota$
$-1$	$-1$	$1$	$-\iota$	$\iota$
$\iota$	$\iota$	$-\iota$	$-1$	$1$
$-\iota$	$-\iota$	$\iota$	$1$	$-1$

Table 2. The Cayley table of $C_4$.

	$1$	$-1$	$\iota$	$-\iota$
$1$	$1$	$-1$	$\iota$	$-\iota$
$-1$	$-1$	$1$	$-\iota$	$\iota$
$\iota$	$\iota$	$-\iota$	$-1$	$1$
$-\iota$	$-\iota$	$\iota$	$1$	$-1$

and a BCF subset $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ of $Ğ$ s.t

${\mu }_{P-Ɓ}(1)=(0.8+\iota 0.7)$, ${\mu }_{N-Ɓ}(1)=(-0.15-\iota 0.2)$, ${\mu }_{P-Ɓ}(-1)=(0.6+\iota 0.55)$, ${\mu }_{N-Ɓ}(-1)=(-0.25-\iota 0.3)$, ${\mu }_{P-Ɓ}(\pm \iota )=(0.4+\iota 0.35)$, ${\mu }_{N-Ɓ}(\pm \iota )=(-0.5-\iota 0.6)$

Then, $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ is a BCF normal subgroup of $Ğ$.

Definition 14.

For two BCF subgroups of Ğ, ${Ɓ}_1=({\mu }_{P-{Ɓ}_1},{\mu }_{N-{Ɓ}_1})=({\mu }_{RP-{Ɓ}_1}+\iota {\mu }_{IP-{Ɓ}_1},{\mu }_{RN-{Ɓ}_1}+\iota {\mu }_{-{Ɓ}_1})$ and ${Ɓ}_2=({\mu }_{P-{Ɓ}_2},{\mu }_{N-{Ɓ}_2})=({\mu }_{RP-{Ɓ}_2}+\iota {\mu }_{IP-{Ɓ}_2},{\mu }_{RN-{Ɓ}_2}+\iota {\mu }_{-{Ɓ}_2})$ we say that ${Ɓ}_1=({\mu }_{P-{Ɓ}_1},{\mu }_{N-{Ɓ}_1})$ is a BCF conjugate of ${Ɓ}_2=({\mu }_{P-{Ɓ}_2},{\mu }_{N-{Ɓ}_2})$ if for any $ӿ\in Ğ$ we have

• ${\mu }_{P-{Ɓ}_1}(ƴ)={\mu }_{P-{Ɓ}_2}({ӿ}^{-1}ƴӿ) \Rightarrow {\mu }_{RP-{Ɓ}_1}(ƴ)={\mu }_{RP-{Ɓ}_2}({ӿ}^{-1}ƴӿ)$ and ${\mu }_{IP-{Ɓ}_1}(ƴ)={\mu }_{IP-{Ɓ}_2}({ӿ}^{-1}ƴӿ)$
• ${\mu }_{N-{Ɓ}_1}(ƴ)={\mu }_{N-{Ɓ}_2}({ӿ}^{-1}ƴӿ) \Rightarrow {\mu }_{RN-{Ɓ}_1}(ƴ)={\mu }_{RN-{Ɓ}_2}({ӿ}^{-1}ƴӿ)$ and ${\mu }_{-{Ɓ}_1}(ƴ)={\mu }_{-{Ɓ}_2}({ӿ}^{-1}ƴӿ)\forall ƴ\in Ğ$.

Definition 15.

The normalizer of a BCF subgroup $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ of Ğ is signified and presented as

$$N(Ɓ)=\left\{ƴ\in Ğ:{\mu }_{P-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{P-Ɓ}(ӿ)\wedge {\mu }_{N-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{N-Ɓ}(ӿ),\forall ӿ\in Ğ\right\}$$

$$\left\{\begin{array}{c}ƴ\in Ğ:{\mu }_{RP-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{RP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{IP-Ɓ}(ӿ),\\ {\mu }_{RN-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{RN-Ɓ}(ӿ)\wedge {\mu }_{-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{-Ɓ}(ӿ)\end{array},\forall ӿ\in Ğ\right\}$$

(6)

Theorem 3.

Suppose that $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ is a BCF subgroup of Ğ, then

• $N(Ɓ)$ is a subgroup of Ğ.
• $Ɓ$ is BCF normal subgroup of Ğ iff $N(Ɓ)=Ğ$.

Proof. 

• Assume that ${ƴ}_1,{ƴ}_2\in N(Ɓ)$, to prove that ${ƴ}_1{ƴ}_2^{-1}\in N(Ɓ)$. Now for any $x\in Ğ$, we have

  $$\begin{gathered} {\mu }_{RP-Ɓ}({({ƴ}_1{ƴ}_2^{-1})}^{-1}ӿ({ƴ}_1{ƴ}_2^{-1}))={\mu }_{RP-Ɓ}(({ƴ}_2{ƴ}_1^{-1})ӿ({ƴ}_1{ƴ}_2^{-1})) \\ {\mu }_{RP-Ɓ}({ƴ}_2({ƴ}_1^{-1}ӿ{ƴ}_1){ƴ}_2^{-1})\ge {\mu }_{RP-Ɓ}({ƴ}_1^{-1}ӿ{ƴ}_1)\ge {\mu }_{RP-Ɓ}(ӿ). \end{gathered}$$

  Similarly,

  $$\begin{gathered} {\mu }_{IP-Ɓ}({({ƴ}_1{ƴ}_2^{-1})}^{-1}ӿ({ƴ}_1{ƴ}_2^{-1}))={\mu }_{IP-Ɓ}(({ƴ}_2{ƴ}_1^{-1})ӿ({ƴ}_1{ƴ}_2^{-1})) \\ {\mu }_{IP-Ɓ}({ƴ}_2({ƴ}_1^{-1}ӿ{ƴ}_1){ƴ}_2^{-1})\ge {\mu }_{IP-Ɓ}({ƴ}_1^{-1}ӿ{ƴ}_1)\ge {\mu }_{IP-Ɓ}(ӿ). \end{gathered}$$

  Next,

  $$\begin{gathered} {\mu }_{RN-Ɓ}({({ƴ}_1{ƴ}_2^{-1})}^{-1}ӿ({ƴ}_1{ƴ}_2^{-1}))={\mu }_{RN-Ɓ}(({ƴ}_2{ƴ}_1^{-1})ӿ({ƴ}_1{ƴ}_2^{-1})) \\ {\mu }_{RN-Ɓ}({ƴ}_2({ƴ}_1^{-1}ӿ{ƴ}_1){ƴ}_2^{-1})\le {\mu }_{RN-Ɓ}({ƴ}_1^{-1}ӿ{ƴ}_1)\le {\mu }_{RP-Ɓ}(ӿ). \end{gathered}$$

  Similarly,

  $$\begin{gathered} {\mu }_{-Ɓ}({({ƴ}_1{ƴ}_2^{-1})}^{-1}ӿ({ƴ}_1{ƴ}_2^{-1}))={\mu }_{-Ɓ}(({ƴ}_2{ƴ}_1^{-1})ӿ({ƴ}_1{ƴ}_2^{-1})) \\ {\mu }_{-Ɓ}({ƴ}_2({ƴ}_1^{-1}ӿ{ƴ}_1){ƴ}_2^{-1})\le {\mu }_{IP-Ɓ}({ƴ}_1^{-1}ӿ{ƴ}_1)\le {\mu }_{IP-Ɓ}(ӿ) \end{gathered}$$

  $\Rightarrow {ƴ}_1{ƴ}_2^{-1}\in N(Ɓ)\Rightarrow N(Ɓ)$ is a subgroup of Ğ.
• First, we assume that $Ɓ$ is the BCF normal subgroup of Ğ and prove that $N(Ɓ)=Ğ$. Suppose $ƴ\in Ğ$, then as $Ɓ$ is BCF normal subgroup of $Ğ$, so $\forall ӿ\in Ğ$, we have

  $${\mu }_{RP-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{RP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{IP-Ɓ}(ӿ),$$

  $${\mu }_{RN-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{RN-Ɓ}(ӿ),{\mu }_{-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{-Ɓ}(ӿ)$$

  $\Rightarrow ƴ\in N(Ɓ)\Rightarrow Ğ\subseteq N(Ɓ)\Rightarrow Ğ=N(Ɓ)$.

Next, conversely assume that $N(Ɓ)=Ğ$, then for any $ӿ,ƴ\in Ğ\Rightarrow ӿ,ƴ\in N(Ɓ)$

$$\Rightarrow {\mu }_{RP-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{RP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{IP-Ɓ}(ӿ),$$

$${\mu }_{RN-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{RN-Ɓ}(ӿ),{\mu }_{-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{-Ɓ}(ӿ)$$

$\Rightarrow Ɓ$ is the BCF normal subgroup of Ğ. □

Definition 16.

