Application of t-intuitionistic fuzzy subgroup to Sylow theory

Abstract

In this paper, we define the notion of a t-intuitionistic fuzzy conjugate element and determine the t-intuitionistic fuzzy conjugacy classes of a t-intuitionistic fuzzy subgroup. We propose the idea of a t-intuitionistic fuzzy $p\mathit{-}$ subgroup and prove the t-intuitionistic fuzzy version of the Cauchy theorem. In addition, we present the idea of a t-intuitionistic fuzzy conjugate subgroup and investigate various fundamental algebraic characteristics of this notion. Furthermore, we provide the idea of the t-intuitionistic fuzzy Sylow $p-$ subgroup and prove the t-intuitionistic fuzzification of Sylow's theorems.

Mathematics Subject Classification: 08A72, 20N25, 03E72.

1. Introduction

The Sylow theorems are an important part of finite group theory, and they have many important uses in classical group theory, especially when it comes to classifying the finite simple group. Sylow applied this theory in the framework of solving an algebraic equation and associated the roots of this equation to the solvability of its Galois group by radicals. One of the main advantages of these theorems is that they establish a partial converse of Lagrange's Theorem in the literature. Moreover, one can easily investigate the existence of a subgroup of certain orders utilizing this technique.

Fuzzy approaches have several advantages over crisp ones: the most important being that they have more flexible decision boundaries and are thus characterized by their higher ability to adjust to a specific domain of application and more accurately reflect its particularities. The importance of fuzzy logic derives from the fact that most modes of human reasoning, especially commonsense reasoning, are approximate in nature. The concept of an $\mathbb{I}\mathbb{F}\mathbb{S}$ is characterized by membership and non-membership functions as much efficient way to cope with uncertainty. The concept of $\mathbb{I}\mathbb{F}\mathbb{S}$ is very useful in providing a flexible model to elaborate on the uncertainty and vagueness involved in decision-making. It is a very useful tool for human consistency in reasoning under imperfectly defined facts and imprecise knowledge.

The concept of fuzzy set ($\mathbb{F}\mathbb{S}$) theory was introduced by Zadeh [1]. Rosenfeld [2] initiated the idea of a fuzzy subgroup($\mathbb{F}\mathbb{S}\mathbb{G}$) and established numerous essential characteristics of this phenomenon. In Ref. [3], Sunderrajan initiated the study of homomorphism and anti-homomorphism of L-fuzzy quotient ℓ-groups. Atanassov [4,5] proposed the study of $\mathbb{I}\mathbb{F}\mathbb{S}$ in 1986. In Ref. [6], Ejegwa et al. presented a concise overview of $\mathbb{I}\mathbb{F}\mathbb{S}$. This theory has received a great deal of attention from researchers in the last two decades as it has been effectively applied in career determination [7,8], pattern recognition [9,10], medical diagnosis [11,12], expert systems [13,14], neural networks [15,16], decision making [17,18], machine learning [19,20], and semantic representations [21,22]. A useful approach was developed to study intuitionistic fuzzy soft rough sets based on decision-making in Refs. [23,24]. Biswas [25] proposed the idea of intuitionistic fuzzy subgroup ($\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$). In Ref. [26], Ahn et al. discussed various types of sublattices of the lattice of intuitionistic fuzzy subgroups. Sharma [27] proved many properties of $\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$. For more development on $\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$, we refer to Refs. [[28], [29], [30], [31], [32]]. In addition, the ideas of an intuitionistic fuzzy ring [33], an intuitionistic fuzzy ideal [34], an intuitionistic fuzzy normed ring [35], an intuitionistic fuzzy soft ring [36], and an intuitionistic fuzzy normal subring [37] were proposed. The theory of complex $\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ was proposed and developed by Husban et al. [38,39]. The theory of t-intuitionistic fuzzy subgroup(t-$\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$) was presented by Sharma in Ref. [40]. Moreover, a comprehensive development of the theory of t-$\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ can be viewed in Refs. [41,42]. In Ref. [43], Kattan et al. proved the interval-valued intuitionistic fuzzy version of Lagrange's theorem.

Real-world data can be challenging due to its imprecision, resulting from uncertainties, vagueness, or conflicting information. Although traditional approaches like as $\mathbb{F}\mathbb{S}$ and $\mathbb{I}\mathbb{F}\mathbb{S}$ are often used in the management of imprecision. The t-$\mathbb{I}\mathbb{F}\mathbb{S}$ theory provides an additional parameterizing component, which makes it possible to express uncertain data in a more comprehensive manner. This parameter, which is denoted by the letter "t" is essential in defining the degree of imprecision that is taken into account during modeling. Analysts can more precisely depict imprecision by modulating the impact of inconsistent or vague information via "t" resulting in more reliable outcomes and decision-making. This adaptability is essential when dealing with varying degrees of imprecision. Consequently, t-$\mathbb{I}\mathbb{F}\mathbb{S}$ enhances the precision of managing imprecise data. In contrast to the theories of classical $\mathbb{F}\mathbb{S}$ and $\mathbb{I}\mathbb{F}\mathbb{S}$, it is an essential tool for addressing data imprecision by introducing a parameterizing factor during the process. It is a very useful technique as it provides a flexible model to counter the uncertainty and vagueness involved in making decisions.

The following discussion describes the motivations of this present study.

• •Introduce t-intuitionistic fuzzy conjugate elements and conjugacy classes (in Section 3). These phenomena help study t-$\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ structure and organization. Analyzing conjugate elements' patterns, connections, and symmetries may reveal a subgroup's structure. Classifying and characterizing t-$\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ is made easier using conjugate elements and their conjugacy classes. These tools help organize elements into subgroups based on their conjugacy, which can provide insight into common properties, patterns, and behaviors.
• •Introduce the notion of the t-intuitionistic fuzzy $\mathrm{p}$-subgroup and prove the t-intuitionistic fuzzy version of the Cauchy theorem (in Section 4). They are critical for classifying finite groups. It assists researchers in determining the likely order of components inside a group, which may subsequently be used to organize the groupings based on their attributes. This categorization aids in the study of qualities by allowing for a better understanding of the similarities and differences between distinct groups.
• •Initiate the ideas of t-intuitionistic fuzzy conjugate subgroups and their classes (in Section 5). These concepts help classify and organize subgroups within a larger group. The study of groups is simplified when subgroups are divided into discrete classes based on shared qualities or characteristics. This categorization is used to establish significant theorems, such as Sylow's theorems, which reveal information about the structure of finite groups. These ideas are strongly related to normal subgroups.
• •Explore the idea of a t-intuitionistic fuzzy Sylow $\mathrm{p}-$ subgroup and prove a t-intuitionistic fuzzy version of Sylow's theorems (in Section 6). To better understand how subgroups work within finite groups, we can use t-intuitionistic logic to fuzzify Sylow's theorem. These are helpful even when dealing with uncertainties and imprecisions in real-life situations. The t-intuitionistic fuzzy version of Sylow's theorem can be applied to practical problems involving uncertainty, vagueness, and imprecision in subgroup analysis. This framework can improve decision-making, data analysis, optimization, and social network analysis. It helps to understand better real-life scenarios that have uncertain group structures.

The remaining sections of the article are arranged in the following order: Section 1 presents the introduction and motivation; Section 2 contains some elementary definitions and results; and conclusion and future scopes are presented in Section 7.

2. Preliminaries

This section contains some basic definitions and results related to them, which are very important to recognize the modernity of this article.

Definition 1

[40] A t-intuitionistic fuzzy set (t-$\mathbb{I}\mathbb{F}\mathbb{S}$) ${\mathrm{A}}_{\mathrm{t}}$ of a universal set $\mathbb{U}$ is defined as:

$$A_{\mathit{t}}\mathit{=}\left\{\left({\mathit{a}}_{\mathit{1}}\mathit{,}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{,}{\mathit{\nu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\right)\mathit{:}{\mathit{a}}_{\mathit{1}}\mathit{\in }\mathbb{U}\right\}\mathit{,}t\mathit{\in }\left[\mathit{0}\mathit{,}\mathit{1}\right]$$

where ${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$ and ${\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$ are functions that assign degrees of membership and non-membership, respectively, to the element $a_{\mathit{1}}$ in $\mathbb{U}$. The functions ${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$ and ${\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$ are defined based on an existing $\mathbb{I}\mathbb{F}\mathbb{S}$ $A\mathit{=}\left({\mathit{\mu }}_{\mathit{A}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{,}{\mathit{\nu }}_{\mathit{A}}\left({\mathit{a}}_{\mathit{1}}\right)\right)$ on $\mathbb{U}$. Specifically,

$${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{=}min{\mathit{\{}\mathit{\mu }}_{\mathit{A}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{,}t\mathit{\}}\text{and }{\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{=}max\left\{{\mathit{\nu }}_{\mathit{A}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{,}\mathit{1}\mathit{-}\mathit{t}\right\}\text{.}$$

The condition is true for each $a_{\mathit{1}}$ in $\mathbb{U}\text{,}$ $\mathit{0}\mathit{\le }{\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{+}{\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{\le }\mathit{1}$.

Definition 2

[40] A $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}{\mathrm{A}}_{\mathrm{t}}$ is called t-intuitionistic fuzzy subgroup$(\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}){\mathrm{A}}_{\mathrm{t}}$ of group $\mathbb{G}$ if it admits the following conditions:

• 1)${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{b}}_{\mathit{1}}\right)\mathit{\ge }min\mathit{}\left\{{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{,}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{b}}_{\mathit{1}}\right)\right\}$.
• 2)${\nu }_{A_t}\left(a_{\mathit{1}}{b}_{\mathit{1}}\right)\mathit{\le }max\mathit{}\left\{{\nu }_{A_t}\left(a_{\mathit{1}}\right)\mathit{,}{\nu }_{A_t}\left(b_{\mathit{1}}\right)\right\}$.
• 3)${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}^{\mathit{-}\mathit{1}}\right)\mathit{=}{\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$.
• 4)${\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}^{\mathit{-}\mathit{1}}\right)\mathit{=}{\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$, $\mathit{\forall }{a}_{\mathit{1}}\mathit{,}{b}_{\mathit{1}}\mathit{\in }\mathbb{G}$.

In ${\mathrm{A}}_{\mathrm{t}}\text{,}$ the terms ${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{b}}_{\mathit{1}}\right)\mathit{,}{\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$ and ${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{b}}_{\mathit{1}}\right)$ represent the degree of membership of the elements $a_{\mathit{1}}{b}_{\mathit{1}}\mathit{,}{a}_{\mathit{1}}\mathit{,}{b}_{\mathit{1}}\text{.}$ On the other hand, the expressions ${\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{b}}_{\mathit{1}}\right)\mathit{,}{\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$ and ${\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{b}}_{\mathit{1}}\right)$ indicate the degree of non-membership degrees to the same elements in $A_{\mathit{t}}$.

Definition 3

[41] Let ${\mathrm{A}}_{\mathrm{t}}$ be a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}$ of a universe $\mathbb{U}$. The $\left(\mathrm{\rho },\mathrm{\eta }\right)-$ cut set of $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}$ is a subset of the universe $\mathbb{U}$, which may be defined as:

$${ℭ}_{\left(\rho \mathit{,}\eta \right)}\left(A_t\right)\mathit{=}\left\{a_{\mathit{1}}\mathit{\in }\mathbb{U}\mathit{:}{\mu }_{A_t}\left(a_{\mathit{1}}\right)\mathit{\ge }\rho \text{and}{\nu }_{A_t}\left(a_{\mathit{1}}\right)\mathit{\le }\eta \right\}$$

where $0\le \mathrm{\rho }\le 1$ and $0\le \mathrm{\eta }\le 1$ such that $0\le \mathrm{\rho }+\mathrm{\eta }\le 1$.

Where ${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$ and ${\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$ indicate the membership and non-membership degrees to the element $a_{\mathit{1}}$ in ${\mathrm{A}}_{\mathrm{t}}$.

Definition 4

[41] The support set ${\mathrm{A}}_{\mathrm{t}}^{*}$ of $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}{\mathrm{A}}_{\mathrm{t}}$ of the universe $\mathbb{U}$ is defined as:

$$A_{\mathit{t}}^{\mathit{*}}\mathit{=}\left\{{\mathit{a}}_{\mathit{1}}\mathit{\in }\mathbb{U}\mathit{:}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{>}\mathit{0}\mathit{and}{\mathit{\nu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{<}\mathit{1}\right\}$$

The functions ${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$ and ${\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$ respectively assign the membership and non-membership degrees to the element $a_{\mathit{1}}$ in ${\mathrm{A}}_{\mathrm{t}}$.

Definition 5

[41] The level set of $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}{\mathrm{A}}_{\mathrm{t}}$ may be defined as:

$$\Lambda \left({\mathit{A}}_{\mathit{t}}\right)\mathit{=}\left\{\left(\mathit{\rho }\mathit{,}\mathit{\eta }\right)\mathit{:}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{=}\mathit{\rho }\mathit{,}{\mathit{\nu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{=}\mathit{\eta }\mathit{,}\mathit{for}\mathit{some}{\mathit{a}}_{\mathit{1}}\mathit{\in }\mathbb{U}\right\}\mathit{.}$$

${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$ and ${\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)$ stand for the degrees of membership and non-membership of the element $a_{\mathit{1}}$ in ${\mathrm{A}}_{\mathrm{t}}$, respectively.

Definition 6

[41] Let ${\mathrm{A}}_{\mathrm{t}}$ be a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of group $\mathbb{G}$. The subgroup ${ℭ}_{\left(\mathit{\rho }\mathit{,}\mathit{\eta }\right)}\left({\mathit{A}}_{\mathit{t}}\right)\rho \mathit{,}\eta \mathit{\in }\left[\mathit{0}\mathit{,}\mathit{1}\right]$ with $0\le \mathrm{\rho }+\mathrm{\eta }\le 1$ is called a level subgroup of ${\mathrm{A}}_{\mathrm{t}}$

The symbol ${ℭ}_{\left(\mathit{\rho }\mathit{,}\mathit{\eta }\right)}\left({\mathit{A}}_{\mathit{t}}\right)$ represent the $\left(\mathrm{\rho },\mathrm{\eta }\right)-$ cut set of $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ ${\mathrm{A}}_{\mathrm{t}}$.