Suppose a BCF subgroup $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ of Ğ, then,

• BCF left coset $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ in Ğ is determined by $ӿ\in Ğ$ is a BCF subset ${Ɓ}_{ӿ}^l=({\mu }_{P-{Ɓ}_{ӿ}^l},{\mu }_{N-{Ɓ}_{ӿ}^l})=({\mu }_{RP-{Ɓ}_{ӿ}^l}+\iota {\mu }_{IP-{Ɓ}_{ӿ}^l},{\mu }_{RN-{Ɓ}_{ӿ}^l}+\iota {\mu }_{-{Ɓ}_{ӿ}^l})$ of Ğ, that is,

  $$\begin{gathered} {\mu }_{RP-{Ɓ}_{ӿ}^l}:Ğ\to \left[0,1\right] \mathrm{described} \mathrm{as} {\mu }_{RP-{Ɓ}_{ӿ}^l}(ƴ)={\mu }_{RP-Ɓ}({ӿ}^{-1}ƴ), \\ {\mu }_{IP-{Ɓ}_{ӿ}^l}:Ğ\to \left[0,1\right] \mathrm{described} \mathrm{as} {\mu }_{IP-{Ɓ}_{ӿ}^l}(ƴ)={\mu }_{IP-Ɓ}({ӿ}^{-1}ƴ), \\ {\mu }_{RN-{Ɓ}_{ӿ}^l}:Ğ\to \left[-1,0\right] \mathrm{described} \mathrm{as} {\mu }_{RN-{Ɓ}_{ӿ}^l}(ƴ)={\mu }_{RN-Ɓ}({ӿ}^{-1}ƴ), \\ \mathrm{and} {\mu }_{-{Ɓ}_{ӿ}^l}:Ğ\to \left[-1,0\right] \mathrm{described} \mathrm{as} {\mu }_{-{Ɓ}_{ӿ}^l}(ƴ)={\mu }_{-Ɓ}({ӿ}^{-1}ƴ). \end{gathered}$$
• BCF right coset $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ in Ğ is determined by $ӿ\in Ğ$ is a BCF subset ${Ɓ}_{ӿ}^r=({\mu }_{P-{Ɓ}_{ӿ}^r},{\mu }_{N-{Ɓ}_{ӿ}^r})=({\mu }_{RP-{Ɓ}_{ӿ}^r}+\iota {\mu }_{IP-{Ɓ}_{ӿ}^r},{\mu }_{RN-{Ɓ}_{ӿ}^r}+\iota {\mu }_{-{Ɓ}_{ӿ}^r})$ of Ğ, that is,

  $$\begin{gathered} {\mu }_{RP-{Ɓ}_{ӿ}^r}:Ğ\to \left[0,1\right] \mathrm{described} \mathrm{as} {\mu }_{RP-{Ɓ}_{ӿ}^r}(ƴ)={\mu }_{RP-Ɓ}(ƴ{ӿ}^{-1}), \\ {\mu }_{IP-{Ɓ}_{ӿ}^r}:Ğ\to \left[0,1\right]\mathrm{described} \mathrm{as} {\mu }_{IP-{Ɓ}_{ӿ}^r}(ƴ)={\mu }_{IP-Ɓ}(ƴ{ӿ}^{-1}), \\ {\mu }_{RN-{Ɓ}_{ӿ}^r}:Ğ\to \left[-1,0\right] \mathrm{described} \mathrm{as} {\mu }_{RN-{Ɓ}_{ӿ}^r}(ƴ)={\mu }_{RN-Ɓ}(ƴ{ӿ}^{-1}), \\ \mathrm{and} {\mu }_{-{Ɓ}_{ӿ}^r}:Ğ\to \left[-1,0\right] \mathrm{described} \mathrm{as} {\mu }_{-{Ɓ}_{ӿ}^r}(ƴ)={\mu }_{-Ɓ}(ƴ{ӿ}^{-1}). \end{gathered}$$

Theorem 4.

A BCF subgroup $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ of Ğ is BCF normal subgroup of Ğ if and only if $\forall ӿ\in Ğ$,

• ${\mu }_{P-{Ɓ}_{ӿ}^l}={\mu }_{P-{Ɓ}_{ӿ}^r}\Rightarrow {\mu }_{RP-{Ɓ}_{ӿ}^l}={\mu }_{RP-{Ɓ}_{ӿ}^r}$ and ${\mu }_{IP-{Ɓ}_{ӿ}^l}={\mu }_{IP-{Ɓ}_{ӿ}^r}$
• ${\mu }_{N-{Ɓ}_{ӿ}^l}={\mu }_{N-{Ɓ}_{ӿ}^r}\Rightarrow {\mu }_{RN-{Ɓ}_{ӿ}^l}={\mu }_{RN-{Ɓ}_{ӿ}^r}$ and ${\mu }_{-{Ɓ}_{ӿ}^l}={\mu }_{-{Ɓ}_{ӿ}^r}$

Proof. 

Firstly, assume that $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ is a normal subgroup of Ğ and to prove that 1 and 2 hold.

• $\forall ӿ\in Ğ$ we have

  $$\begin{gathered} {\mu }_{RP-{Ɓ}_{ӿ}^l}(ƴ)={\mu }_{RP-Ɓ}({ӿ}^{-1}ƴ)={\mu }_{RP-Ɓ}({ƴ}^{-1}ƴ{ӿ}^{-1}ƴ) \\ {\mu }_{RP-Ɓ}({ƴ}^{-1}(ƴ{ӿ}^{-1})ƴ)\ge {\mu }_{RP-Ɓ}(ƴ{ӿ}^{-1})={\mu }_{RP-{Ɓ}_{ӿ}^r}(ƴ) \\ \Rightarrow {\mu }_{RP-{Ɓ}_{ӿ}^l}(ƴ)\ge {\mu }_{RP-{Ɓ}_{ӿ}^r}(ƴ). \end{gathered}$$

  Similarly,

  $$\begin{gathered} {\mu }_{IP-{Ɓ}_{ӿ}^l}(ƴ)={\mu }_{IP-Ɓ}({ӿ}^{-1}ƴ)={\mu }_{IP-Ɓ}({ƴ}^{-1}ƴ{ӿ}^{-1}ƴ) \\ {\mu }_{IP-Ɓ}({ƴ}^{-1}(ƴ{ӿ}^{-1})ƴ)\ge {\mu }_{IP-Ɓ}(ƴ{ӿ}^{-1})={\mu }_{IP-{Ɓ}_{ӿ}^r}(ƴ) \\ \Rightarrow {\mu }_{IP-{Ɓ}_{ӿ}^l}(ƴ)\ge {\mu }_{IP-{Ɓ}_{ӿ}^r}(ƴ). \end{gathered}$$

  Now again for every $ӿ\in Ğ$

  $$\begin{gathered} {\mu }_{RP-{Ɓ}_{ӿ}^r}(ƴ)={\mu }_{RP-Ɓ}(ƴ{ӿ}^{-1})={\mu }_{RP-Ɓ}(ƴ{ӿ}^{-1}ƴ{ƴ}^{-1}) \\ {\mu }_{RP-Ɓ}(ƴ({ӿ}^{-1}ƴ){ƴ}^{-1})\ge {\mu }_{RP-Ɓ}({ӿ}^{-1}ƴ)={\mu }_{RP-{Ɓ}_{ӿ}^l}(ƴ) \\ \Rightarrow {\mu }_{RP-{Ɓ}_{ӿ}^r}(ƴ)\ge {\mu }_{RP-{Ɓ}_{ӿ}^l}(ƴ). \end{gathered}$$

  Similarly,

  $$\begin{gathered} {\mu }_{IP-{Ɓ}_{ӿ}^r}(ƴ)={\mu }_{IP-Ɓ}(ƴ{ӿ}^{-1})={\mu }_{IP-Ɓ}(ƴ{ӿ}^{-1}ƴ{ƴ}^{-1}) \\ {\mu }_{IP-Ɓ}(ƴ({ӿ}^{-1}ƴ){ƴ}^{-1})\ge {\mu }_{IP-Ɓ}({ӿ}^{-1}ƴ)={\mu }_{IP-{Ɓ}_{ӿ}^l}(ƴ) \\ \Rightarrow {\mu }_{IP-{Ɓ}_{ӿ}^r}(ƴ)\ge {\mu }_{IP-{Ɓ}_{ӿ}^l}(ƴ) \\ \Rightarrow {\mu }_{P-{Ɓ}_{ӿ}^l}={\mu }_{P-{Ɓ}_{ӿ}^r} \end{gathered}$$
• For every $ӿ\in Ğ$,

  $$\begin{gathered} {\mu }_{RN-{Ɓ}_{ӿ}^l}(ƴ)={\mu }_{RN-Ɓ}({ӿ}^{-1}ƴ)={\mu }_{RN-Ɓ}({ƴ}^{-1}ƴ{ӿ}^{-1}ƴ) \\ {\mu }_{RN-Ɓ}({ƴ}^{-1}(ƴ{ӿ}^{-1})ƴ)\le {\mu }_{RN-Ɓ}(ƴ{ӿ}^{-1})={\mu }_{RN-{Ɓ}_{ӿ}^r}(ƴ) \\ \Rightarrow {\mu }_{RN-{Ɓ}_{ӿ}^l}(ƴ)\le {\mu }_{RN-{Ɓ}_{ӿ}^r}(ƴ). \end{gathered}$$