Definition 7

[40] A $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}{\mathrm{A}}_{\mathrm{t}}$ is called t-intuitionistic fuzzy normal subgroup ($\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{N}\mathbb{S}\mathbb{G}$) of $\mathbb{G}$ if it satisfies the following conditions:

• 1)${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{b}}_{\mathit{1}}\right)\mathit{=}{\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{b}}_{\mathit{1}}{\mathit{a}}_{\mathit{1}}\right)$
• 2)${\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{b}}_{\mathit{1}}\right)\mathit{=}{\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{b}}_{\mathit{1}}{\mathit{a}}_{\mathit{1}}\right)\mathit{,}\mathit{\forall }{a}_{\mathit{1}}\mathit{,}{b}_{\mathit{1}}\mathit{\in }\mathbb{G}$.

The terms ${\mu }_{A_t}\left(a_1{b}_1\right)$ and ${\nu }_{A_t}\left(a_1{b}_1\right)$ respectively denote the membership and non-membership degrees of the element $″a_1{b}_1″$ in $A_t\text{.}$ Likewise, the terminologies ${\mu }_{A_t}\left(b_1{a}_1\right)$ and ${\nu }_{A_t}\left(b_1{a}_1\right)$ respectively characterize the membership and non-membership degrees of the element $″b_1{a}_1″$ in $A_t$.

Theorem 1

[41] A $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}{A}_t$ of a group $\mathbb{G}$ is $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ if and only if its each level set is a subgroup of group $\mathbb{G}$.

Theorem 2

[41] A $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}{A}_t$ of a group $\mathbb{G}$ is $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$ iff ${\mathfrak{C}}_{(\rho ,\eta )}\left(A_t\right)$ is a normal subgroup of $\mathbb{G}$ where $\rho$ and $\eta$ are positive real numbers such that their sum lies in the closed unit interval.

Theorem 3

[41] Consider a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}{A}_t$ of a group $\mathbb{G}$ and

$$\Lambda \left(A_t\right)=\left\{\left(\rho ,\eta \right):{\mu }_{A_t}\left(a_1\right)=\rho ,{\nu }_{A_t}\left(a_1\right)=\eta \mathrm{for}\mathrm{some}{a}_1\in \mathbb{U}\right\}\text{.}$$

Then the family of level subgroups ${{\mathfrak{I}}_A}_t=\left\{{\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(A_t\right):0\le i\le r\right\}$ constitutes all the level subgroups of $A_t$.

Definition 8

[41] Consider a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_t$ of a finite group $\mathbb{G}$ and $b_1\in \mathbb{G}$. Then the t-intuitionistic fuzzy order of the element $b_1\in A_t$ is denoted by $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(b_1\right)$ and is defined as:

$$t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(b_1\right)=\left|\mathbb{H}\left({\mathrm{b}}_1\right)\right|$$

Where $ℍ\left(b_{\mathit{1}}\right)\mathit{=}\left\{a_{\mathit{1}}\mathit{\in }\mathbb{G}\mathit{:}{\mu }_{A_t}\left(a_{\mathit{1}}\right)\mathit{\ge }{\mu }_{A_t}\left(b_{\mathit{1}}\right)\mathit{,}{\nu }_{A_t}\left(a_{\mathit{1}}\right)\mathit{\le }{\nu }_{A_t}\left(b_{\mathit{1}}\right)\right\}\mathit{.}$

The set $\mathbb{H}\left(b_1\right)$ is calculated by the elements ${{″b}_1}^{″}$ in $\mathbb{G}\text{.}$ The set $\mathbb{H}\left(b_1\right)$ comprises of elements of “$a_1$” that belong to the group $\mathbb{G}$ and have a membership degree ${\mu }_{A_t}$ in $A_t$ greater than or equal to that element “$b_1$” and a non-membership degree ${\nu }_{A_t}$ in $A_t$ less than that element “$b_1$”. Here, ${\mu }_{A_t}\left(a_1\right),{\mu }_{A_t}\left(b_1\right)$ and ${\nu }_{A_t}\left(a_1\right),{\nu }_{A_t}\left(b_1\right)$, respectively, represents the membership and non-membership degrees of elements $a_1$ and $b_1$ in $A_t$.

Theorem 4

[42] Any subgroup $\mathbb{H}$ of a group $\mathbb{G}$ can be visualized as a level subgroup of some $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$.

Theorem 5

[42] If $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)=n$ then $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1^m\right)=\frac{t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)}{(n,m)}$, for some integer m.

Theorem 6

[42] Let $\mathbb{G}$ be a finite group and $A_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$. Then $\left[\mathbb{G}:A_t\right]$ divides $O\left(\mathbb{G}\right)\text{.}$

3. Conjugacy class of t-intuitionistic fuzzy subgroup of a group

In this section, we define the notion of a t-intuitionistic fuzzy conjugate element of a finite group $\mathbb{G}$ and establish the classification of these newly defined elements. We also discuss the class equation of the $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of the finite group $\mathbb{G}$.

Definition 9

Let $A_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$ and $a_1,b_1\in \mathbb{G}\text{.}$ Then $a_{\mathit{1}}$ is t-intuitionistic fuzzy conjugate ($t-\mathbb{I}\mathbb{F}\mathbb{C}$) to $b_1$ (written as $a_1{\sim }_{A_t}{b}_1)$ if there exists a non-identity element $g_1\in \mathbb{G}$ such that ${\mu }_{A_t}\left(a_1\right)={\mu }_{A_t}\left(g_1^{-1}{b}_1{g}_1\right)$ and ${\nu }_{A_t}\left(a_1\right)={\nu }_{A_t}\left(g_1^{-1}{b}_1{g}_1\right)\text{.}$

Example 1

The dihedral group $D_3$ of degree 3 is defined as:

$$D_3=⟨r,s:r^3=s^2=1,sr=r^{2}s⟩\text{.}$$

The $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_t$ of $D_{\mathit{3}}$ for $t=0.70$ is defined as follows: ${\mu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.70\\ 0.50\\ 0.40\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1=s\\ if{z}_1\in \left\{r,r^2,rs,r^{2}s\right\}\end{array}\end{array}$and ${\nu }_{0.70}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.30\\ 0.40\\ 0.50\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1=s\\ if{z}_1\in \{r,r^2,rs,r^{2}s\}\end{array}\end{array}$.

In the light of definition (9), we obtain that $s{\sim }_{A_{0.70}}s\text{,}$ $r{\sim }_{A_{0.70}}{r}^2$ and $rs{\sim }_{A_{0.70}}{r}^{2}s$.

Theorem 7

If $a_1$ is $t-\mathbb{I}\mathbb{F}\mathbb{C}$ to $b_1$ then $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)=t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(b_1\right)\text{,}$ where $a_1,b_1\in \mathbb{G}\text{.}$

Proof. Since $a_1$ and $b_1$ are $t-\mathbb{I}\mathbb{F}\mathbb{C}$ then there exists $g_1\in \mathbb{G}$ such that

${\mu }_{A_t}\left(a_1\right)={\mu }_{A_t}\left(g_1^{-1}{b}_1{g}_1\right)$ and ${\nu }_{A_t}\left(a_1\right)={\nu }_{A_t}\left(g_1^{-1}{b}_1{g}_1\right)$.

Consider

$${\mu }_{A_t}\left(a_1^2\right)={\mu }_{A_t}\left(g_1^{-1}{b}_1{g}_1·g_1^{-1}{b}_1{g}_1\right)={\mu }_{A_t}\left(g_1^{-1}{b}_1^2{g}_1\right)$$

and

$${\nu }_{A_t}\left(a_1^2\right)={\nu }_{A_t}\left(g_1^{-1}{b}_1{g}_1·g_1^{-1}{b}_1{g}_1\right)={\nu }_{A_t}\left(g_1^{-1}{b}_1^2{g}_1\right)\text{.}$$

The application of the method of induction in the above relation gives us:

$${\mu }_{A_t}\left(a_1^n\right)={\mu }_{A_t}\left(g_1^{-1}{b}_1^n{g}_1\right),{\nu }_{A_t}\left(a_1^n\right)={\nu }_{A_t}\left(g_1^{-1}{b}_1^n{g}_1\right)\text{,}$$

and

$${\mu }_{A_t}\left(b_1^n\right)={\mu }_{A_t}\left(g_1^{-1}{a}_1^n{g}_1\right),{\nu }_{A_t}\left(b_1^n\right)={\nu }_{A_t}\left(g_1^{-1}{a}_1^n{g}_1\right)\text{.}$$

Consider $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)=m_1$ and $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(b_1\right)=m_2$. This implies that

$${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{b}}_{\mathit{1}}^{{\mathit{m}}_{\mathit{1}}}\right)\mathit{=}{\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{g}}_{\mathit{1}}^{\mathit{-}\mathit{1}}{\mathit{a}}_{\mathit{1}}^{{\mathit{m}}_{\mathit{1}}}{\mathit{g}}_{\mathit{1}}\right)\mathit{=}{\mu }_{{\mathit{A}}_{\mathit{t}}}\left(\mathit{e}\right)\mathit{.}$$

In view of Theorem (5), we have

$$m_1|m_2\text{.}$$ (1)

Moreover, ${\mu }_{A_t}\left(a_1^{m_2}\right)={\mu }_{A_t}\left(g_1^{-1}{b}_1^{m_2}{g}_1\right)={\mu }_{A_t}\left(e\right)$.

The application of Theorem (5) gives that

$$m_2|m_1\text{.}$$ (2)

From relations (1) and (2), we have the required result.

Definition 10

Let $A_t$ be a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$. The t-intuitionistic fuzzy conjugacy class ($\mathrm{t}-\mathbb{I}\mathbb{F}\mathrm{ℂ}\mathrm{ℂ}\mathcal{l})$ of an element $a_1\in \mathbb{G}$ is defined as:

$$\mathcal{C}{l}_{A_t}\left(a_1\right)=\left\{b_1\in \mathbb{G}:b_1{\sim }_{A_t}{a}_1\right\}\mathit{=}\left\{b_{\mathit{1}}\mathit{\in }\mathbb{G}\mathit{:}{\mu }_{A_t}\left(b_{\mathit{1}}\right)\mathit{=}{\mu }_{A_t}\left(g_{\mathit{1}}^{\mathit{-}\mathit{1}}{a}_{\mathit{1}}{g}_{\mathit{1}}\right)\mathrm{and}{\nu }_{A_t}\left(b_{\mathit{1}}\right)\mathit{=}{\nu }_{A_t}\left(g_{\mathit{1}}^{\mathit{-}\mathit{1}}{a}_{\mathit{1}}{g}_{\mathit{1}}\right)\mathit{,}{g}_{\mathit{1}}\mathit{\in }\mathbb{G}\right\}\mathit{.}$$

Example 2

The finite presentation of the dihedral group $D_4$ of degree 4 is defined as: $D_4=⟨r,s:r^4=s^2=1,sr=r^{3}s⟩$.

The $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_t$ of $D_4$ correspond to the value $t=0.70$ is given as: ${\mu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.70\\ 0.65\\ 0.60\\ 0.50\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1=s\\ if{z}_1\in \left\{r^2,r^{2}s\right\}\\ if{z}_1\in \left\{r,r^3,rs,r^{3}s\right\}\end{array}\end{array}$ and ${\nu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.30\\ 0.35\\ 0.40\\ 0.50\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1=s\\ if{z}_1\in \left\{r^2,r^{2}s\right\}\\ if{z}_1\in \left\{r,r^3,rs,r^{3}s\right\}\end{array}\end{array}$.

The application of definition (10) yields that $\mathcal{C}{l}_{A_{0.70}}\left(1\right)=\left\{1\right\}$, $\mathcal{C}{l}_{A_{0.70}}\left(r\right)=\left\{r,r^3\right\}$, $\mathcal{C}{l}_{A_{0.70}}\left(r^2\right)=\left\{r^2\right\}$, $\mathcal{C}{l}_{A_{0.70}}\left(s\right)=\left\{s\right\}$, $\mathcal{C}{l}_{A_{0.70}}\left(r^{2}s\right)=\left\{r^{2}s\right\}$ and $\mathcal{C}{l}_{A_{0.70}}\left(rs\right)=\left\{{rs,r}^{3}s\right\}$.

The following theorem describes that the $t-\mathbb{I}\mathbb{F}\mathbb{C}$ relation between elements of $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ in a group is an equivalence relation.

Proposition 1

The t-intuitionistic fuzzy conjugacy between elements $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}{A}_t$ of a group $\mathbb{G}$ is an equivalence relation.

Proof. Reflexivity: The application of the definition (9) to any element $a_1\in \mathbb{G}\text{,}$ we have ${\mu }_{A_t}\left(a_1\right)={\mu }_{A_t}\left(e^{-1}{a}_{1}e\right)$ and ${\nu }_{A_t}\left(a_1\right)={\nu }_{A_t}\left(e^{-1}{a}_{1}e\right)\text{,}$ where $e\in \mathbb{G}$.

It follows that $a_1{\sim }_{A_t}{a}_1$.

Symmetry: Consider $a_1{\sim }_{A_t}{b}_1$ so that there exists an element $g_1\in \mathbb{G}$ such that ${\mu }_{A_t}\left(a_1\right)={\mu }_{A_t}\left(g_1^{-1}{b}_1{g}_1\right)$ and ${\nu }_{A_t}\left(a_1\right)={\nu }_{A_t}\left(g_1^{-1}{b}_1{g}_1\right),a_1,b_1\in \mathbb{G}$.

This implies that ${\mu }_{A_t}\left(g_1{a}_1{g}_1^{-1}\right)={\mu }_{A_t}\left(g_1{g}_1^{-1}{b}_1{g}_1{g}_1^{-1}\right)$ and ${\nu }_{A_t}\left(g_1{a}_1{g}_1^{-1}\right)={\nu }_{A_t}\left(g_1{g}_1^{-1}{b}_1{g}_1{g}_1^{-1}\right)$.

This further implies that ${\mu }_{A_t}\left(g_1^{-1}{a}_1{g}_1\right)={\mu }_{A_{\mathrm{t}}}\left(b_1\right)$ and ${\nu }_{A_t}\left(g_1^{-1}{a}_1{g}_1\right)={\nu }_{A_t}\left(b_1\right)$

This shows that $b_1{\sim }_{A_t}{a}_1$.