  Similarly,

  $$\begin{gathered} {\mu }_{-{Ɓ}_{ӿ}^l}(ƴ)={\mu }_{-Ɓ}({ӿ}^{-1}ƴ)={\mu }_{-Ɓ}({ƴ}^{-1}ƴ{ӿ}^{-1}ƴ) \\ {\mu }_{-Ɓ}({ƴ}^{-1}(ƴ{ӿ}^{-1})ƴ)\le {\mu }_{-Ɓ}(ƴ{ӿ}^{-1})={\mu }_{-{Ɓ}_{ӿ}^r}(ƴ) \\ \Rightarrow {\mu }_{-{Ɓ}_{ӿ}^l}(ƴ)\le {\mu }_{-{Ɓ}_{ӿ}^r}(ƴ). \end{gathered}$$

  Now again for every $ӿ\in Ğ$

  $$\begin{gathered} {\mu }_{RN-{Ɓ}_{ӿ}^r}(ƴ)={\mu }_{RN-Ɓ}(ƴ{ӿ}^{-1})={\mu }_{NP-Ɓ}(ƴ{ӿ}^{-1}ƴ{ƴ}^{-1}) \\ {\mu }_{RN-Ɓ}(ƴ({ӿ}^{-1}ƴ){ƴ}^{-1})\le {\mu }_{RN-Ɓ}({ӿ}^{-1}ƴ)={\mu }_{RN-{Ɓ}_{ӿ}^l}(ƴ) \\ \Rightarrow {\mu }_{RN-{Ɓ}_{ӿ}^r}(ƴ)\le {\mu }_{RN-{Ɓ}_{ӿ}^l}(ƴ). \end{gathered}$$

  Similarly,

  $$\begin{gathered} {\mu }_{-{Ɓ}_{ӿ}^r}(ƴ)={\mu }_{-Ɓ}(ƴ{ӿ}^{-1})={\mu }_{IP-Ɓ}(ƴ{ӿ}^{-1}ƴ{ƴ}^{-1}) \\ {\mu }_{-Ɓ}(ƴ({ӿ}^{-1}ƴ){ƴ}^{-1})\le {\mu }_{-Ɓ}({ӿ}^{-1}ƴ)={\mu }_{-{Ɓ}_{ӿ}^l}(ƴ) \\ \Rightarrow {\mu }_{-{Ɓ}_{ӿ}^r}(ƴ)\le {\mu }_{-{Ɓ}_{ӿ}^l}(ƴ) \\ \Rightarrow {\mu }_{N-{Ɓ}_{ӿ}^l}={\mu }_{N-{Ɓ}_{ӿ}^r} \end{gathered}$$

  Secondly, let $\forall ӿ\in Ğ$, 1, and 2 hold to prove that $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ is a BCF normal subgroup of Ğ. For some $ӿ,ƴ\in Ğ$, we have

  $$\begin{gathered} {\mu }_{RP-Ɓ}({ӿ}^{-1}ƴӿ)={\mu }_{RP-{Ɓ}_{ӿ}^l}(ƴӿ)={\mu }_{RP-{Ɓ}_{ӿ}^r}(ƴӿ)={\mu }_{RP-Ɓ}(ƴӿ{ӿ}^{-1})={\mu }_{RP-Ɓ}(ƴ) \\ \Rightarrow {\mu }_{RP-Ɓ}({ӿ}^{-1}ƴӿ)\ge {\mu }_{RP-Ɓ}(ƴ) \\ {\mu }_{IP-Ɓ}({ӿ}^{-1}ƴӿ)={\mu }_{IP-{Ɓ}_{ӿ}^l}(ƴӿ)={\mu }_{IP-{Ɓ}_{ӿ}^r}(ƴӿ)={\mu }_{IP-Ɓ}(ƴӿ{ӿ}^{-1})={\mu }_{IP-Ɓ}(ƴ) \\ \Rightarrow {\mu }_{IP-Ɓ}({ӿ}^{-1}ƴӿ)\ge {\mu }_{IP-Ɓ}(ƴ) \end{gathered}$$

  Next,

  $$\begin{gathered} {\mu }_{RN-Ɓ}({ӿ}^{-1}ƴӿ)={\mu }_{RN-{Ɓ}_{ӿ}^l}(ƴӿ)={\mu }_{RN-{Ɓ}_{ӿ}^r}(ƴӿ)={\mu }_{RN-Ɓ}(ƴӿ{ӿ}^{-1})={\mu }_{RN-Ɓ}(ƴ) \\ \Rightarrow {\mu }_{RN-Ɓ}({ӿ}^{-1}ƴӿ)\le {\mu }_{RN-Ɓ}(ƴ) \\ {\mu }_{-Ɓ}({ӿ}^{-1}ƴӿ)={\mu }_{-{Ɓ}_{ӿ}^l}(ƴӿ)={\mu }_{-{Ɓ}_{ӿ}^r}(ƴӿ)={\mu }_{-Ɓ}(ƴӿ{ӿ}^{-1})={\mu }_{-Ɓ}(ƴ) \\ \Rightarrow {\mu }_{-Ɓ}({ӿ}^{-1}ƴӿ)\le {\mu }_{-Ɓ}(ƴ) \end{gathered}$$

  This completes the proof. □

Definition 17.

A BCF subgroup $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ of Ğ is called BCF abelian subgroup of Ğ iff $\forall ӿ,ƴ\in Ğ$

• ${\mu }_{P-Ɓ}(ӿƴ)={\mu }_{P-Ɓ}(ƴӿ)\Rightarrow {\mu }_{RP-Ɓ}(ӿƴ)={\mu }_{RP-Ɓ}(ƴӿ)$ and ${\mu }_{IP-Ɓ}(ӿƴ)={\mu }_{IP-Ɓ}(ƴӿ)$
• ${\mu }_{N-Ɓ}(ӿƴ)={\mu }_{N-Ɓ}(ƴӿ)\Rightarrow {\mu }_{RN-Ɓ}(ӿƴ)={\mu }_{RN-Ɓ}(ƴӿ)$ and ${\mu }_{-Ɓ}(ӿƴ)={\mu }_{-Ɓ}(ƴӿ)$

Example 3.

Consider$Ğ=Q_8=\left\{\pm 1,\pm i,\pm j,\pm k\right\}$ and a BCF subset $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ of $Ğ$ s.t

${\mu }_{P-Ɓ}(1)=(0.81+\iota 0.68)$, ${\mu }_{N-Ɓ}(1)=(-0.13-\iota 0.19)$, ${\mu }_{P-Ɓ}(-1)=(0.7+\iota 0.5)$, ${\mu }_{N-Ɓ}(-1)=(-0.2-\iota 0.27)$, ${\mu }_{P-Ɓ}(\pm i)={\mu }_{P-Ɓ}(\pm j)={\mu }_{P-Ɓ}(\pm k)=(0.52+\iota 0.4)$, ${\mu }_{N-Ɓ}(\pm i)={\mu }_{N-Ɓ}(\pm j)={\mu }_{N-Ɓ}(\pm k)=(-0.6-\iota 0.7)$

Then, $Ɓ$ is the BCF abelian subgroup of $Ğ$.

Theorem 5.

Suppose that $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ is a BCF subgroup of Ğ, then the below-given axioms are equivalent

• $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ is BCF abelian subgroup of Ğ
• $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ is BCF normal subgroup of Ğ.

Proof. 

$1\Rightarrow 2$, let $ӿ,ƴ\in Ğ$. To prove that

A.

${\mu }_{P-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{P-Ɓ}(ӿ)$ $\Rightarrow {\mu }_{RP-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{RP-Ɓ}(ӿ)$ and ${\mu }_{IP-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{IP-Ɓ}(ӿ)$

B.