Transitivity: For any $a_1,b_1,c_1\in \mathbb{G}$. Consider $a_1{\sim }_{A_t}{b}_1$ and $b_1{\sim }_{A_t}{c}_1$ there exist two elements $g_1,g_2\in \mathbb{G}$ such that

$${\mu }_{A_t}\left(a_1\right)={\mu }_{A_t}\left(g_1^{-1}{b}_1{g}_1\right),{\nu }_{A_t}\left(a_1\right)={\nu }_{A_t}\left(g_1^{-1}{b}_1{g}_1\right)$$

and ${\mu }_{A_t}\left(b_1\right)={\mu }_{A_t}\left(g_2^{-1}{c}_1{g}_2\right)\text{,}$ ${\nu }_{A_t}\left(b_1\right)={\nu }_{A_t}\left(g_2^{-1}{c}_1{g}_2\right)$ where $a_1,b_1,c_1\in \mathbb{G}$.

This implies that ${\mu }_{A_t}\left(a_1\right)={\mu }_{A_t}\left(g_1^{-1}{g_2^{-1}c}_1{g}_2{g}_1\right)$ and ${\nu }_{A_t}\left(a_1\right)={\nu }_{A_t}\left(g_1^{-1}{g_2^{-1}c}_1{g}_2{g}_1\right)$

This shows that ${\mu }_{A_t}\left(a_1\right)={\mu }_{A_t}\left({\left(g_2{g}_1\right)}^{-1}{c}_1\left(g_2{g}_1\right)\right)$ and ${\nu }_{A_t}\left(a_1\right)={\nu }_{A_t}\left({\left(g_2{g}_1\right)}^{-1}{c}_1\left(g_2{g}_1\right)\right)$

This means that $a_1{\sim }_{A_t}{c}_1\text{.}$ Consequently, this proves the equivalence properties of the t-intuitionistic fuzzy conjugacy relation.

Remark 1

• 1)In view of the above theorem, the group $\mathbb{G}$ is partitioned into equivalence classes, such classes are called $\mathrm{t}-\mathbb{I}\mathbb{F}\mathrm{ℂ}\mathrm{ℂ}\mathcal{l}$ of the group $\mathbb{G}$. It is important to note that one can obtain a different partition of $\mathbb{G}$ corresponding to different $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ defined on it. Whereas there is only one partition of group $\mathbb{G}$ in classical group theory. The significance of this approach of getting many partitions of a group $\mathbb{G}$ is to obtain an economic solution of a decision-making problem under consideration.
• 2)The $\mathrm{t}-\mathbb{I}\mathbb{F}\mathrm{ℂ}\mathrm{ℂ}\mathcal{l}$ of an element of a finite abelian group $\mathbb{G}$ is always a singleton set.

The following result indicates the conditions under which the $\mathrm{t}-\mathbb{I}\mathbb{F}\mathrm{ℂ}\mathrm{ℂ}\mathcal{l}$ of two elements are equal.

Theorem 8

The $\mathrm{t}-\mathbb{I}\mathbb{F}\mathrm{ℂ}\mathrm{ℂ}\mathcal{l}$ of the elements $a_{\mathit{1}}$ and $b_{\mathit{1}}$ are equal if and only if $a_{\mathit{1}}$ and $b_{\mathit{1}}$ are $\mathrm{t}-\mathbb{I}\mathbb{F}\mathrm{ℂ}$ to each other.

Proof. Suppose that $a_1$ and $b_1$ are $t-\mathbb{I}\mathbb{F}\mathbb{C}$ to each other. Consider $g_1\in \mathcal{C}{l}_{A_t}\left(a_1\right)\text{,}$ then by using definition (10), we have $g_1{\sim }_{A_t}{a}_1\text{.}$ Since $g_1{\sim }_{A_t}{a}_1$ and $a_1{\sim }_{A_t}{b}_1$. By the transitive property, we have $g_1{\sim }_{{\mathrm{A}}_t}{b}_1\text{.}$ Thus $g_1\in \mathcal{C}{l}_{A_t}\left(b_1\right)\text{.}$ This shows that $\mathcal{C}{l}_{A_t}\left(a_1\right)\subseteq \mathcal{C}{l}_{A_t}\left(b_1\right)\text{.}$ In the same way, we obtain $\mathcal{C}{l}_{A_t}\left(b_1\right)\subseteq \mathcal{C}{l}_{A_t}\left(a_1\right)\text{.}$ Consequently, $\mathcal{C}{l}_{A_t}\left(a_1\right)=\mathcal{C}{l}_{A_t}\left(b_1\right)$.

Conversely, let $\mathcal{C}{l}_{A_t}\left(a_1\right)=\mathcal{C}{l}_{A_t}\left(b_1\right)\text{.}$ This implies that $g_1{\sim }_{A_t}{a}_1$ and $g_1{\sim }_{A_t}{b}_1,g_1\in \mathbb{G}$. Consequently, $a_1{\sim }_{A_t}{b}_1$.

Definition 11

If $A_{\mathit{t}}$ is a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$, the centralizer $\mathrm{Ć}\left(A_t\right)$ of $A_{\mathit{t}}$ in $\mathbb{G}$ is defined as: $Ć\left(A_t\right)\mathit{=}\left\{a_{\mathit{1}}\mathit{\in }\mathbb{G}\mathit{:}{\mu }_{A_t}\left(a_{\mathit{1}}{g}_{\mathit{1}}\right)\mathit{=}{\mu }_{A_t}\left(g_{\mathit{1}}{a}_{\mathit{1}}\right)\mathit{and}{\nu }_{A_t}\left(a_{\mathit{1}}{g}_{\mathit{1}}\right)\mathit{=}{\nu }_{A_t}\left(g_{\mathit{1}}{a}_{\mathit{1}}\right)\mathit{,}\mathit{\forall }{g}_{\mathit{1}}\mathit{\in }\mathbb{G}\right\}$

Lemma 1

Let $A_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ and $\mathbb{T}\mathit{=}\left\{{\mathit{a}}_{\mathit{1}}\mathit{\in }\mathbb{G}\mathit{:}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{g}}_{\mathit{1}}{\mathit{a}}_{\mathit{1}}^{\mathit{-}\mathit{1}}{\mathit{g}}_{\mathit{1}}^{\mathit{-}\mathit{1}}\right)\mathit{=}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left(\mathit{e}\right)\mathit{and}{\mathit{\nu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{g}}_{\mathit{1}}{\mathit{a}}_{\mathit{1}}^{\mathit{-}\mathit{1}}{\mathit{g}}_{\mathit{1}}^{\mathit{-}\mathit{1}}\right)\mathit{=}{\mathit{\nu }}_{{\mathit{A}}_{\mathit{t}}}\left(\mathit{e}\right)\mathit{,}\mathit{\forall }{\mathit{g}}_{\mathit{1}}\mathit{\in }\mathbb{G}\right\}$ then $\mathbb{T}=\mathbb{Ć}\left(A_t\right)$.

Proof. Let $a_1\in \mathbb{T}$ then for all $g_1\in \mathbb{G}\text{,}$ we get

$${\mu }_{A_t}\left(a_1{g}_1{\left(g_1{a}_1\right)}^{-1}\right)={\mu }_{A_t}\left(a_1{g}_1{g}_1^{-1}{a}_1^{-1}\right)={\mu }_{A_t}\left(e\right)$$

and

$${\nu }_{A_t}\left(a_1{g}_1{\left(g_1{a}_1\right)}^{-1}\right)={\nu }_{A_t}\left(a_1{g}_1{g}_1^{-1}{a}_1^{-1}\right)={\nu }_{A_t}\left(e\right)\text{.}$$

This implies that ${\mu }_{A_t}\left(a_1{g}_1\right)={\mu }_{A_t}\left(g_1{a}_1\right)$ and${\nu }_{A_t}\left(g_1{a}_1\right)={\nu }_{A_t}\left(a_1{g}_1\right),\forall g_1\in \mathbb{G}\text{.}$ Thus $\mathbb{T}\subseteq \mathbb{Ć}\left(A_t\right)$.

Furthermore, if $a_1\in \mathrm{Ć}\left(A_t\right)$ then ${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{g}}_{\mathit{1}}\right)\mathit{=}{\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{g}}_{\mathit{1}}{\mathit{a}}_{\mathit{1}}\right)\mathrm{and}{\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{g}}_{\mathit{1}}{\mathit{a}}_{\mathit{1}}\right)\mathit{=}{\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{g}}_{\mathit{1}}\right)$.

This implies that ${\mu }_{A_t}\left(a_1{g}_1{a}_1^{-1}{g}_1^{-1}\right)={\mu }_{A_t}\left(e\right)$ and ${\nu }_{A_t}\left(a_1{g}_1{a}_1^{-1}{g}_1^{-1}\right)={\nu }_{A_t}\left(e\right),\forall a_1,g_1\in \mathbb{G}$.

So, $a_1\in \mathbb{T}\text{.}$ Thus $\mathbb{T}\supseteq \mathbb{Ć}\left(A_t\right)\text{.}$ Hence $\mathbb{T}=\mathbb{Ć}\left(A_t\right)$.

Definition 12

If $A_t$ is a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$, the centralizer of an element of $A_t$ in $\mathbb{G}$ (written as ${\mathrm{Ĉ}}_{A_t}\left(a_1\right))$ is defined as: ${\mathrm{Ĉ}}_{A_t}\left(a_1\right)=\left\{g_1\in \mathbb{G}:{\mu }_{A_t}\left(a_1{g}_1{a}_1^{-1}{g}_1^{-1}\right)={\mu }_{A_t}\left(e\right)\mathrm{and}{\nu }_{A_t}\left(a_1{g}_1{a}_1^{-1}{g}_1^{-1}\right)={\nu }_{A_t}\left(e\right)\right\}$.

Example 3

Consider $0.80-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}{A}_{0.80}$ of dihedral group $D_3$ as follows: ${\mu }_{A_{\mathrm{0,80}}}\left(z_1\right)=\{\begin{array}{cc}0.80& if{z}_1\in \left\{{1,r,r}^2\right\}\\ 0.50& if{z}_1\in \left\{s,r^{2}s,rs\right\}\end{array}$ and ${\nu }_{A_{0.20}}\left(z_1\right)=\{\begin{array}{cc}0.20& if{z}_1\in \left\{{1,r,r}^2\right\}\\ 0.45& if{z}_1\in \left\{s,r^{2}s,rs\right\}\end{array}$.

In the light of definitions (11) and (12), we acquire:

$$\mathrm{Ć}\left(A_{0.80}\right)=D_3\text{.}$$

$${\mathrm{Ĉ}}_{A_{0.80}}\left(1\right)=D_3\text{,}$$

$${\mathrm{Ĉ}}_{A_{0.80}}\left(r\right)={\mathrm{Ĉ}}_{A_{0.80}}\left(r^2\right)=\left\{1,r,r^2\right\}$$

and

$${\mathrm{Ĉ}}_{A_{0.80}}\left(s\right)={\mathrm{Ĉ}}_{A_{0.80}}\left(rs\right)={\mathrm{Ĉ}}_{A_{0.80}}\left(r^{2}s\right)=D_3$$

Theorem 9

The t-intuitionistic fuzzy centralizer of $A_t$ is a subgroup of $\mathbb{G}$.

Proof. For any two elements $g_1,g_2\in \mathrm{Ć}\left(A_t\right)$, we have

$$A_t\left(g_1{a}_1{g}_1^{-1}{a}_1^{-1}\right)=A_t\left(e\right),\forall a_1\in \mathbb{G}$$ (3)

$$A_t\left(g_2{b}_1{g}_2^{-1}{b}_1^{-1}\right)=A_t\left(e\right),\forall b_1\in \mathbb{G}$$ (4)

Consider

$$A_t\left(\left(g_1{g}_2\right)c_1{\left(g_1{g}_2\right)}^{-1}{c}_1^{-1}\right)=A_t\left(g_1{g}_2{c}_1{g}_2^{-1}{g}_1^{-1}{c}_1^{-1}\right)$$

$$=A_t\left(g_1{g}_2{c}_1{g}_2^{-1}{c}_1^{-1}{c}_1{g}_1^{-1}{c}_1^{-1}\right)$$

$$=A_t\left(g_1{c}_1{g}_1^{-1}{c}_1^{-1}\right)$$

$$=A_t\left(e\right)\text{.}$$

This shows that $g_1{g}_2\in \mathrm{Ć}\left(A_t\right)$.

Moreover, consider $A_t\left(g_1^{-1}{a}_1{g}_1{a}_1^{-1}\right)$.

Substituting $a_1=g_1{d}_1$ in the above relation, we get

$$A_t\left(g_1^{-1}{a}_1{g}_1{a}_1^{-1}\right)=A_t\left(g_1^{-1}{g}_1{d}_1{g}_1{d}_1^{-1}{g}_1^{-1}\right)$$

$$=A_t\left(d_1{g}_1{d}_1^{-1}{g}_1^{-1}\right)$$

$$=A_t\left(e\right)$$

Thus, $g_1^{-1}\in \mathrm{Ć}\left(A_t\right)$.

Consequently, $\mathrm{Ć}\left(A_t\right)$ is a subgroup of $\mathbb{G}$.

Lemma 2

Let $A_{t}beat-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}\text{.}$ Then $a_1\in \mathrm{Ć}\left(A_t\right)$ if

$${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{g}}_{\mathit{1}}{\mathit{g}}_{\mathit{2}}\mathit{\dots }{\mathit{g}}_{\mathit{s}}\right)\mathit{=}{\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{g}}_{\mathit{1}}{\mathit{a}}_{\mathit{1}}{\mathit{g}}_{\mathit{2}}\mathit{\dots }{\mathit{g}}_{\mathit{s}}\right)\mathit{=}\mathit{\dots }{\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{g}}_{\mathit{1}}{\mathit{g}}_{\mathit{2}}\mathit{\dots }{\mathit{g}}_{\mathit{s}}{\mathit{a}}_{\mathit{1}}\right)$$

and

$${\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{g}}_{\mathit{1}}{\mathit{g}}_{\mathit{2}}\mathit{\dots }{\mathit{g}}_{\mathit{s}}\right)\mathit{=}{\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{g}}_{\mathit{1}}{\mathit{a}}_{\mathit{1}}{\mathit{g}}_{\mathit{2}}\mathit{\dots }{\mathit{g}}_{\mathit{s}}\right)\mathit{=}\mathit{\dots }{\nu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{g}}_{\mathit{1}}{\mathit{g}}_{\mathit{2}}\mathit{\dots }{\mathit{g}}_{\mathit{s}}{\mathit{a}}_{\mathit{1}}\right)\mathit{,}\mathit{\forall }{g}_{\mathit{1}}\mathit{,}{g}_{\mathit{2}}\mathit{,}\mathit{\dots }{g}_{\mathit{s}}\mathit{\in }\mathbb{G}\mathit{.}$$

Proof. We prove by induction to $s$. Suppose $a_1\in \mathbb{C}\left(A_t\right)\text{.}$ Then

$${\mu }_{A_t}\left({a_{1}g}_1{g}_2\right)={\mu }_{A_t}\left(g_1{g}_2{a}_1\right),\forall g_1,g_2\in \mathbb{G}\text{.}$$

Assume that ${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{g}}_{\mathit{1}}{\mathit{g}}_{\mathit{2}}\mathit{\dots }{\mathit{g}}_{\mathit{s}}\right)\mathit{=}{\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{g}}_{\mathit{1}}{\mathit{a}}_{\mathit{1}}{\mathit{g}}_{\mathit{2}}\mathit{\dots }{\mathit{g}}_{\mathit{s}}\right)\mathit{=}\mathit{\dots }{\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{g}}_{\mathit{1}}{\mathit{g}}_{\mathit{2}}\mathit{\dots }{\mathit{g}}_{\mathit{s}}{\mathit{a}}_{\mathit{1}}\right)$.

Then

$${\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}{\mathit{g}}_{\mathit{1}}{\mathit{g}}_{\mathit{2}}\mathit{\dots }\left({\mathit{g}}_{\mathit{s}}{\mathit{g}}_{\mathit{s}\mathit{+}\mathit{1}}\right)\right)\mathit{=}{\mu }_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{g}}_{\mathit{1}}{\mathit{a}}_{\mathit{1}}{\mathit{g}}_{\mathit{2}}\mathit{\dots }\left({\mathit{g}}_{\mathit{s}}{\mathit{g}}_{\mathit{s}\mathit{+}\mathit{1}}\right)\right)$$

$$⋮$$

$$={\mu }_{A_t}\left(g_1{g}_2\dots a_1\left(g_s{g}_{s+1}\right)\right)$$

$$={\mu }_{A_t}\left(g_1{g}_2\dots \left(g_s{g}_{s+1}\right)a_1\right)$$

and

$${\mu }_{A_t}\left(a_1\left(g_1{g}_2\right)\dots g_s{g}_{s+1}\right)={\mu }_{A_t}\left(\left(g_1{g}_2\right)a_1\dots g_s{g}_{s+1}\right)$$

$$⋮$$

$$={\mu }_{A_t}\left(\left(g_1{g}_2\right)\dots g_s{a}_1{g}_{s+1}\right)$$

$$={\mu }_{A_t}\left(\left(g_1{g}_2\right)a_1\dots .g_s{g}_{s+1}{a}_1\right),\forall g_1,g_2,\dots g_s\in \mathbb{G}$$

Now, we obtain the required result by applying the similar arguments for non-membership function ${\nu }_{{\mathit{A}}_{\mathit{t}}}$.

This completes the proof.

Corollary 1

Let $A_{\mathit{t}}$ be a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$, then

• 1)If $A_{\mathit{t}}$ is a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathrm{\mathbb{N}}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ then $\mathrm{Ć}\left(A_t\right)⊴\mathbb{G}$.
• 2)If $A_{\mathit{t}}$ is a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of an abelian group $\mathbb{G}$ then $\mathrm{Ć}\left(A_t\right)=\mathbb{G}$.

Theorem 10

Let $A_t$ be a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathrm{\mathbb{N}}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$ then

$$\left|\mathcal{C}{l}_{A_t}\left(a_1\right)\right|=\frac{\left|\mathbb{G}\right|}{\left|{\mathrm{Ĉ}}_{A_t}\left(a_1\right)\right|},a_1\in \mathbb{G}\text{.}$$

Proof. Consider the collection $\mathbb{H}$ of all disjoint cosets of ${\mathrm{Ĉ}}_{A_t}\left(a_1\right)$ in $\mathbb{G}$ is given by:

$$\mathrm{ℍ}\mathit{=}\left\{{\mathit{g}}_{\mathit{1}}{\mathit{Ĉ}}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{,}{\mathit{g}}_{\mathit{2}}{\mathit{Ĉ}}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{,}{\mathit{g}}_{\mathit{3}}{\mathit{Ĉ}}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{,}\mathit{\dots }\mathit{,}{\mathit{g}}_{\mathit{s}}{\mathit{Ĉ}}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{:}{\mathit{g}}_{\mathit{s}}\mathit{\in }\mathbb{G}\right\}$$

Moreover, the left decomposition of $\mathbb{G}$ as a disjoint union of cosets of ${\mathrm{Ĉ}}_{A_t}\left(a_1\right)$ in $\mathbb{G}$ is given by: $\mathbb{G}=\underset{i=1}{\overset{s}{\cup }}{g}_i{\mathrm{Ĉ}}_{A_t}\left(a_1\right)$, $g_i\in \mathbb{G}$.

It follows that $O\left(\mathbb{G}\right)=s·O\left({\mathrm{Ĉ}}_{A_t}\left(a_1\right)\right)\text{.}$ Define a mapping $\phi :\mathbb{H}\to \mathcal{C}{l}_{A_t}\left(a_1\right)$ by

$$\phi \left(g_1{\mathrm{Ĉ}}_{A_t}\left(a_1\right)\right)=A_t\left(b_1^{-1}{a}_1{b}_1\right)\text{.}$$

As $\phi$ is well defined because for any two elements $b_1,c_1\in \mathbb{G}\text{,}$ we have $b_1{\mathrm{Ĉ}}_{A_t}\left(a_1\right)=c_1{\mathrm{Ĉ}}_{A_t}\left(a_1\right)$ this implies that $b_1^{-1}{c}_1\in {\mathrm{Ĉ}}_{A_t}\left(a_1\right)$.

By using definition (12), we have

$$A_t\left(a_1\left(b_1^{-1}{c}_1\right)a_1^{-1}{\left(b_1^{-1}{c}_1\right)}^{-1}\right)=A_t\left(e\right)$$

This shows that $A_t\left(b_1^{-1}{a}_1{b}_1\right)=A_t\left(c_1^{-1}{a}_1{c}_1\right)$. Consequently, $\phi \left(b_1{C}_{A_t}\left(a_1\right)\right)=\phi \left(c_1{C}_{A_t}\left(a_1\right)\right)$.

As $\phi$ is injective as for any two elements $b_1,c_1\in \mathbb{G}$, we have $\phi \left(b_1{C}_{A_t}\left(a_1\right)\right)=\phi \left(c_1{C}_{A_t}\left(a_1\right)\right)$.

This implies that $A_t\left(b_1^{-1}{a}_1{b}_1\right)=A_t\left(c_1^{-1}{a}_1{c}_1\right)\text{.}$ This further implies that

$$A_t\left(a_1\left(b_1^{-1}{c}_1\right)a_1^{-1}{\left(b_1^{-1}{c}_1\right)}^{-1}\right)=A_t\left(e\right)$$

This means that $b_1^{-1}{c}_1\in {\mathrm{Ĉ}}_{A_t}\left(a_1\right)\text{.}$ Thus $b_1{\mathrm{Ĉ}}_{A_t}\left(a_1\right)=c_1{\mathrm{Ĉ}}_{A_t}\left(a_1\right)$.

Moreover, one can easily prove that $\phi$ is surjective.

Thus, there is a one-to-one correspondence between $\mathbb{H}$ and $\mathcal{C}{l}_{A_t}\left(a_1\right)$.Hence $O\left(\mathbb{H}\right)=O\left(\mathcal{C}{l}_{A_t}\left(a_1\right)\right)\text{.}$ Consequently, $\left|\mathcal{C}{l}_{A_t}\left(a_1\right)\right|=\frac{\left|\mathbb{G}\right|}{\left|{\mathrm{Ĉ}}_{A_t}\left(a_1\right)\right|}$.

Corollary 2

The cardinality of the $\mathrm{t}-\mathbb{I}\mathbb{F}\mathrm{ℂ}\mathrm{ℂ}\mathcal{l}$ of an element of the finite group $\mathbb{G}$ divides the order of $\mathbb{G}$.

Proof. Since ${\mathrm{Ĉ}}_{A_t}\left(a_1\right)$ is a subgroup of $\mathbb{G}$, its order and index divide the order of $\mathbb{G}$ by Lagrange’s Theorem. However, the index of ${\mathrm{Ĉ}}_{A_t}\left(a_1\right)$ is equal to the number of elements in $\mathcal{C}{l}_{A_t}\left(a_1\right)$ which therefore divides the order of $\mathbb{G}$.

Definition 13

If $A_t$ is a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$, the t-intuitionistic fuzzy normalizer $\mathcal{N}\left({\mathit{A}}_{\mathit{t}}\right)$ of $A_t$ in $\mathbb{G}$ is defined as:

$$\mathcal{N}\left(A_t\right)=\left\{g_1\in \mathbb{G}:{\mu }_{A_t}\left(a_1\right)={\mu }_{A_t}\left(g_1^{-1}{a}_1{g}_1\right)\mathrm{and}{\nu }_{A_t}\left(a_1\right)={\nu }_{A_t}\left(g_1^{-1}{a}_1{g}_1\right),\forall a_1\in \mathbb{G}\right\}\text{.}$$

Example 4

Consider the $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_t$ of dihedral group $D_3$ for the value $t=0.70$ as follows: ${\mu }_{A_{\mathrm{0,70}}}\left(z_1\right)=\{\begin{array}{cc}0.70& if{z}_1\in \left\{{1,r,r}^2\right\}\\ 0.40& if{z}_1\in \left\{s,r^{2}s,rs\right\}\end{array}$ and ${\nu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}0.30& if{z}_1\in \left\{{1,r,r}^2\right\}\\ 0.25& if{z}_1\in \left\{s,r^{2}s,rs\right\}\end{array}$.

With reference to definition (13), we obtain $\mathcal{N}\left(A_{0.70}\right)=\{1,r,r^2\}$

Corollary 3

Let $A_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ then:

• 1)If $A_t$ is a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ then $\mathcal{N}\left(A_t\right)\le \mathbb{G}$.
• 2)If $A_t$ is a $t-\mathbb{I}\mathbb{F}\mathbb{N}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ then $\mathcal{N}\left(A_t\right)=\mathbb{G}$.
• 3)$\mathrm{Ć}\left(A_t\right)$ is a normal subgroup of $\mathcal{N}\left(A_t\right)$.

Definition 14

The class equation of $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_t$ of the finite group $\mathbb{G}$ is defined as:

$$\left|\mathbb{G}\right|=\sum _{a_1\in \mathbb{G}}\left|\mathcal{C}{l}_{A_t}\left(a_1\right)\right|$$

where $\mathcal{C}{l}_{A_t}\left(a_1\right)$ is the $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{C}\mathcal{l}$ of an element of $A_t$ of $\mathbb{G}$.

Definition 15

The Class equation of $t-\mathbb{I}\mathbb{F}\mathbb{N}\mathbb{S}\mathbb{G}$ $A_t$ of the finite group $\mathbb{G}$ is defined as:

$$\left|\mathbb{G}\right|=\sum _{a_1\in \mathbb{G}}\left|\left[\mathbb{G}:{\mathrm{Ĉ}}_{A_t}\left(a_1\right)\right]\right|$$

The following examples demonstrate the aforementioned algebraic facts.

Example 5

The class equation of $0.70-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}{A}_{0.70}$ of $D_4$ is $\left|D_4\right|=1+2+1+1+1+2=8$ (See example 2).

Example 6

Consider the $0.70-\mathbb{I}\mathbb{F}\mathbb{N}\mathbb{S}\mathbb{G}{A}_{0.70}$ of dihedral group $D_{10}$ as follows:

$${\mu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.70\\ 0.65\\ 0.45\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1\in \left\{r,r^2,r^3,r^4\right\}\\ if{z}_1\in \left\{s,rs,r^{2}s,r^{3}s,r^{4}s\right\}\end{array}\end{array}$$

and

$${\nu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.30\\ 0.35\\ 0.50\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1\in \left\{r,r^2,r^3,r^4\right\}\\ if{z}_1\in \left\{s,rs,r^{2}s,r^{3}s,r^{4}s\right\}\end{array}\end{array}$$

In accordance with the definition (12), we have

$${\mathrm{Ĉ}}_{A_{0.70}}\left(1\right)=D_{10},{\mathrm{Ĉ}}_{A_{0.70}}\left(r\right)=\left\{1,r,r^2,r^3,r^4\right\},{\mathrm{Ĉ}}_{A_{0.70}}\left(s\right)=\left\{1,s\right\},{\mathrm{Ĉ}}_{A_{0.70}}\left(rs\right)=\left\{1,rs\right\}\text{,}$$

${\mathrm{Ĉ}}_{A_{0.70}}\left(r^{2}s\right)=\left\{1,r^{2}s\right\},{\mathrm{Ĉ}}_{A_{0.70}}\left(r^{3}s\right)=\left\{1,r^{3}s\right\}\text{,}$ and ${\mathrm{Ĉ}}_{A_{0.70}}\left(r^{4}s\right)=\left\{1,r^{4}s\right\}$.

The class equation of $0.70-\mathbb{I}\mathbb{F}\mathbb{N}\mathbb{S}\mathbb{G}{A}_{0.70}$ of $D_{10}$ is

$$\left|\mathbb{G}\right|=5+2+2+1=10\text{.}$$

4. Algebraic characteristics of t-intuitionistic fuzzy $\mathbf{p}-$ subgroup

In this section, we initiate the idea of the t-intuitionistic fuzzy $p-$ subgroup of $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ and establish the various algebraic properties of this phenomenon. Furthermore, we prove the t-intuitionistic fuzzy version of the Cauchy Theorem.

Definition 16

A $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_t$ of a group, $\mathbb{G}$ is a t-intuitionistic fuzzy p-subgroup if $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)$ is a power of prime $p,\forall a_1\in \mathbb{G}\text{.}$

Theorem 11

[42]. Let $A_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{N}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$, then the set $G_{\mathit{t}}\mathit{=}\left\{{\mathit{a}}_{\mathit{1}}\mathit{\in }\mathbb{G}\mathit{:}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{=}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left(\mathit{e}\right)\mathit{and}{\mathit{\nu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{=}{\mathit{\nu }}_{{\mathit{A}}_{\mathit{t}}}\left(\mathit{e}\right)\right\}$ is a normal subgroup of $\mathbb{G}$.