${\mu }_{N-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{N-Ɓ}(ӿ)$  $\Rightarrow {\mu }_{RN-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{RN-Ɓ}(ӿ)$ and ${\mu }_{-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{-Ɓ}(ӿ)$

A. $\forall ӿ,ƴ\in Ğ$

$$\begin{gathered} {\mu }_{RP-Ɓ}({ƴ}^{-1}ӿƴ)={\mu }_{RP-Ɓ}(({ƴ}^{-1})(ӿƴ))={\mu }_{RP-Ɓ}((ӿƴ)({ƴ}^{-1}))={\mu }_{RP-Ɓ}(ӿƴ{ƴ}^{-1}) \\ {\mu }_{RP-Ɓ}(ӿ)\Rightarrow {\mu }_{RP-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{RP-Ɓ}(ӿ) \end{gathered}$$

similarly,

$${\mu }_{IP-Ɓ}({ƴ}^{-1}ӿƴ)\ge {\mu }_{IP-Ɓ}(ӿ)$$

B. $\forall ӿ,ƴ\in Ğ$

$$\begin{gathered} {\mu }_{RN-Ɓ}({ƴ}^{-1}ӿƴ)={\mu }_{RN-Ɓ}(({ƴ}^{-1})(ӿƴ))={\mu }_{RN-Ɓ}((ӿƴ)({ƴ}^{-1}))={\mu }_{RN-Ɓ}(ӿƴ{ƴ}^{-1}) \\ {\mu }_{RN-Ɓ}(ӿ)\Rightarrow {\mu }_{RN-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{RN-Ɓ}(ӿ). \end{gathered}$$

Similarly,

$${\mu }_{-Ɓ}({ƴ}^{-1}ӿƴ)\le {\mu }_{-Ɓ}(ӿ)$$

$2\Rightarrow 1$, let $ӿ,ƴ\in Ğ$. To prove that

A.

${\mu }_{P-Ɓ}(ӿƴ)={\mu }_{P-Ɓ}(ƴӿ)\Rightarrow {\mu }_{RP-Ɓ}(ӿƴ)={\mu }_{RP-Ɓ}(ƴӿ)$ and ${\mu }_{IP-Ɓ}(ӿƴ)={\mu }_{IP-Ɓ}(ƴӿ)$

B.

${\mu }_{N-Ɓ}(ӿƴ)={\mu }_{N-Ɓ}(ƴӿ)\Rightarrow {\mu }_{RN-Ɓ}(ӿƴ)={\mu }_{RN-Ɓ}(ƴӿ)$ and ${\mu }_{-Ɓ}(ӿƴ)={\mu }_{-Ɓ}(ƴӿ)$

• A. We have

$${\mu }_{RP-Ɓ}(ӿƴ)={\mu }_{RP-Ɓ}(ӿƴӿ{ӿ}^{-1})={\mu }_{RP-Ɓ}(ӿ(ƴӿ){ӿ}^{-1})\ge {\mu }_{RP-Ɓ}(ƴӿ)\Rightarrow {\mu }_{RP-Ɓ}(ӿƴ)\ge {\mu }_{RP-Ɓ}(ƴӿ)$$

Next,

$$\begin{gathered} {\mu }_{RP-Ɓ}(ƴӿ)={\mu }_{RP-Ɓ}(ƴӿƴ{ƴ}^{-1})={\mu }_{RP-Ɓ}(ƴ(ӿƴ){ƴ}^{-1})\ge {\mu }_{RP-Ɓ}(ӿƴ)\Rightarrow {\mu }_{RP-Ɓ}(ƴӿ)\ge {\mu }_{RP-Ɓ}(ӿƴ) \\ \Rightarrow {\mu }_{RP-Ɓ}(ӿƴ)={\mu }_{RP-Ɓ}(ƴӿ) \end{gathered}$$

Likewise, we can prove that ${\mu }_{IP-Ɓ}(ӿƴ)={\mu }_{IP-Ɓ}(ƴӿ)$. This implies that ${\mu }_{P-Ɓ}(ӿƴ)={\mu }_{P-Ɓ}(ƴӿ).$

• B. We have

$${\mu }_{RN-Ɓ}(ӿƴ)={\mu }_{RN-Ɓ}(ӿƴӿ{ӿ}^{-1})={\mu }_{RN-Ɓ}(ӿ(ƴӿ){ӿ}^{-1})\le {\mu }_{RN-Ɓ}(ƴӿ)\Rightarrow {\mu }_{RN-Ɓ}(ӿƴ)\ge {\mu }_{RN-Ɓ}(ƴӿ)$$

Next,

$$\begin{gathered} {\mu }_{RN-Ɓ}(ƴӿ)={\mu }_{RN-Ɓ}(ƴӿƴ{ƴ}^{-1})={\mu }_{RN-Ɓ}(ƴ(ӿƴ){ƴ}^{-1})\le {\mu }_{RN-Ɓ}(ӿƴ)\Rightarrow {\mu }_{RN-Ɓ}(ƴӿ)\le {\mu }_{RN-Ɓ}(ӿƴ) \\ \Rightarrow {\mu }_{RN-Ɓ}(ӿƴ)={\mu }_{RN-Ɓ}(ƴӿ) \end{gathered}$$

Likewise, we can prove that ${\mu }_{-Ɓ}(ӿƴ)={\mu }_{-Ɓ}(ƴӿ)$. This implies that ${\mu }_{N-Ɓ}(ӿƴ)={\mu }_{N-Ɓ}(ƴӿ).$

• $\Rightarrow$ $Ɓ$ is a BCF abelian subgroup of Ğ. □

Remark 3.

Suppose a BCF subgroup $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ of Ğ, then

$$N(Ɓ)=\left\{ƴ\in Ğ,{\mu }_{P-Ɓ}(ӿƴ)={\mu }_{P-Ɓ}(ƴӿ)\wedge {\mu }_{N-Ɓ}(ӿƴ)={\mu }_{N-Ɓ}(ƴӿ)\forall ӿ\in Ğ\right\}$$

Theorem 6.

Suppose a BCF subgroup $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ of Ğ, then the cardinality of $\left\{{Ɓ}^w\vee w\in Ğ\right\}$ where, ${Ɓ}^w$ is BCF subgroup of Ğ conjugate to $Ɓ$ is equal to the index of $N(Ɓ)$.

Proof. 

Suppose $D=\left\{{Ɓ}^w\vee w\in Ğ\right\}$ and $D^{*}=\left\{wN(Ɓ)\vee w\in Ğ\right\}$. Now we have to illustrate that $D$ and $D^{*}$ are equivalent. Define $\Omega :D\to D^{*}$ by $\Omega ({Ɓ}^w)=wN(Ɓ)$. For $w_1,w_2\in Ğ$, we have

$$\begin{gathered} {Ɓ}^{w_1}={Ɓ}^{w_2}\Rightarrow {Ɓ}^{w_1}(ӿ)={Ɓ}^{w_2}(ӿ) \\ \iff {\mu }_{P-Ɓ}^{w_1}(ӿ)={\mu }_{P-Ɓ}^{w_2}(ӿ) \mathrm{and} {\mu }_{N-Ɓ}^{w_1}(ӿ)={\mu }_{N-Ɓ}^{w_2}(ӿ)\forall ӿ\in Ğ \end{gathered}$$

First, we consider this ${\mu }_{P-Ɓ}^{w_1}(ӿ)={\mu }_{P-Ɓ}^{w_2}(ӿ)\forall ӿ\in Ğ$

$$\begin{gathered} \iff {\mu }_{RP-Ɓ}^{w_1}(ӿ)={\mu }_{RP-Ɓ}^{w_2}(ӿ) \mathrm{and} {\mu }_{IP-Ɓ}^{w_1}(ӿ)={\mu }_{IP-Ɓ}^{w_2}(ӿ)\forall ӿ\in Ğ \\ \iff {\mu }_{RP-Ɓ}(w_1^{-1}ӿw_1)={\mu }_{RP-Ɓ}(w_2^{-1}ӿw_2) \mathrm{and} {\mu }_{IP-Ɓ}(w_1^{-1}ӿw_1)={\mu }_{IP-Ɓ}(w_2^{-1}ӿw_2)\forall ӿ\in Ğ \\ \iff {\mu }_{RP-Ɓ}(ӿw_2^{-1}{w}_1)={\mu }_{RP-Ɓ}(w_2^{-1}{w}_1ӿ) \mathrm{and} {\mu }_{IP-Ɓ}(w_1^{-1}ӿw_1)={\mu }_{IP-Ɓ}(w_2^{-1}ӿw_2)\forall ӿ\in Ğ \\ \iff w_2^{-1}{w}_1\in N(Ɓ)\iff w_2^{-1}{w}_{1}N(Ɓ)=N(Ɓ)\iff w_{1}N(Ɓ)=w_{2}N(Ɓ) \end{gathered}$$

Now consider ${\mu }_{N-Ɓ}^{w_1}(ӿ)={\mu }_{N-Ɓ}^{w_2}(ӿ)$ $\forall ӿ\in Ğ$

$$\begin{gathered} \iff {\mu }_{RN-Ɓ}^{w_1}(ӿ)={\mu }_{RN-Ɓ}^{w_2}(ӿ) \mathrm{and} {\mu }_{-Ɓ}^{w_1}(ӿ)={\mu }_{-Ɓ}^{w_2}(ӿ)\forall ӿ\in Ğ \\ \iff {\mu }_{RN-Ɓ}(w_1^{-1}ӿw_1)={\mu }_{RN-Ɓ}(w_2^{-1}ӿw_2) \mathrm{and} {\mu }_{-Ɓ}(w_1^{-1}ӿw_1)={\mu }_{-Ɓ}(w_2^{-1}ӿw_2)\forall ӿ\in Ğ \\ \iff {\mu }_{RN-Ɓ}(ӿw_2^{-1}{w}_1)={\mu }_{RN-Ɓ}(w_2^{-1}{w}_1ӿ) \mathrm{and} {\mu }_{-Ɓ}(w_1^{-1}ӿw_1)={\mu }_{-Ɓ}(w_2^{-1}ӿw_2)\forall ӿ\in Ğ \\ \iff w_2^{-1}{w}_1\in N(Ɓ)\iff w_2^{-1}{w}_{1}N(Ɓ)=N(Ɓ)\iff w_{1}N(Ɓ)=w_{2}N(Ɓ) \end{gathered}$$

$\Rightarrow \Omega$ is bijective from $D$ to $D^{*}$. This completes the proof. □

Remark 4.

By way of the typical meanings, $1_{ӿ}\circ {\mu }_{P-Ɓ}={\mu }_{P-{Ɓ}_{ӿ}}\Rightarrow 1_{ӿ}\circ {\mu }_{RP-Ɓ}={\mu }_{RP-{Ɓ}_{ӿ}}$ and $1_{ӿ}\circ {\mu }_{IP-Ɓ}={\mu }_{IP-{Ɓ}_{ӿ}}$, $1_{ӿ}\circ {\mu }_{N-Ɓ}={\mu }_{N-{Ɓ}_{ӿ}}\Rightarrow 1_{ӿ}\circ {\mu }_{RN-Ɓ}={\mu }_{RN-{Ɓ}_{ӿ}}$ and $1_{ӿ}\circ {\mu }_{-Ɓ}={\mu }_{-{Ɓ}_{ӿ}}$, and ${\mu }_{P-Ɓ}\circ 1_{ӿ}={\mu }_{P-{Ɓ}_{ӿ}}\Rightarrow {\mu }_{RP-Ɓ}\circ 1_{ӿ}={\mu }_{RP-{Ɓ}_{ӿ}}$ and ${\mu }_{IP-Ɓ}\circ 1_{ӿ}={\mu }_{IP-{Ɓ}_{ӿ}}$, ${\mu }_{N-Ɓ}\circ 1_{ӿ}={\mu }_{N-{Ɓ}_{ӿ}}\Rightarrow {\mu }_{RN-Ɓ}\circ 1_{ӿ}={\mu }_{RN-{Ɓ}_{ӿ}}$ and ${\mu }_{-Ɓ}\circ 1_{ӿ}={\mu }_{-{Ɓ}_{ӿ}}$

Theorem 7.

For a BCF fuzzy normal subgroup of Ğ $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ and ${ӿ}_1,{ӿ}_2\in Ğ$, we have

• ${\mu }_{P-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{P-{Ɓ}_{{ӿ}_2}^r}={\mu }_{P-{Ɓ}_{{ӿ}_1{ӿ}_2}^r}\Rightarrow {\mu }_{RP-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{RP-{Ɓ}_{{ӿ}_2}^r}={\mu }_{RP-{Ɓ}_{{ӿ}_1{ӿ}_2}^r}$, ${\mu }_{IP-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{IP-{Ɓ}_{{ӿ}_2}^r}={\mu }_{IP-{Ɓ}_{{ӿ}_1{ӿ}_2}^r}$ and ${\mu }_{N-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{N-{Ɓ}_{{ӿ}_2}^r}={\mu }_{N-{Ɓ}_{{ӿ}_1{ӿ}_2}^r}\Rightarrow {\mu }_{RN-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{RN-{Ɓ}_{{ӿ}_2}^r}={\mu }_{RN-{Ɓ}_{{ӿ}_1{ӿ}_2}^r}$ and ${\mu }_{-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{-{Ɓ}_{{ӿ}_2}^r}={\mu }_{-{Ɓ}_{{ӿ}_1{ӿ}_2}^r}$
• ${\mu }_{P-{Ɓ}_{{ӿ}_1}^l}\circ {\mu }_{P-{Ɓ}_{{ӿ}_2}^l}={\mu }_{P-{Ɓ}_{{ӿ}_1{ӿ}_2}^l}\Rightarrow {\mu }_{RP-{Ɓ}_{{ӿ}_1}^l}\circ {\mu }_{RP-{Ɓ}_{{ӿ}_2}^l}={\mu }_{RP-{Ɓ}_{{ӿ}_1{ӿ}_2}^l}$, ${\mu }_{IP-{Ɓ}_{{ӿ}_1}^l}\circ {\mu }_{IP-{Ɓ}_{{ӿ}_2}^l}={\mu }_{IP-{Ɓ}_{{ӿ}_1{ӿ}_2}^l}$ and ${\mu }_{N-{Ɓ}_{{ӿ}_1}^l}\circ {\mu }_{N-{Ɓ}_{{ӿ}_2}^l}={\mu }_{N-{Ɓ}_{{ӿ}_1{ӿ}_2}^l}\Rightarrow {\mu }_{RN-{Ɓ}_{{ӿ}_1}^l}\circ {\mu }_{RN-{Ɓ}_{{ӿ}_2}^l}={\mu }_{RN-{Ɓ}_{{ӿ}_1{ӿ}_2}^l}$ and ${\mu }_{-{Ɓ}_{{ӿ}_1}^l}\circ {\mu }_{-{Ɓ}_{{ӿ}_2}^l}={\mu }_{-{Ɓ}_{{ӿ}_1{ӿ}_2}^l}$

Proof. 

1. ${\mu }_{P-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{P-{Ɓ}_{{ӿ}_2}^r}\Rightarrow {\mu }_{RP-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{RP-{Ɓ}_{{ӿ}_2}^r}$ and ${\mu }_{IP-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{IP-{Ɓ}_{{ӿ}_2}^r}$. Now

$$\begin{gathered} {\mu }_{RP-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{RP-{Ɓ}_{{ӿ}_2}^r}=({\mu }_{RP-Ɓ}\circ 1_{{ӿ}_1})\circ ({\mu }_{RP-Ɓ}\circ 1_{{ӿ}_2})={\mu }_{RP-Ɓ}\circ (1_{{ӿ}_1}\circ {\mu }_{RP-Ɓ})\circ 1_{{ӿ}_2} \\ {\mu }_{RP-Ɓ}\circ ({\mu }_{RP-Ɓ}\circ 1_{{ӿ}_1})\circ 1_{{ӿ}_1}=({\mu }_{RP-Ɓ}\circ {\mu }_{RP-Ɓ})\circ (1_{{ӿ}_1}\circ 1_{{ӿ}_2}) \\ {\mu }_{RP-Ɓ}\circ (1_{{ӿ}_1}\circ 1_{{ӿ}_2})={\mu }_{RP-{Ɓ}_{{ӿ}_{1{ӿ}_2}}^r} \end{gathered}$$

Similarly, we one can show that ${\mu }_{IP-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{IP-{Ɓ}_{{ӿ}_2}^r}={\mu }_{IP-{Ɓ}_{{ӿ}_{1{ӿ}_2}}^r}\Rightarrow {\mu }_{P-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{P-{Ɓ}_{{ӿ}_2}^r}={\mu }_{P-{Ɓ}_{{ӿ}_1{ӿ}_2}^r}$. Next,

$$\begin{gathered} {\mu }_{RN-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{RN-{Ɓ}_{{ӿ}_2}^r}=({\mu }_{RN-Ɓ}\circ 1_{{ӿ}_1})\circ ({\mu }_{RN-Ɓ}\circ 1_{{ӿ}_2})={\mu }_{RN-Ɓ}\circ (1_{{ӿ}_1}\circ {\mu }_{RN-Ɓ})\circ 1_{{ӿ}_2} \\ {\mu }_{RN-Ɓ}\circ ({\mu }_{RN-Ɓ}\circ 1_{{ӿ}_1})\circ 1_{{ӿ}_1}=({\mu }_{RN-Ɓ}\circ {\mu }_{RN-Ɓ})\circ (1_{{ӿ}_1}\circ 1_{{ӿ}_2}) \\ {\mu }_{RN-Ɓ}\circ (1_{{ӿ}_1}\circ 1_{{ӿ}_2})={\mu }_{RN-{Ɓ}_{{ӿ}_{1{ӿ}_2}}^r} \end{gathered}$$

Similarly, we one can show that ${\mu }_{-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{-{Ɓ}_{{ӿ}_2}^r}={\mu }_{-{Ɓ}_{{ӿ}_{1{ӿ}_2}}^r}$. $\Rightarrow {\mu }_{N-{Ɓ}_{{ӿ}_1}^r}\circ {\mu }_{N-{Ɓ}_{{ӿ}_2}^r}={\mu }_{N-{Ɓ}_{{ӿ}_1{ӿ}_2}^r}$.