In the following result, we establish a condition under which a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ is a t-intuitionistic fuzzy p- subgroup.

Theorem 12

Consider a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_{\mathit{t}}$ of a finite group $\mathbb{G}$ such that

$$G_{\mathit{t}}\mathit{=}\left\{{\mathit{a}}_{\mathit{1}}\mathit{\in }\mathbb{G}\mathit{:}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{=}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left(\mathit{e}\right)\mathit{and}{\mathit{\nu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{=}{\mathit{\nu }}_{{\mathit{A}}_{\mathit{t}}}\left(\mathit{e}\right)\right\}$$

is a normal subgroup of $\mathbb{G}$ then $A_t$ is a t-intuitionistic fuzzy p- subgroup if and only if $\mathbb{G}/{\mathrm{G}}_t$ is a $p-$ group.

Proof. In view of definition (16) and for any element $a_1$ in $\mathbb{G}$, we have $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)=p^s$ for some non-negative integer $s$ and so $a_1^{p^s}\in {\mathrm{G}}_t\text{.}$ Thus $\mathbb{G}/{\mathrm{G}}_t$ is a $p-$ group. Conversely, let $\mathbb{G}/{\mathrm{G}}_t$ is a $p-$ group. If $a_1\in \mathbb{G}$ then $a_1^{p^s}\in {\mathrm{G}}_t$ for some nonnegative integer $s$ and so $A_t\left(a_1^{p^s}\right)=A_t\left(e\right)\text{.}$ Consequently, $A_t$ is a t-intuitionistic fuzzy p-subgroup.

Theorem 13

If $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)=m_1{n}_1$ for some coprime positive integer $m_1$ and $n_1$, then there exist $b_1,b_2\in \mathbb{G}$ such that $a_1=b_1{b}_2=b_2{b}_1,t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(b_1\right)=m_1$ and $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(b_2\right)=n_1\text{.}$

Proof. Assume that $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)=m_1{n}_1$. Since $(m_1,n_1)=1$ then there exist integers $p_1$ and $q_1$ such that $m_1{p}_1+n_1{q}_1=1$. Here $\left(m_1,q_1\right)=\left(n_1,p_1\right)=1\text{.}$ Let $b_1=a_1^{n_1{p}_1}$ and $b_1=a_1^{m_1{q}_1}$ then $a_1=b_1{b}_2=b_2{b}_1$.

By using Theorem (5), we have

$$t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(b_1\right)=t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1^{n_1{p}_1}\right)=m_1$$

and

$$t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(b_2\right)=t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1^{m_1{q}_1}\right)=n_1\text{.}$$

Theorem 14

(t-Intuitionistic Fuzzification of Cauchy Theorem). Let $A_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$ and $t-\mathbb{I}\mathbb{F}\mathbb{O}\left(A_t\right)=p_1^r{m}_1\text{,}$ where $p$ is prime and $(p_1,m_1)=1$, then there is an element $a_1\in \mathbb{G}$ such that $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)=p_1^s$, for each nonnegative integer $s\le r\text{.}$

Proof. Since $t-\mathbb{I}\mathbb{F}\mathbb{O}\left(A_t\right)$ is the greatest common divisor of $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)$, where $a_1\in \mathbb{G}$ there is an element $a_1$ in $\mathbb{G}$ such that $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)=p_1^s\text{.}$ By using the induction method on $s$ and the Cauchy Theorem in classical group theory we have an element $a_1\in \mathbb{G}$ such that $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)=p_1^s$.

Corollary 4

If $A_t$ is a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of an abelian group $\mathbb{G}$ and $t-\mathbb{I}\mathbb{F}\mathbb{O}\left(A_t\right)=m_1{n}_1$ for some $m_1,n_1\in \mathbb{Z}\text{,}$ then there is an element $a_1$ in $\mathbb{G}$ such that $t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right)=m_1\text{.}$

Remark 2

Let $A_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ satisfying the following conditions for some prime $p_1$ then ${\mathbb{H}}_{p_1}=\{a_1\in \mathbb{G}:(t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1\right),p_1)=1\}$ and ${\mathbb{L}}_{p_1}=\left\{a_1\in \mathbb{G}:t-{\mathbb{I}\mathbb{F}\mathbb{O}}_{A_t}\left(a_1^{p_1}\right)\right\}$ are constitute a subgroup of $\mathbb{G}$.

Theorem 15

Let $A_t$ be any $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ such that the t-intuitionistic fuzzy index of $A_t$ is $p\text{,}$ where $p$ is the smallest prime $p$ divisor the order of $\mathbb{G}$ then $A_t$ is a $t-\mathbb{I}\mathbb{F}\mathbb{N}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$.

Proof. Define a subgroup $\mathbb{H}$ of index $p$ as follows:

$$\mathbb{H}=\left\{{\mathit{z}}_{\mathbf{1}}\in \mathbb{G}:{\mathit{A}}_{\mathit{t}}\left({\mathit{z}}_{\mathbf{1}}\right)={\mathit{A}}_{\mathit{t}}\left(\mathit{e}\right)\right\}\text{.}$$

Then the group $\mathbb{G}$ acts on the left cosets $\frac{\mathbb{G}}{\mathbb{H}}$ by left multiplication as:

$$\left\{{\mathit{a}}_{\mathit{i}}\mathbb{H}:\mathbf{1}\le \mathit{i}\le {\mathit{p}}_{\mathbf{1}}\right\}$$

Now, consider the permutation representation of $\mathbb{G}$ on the cosets of $\mathbb{H}$ given by the map

$$\mathit{\rho }:\mathbb{G}\to {\mathit{\rho }}_{\mathit{a}}$$

where

$${\rho }_a:a_i\mathbb{H}\to a{a}_i\mathbb{H},1\le i\le p_1\text{.}$$

As is well known, $\rho$ is a homomorphism of $\mathbb{G}$ into the symmetric group. Further, the kernel of the map $\rho$ is the core of $\mathbb{H}$. By the First Isomorphism Theorem of groups, the quotient group $\mathbb{G}/Core\left(\mathbb{H}\right)$ is isomorphic to the subgroup of the symmetric group, thus $O\left(\mathbb{G}/Core\left(\mathbb{H}\right)\right)$ divides $p_1!$ by Lagrange’s Theorem. Since $O\left(\mathbb{G}/\mathbb{H}\right)$ is $p\text{,}$ it follows that the $O\left(\mathbb{H}/Core\left(\mathbb{H}\right)\right)$ divides $(p_1-1)!\text{.}$ Now since the $O\left(\mathbb{H}\right)$ divides the $O\left(\mathbb{G}\right)\text{,}$ we obtain $\mathbb{H}=Core\left(\mathbb{H}\right)\text{;}$ otherwise, we get a contradiction to the fact that $p_1$ is the smallest prime divisor the $O\left(\mathbb{G}\right)\text{.}$ As $Core\left(\mathbb{H}\right)$ is always a normal subgroup, it follows that $\mathbb{H}$ is a normal subgroup of $\mathbb{G}$. Moreover, $\mathbb{G}/\mathbb{H}$ is abelian. It follows that $a_1{b}_1=b_1{a}_1{h}_1$ for some $h_1\in \mathbb{H}$ implies that

$${\mu }_{A_t}\left(a_1{b}_1\right)={\mu }_{A_t}\left(b_1{a}_1{h}_1\right)$$

In view of definition (2), we have

$$\ge min\left\{{\mu }_{A_t}\left(b_1{a}_1\right),{\mu }_{{\mathrm{A}}_t}\left(h_1\right)\right\}$$

$$\mathit{=}min\left\{{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{b}}_{\mathit{1}}{\mathit{a}}_{\mathit{1}}\right)\mathit{,}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left(\mathit{e}\right)\right\}$$

Consequently, ${\mu }_{A_t}\left(a_1{b}_1\right)\ge {\mu }_{A_t}\left(b_1{a}_1\right)$.

Similarly, ${\mu }_{A_t}\left(a_1{b}_1\right)\le {\mu }_{A_t}\left(b_1{a}_1\right)$.

Thus ${\mu }_{A_t}\left(a_1{b}_1\right)={\mu }_{A_t}\left(b_1{a}_1\right),\forall a_1,b_1\in \mathbb{G}$.

The same procedure is applied to get ${\nu }_{A_t}\left(a_1{b}_1\right)={\nu }_{A_t}\left(b_1{a}_1\right),\forall a_1,b_1\in \mathbb{G}$.

This concludes that $A_t$ is $t-\mathbb{I}\mathbb{F}\mathbb{N}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$.

Corollary 5

Let $A_t$ be any $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ such that the t-intuitionistic fuzzy index of $A_t$ is 2, then $A_t$ is $t-\mathbb{I}\mathbb{F}\mathbb{N}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$.

Definition 17

Let $A_t$ be any $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$ and

$$\mathrm{ℍ}=\left\{{\mathit{a}}_{\mathit{1}}\mathit{\in }\mathbb{G}\mathit{:}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{=}{\mathit{\mu }}_{{\mathit{A}}_{\mathit{t}}}\left(\mathit{e}\right)\mathit{and}{\mathit{\nu }}_{{\mathit{A}}_{\mathit{t}}}\left({\mathit{a}}_{\mathit{1}}\right)\mathit{=}{\mathit{\nu }}_{{\mathit{A}}_{\mathit{t}}}\left(\mathit{e}\right)\right\}$$

Then $A_t$ is t-intuitionistic fuzzy abelian if $\mathbb{H}$ is an abelian subgroup of $\mathbb{G}$.

Theorem 16

A $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_t$ is t-intuitionistic fuzzy abelian if $t-\mathbb{I}\mathbb{F}\mathbb{O}\left(A_t\right)=p_1^2\text{,}$ where $p_1$ a is prime.

Proof. The result is obtained by a simple application of the definition (16).

5. t-Intuitionistic fuzzy conjugate subgroup of t-intuitionistic fuzzy subgroups

In this section, we define the notions of equivalent $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ and $t$-intuitionistic fuzzy conjugate subgroup of $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$. We also establish the various algebraic aspects of this phenomenon. Moreover, we establish the relation under which a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ is a $t$-intuitionistic fuzzy conjugate subgroup.

Theorem 17

If $A_t$ is a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$ and $\left({\rho }_i,{\eta }_i\right),\left({\rho }_j,{\eta }_j\right)\in \Lambda \left(A_t\right)$ such that ${\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(A_t\right)={\mathfrak{C}}_{\left({\rho }_j,{\eta }_j\right)}\left(A_t\right)$, then $\left({\rho }_i,{\eta }_i\right)=\left({\rho }_j,{\eta }_j\right)$.

Proof. Let $a_i\in {\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(A_t\right)$ and $a_j\in {\mathfrak{C}}_{\left({\rho }_j,{\eta }_j\right)}\left(A_t\right)$. As $a_i\in {\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(A_t\right)$, we have

$$A_t\left(a_i\right)=\left({\rho }_i,{\eta }_i\right)\ge \left({\rho }_j,{\eta }_j\right)$$

Similarly, $\left({\rho }_j,{\eta }_j\right)\ge \left({\rho }_i,{\eta }_i\right)$. Consequently, $\left({\rho }_i,{\eta }_i\right)=\left({\rho }_j,{\eta }_j\right)$.

Theorem 18

Let $A_t$ and $B_t$ be two $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$ having an identical family of level subgroups. If

$$\Lambda \left(A_t\right)=\left\{\left({\rho }_0,{\eta }_0\right),\left({\rho }_1,{\eta }_1\right),\dots ,\left({\rho }_r,{\eta }_r\right)\right\}$$

and

$$\Lambda \left(B_t\right)=\left\{\left({\rho }_0^{\text{'}},{\eta }_0^{\text{'}}\right),\left({\rho }_1^{\text{'}},{\eta }_1^{\text{'}}\right),\dots ,\left({\rho }_s^{\text{'}},{\eta }_s^{\text{'}}\right)\right\}$$

where $1={\rho }_0\ge {\rho }_1\ge \dots \ge {\rho }_r,0={\eta }_0\le {\eta }_1\le \dots \le {\eta }_r$ and $1={\rho }_0^{\prime }\ge {\rho }_1^{\prime }\ge \dots \ge {\rho }_s^{\prime }\text{,}$ $0={\eta }_0^{\prime }\le {\eta }_1^{\prime }\le \dots \le {\eta }_s^{\prime }$ then

• 1)$r=s$
• 2)${\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(A_t\right)={\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(B_t\right),0\le i\le r$
• 3)For any element $a_1\in \mathbb{G}$ such that $A_t\left(a_1\right)=\left({\rho }_i,{\eta }_i\right)$ then $B_t\left(a_1\right)=\left({\rho }_i^{\prime },{\eta }_i^{\prime }\right),0\le i\le r$.

Proof. (1) By using Theorem (3), the only level subgroups of $A_t$ and $B_t$ have the same families

$${\mathfrak{I}}_{A_t}={\mathfrak{I}}_{B_t}\text{.}$$

As $A_t$ and $B_t$ have the same family of level subgroups, it follows that $r=s$.

(2) There are two chains of level subgroups, which can be identified using part (1) of the theorem (18) and the theorem (3)

$${\mathfrak{C}}_{\left({\rho }_0,{\eta }_0\right)}\left(A_t\right)\subseteq {\mathfrak{C}}_{\left({\rho }_1,{\eta }_1\right)}\left(A_t\right)\subseteq ...\subseteq {\mathfrak{C}}_{\left({\rho }_r,{\eta }_r\right)}\left(A_t\right)=\mathbb{G}$$

and

$${\mathfrak{C}}_{\left({\rho }_0^{\text{'}},{\eta }_0^{\text{'}}\right)}\left(B_t\right)\subseteq {\mathfrak{C}}_{\left({\rho }_1^{\text{'}},{\eta }_1^{\text{'}}\right)}\left(B_t\right)\subseteq ...\subseteq {\mathfrak{C}}_{\left({\rho }_s^{\text{'}},{\eta }_s^{\text{'}}\right)}\left(B_t\right)=\mathbb{G}$$

From this, it now follows that if: $\left({\rho }_i,{\eta }_i\right),\left({\rho }_j,{\eta }_j\right)\in \Lambda \left(A_t\right)$ such that $({\rho }_i,{\eta }_i)>({\rho }_j,{\eta }_j)\text{,}$ then

$${\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(A_t\right)\subseteq {\mathfrak{C}}_{\left({\rho }_j,{\eta }_j\right)}\left(A_t\right)$$ (5)

If $\left({\rho }_i^{\text{'}},{\eta }_i^{\text{'}}\right),\left({\rho }_j^{\text{'}},{\eta }_j^{\text{'}}\right)\in \Lambda \left(B_t\right)$ such that $\left({\rho }_i^{\prime },{\eta }_i^{\prime }\right)>\left({\rho }_j^{\prime },{\eta }_j^{\prime }\right)$.

then

$${\mathfrak{C}}_{({\rho }_i^{\prime },{\eta }_i^{\prime })}\left(B_t\right)\subseteq {\mathfrak{C}}_{({\rho }_j^{\prime },{\eta }_j^{\prime })}\left(B_t\right)$$ (6)

Since the two families of level subgroups are identical, it is clear that ${\mathfrak{C}}_{({\rho }_0,{\eta }_0)}\left(A_t\right)={\mathfrak{C}}_{({\rho }_0^{\prime },{\eta }_0^{\prime })}\left(B_t\right)$.

Now by hypothesis ${\mathfrak{C}}_{({\rho }_1,{\eta }_1)}\left(A_t\right)={\mathfrak{C}}_{({\rho }_j^{\prime },{\eta }_j^{\prime })}\left(B_t\right)$ for some $j>0\text{.}$ Suppose that ${\mathfrak{C}}_{({\rho }_1,{\eta }_1)}\left(A_t\right)={\mathfrak{C}}_{({\rho }_j^{\prime },{\eta }_j^{\prime })}\left(B_t\right)$ for some $j>1\text{.}$ Again, we have that ${\mathfrak{C}}_{({\rho }_i,{\eta }_i)}\left(A_t\right)={\mathfrak{C}}_{({\rho }_1^{\prime },{\eta }_1^{\prime })}\left(B_t\right)$ for some (${\rho }_i,{\eta }_i)<\left({\rho }_1,{\eta }_1\right)\text{.}$ It is clear that $({\rho }_i,{\eta }_i)\ne ({\rho }_1,{\eta }_1)$.

According to relation (5), we obtain

$${\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(A_t\right)<{\mathfrak{C}}_{\left({\rho }_j^{\prime },{\eta }_j^{\prime }\right)}\left(B_t\right)\text{.}$$ (7)

The application of relation (6) yields that

$${\mathfrak{C}}_{\left({\rho }_j^{\prime },{\eta }_j^{\prime }\right)}\left(B_t\right)<{\mathfrak{C}}_{({\rho }_i,{\eta }_i)}\left(A_t\right)\text{.}$$ (8)

Relations (7) and (8) allow us to derive that

$${\mathfrak{C}}_{({\rho }_1,{\eta }_1)}\left(A_t\right)={\mathfrak{C}}_{({\rho }_1^{\prime },{\eta }_1^{\prime })}\left(B_t\right)\text{.}$$

By using the induction method on $i$, we obtain

$${\mathfrak{C}}_{({\rho }_i,{\eta }_i)}\left(A_t\right)={\mathfrak{C}}_{({\rho }_i^{\prime },{\eta }_i^{\prime })}\left(B_t\right),0\le i\le r\text{.}$$

(3) Consider any non-zero element $a_1\in \mathbb{G}$ such that $A_t\left(a_1\right)=({\rho }_i,{\eta }_i)$ and $B_t\left(a_1\right)=({\rho }_j^{\prime },{\eta }_j^{\prime })$.

By using part (2) of Theorem (18), we have ${\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(A_t\right)={\mathfrak{C}}_{\left({\rho }_i^{\prime },{\eta }_i^{\prime }\right)}\left(B_t\right)\text{.}$ So, $a_1\in {\mathfrak{C}}_{({\rho }_i^{\prime },{\eta }_i^{\prime })}\left(B_t\right)$ implies that $B_t\left(a_1\right)=({\rho }_j^{\prime },{\eta }_j^{\prime })$ such that $({\rho }_j^{\prime },{\eta }_j^{\prime })\ge ({\rho }_i^{\prime },{\eta }_i^{\prime })$.

By applying relation (8), we get

$${\mathfrak{C}}_{({\rho }_j^{\prime },{\eta }_j^{\prime })}\left(B_t\right)\le {\mathfrak{C}}_{({\rho }_i^{\prime },{\eta }_i^{\prime })}\left(B_t\right)\text{.}$$

Again, by using part (2) of Theorem (18), we have

$${\mathfrak{C}}_{({\rho }_j,{\eta }_j)}\left(A_t\right)={\mathfrak{C}}_{({\rho }_j^{\prime },{\eta }_j^{\prime })}\left(B_t\right)\text{.}$$

So, as $a_1\in {\mathfrak{C}}_{\left({\rho }_j^{\prime },{\eta }_j^{\prime }\right)}\left(B_t\right)\text{,}$ we have that $a_1\in {\mathfrak{C}}_{\left({\rho }_j,{\eta }_j\right)}\left(A_t\right)$ and so $A_t\left(a_1\right)=\left({\rho }_i,{\eta }_i\right)\ge \left({\rho }_j,{\eta }_j\right)\text{.}$.

Now, from relation (5), we get

$${\mathfrak{C}}_{({\rho }_i,{\eta }_i)}\left(A_t\right)\le {\mathfrak{C}}_{({\rho }_j,{\eta }_j)}\left(A_t\right)\text{.}$$

But by using part (2) of Theorem (18), we have ${\mathfrak{C}}_{({\rho }_i,{\eta }_i)}\left(A_t\right)={\mathfrak{C}}_{({\rho }_i^{\prime },{\eta }_i^{\prime })}\left(B_t\right)$ and ${\mathfrak{C}}_{({\rho }_j,{\eta }_j)}\left(A_t\right)={\mathfrak{C}}_{({\rho }_j^{\prime },{\eta }_j^{\prime })}\left(B_t\right)$.

Consequently, we have that ${\mathfrak{C}}_{({\rho }_i^{\prime },{\eta }_i^{\prime })}\left(B_t\right)\le {\mathfrak{C}}_{({\rho }_j^{\prime },{\eta }_{\mathrm{j}}^{\prime })}\left(B_t\right)$ which contradicts the fact ${\mathfrak{C}}_{({\rho }_j^{\prime },{\eta }_j^{\prime })}\left(B_t\right)\le {\mathfrak{C}}_{({\rho }_i^{\prime },{\eta }_i^{\prime })}\left(B_t\right)$ obtained earlier unless we have that ${\mathfrak{C}}_{\left({\rho }_j^{\prime },{\eta }_j^{\prime }\right)}\left(B_t\right)={\mathfrak{C}}_{\left({\rho }_i^{\prime },{\eta }_i^{\prime }\right)}\left(B_t\right)\text{.}$.

Using Theorem (17), it follows that $({\rho }_j^{\prime },{\eta }_j^{\prime })=({\rho }_i^{\prime },{\eta }_i^{\prime })$.

Theorem 19

Let $A_t$ and $B_t$ be any two $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$. Then $A_t=B_t$ if and only if $\Lambda \left(A_t\right)=\Lambda \left(B_t\right)\text{.}$

Proof. If $A_t=B_t$ then obviously $\Lambda \left(A_t\right)=\Lambda \left(B_t\right)\text{.}$ Conversely, assume that $\Lambda \left(A_t\right)=\Lambda \left(B_t\right)\text{.}$ For convenience, let us denote $\Lambda \left(A_t\right)=\left\{\left({\rho }_0,{\eta }_0\right),\left({\rho }_1,{\eta }_1\right),...,\left({\rho }_r,{\eta }_r\right)\right\}$ where $1={\rho }_0\ge {\rho }_1\ge ...\ge {\rho }_r$, $0={\eta }_0\le {\eta }_1\le ...\le {\eta }_r$ and $\Lambda \left(B_t\right)=\left\{\left({\rho }_0^{\text{'}},{\eta }_0^{\text{'}}\right),\left({\rho }_1^{\text{'}},{\eta }_1^{\text{'}}\right),...,\left({\rho }_r^{\text{'}},{\eta }_r^{\text{'}}\right)\right\}$ where $1={\rho }_0^{\prime }\ge {\rho }_1^{\prime }\ge ...\ge {\rho }_r^{\prime }$, $0={\eta }_0^{\prime }\le {\eta }_1^{\prime }\le ...\le {\eta }_r^{\prime }\text{.}$ Therefore $({\rho }_{k_0},{\eta }_{k_0})=({\rho }_s^{\prime },{\eta }_s^{\prime })$ for some $k_0\text{.}$ Suppose if possible $\left({\rho }_{k_0},{\eta }_{k_0}\right)\ne ({\rho }_0,{\eta }_0)\text{.}$ So, $({\rho }_{k_0},{\eta }_{k0})<({\rho }_1,{\eta }_1)\text{.}$ Now, $\left({\rho }_1^{\text{'}},{\eta }_1^{\text{'}}\right)\in \Lambda \left(B_t\right)$ and so $({\rho }_1^{\prime },{\eta }_1^{\prime })=({\rho }_{k_1},{\eta }_{k_1})$ for some $k_1\text{.}$ We have $({\rho }_0^{\prime },{\eta }_0^{\prime })>({\rho }_1^{\prime },{\mathrm{\eta }}_1^{\prime })$ and $({\rho }_{k_0},{\eta }_{k_0})>({\rho }_{k_1},{\eta }_{k_1})\text{.}$ Proceeding in this way, we have

$$({\rho }_{k_0},{\eta }_{k_0})>({\rho }_{k_1},{\eta }_{k_1})>...>({\rho }_{k_r},{\eta }_{k_r})$$ (9)

where

$$({\rho }_0^{\prime },{\eta }_0^{\prime })=({\rho }_{k_0},{\eta }_{k_0})>({\rho }_0,{\eta }_0)\text{.}$$ (10)

However, relations (9) and (10) contradict the fact that

$$\Lambda \left(A_t\right)=\Lambda \left(B_t\right)\text{.}$$

Hence, we have $({\rho }_0,{\eta }_0)=({\rho }_0^{\prime },{\eta }_0^{\prime })\text{.}$ Arranging in this manner, we obtain that $({\rho }_i,{\eta }_i)=({\rho }_i^{\prime },{\eta }_i^{\prime }),0\le i\le r\text{.}$ Now, let $b_0,b_1,...,b_r$ be distinct elements of $\mathbb{G}$ such that $A_t\left(b_i\right)=({\rho }_i,{\eta }_i),0\le i\le r$.

Then by Theorem (18), we have that

$$B_t\left(b_i\right)=({\rho }_i^{\prime },{\eta }_i^{\prime }),0\le i\le r\text{.}$$

Since $({\rho }_i,{\eta }_i)=({\rho }_i^{\prime },{\eta }_i^{\prime })\text{,}$ it follows that $A_t\left(b\right)=B_t\left(b\right),\forall b\in \mathbb{G}\text{.}$ Hence, $A_t=B_t$.

Definition 18

Let $A_t$ and $B_t$ be any two $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ then $A_t$ and $B_t$ are said to be equivalent (denoted by $A_t\leftrightarrow B_t)\text{,}$ if ${\mathfrak{I}}_{A_t}={\mathfrak{I}}_{B_t}$.

Remark 3

Let ℱ denotes the collection of $\mathit{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$. Define a relation on ℱ as follows: ${\mathit{A}}_{\mathit{t}}\leftrightarrow {\mathit{B}}_{\mathit{t}}$ if and only if ${\mathit{A}}_{\mathit{t}}$ and ${\mathit{B}}_{\mathit{t}}$ have the same family of level subgroups for any ${\mathit{A}}_{\mathit{t}}$ and ${\mathit{B}}_{\mathit{t}}$ in ℱ.

Lemma 3

The relation ‘$\leftrightarrow$’ is an equivalence relation.

Proof. Let $\left[A_t\right]$ denotes the equivalence class containing $A_t$, where $A_t\in \mathcal{F}$. As group $\mathbb{G}$ is finite, the number of possible distinct level subgroups of $\mathbb{G}$ is finite since each level subgroup is a subgroup of group $\mathbb{G}$. From Theorem (3), it follows that the number of possible chains of level subgroups is also finite. As each equivalence class is characterized completely by its chain of level subgroups.

Definition 19

Let $A_t$ and $B_t$ be any two $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$. We say that $A_t$ is t-intuitionistic fuzzy conjugate ($t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G})$ to $B_t$ if there exists $a_1\in \mathbb{G}$ such that ${\mu }_{A_t}\left(b_1\right)={\mu }_{B_t}\left(a_1^{-1}{b}_1{a}_1\right)$ and ${\nu }_{A_t}\left(b_1\right)={\nu }_{B_t}\left(a_1^{-1}{b}_1{a}_1\right),\forall b_1\in \mathbb{G}$.

It is interesting to note that the relation of conjugacy between $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ is an equivalence. Consequently, the family of $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ is a union of pairwise disjoint classes of $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ each consisting of equivalent $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$.

Theorem 20

Let $A_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ and $B_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{S}$ of a group $\mathbb{G}$. If $A_t$ and $B_t$ are $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$ then $B_t$ is a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of group $\mathbb{G}$.

Proof. Since $A_t$ and $B_t$ are $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$. In view of the definition (18) and for some $a_1\in \mathbb{G}$, we have ${\mu }_{B_t}\left(b_1\right)={\mu }_{A_t}\left(a_1^{-1}{b}_1{a}_1\right)$ and ${\nu }_{B_t}\left(b_1\right)={\nu }_{A_t}\left(a_1^{-1}{b}_1{a}_1\right),\forall b_1\in \mathbb{G}$.