2. Likewise 1, so we are omitting the proof. □

Theorem 8.

Let a BCF normal subgroup $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ of Ğ. Then, the set signified and described by $Ğ/Ɓ=\left\{{Ɓ}_{ӿ}^l=({\mu }_{P-{Ɓ}_{ӿ}^l},{\mu }_{N-{Ɓ}_{ӿ}^l})=({\mu }_{RP-{Ɓ}_{ӿ}^l}+\iota {\mu }_{IP-{Ɓ}_{ӿ}^l},{\mu }_{RN-{Ɓ}_{ӿ}^l}+\iota {\mu }_{-{Ɓ}_{ӿ}^l}):ӿ\in Ğ\right\}$ is a group (This group is said to be a quotient or factor group).

Proof. 

Let ${Ɓ}_{{ӿ}_1}^l=\left({\mu }_{P-{Ɓ}_{{ӿ}_1}^l},{\mu }_{N-{Ɓ}_{{ӿ}_1}^l}\right)=\left({\mu }_{RP-{Ɓ}_{{ӿ}_1}^l}+\iota {\mu }_{IP-{Ɓ}_{{ӿ}_1}^l},{\mu }_{RN-{Ɓ}_{{ӿ}_1}^l}+\iota {\mu }_{-{Ɓ}_{{ӿ}_1}^l}\right)$, ${Ɓ}_{{ӿ}_2}^l=\left({\mu }_{P-{Ɓ}_{{ӿ}_2}^l},{\mu }_{N-{Ɓ}_{{ӿ}_2}^l}\right)=\left({\mu }_{RP-{Ɓ}_{{ӿ}_2}^l}+\iota {\mu }_{IP-{Ɓ}_{{ӿ}_2}^l},{\mu }_{RN-{Ɓ}_{{ӿ}_2}^l}+\iota {\mu }_{-{Ɓ}_{{ӿ}_2}^l}\right)\in Ğ/Ɓ$, we define

$$\begin{gathered} {Ɓ}_{{ӿ}_1}^l\circ {Ɓ}_{{ӿ}_2}^l=\left({\mu }_{P-{Ɓ}_{{ӿ}_1}^l}\circ {\mu }_{P-{Ɓ}_{{ӿ}_2}^l},{\mu }_{N-{Ɓ}_{{ӿ}_1}^l}\circ {\mu }_{N-{Ɓ}_{{ӿ}_2}^l}\right) \\ ({\mu }_{RP-{Ɓ}_{{ӿ}_1}^l}\circ {\mu }_{RP-{Ɓ}_{{ӿ}_2}^l}+\iota \left({\mu }_{IP-{Ɓ}_{{ӿ}_1}^l}\circ {\mu }_{IP-{Ɓ}_{{ӿ}_2}^l}\right),{\mu }_{RN-{Ɓ}_{{ӿ}_1}^l}\circ {\mu }_{RN-{Ɓ}_{{ӿ}_2}^l}+\iota \left({\mu }_{-{Ɓ}_{{ӿ}_1}^l}\circ {\mu }_{-{Ɓ}_{{ӿ}_2}^l}\right)) \\ \left({\mu }_{RP-{Ɓ}_{{ӿ}_3}^l}+\iota {\mu }_{IP-{Ɓ}_{{ӿ}_3}^l},{\mu }_{RN-{Ɓ}_{{ӿ}_3}^l}+\iota {\mu }_{-{Ɓ}_{{ӿ}_3}^l}\right)={Ɓ}_{{ӿ}_3}^l\in Ğ/Ɓ \end{gathered}$$

$\Rightarrow Ğ/Ɓ$ is closed.

Since $\circ$ is associative in $Ğ/Ɓ$. As Ğ is a group and contains identity, assume that $e$ is the identity in Ğ, then ${Ɓ}_e^l=\left({\mu }_{P-{Ɓ}_e^l},{\mu }_{N-{Ɓ}_e^l}\right)=\left({\mu }_{RP-{Ɓ}_e^l}+\iota {\mu }_{IP-{Ɓ}_e^l},{\mu }_{RN-{Ɓ}_e^l}+\iota {\mu }_{-{Ɓ}_e^l}\right)\in Ğ/Ɓ$ $\forall {Ɓ}_{ӿ}^l=({\mu }_{P-{Ɓ}_{ӿ}^l},{\mu }_{N-{Ɓ}_{ӿ}^l})=({\mu }_{RP-{Ɓ}_{ӿ}^l}+\iota {\mu }_{IP-{Ɓ}_{ӿ}^l},{\mu }_{RN-{Ɓ}_{ӿ}^l}+\iota {\mu }_{-{Ɓ}_{ӿ}^l})$

$$\begin{gathered} {Ɓ}_e^l\circ {Ɓ}_{ƴ}^l=\left({\mu }_{P-{Ɓ}_e^l}\circ {\mu }_{P-{Ɓ}_{ӿ}^l},{\mu }_{N-{Ɓ}_e^l}\circ {\mu }_{N-{Ɓ}_{ӿ}^l}\right) \\ \left({\mu }_{RP-{Ɓ}_e^l}\circ {\mu }_{RP-{Ɓ}_{ӿ}^l}+\iota \left({\mu }_{IP-{Ɓ}_e^l}\circ {\mu }_{IP-{ӿ}_{ƴ}^l}\right),{\mu }_{RN-{Ɓ}_e^l}\circ {\mu }_{RN-{Ɓ}_{ӿ}^l}+\iota \left({\mu }_{-{Ɓ}_e^l}\circ {\mu }_{-{Ɓ}_{ӿ}^l}\right)\right) \\ \left({\mu }_{RP-{Ɓ}_{eӿ}^l}+\iota {\mu }_{IP-{Ɓ}_{eӿ}^l},{\mu }_{RN-{Ɓ}_{eӿ}^l}+\iota {\mu }_{-{Ɓ}_{eӿ}^l}\right)=\left({\mu }_{RP-{Ɓ}_{ӿ}^l}+\iota {\mu }_{IP-{Ɓ}_{ӿ}^l},{\mu }_{RN-{Ɓ}_{ӿ}^l}+\iota {\mu }_{-{Ɓ}_{ӿ}^l}\right)={Ɓ}_{ӿ}^l \end{gathered}$$

Similarly, one can prove that ${Ɓ}_{ӿ}^l\circ {Ɓ}_e^l$. Thus, ${Ɓ}_e^l$ is the identity in $Ğ/Ɓ$.

For each ${Ɓ}_{ӿ}^l=({\mu }_{P-{Ɓ}_{ӿ}^l},{\mu }_{N-{Ɓ}_{ӿ}^l})=({\mu }_{RP-{Ɓ}_{ӿ}^l}+\iota {\mu }_{IP-{Ɓ}_{ӿ}^l},{\mu }_{RN-{Ɓ}_{ӿ}^l}+\iota {\mu }_{-{Ɓ}_{ӿ}^l})$ $\exists$ ${Ɓ}_{{ӿ}^{-1}}^l=({\mu }_{P-{Ɓ}_{{ӿ}^{-1}}^l},{\mu }_{N-{Ɓ}_{{ӿ}^{-1}}^l})=({\mu }_{RP-{Ɓ}_{{ӿ}^{-1}}^l}+\iota {\mu }_{IP-{Ɓ}_{{ӿ}^{-1}}^l},{\mu }_{RN-{Ɓ}_{{ӿ}^{-1}}^l}+\iota {\mu }_{-{Ɓ}_{{ӿ}^{-1}}^l})$ s.t.

$$\begin{gathered} {Ɓ}_{{ӿ}^{-1}}^l\circ {ӿ}_{ƴ}^l=\left({\mu }_{P-{Ɓ}_{{ӿ}^{-1}}^l}\circ {\mu }_{P-{Ɓ}_{ӿ}^l},{\mu }_{N-{Ɓ}_{{ӿ}^{-1}}^l}\circ {\mu }_{N-{Ɓ}_{ӿ}^l}\right) \\ \left({\mu }_{RP-{Ɓ}_{{ӿ}^{-1}}^l}\circ {\mu }_{RP-{Ɓ}_{ӿ}^l}+\iota \left({\mu }_{IP-{Ɓ}_{{ӿ}^{-1}}^l}\circ {\mu }_{IP-{ӿ}_{ƴ}^l}\right),{\mu }_{RN-{Ɓ}_{{ӿ}^{-1}}^l}\circ {\mu }_{RN-{Ɓ}_{ӿ}^l}+\iota \left({\mu }_{-{Ɓ}_{{ӿ}^{-1}}^l}\circ {\mu }_{-{Ɓ}_{ӿ}^l}\right)\right) \\ \left({\mu }_{RP-{Ɓ}_{{ӿ}^{-1}ӿ}^l}+\iota {\mu }_{IP-{Ɓ}_{{ӿ}^{-1}ӿ}^l},{\mu }_{RN-{Ɓ}_{{ӿ}^{-1}ӿ}^l}+\iota {\mu }_{-{Ɓ}_{{ӿ}^{-1}ӿ}^l}\right)=\left({\mu }_{RP-{Ɓ}_e^l}+\iota {\mu }_{IP-{Ɓ}_e^l},{\mu }_{RN-{Ɓ}_e^l}+\iota {\mu }_{-{Ɓ}_e^l}\right)={Ɓ}_e^l \end{gathered}$$

Similarly, one can prove that ${Ɓ}_{ӿ}^l\circ {Ɓ}_{{ӿ}^{-1}}^l$. Thus, for each ${Ɓ}_{ӿ}^l\in Ğ/Ɓ$ $\exists$ ${Ɓ}_{{ӿ}^{-1}}^l\in Ğ/Ɓ$ and act as the inverse of ${Ɓ}_{ӿ}^l$.

$\Rightarrow Ğ/Ɓ$ is a group.□

Definition 18.

Let ${Ğ}_1$, ${Ğ}_2$ be two groups, $\Gamma :{Ğ}_1\to {Ğ}_2$ be a function and ${Ɓ}_1=({\mu }_{P-{Ɓ}_1},{\mu }_{N-{Ɓ}_1})$, ${Ɓ}_2=({\mu }_{P-{Ɓ}_2},{\mu }_{N-{Ɓ}_2})$ be two BCF subsets of ${Ğ}_1$ and ${Ğ}_2$, respectively. Then, the image $\Gamma ({Ɓ}_1)=(\Gamma ({\mu }_{P-{Ɓ}_1}),\Gamma ({\mu }_{N-{Ɓ}_1}))$ of ${Ɓ}_1$ is a BCF subset ${Ğ}_2$ described as

$$\begin{gathered} \Gamma ({\mu }_{P-{Ɓ}_1})(ƴ)=\{\begin{array}{c}max\left\{{\mu }_{P-{Ɓ}_1}(ӿ):ӿ\in {\Gamma }^{-1}(ӿ)\right\},if{\Gamma }^{-1}(ƴ)\ne \varnothing \\ 0,Otherwise\end{array} \\ \{\begin{array}{c}\left\{max{\mu }_{RP-{Ɓ}_1}(ӿ):ӿ\in {\Gamma }^{-1}(ӿ)\right\}+\iota \left\{max{\mu }_{IP-{Ɓ}_1}(ӿ):ӿ\in {\Gamma }^{-1}(ӿ)\right\}if{\Gamma }^{-1}(ƴ)\ne \varnothing \\ 0,Otherwise\end{array} \\ \Gamma ({\mu }_{N-{Ɓ}_1})(ƴ)=\{\begin{array}{c}min\left\{{\mu }_{N-{Ɓ}_1}(ӿ):ӿ\in {\Gamma }^{-1}(ӿ)\right\},if{\Gamma }^{-1}(ƴ)\ne \varnothing \\ 0,Otherwise\end{array} \\ \{\begin{array}{c}\left\{min{\mu }_{RN-{Ɓ}_1}(ӿ)+\iota min{\mu }_{-{Ɓ}_1}(ӿ):ӿ\in {\Gamma }^{-1}(ӿ)\right\}if{\Gamma }^{-1}(ƴ)\ne \varnothing \\ 0,Otherwise\end{array} \end{gathered}$$

and the pre-image ${\Gamma }^{-1}({Ɓ}_2)=({\Gamma }^{-1}({\mu }_{P-{Ɓ}_2}),{\Gamma }^{-1}({\mu }_{N-{Ɓ}_2}))$ of ${Ɓ}_2$ is the BCF subset of ${Ğ}_1$ described as

$${\Gamma }^{-1}({\mu }_{P-{Ɓ}_2})(ӿ)={\mu }_{P-{Ɓ}_2}(\Gamma (ӿ))={\mu }_{RP-{Ɓ}_2}(\Gamma (ӿ))+\iota {\mu }_{IP-{Ɓ}_2}(\Gamma (ӿ))$$

and

$${\Gamma }^{-1}({\mu }_{N-{Ɓ}_2})(ӿ)={\mu }_{N-{Ɓ}_2}(\Gamma (ӿ))={\mu }_{RN-{Ɓ}_2}(\Gamma (ӿ))+\iota {\mu }_{-{Ɓ}_2}(\Gamma (ӿ))$$

We consider that the image $\Gamma ({Ɓ}_1)=(\Gamma ({\mu }_{P-{Ɓ}_1}),\Gamma ({\mu }_{N-{Ɓ}_1}))$ of ${Ɓ}_1$ is a homomorphic image of ${Ɓ}_1$ if $\Gamma$ is a homomorphism. Likewise, We consider that the image $\Gamma ({Ɓ}_1)=(\Gamma ({\mu }_{P-{Ɓ}_1}),\Gamma ({\mu }_{N-{Ɓ}_1}))$ of ${Ɓ}_1$ is an isomorphic image of ${Ɓ}_1$ if $\Gamma$ is an isomorphism. Obviously, over the class of all BCF subgroups, isomorphism is an equivalence relation. Thus, for two BCF subgroups ${Ɓ}_1$ and ${Ɓ}_2$ of ${Ğ}_1$ and ${Ğ}_2$, respectively, if there is an isomorphism from ${Ğ}_1$ to ${Ğ}_2$ or from ${Ğ}_2$ to ${Ğ}_1$, then we say that ${Ɓ}_1$ and ${Ɓ}_2$ are isomorphic to one another.

Theorem 9.

Suppose $\Gamma :{Ğ}_1\to {Ğ}_2$ are a homomorphism and $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ is BCF subgroup of ${Ğ}_1$, then $\Gamma (Ɓ)=(\Gamma ({\mu }_{P-Ɓ}),\Gamma ({\mu }_{N-Ɓ}))$ is a BCF subgroup of ${Ğ}_2$.

Proof. 

Let $\Gamma (Ɓ)=(\Gamma ({\mu }_{P-Ɓ}),\Gamma ({\mu }_{N-Ɓ}))$ is a BCF subgroup of ${Ğ}_1$. As $\Gamma$ is homomorphism so for each $u,v\in {Ğ}_2$ there are $ӿ,ƴ\in {Ğ}_1$ such that $\Gamma (ӿ)=u$ and $\Gamma (ƴ)=v$, consequently $ӿƴ\in {\Gamma }^{-1}(uv)$. Next,

$$\begin{gathered} \Gamma ({\mu }_{P-Ɓ})(uv)=max\left\{{\mu }_{P-Ɓ}(ȥ):ȥ=ӿƴ\in {\Gamma }^{-1}(uv)\right\} \\ \ge max\left\{{\mu }_{P-Ɓ}(ӿƴ):ӿ\in {\Gamma }^{-1}(u)\wedge ƴ\in {\Gamma }^{-1}(v)\right\} \\ \left\{max{\mu }_{RP-Ɓ}(ӿƴ):ӿ\in {\Gamma }^{-1}(u)\wedge ƴ\in {\Gamma }^{-1}(v)\right\} \\ +\iota \left\{max{\mu }_{IP-Ɓ}(ӿƴ):ӿ\in {\Gamma }^{-1}(v)\wedge ƴ\in {\Gamma }^{-1}(v)\right\} \\ \ge \left(\begin{array}{c}\left\{max(min\left\{{\mu }_{RP-Ɓ}(ӿ),{\mu }_{RP-Ɓ}(ƴ)\right\}):ӿ\in {\Gamma }^{-1}(u)\wedge ƴ\in {\Gamma }^{-1}(v)\right\}\\ +\iota \left\{max(min\left\{{\mu }_{IP-Ɓ}(ӿ),{\mu }_{IP-Ɓ}(ƴ)\right\}):ӿ\in {\Gamma }^{-1}(u)\wedge ƴ\in {\Gamma }^{-1}(v)\right\}\end{array}\right) \\ \left(\begin{array}{c}min(max\left\{{\mu }_{RP-Ɓ}(ӿ):ӿ\in {\Gamma }^{-1}(u)\right\},max\left\{{\mu }_{RP-Ɓ}(ƴ):ƴ\in {\Gamma }^{-1}(v)\right\})\\ +\iota min(max\left\{{\mu }_{IP-Ɓ}(ӿ):ӿ\in {\Gamma }^{-1}(u)\right\},max\left\{{\mu }_{IP-Ɓ}(ƴ):ƴ\in {\Gamma }^{-1}(v)\right\})\end{array}\right) \\ \left(min(\begin{array}{c}max\left\{{\mu }_{RP-Ɓ}(ӿ):ӿ\in {\Gamma }^{-1}(u)+\iota max{\mu }_{IP-Ɓ}(ӿ):ӿ\in {\Gamma }^{-1}(u)\right\},\\ max\left\{{\mu }_{RP-Ɓ}(ƴ):ƴ\in {\Gamma }^{-1}(v)+\iota max{\mu }_{IP-Ɓ}(ƴ):ƴ\in {\Gamma }^{-1}(v)\right\}\end{array})\right) \\ min(max\left\{{\mu }_{P-Ɓ}(ӿ):ӿ\in {\Gamma }^{-1}(u)\right\},max\left\{{\mu }_{P-Ɓ}(ƴ):y\in {\Gamma }^{-1}(v)\right\}) \\ min\left\{\Gamma ({\mu }_{P-Ɓ})(u),\Gamma ({\mu }_{P-Ɓ})(v)\right\} \end{gathered}$$