Consider

$${\mu }_{B_t}\left(b_1\right)={\mu }_{B_t}\left(e_1{b}_1{e}_1\right)$$

$$={\mu }_{B_t}\left(a_1^{-1}{a}_1{b}_1{a}_1^{-1}{a}_1\right)$$

$$={\mu }_{A_t}\left(a_1^{-1}{b}_1{a}_1\right)\text{.}$$

Consider

$${\mu }_{B_t}\left({a_{1}b}_1\right)={\mu }_{B_t}\left(e_1{a}_1{e}_1{b}_1{e}_1\right)$$

$$={\mu }_{B_t}\left(c_1^{-1}{c}_1{a}_1{c}_1^{-1}{c}_1{b}_1{c}_1^{-1}{c}_1\right)$$

$$={\mu }_{A_t}\left(c_1{a}_1{c}_1^{-1}{c}_1{b}_1{c}_1^{-1}\right)$$

Since $A_t$ is a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of the group $\mathbb{G}$.

$$\ge \mathit{min}\{{\mu }_{A_t}\left(c_1{a}_1{c}_1^{-1}\right),{\mu }_{A_t}\left(c_1{b}_1{c}_1^{-1}\right)\}$$

$$=\mathit{min}\{{\mu }_{B_t}\left(a_1\right),{\mu }_{B_t}\left(b_1\right)\}$$

Consequently, ${\mu }_{B_t}\left(a_1{b}_1\right)\ge \mathit{min}\left\{{\mu }_{B_t}\left(a_1\right),{\mu }_{B_t}\left(b_1\right)\right\}$.

Also ${\mu }_{B_t}\left(b_1^{-1}\right)={\mu }_{A_t}\left(a_1{b}_1^{-1}{a}_1^{-1}\right)$.

$$={\mu }_{A_t}\left(a_1{b}_1{a}_1^{-1}\right)$$

$$={\mu }_{B_t}\left(b_1\right)\text{.}$$

The same procedure is applied for ${\nu }_{B_t}$.

This shows that $B_t$ is a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of group $\mathbb{G}$.

Theorem 21

Let $A_t$ and $B_t$ be any two $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ then $A_t$ and $B_t$ are $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$ if and only if $A_t=B_t\text{.}$

Proof. Assume that $A_t$ and $B_t$ are $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$. With reference to definition (18), we have ${\mu }_{B_t}\left(b_1\right)={\mu }_{A_t}\left(a_1^{-1}{b}_1{a}_1\right)$ and ${\nu }_{B_t}\left(b_1\right)={\nu }_{A_t}\left(a_1^{-1}{b}_1{a}_1\right),\forall b_1\in \mathbb{G}$.

Consider $a_1{b}_1=b_1{a}_1\in \mathbb{G}$ then ${\mu }_{A_t}\left(a_1{b}_1\right)={\mu }_{B_t}\left(a_1^{-1}{b}_1{a}_1{a}_1\right)$. This implies that

$${\mu }_{A_t}\left(a_1{b}_1\right)={\mu }_{B_t}\left(b_1{a}_1\right)\text{.}$$

For some $a_1=e_1\in \mathbb{G}\text{,}$ we have ${\mu }_{A_t}\left(e_1{b}_1\right)={\mu }_{B_t}\left(b_1{e}_1\right)\text{.}$ This further implies that

$${\mu }_{A_t}\left(b_1\right)={\mu }_{B_t}\left(b_1\right)\text{.}$$

Consequently ${\mu }_{A_t}={\mu }_{B_t}\text{.}$ Similarly, ${\nu }_{A_t}={\nu }_{B_t}\text{.}$ This shows that $A_t=B_t$.

Conversely, suppose that $A_t=B_t\text{.}$ Let ${\mu }_{A_t}={\mu }_{B_t}\text{.}$ This implies that ${\mu }_{A_t}\left(b_1\right)={\mu }_{B_t}\left(b_1\right)$.

This means that ${\mu }_{A_t}\left(b_1\right)={\mu }_{B_t}\left(e_1^{-1}{b}_1{e}_1\right),\forall b_1,e_1\in \mathbb{G}$.

In the same way, we have ${\nu }_{A_t}\left(b_1\right)={\nu }_{B_t}\left(e_1^{-1}{b}_1{e}_1\right),\forall b_1,e_1\in \mathbb{G}$.

Consequently, $A_t$ and $B_t$ are $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$.

Corollary 6

If $A_t$ and $B_t$ are any two $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ then

$$t-\mathbb{I}\mathbb{F}\mathbb{O}\left(A_t\right)=t-\mathbb{I}\mathbb{F}\mathbb{O}\left(B_t\right)\text{.}$$

Theorem 22

Let $A_t$ be any $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$. For any element $g_1$ in $\mathbb{G}$ define a map ${\mathcal{A}}_t^{g_1}:\mathbb{G}\to \left[\mathrm{0,1}\right]$ by ${\mu }_{{\mathcal{A}}_t^{g_1}}\left(a_1\right)={\mu }_{A_t}\left(g_1^{-1}{a}_1{g}_1\right)$ and ${\nu }_{{\mathcal{A}}_t^{g_1}}\left(a_1\right)={\nu }_{A_t}\left(g_1^{-1}{a}_1{g}_1\right),\forall a_1\in \mathbb{G}$. Then the following statements are true:

• 1)${\mathcal{A}}_t^{g_1}$ is a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ and ${\mathcal{A}}_t^{g_1}$ is a $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$ determined by $A_t$ and $g_1$ in $\mathbb{G}$.
• 2)If $\left|\Lambda \left(A_t\right)\right|=2$, then ${\mathcal{A}}_t^{g_1}=\left\{g_1{\mathfrak{C}}_{\left({\rho }_0,{\eta }_0\right)}\left(A_t\right)g_1^{-1},\mathbb{G}\right\}$, where ${\rho }_0={\mu }_{A_t}\left(e\right)$ and ${\eta }_0={\nu }_{A_t}\left(e\right)$.

Proof. (1). Define a map $I_{g_1}:\mathbb{G}\to \mathbb{G}$ by $I_{g_1}\left(a_1\right)=g_1^{-1}{a}_1{g}_1,\forall a_1\in \mathbb{G}\text{.}$ Then $I_{g_1}\in Aut\mathbb{G}$ and ${\mathcal{A}}_t^{g_1}=A_t\circ I_{g_1}=I_{g_1}^{-1}\left(A_t\right)$. Since the inverse image of a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ by a group homomorphism is always a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$, it follows that ${\mathcal{A}}_t^{g_1}$ is a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$. Thus ${\mathcal{A}}_t^{g_1}$ is a $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ of group $\mathbb{G}$ determined by $A_t$ and $g_1$ in $\mathbb{G}$ is trivial.

(2). Since ${\mathcal{A}}_t^{g_1}=I_{g_1}^{-1}\left(A_t\right)$, it follows that $\Lambda \left({\mathcal{A}}_t^{g_1}\right)=\Lambda \left(A_t\right)\text{.}$ Let $\Lambda \left(A_t\right)=\{({\rho }_i,{\eta }_i):0\le i\le 1\}$In view of definition (3), we have

$$b_1\in {\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left({\mathcal{A}}_t^{g_1}\right)$$

$\iff {\mu }_{{\mathcal{A}}_t^{g_1}}\left(b_1\right)\ge {\rho }_i$ and ${\nu }_{{\mathcal{A}}_t^{g_1}}\left(b_1\right)\le {\eta }_i$.

$\iff {\mu }_{A_t}\left(g_1^{-1}{b}_1{g}_1\right)\ge {\rho }_{i}and{\nu }_{A_t}\left(g_1^{-1}{b}_1{g}_1\right)\le {\eta }_i$.

$$\iff g_1^{-1}{b}_1{g}_1\in {\mathfrak{C}}_{({\rho }_i,{\eta }_i)}\left(A_t\right)$$

$$\iff b_1\in g_1{\mathfrak{C}}_{({\rho }_i,{\eta }_i)}\left(A_t\right)g_1^{-1}$$

Thus ${\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left({\mathcal{A}}_t^{g_1}\right)=g_1{\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(A_t\right)g_1^{-1}$.

Consequently, we obtain that

$${\mathfrak{I}}_{{\mathcal{A}}_t^{g_1}}=\left\{{\mathfrak{C}}_{\left({\rho }_0,{\eta }_0\right)}\left({\mathcal{A}}_t^{g_1}\right),{\mathfrak{C}}_{\left({\rho }_1,{\eta }_1\right)}\left({\mathcal{A}}_t^{g_1}\right)\right\}=\left\{g_1{\mathfrak{C}}_{\left({\rho }_0,{\eta }_0\right)}\left(A_t\right)g_1^{-1},\mathbb{G}\right\}\text{.}$$

Theorem 23

If the chain of the level subgroups of $\mathit{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}{\mathit{A}}_{\mathit{t}}$ is given by:

$${\mathfrak{C}}_{\left({\rho }_0,{\eta }_0\right)}\left(A_t\right)\subseteq {\mathfrak{C}}_{\left({\rho }_1,{\eta }_1\right)}\left(A_t\right)\subseteq \dots \subseteq {\mathfrak{C}}_{\left({\rho }_r,{\eta }_r\right)}\left(A_t\right)=\mathbb{G}$$

then the chain of level subgroups $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}{\mathcal{A}}_t^{g_1}$, the $\mathit{t}-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ ${\mathcal{A}}_t^{g_1}$ of $\mathbb{G}$ is determined by ${\mathit{A}}_{\mathit{t}}$ and ${\mathit{g}}_{\mathbf{1}}\in \mathbb{G}$ is given by

$$g_1{\mathfrak{C}}_{\left({\rho }_0,{\eta }_0\right)}\left(A_t\right)g_1^{-1}\subseteq g_1{\mathfrak{C}}_{\left({\rho }_1,{\eta }_1\right)}\left(A_t\right)g_1^{-1}\subseteq \dots \subseteq g_1{\mathfrak{C}}_{({\rho }_r,{\eta }_r)}\left(A_t\right)g_1^{-1}=\mathbb{G}\text{.}$$

Proof. In view of Theorem (22), we have ${\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left({\mathcal{A}}_t^{g_1}\right)=g_1{\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(A_t\right)g_1^{-1}$.

From this, we conclude that the chain of level subgroups of ${\mathcal{A}}_t^{g_1}$ is given by

$$g_{\mathit{1}}{ℭ}_{\left({\mathit{\rho }}_{\mathit{0}}\mathit{,}{\mathit{\eta }}_{\mathit{0}}\right)}\left({\mathit{A}}_{\mathit{t}}\right)g_{\mathit{1}}^{\mathit{-}\mathit{1}}\mathit{\subseteq }{g}_{\mathit{1}}{ℭ}_{\left({\mathit{\rho }}_{\mathit{1}}\mathit{,}{\mathit{\eta }}_{\mathit{1}}\right)}\left({\mathit{A}}_{\mathit{t}}\right)g_{\mathit{1}}^{\mathit{-}\mathit{1}}\mathit{\subseteq }\mathit{.}\mathit{.}\mathit{.}\mathit{\subseteq }{g}_{\mathit{1}}{ℭ}_{\left({\mathit{\rho }}_{\mathit{r}}\mathit{,}{\mathit{\eta }}_{\mathit{r}}\right)}\left({\mathit{A}}_{\mathit{t}}\right)g_{\mathit{1}}^{\mathit{-}\mathit{1}}\mathit{=}\mathbb{G}$$

In the following result, we develop a mechanism to obtain a $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ corresponding to a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ in the framework of the concept of level subgroups.

Theorem 24

If the chain of the level subgroups of $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_t$ is given by

$${\mathfrak{C}}_{({\rho }_0,{\eta }_0)}\left(A_t\right)\subseteq {\mathfrak{C}}_{({\rho }_1,{\eta }_1)}\left(A_t\right)\subseteq ...\subseteq {\mathfrak{C}}_{({\rho }_r,{\eta }_r)}\left(A_t\right)=\mathbb{G}$$

then there exists a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $B_t$ of $\mathbb{G}$ whose chain of level subgroups is given by

$${g_1\mathfrak{C}}_{\left({\rho }_0,{\eta }_0\right)}\left(B_t\right)g_1^{-1}\subseteq g_1{\mathfrak{C}}_{\left({\rho }_1,{\eta }_1\right)}\left(B_t\right)g_1^{-1}\subseteq \dots \subseteq g_1{\mathfrak{C}}_{({\rho }_r,{\eta }_r)}\left(B_t\right)g_1^{-1}=\mathbb{G}$$

where $g_1\in \mathbb{G}$ and $B_t={\mathcal{A}}_t^{g_1}$.

Proof. Consider the $t-\mathbb{I}\mathbb{F}\mathbb{S}{B}_t$ of the group $\mathbb{G}$ as follows:

$${\mu }_{B_t}\left(a_1\right)=\{\begin{array}{cc}{\rho }_0& if{a}_1\in g_1{\mathfrak{C}}_{\left({\rho }_0,{\eta }_0\right)}\left(B_t\right)g_1^{-1}\\ {\rho }_r& if{a}_1\in g_1{\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(B_t\right)g_1^{-1}\leftrightarrow g_1{\mathfrak{C}}_{\left({\rho }_{i-1},{\eta }_{i-1}\right)}\left(B_t\right)g_1^{-1}\end{array}$$

and

$${\nu }_{B_t}\left(a_1\right)=\{\begin{array}{cc}{\eta }_0& if{a}_1\in g_1{\mathfrak{C}}_{\left({\rho }_0,{\eta }_0\right)}\left(B_t\right)g_1^{-1}\\ {\eta }_r& if{a}_1\in g_1{\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(B_t\right)g_1^{-1}\leftrightarrow g_1{\mathfrak{C}}_{\left({\rho }_{i-1},{\eta }_{i-1}\right)}\left(B_t\right)g_1^{-1}\end{array},i=\mathrm{1,2},\dots ,r\text{.}$$