Thus, $\Gamma ({\mu }_{P-Ɓ})(uv)\ge min\left\{\Gamma ({\mu }_{P-Ɓ})(u),\Gamma ({\mu }_{P-Ɓ})(v)\right\}$. Next

$$\begin{gathered} \Gamma ({\mu }_{P-Ɓ})(u^{-1})=max\left\{{\mu }_{P-Ɓ}(ӿ):ӿ\in {\Gamma }^{-1}(u^{-1})\right\} \\ \left\{max{\mu }_{RP-Ɓ}(ӿ):ӿ\in {\Gamma }^{-1}(u^{-1})\right\}+\iota \left\{max{\mu }_{IP-Ɓ}(ӿ):ӿ\in {\Gamma }^{-1}(u^{-1})\right\} \\ \left\{max{\mu }_{RP-Ɓ}({ӿ}^{-1}):{ӿ}^{-1}\in {\Gamma }^{-1}(u)\right\}+\iota \left\{max{\mu }_{IP-Ɓ}({ӿ}^{-1}):{ӿ}^{-1}\in {\Gamma }^{-1}(u)\right\} \\ max\left\{{\mu }_{P-Ɓ}({ӿ}^{-1}):{ӿ}^{-1}\in {\Gamma }^{-1}(u)\right\}=\Gamma ({\mu }_{P-Ɓ})(u) \end{gathered}$$

Likewise, one can prove that $\Gamma ({\mu }_{N-Ɓ})(uv)\le max\left\{\Gamma ({\mu }_{N-Ɓ})(u),\Gamma ({\mu }_{N-Ɓ})(v)\right\}$ and $\Gamma ({\mu }_{N-Ɓ})(u^{-1})=\Gamma ({\mu }_{N-Ɓ})(u)$. Thus, $\Gamma (Ɓ)=(\Gamma ({\mu }_{P-Ɓ}),\Gamma ({\mu }_{N-Ɓ}))$ is a BCF subgroup of ${Ğ}_2$.□

Theorem 10.

Suppose $\Gamma :{Ğ}_1\to {Ğ}_2$ are a homomorphism and $Ɓ=({\mu }_{P-Ɓ},{\mu }_{N-Ɓ})=({\mu }_{RP-Ɓ}+\iota {\mu }_{IP-Ɓ},{\mu }_{RN-Ɓ}+\iota {\mu }_{-Ɓ})$ is BCF subgroup of ${Ğ}_2$, then ${\Gamma }^{-1}(Ɓ)=({\Gamma }^{-1}({\mu }_{P-Ɓ}),{\Gamma }^{-1}({\mu }_{N-Ɓ}))$ is a BCF subgroup of ${Ğ}_1$.

Proof. 

Let ${\Gamma }^{-1}(Ɓ)=({\Gamma }^{-1}({\mu }_{P-Ɓ}),{\Gamma }^{-1}({\mu }_{N-Ɓ}))$ is a BCF subgroup of ${Ğ}_2$. Now

$$\begin{gathered} ({\Gamma }^{-1}({\mu }_{P-Ɓ}))(ӿƴ)={\mu }_{P-Ɓ}(\Gamma (ӿƴ))={\mu }_{RP-Ɓ}(\Gamma (ӿƴ))+\iota {\mu }_{IP-Ɓ}(\Gamma (ӿƴ)) \\ {\mu }_{RP-Ɓ}(\Gamma (ӿ)\Gamma (ƴ))+\iota {\mu }_{IP-Ɓ}(\Gamma (ӿ)\Gamma (ƴ)) \\ \ge min\left\{{\mu }_{RP-Ɓ}(\Gamma (ӿ)),{\mu }_{RP-Ɓ}(\Gamma (ƴ))\right\}+\iota min\left\{{\mu }_{IP-Ɓ}(\Gamma (ӿ)),{\mu }_{IP-Ɓ}(\Gamma (ƴ))\right\} \\ min\left\{{\mu }_{P-Ɓ}(\Gamma (ӿ)),{\mu }_{P-Ɓ}(\Gamma (ƴ))\right\}=min\left\{({\Gamma }^{-1}({\mu }_{P-Ɓ}))(ӿ),({\Gamma }^{-1}({\mu }_{P-Ɓ}))(ƴ)\right\} \end{gathered}$$

Next,

$$\begin{gathered} ({\Gamma }^{-1}({\mu }_{P-Ɓ}))({ӿ}^{-1})={\mu }_{P-Ɓ}(\Gamma ({ӿ}^{-1}))={\mu }_{RP-Ɓ}(\Gamma ({ӿ}^{-1}))+\iota {\mu }_{IP-Ɓ}(\Gamma ({ӿ}^{-1})) \\ {\mu }_{RP-Ɓ}(\Gamma {(ӿ)}^{-1})+\iota {\mu }_{IP-Ɓ}(\Gamma {(ӿ)}^{-1})={\mu }_{RP-Ɓ}(\Gamma (ӿ))+\iota {\mu }_{IP-Ɓ}(\Gamma (ӿ)) \\ {\mu }_{P-Ɓ}(\Gamma (ӿ))=({\Gamma }^{-1}({\mu }_{P-Ɓ}))(ӿ) \end{gathered}$$

• Likewise, one can prove that $({\Gamma }^{-1}({\mu }_{N-Ɓ}))(ӿƴ)\le max\left\{({\Gamma }^{-1}({\mu }_{N-Ɓ}))(ӿ),({\Gamma }^{-1}({\mu }_{N-Ɓ}))(ƴ)\right\}$ and $({\Gamma }^{-1}({\mu }_{N-Ɓ}))({ӿ}^{-1})=({\Gamma }^{-1}({\mu }_{N-Ɓ}))(ӿ)$.□

4. Conclusions

In this present article, we interpreted the level set, support, kernel for the BCF set, and bipolar complex characteristic function. Then, we explored the BCF point or BCF singleton set. Moreover, we investigated the BCF subgroup with an example, BCF normal subgroup with an example, and BCF abelian subgroup with an example. We also investigated the concept of the BCF conjugate, normalizer for BCF subgroup, cosets, and BCF factor group. Additionally, we presented the associated theorems along with their proofs. We described the image and pre-image of BCF subgroups under homomorphism and proved the related theorems. Further, one of the biggest advantages of our initiated work is to generalize the conception of fuzzy subgroups, bipolar fuzzy subgroups, and complex fuzzy subgroups. Further, the investigated work also generalized other concepts such as level set, kernel, characteristic function, fuzzy point, etc. As we know, group symmetry has a significant role in the assessment of molecule structures. The fuzzy intellect appears in it because the isotope molecules decay with a specific ratio. If this specific ratio of decay follows the conditions of the BCF set, then none of the prevailing theories can be utilized to study the form of the isotope at a specific time. Thus, only the investigated work can be useful in such circumstances. However, the investigated work has certain limitations as well, such as if the fuzzy intellect appears in the structure of intuitionistic FS, picture FS, spherical FS, etc., then the investigated work cannot be employed for the examination of molecule structures, BCF subgroups cannot generalized intuitionistic fuzzy subgroups, picture fuzzy subgroups, soft groups, etc. In the future, we aim to expand this research to bipolar complex fuzzy soft sets [26,27], and complex bipolar fuzzy N-soft sets [28]. We are hopeful that these notions would be the foundation for innovative research on subgroups.