Then $B_t\left(a_1\right)={\mathcal{A}}_t^{g_1}\left(a_1\right),\forall a_1\in \mathbb{G}$ because $\mathit{i}=\mathbf{1,2},\dots ,\mathit{r}\text{,}$ we have

$$B_t\left(a_1\right)=\left({\rho }_i,{\eta }_i\right)$$

$$\iff a_1\in g_1{\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(B_t\right)g_1^{-1}\leftrightarrow g_1{\mathfrak{C}}_{\left({\rho }_{i-1},{\eta }_{i-1}\right)}\left(B_t\right)g_1^{-1}$$

$$\iff g_1^{-1}{a}_1{g}_1\in {\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(B_t\right)\leftrightarrow {\mathfrak{C}}_{\left({\rho }_{i-1},{\eta }_{i-1}\right)}\left(B_t\right)$$

$$\iff A_t\left(g_1^{-1}{a}_1{g}_1\right)=\left({\rho }_i,{\eta }_i\right)$$

$$\iff {\mathcal{A}}_{\mathit{t}}^{{\mathit{g}}_{\mathbf{1}}}\left({\mathit{a}}_{\mathbf{1}}\right)=({\mathit{\rho }}_{\mathit{i}},{\mathit{\eta }}_{\mathit{i}})$$

and

$${\mathit{B}}_{\mathit{t}}\left({\mathit{a}}_{\mathbf{1}}\right)=\left({\mathit{\rho }}_{\mathbf{0}},{\mathit{\eta }}_{\mathbf{0}}\right)$$

$$\iff {\mathit{a}}_{\mathbf{1}}\in {\mathit{g}}_{\mathbf{1}}{\mathfrak{C}}_{\left({\mathit{\rho }}_{\mathbf{0}},{\mathit{\eta }}_{\mathbf{0}}\right)}\left({\mathit{B}}_{\mathit{t}}\right){\mathit{g}}_{\mathbf{1}}^{-\mathbf{1}}$$

$$\iff {\mathit{g}}_{\mathbf{1}}^{-\mathbf{1}}{\mathit{a}}_{\mathbf{1}}{\mathit{g}}_{\mathbf{1}}\in {\mathfrak{C}}_{\left({\mathit{\rho }}_{\mathbf{0}},{\mathit{\eta }}_{\mathbf{0}}\right)}\left({\mathit{B}}_{\mathit{t}}\right)$$

$$\iff A_t\left(g_1^{-1}{a}_1{g}_1\right)=\left({\rho }_0,{\eta }_0\right)$$

$$\iff {\mathcal{A}}_{\mathit{t}}^{{\mathit{g}}_{\mathbf{1}}}\left({\mathit{a}}_{\mathbf{1}}\right)=({\mathit{\rho }}_{\mathbf{0}},{\mathit{\eta }}_{\mathbf{0}})$$

Next, we define a map $I_{g_1}:\mathbb{G}\to \mathbb{G}$ by $I_{g_1}\left(a_1\right)=g_1^{-1}{a}_1{g}_1,\forall a_1\in \mathbb{G}\text{.}$ Then $I_{g_1}\in Aut\mathbb{G}\text{.}$ Since $A_t$ is a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$, it follows that ${\mathcal{A}}_t^{g_1}=A_t\circ I_{g_1}=I_{g_1}^{-1}\left(A_t\right)$ is also a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$, as the inverse image of a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ by a group homomorphism is a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$. Hence $B_t={\mathcal{A}}_t^{g_1}$ is also a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$. Moreover, it is clear that $\Lambda \left({\mathcal{A}}_t^{g_1}\right)=\Lambda \left(A_t\right)$ and ${\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left({\mathcal{A}}_t^{g_1}\right)=g_1{\mathfrak{C}}_{\left({\rho }_i,{\eta }_i\right)}\left(A_t\right)g_1^{-1}$.

Thus, the chain of level subgroups of ${\mathcal{A}}_t^{g_1}$ and hence of $B_t$ is given by

$$g_1{\mathfrak{C}}_{({\rho }_0,{\eta }_0)}\left(A_t\right)g_1^{-1}\subseteq g_1{\mathfrak{C}}_{({\rho }_1,{\eta }_1)}\left(A_t\right)g_1^{-1}\subseteq ...\subseteq g_1{\mathfrak{C}}_{({\rho }_r,{\eta }_r)}\left(A_t\right)g_1^{-1}=\mathbb{G}$$

Example 7

Consider the $0.70-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_{0.70}$ of $D_4$ as follows:

$${\mu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.70\\ 0.65\\ 0.45\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1\in \left\{rs\right\}\\ if{z}_1\in \left\{s,r,r^2,r^3,r^{2}s,r^{3}s\right\}\end{array}\end{array}$$

and

$${\nu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.30\\ 0.35\\ 0.45\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1\in \left\{rs\right\}\\ if{z}_1\in \left\{s,r,r^2,r^3,r^{2}s,r^{3}s\right\}\end{array}.\end{array}$$

In view of definition (3), we have

$${\mathfrak{C}}_{\left(\mathrm{0.70,0.30}\right)}\left(A_{0.70}\right)=\left\{1\right\}$$

${\mathfrak{C}}_{\left(\mathrm{0.65,0.35}\right)}\left(A_{0.70}\right)=\{1,rs\}$ and

$${\mathfrak{C}}_{\left(\mathrm{0.45,0.45}\right)}\left(A_{0.70}\right)=D_4\text{.}$$

Consider the $0.7-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $B_{0.7}$ of $D_4$ as follows:

$${\mu }_{B_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.70\\ 0.65\\ 0.45\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1\in \left\{r^{3}s\right\}\\ if{z}_1\in \left\{s,r,r^2,r^3,rs,r^{2}s\right\}\end{array}\end{array}$$

and

$${\nu }_{B_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.30\\ 0.35\\ 0.45\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1\in \left\{r^{3}s\right\}\\ if{z}_1\in \left\{s,r,r^2,r^3,rs,r^{2}s\right\}\end{array}.\end{array}$$

In view of definition (3), we have

$${\mathfrak{C}}_{\left(\mathrm{0.70,0.30}\right)}\left(A_{0.70}\right)=\left\{1\right\}\text{,}$$

$${\mathfrak{C}}_{\left(\mathrm{0.65,0.35}\right)}\left(A_{0.70}\right)=\{1,rs\}$$

and

$${\mathfrak{C}}_{\left(\mathrm{0.45,0.45}\right)}\left(A_{0.70}\right)=D_4\text{.}$$

Similarly, ${\mathfrak{C}}_{\left(\mathrm{0.70,0.30}\right)}\left(B_{0.70}\right)=\left\{1\right\}$, ${\mathfrak{C}}_{\left(\mathrm{0.65,0.35}\right)}\left(B_{0.70}\right)=\{1,r^{3}s\}$ and

$${\mathfrak{C}}_{\left(\mathrm{0.45,0.45}\right)}\left(B_{0.70}\right)=D_4\text{.}$$

The chain of level subgroups of $0.70-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_{0.70}$ and $B_{0.70}$ of $D_4$ is given as:

$${\mathfrak{C}}_{\left(\mathrm{0.70,0.30}\right)}\left(A_{0.70}\right)\subseteq {\mathfrak{C}}_{\left(\mathrm{0.65,0.35}\right)}\left(A_{0.70}\right)\subseteq {\mathfrak{C}}_{\left(\mathrm{0.45,0.45}\right)}\left(A_{0.70}\right)$$

and

$${\mathfrak{C}}_{\left(\mathrm{0.70,0.30}\right)}\left(B_{0.70}\right)\subseteq {\mathfrak{C}}_{\left(\mathrm{0.65,0.35}\right)}\left(B_{0.70}\right)\subseteq {\mathfrak{C}}_{\left(\mathrm{0.45,0.45}\right)}\left(B_{0.70}\right)$$

In view of Theorem (24), we have

$$a{\mathfrak{C}}_{\left(\mathrm{0.70,0.30}\right)}\left(B_{0.7}\right)a^{-1}\subseteq a{\mathfrak{C}}_{\left(\mathrm{0.65,0.35}\right)}\left(B_{0.7}\right)a^{-1}\subseteq a{\mathfrak{C}}_{\left(\mathrm{0.45,0.45}\right)}\left(B_{0.7}\right)a^{-1}\text{.}$$

Thus $B_{0.7}={\mathcal{A}}_{0.7}^a$.

Theorem 25

Let $A_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $B_t$ of a finite group $\mathbb{G}$ then the number of distinct $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ of $A_t$ is equal to the index of its normalizer in $\mathbb{G}$.

Proof. Let $A_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$. Consider the decomposition of $\mathbb{G}$ as a union of left cosets of $\mathcal{N}\left(A_t\right)$:

$$\mathbb{G}=a_1\mathcal{N}\left(A_t\right)\cup a_2\mathcal{N}\left(A_t\right)\cup ...\cup a_k\mathcal{N}\left(A_t\right)$$ (11)

where k is the number of distinct left cosets, that is, the index of $\mathcal{N}\left(A_t\right)$ in $\mathbb{G}$.

Let $a\in \mathcal{N}\left(A_t\right)$ and choose i such that $1\le i\le k\text{.}$ Then for any $a_i\in \mathbb{G}$.

$${\mu }_{A_t}^{a_{i}a}\left(b_1\right)={\mu }_{A_t}\left({\left(a_{i}a\right)}^{-1}{b}_1\left(a_{i}a\right)\right)$$

$$={\mu }_{A_t}\left(a^{-1}\left(a_i^{-1}{b}_1{a}_i\right)a\right)$$

$$={\mu }_{A_t}^a\left(a_i^{-1}{b}_1{a}_i\right)$$

$$={\mu }_{A_t}\left(a_i^{-1}{b}_1{a}_i\right)$$

$$={\mu }_{A_t}^{a_i}\left(b_1\right)$$

This means that ${\mu }_{A_t}^{a_{i}a}={\mu }_{A_t}^{a_i}\forall a\in N\left(A_t\right),1\le i\le k$.

Similarly, ${\nu }_{A_t}^{a_{i}a}={\nu }_{A_t}^{a_i}\forall a\in N\left(A_t\right),1\le i\le k$.

So, any two elements of $\mathbb{G}$ which lies in the same cosets $a_i\mathcal{N}\left(A_t\right)$ give rise to the same $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}{A}_t^{a_i}$ of $A_t\text{.}$ Now we show that two distinct $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ of $A_t\text{.}$ Suppose that

$${\mathit{A}}_{\mathit{t}}^{{\mathit{a}}_{\mathit{i}}}={\mathit{A}}_{\mathit{t}}^{{\mathit{a}}_{\mathit{j}}}$$ (12)

where $i\ne j$ and $1\le i,j\le k\text{.}$ Now relation (12) implies that

$$A_t^{a_i}\left(b_1\right)=A_t^{a_j}\left(b_1\right)\forall b_1\in \mathbb{G}$$

This implies that $A_t\left(a_i^{-1}{b}_1{a}_i\right)=A_t\left(a_j^{-1}{b}_1{a}_j\right),\forall b_1\in \mathbb{G}$.

Substituting $b_1=a_j{c}_1{a}_j^{-1}$ in the above relation, we have

$$A_t\left(a_i^{-1}{a}_j{c}_1{a}_j^{-1}{a}_i\right)=A_t\left(c_1\right),\forall c_1\in \mathbb{G}$$

This further implies that $A_t\left(\left(a_i^{-1}{a}_j\right)c_1\left(a_j^{-1}{a}_i\right)\right)=A_t\left(c_1\right),\forall c_1\in \mathbb{G}$.

This further implies that $A_t^{a_j^{-1}{a}_i}\left(c_1\right)=A_t\left(c_1\right),\forall c_1\in \mathbb{G}\text{.}$ This means that $a_j^{-1}{a}_i\in \mathcal{N}\left(A_t\right)$.

Consequently, $a_i\mathcal{N}\left(A_t\right)=a_j\mathcal{N}\left(A_t\right)\text{.}$ However, if $i\ne j$ this is not possible as (11) represents a partition of $\mathbb{G}$ into pairwise disjoint cosets. Hence the number of distinct conjugates of $A_t$ is equal to the index of $\mathcal{N}\left(A_t\right)$ in $\mathbb{G}$.

Definition 20

Let ${\mathit{A}}_{\mathit{t}}$ be a $\mathit{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ and ${\mathit{a}}_{\mathbf{1}}\in \mathbb{G}\text{.}$ Then $a_1^{-1}{\mathfrak{C}}_{\left(\rho ,\eta \right)}\left(A_t\right)a_1$ forms a $\mathit{t}-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ $\mathbb{G}$. This $\mathit{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ is called $\mathit{t}-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ to ${\mathit{A}}_{\mathit{t}}\text{.}$ The set

$$C\mathcal{l}\left(A_t\right)=\{a_1^{-1}{\mathfrak{C}}_{(\rho ,\eta )}\left(A_t\right)a_1:a_1\in \mathbb{G}\}$$

is called the class of $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ to $A_t$.

Example 8

Consider the $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of the dihedral group $D_4$ for the value $t=0.70$ as follows:

$${\mu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.70\\ 0.50\\ 0.40\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1=rs\\ if{z}_1\in \left\{r,r^2,r^3,s,r^{2}s,r^{3}s\right\}\end{array}\end{array}$$

and

$${\nu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.30\\ 0.40\\ 0.50\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1=rs\\ if{z}_1\in \{r,r^2,r^3,s,r^{2}s,r^{3}s\}\end{array}.\end{array}$$

In the light of definition (2), we acquire

$${\mathfrak{C}}_{\left(\mathrm{0.50,0.40}\right)}\left(A_{0.70}\right)=\left\{1,rs\right\}$$

The required class of $0.70-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ to $A_{0.70}$ is obtained as

$$C\mathcal{l}\left(A_{0.70}\right)=\left\{{\mathbb{H}}_1,{\mathbb{H}}_2,{\mathbb{H}}_3,{\mathbb{H}}_4\right\}\text{,}$$

where ${\mathbb{H}}_1=\left\{1,rs\right\},{\mathbb{H}}_2=\left\{1,s\right\},{\mathbb{H}}_3=\{1,r^{2}s\}$ and ${\mathbb{H}}_4=\left\{1,r^{3}s\right\}$.

Corollary 7

Let $a_1^{-1}{\mathfrak{C}}_{(\rho ,\eta )}\left(A_t\right)a_1$ be $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ to $A_t$ then $O\left[C\mathcal{l}\left(A_t\right)\right]=\frac{O\left(\mathbb{G}\right)}{O\left[\mathcal{N}\left(A_t\right)\right]}$.

The following illustration establishes the algebraic facts discussed in the above result.

Example 9

The application of the definition (13) and corollary (7) gives that: $O\left[C\mathcal{l}\left(A_{0.70}\right)\right]=\frac{8}{2}=4\text{.}$ (See example 8).

6. t-Intuitionistic fuzzification of Sylow's theorems

In this section, we define the notion of t-intuitionistic fuzzy Sylow $p-$ subgroup of finite group $\mathbb{G}$. Moreover, we prove the t-intuitionistic fuzzy version of Sylow's Theorems.

Definition 21

A $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_t$ of a group, $\mathbb{G}$ is called a t-intuitionistic fuzzy Sylow's $p-$ subgroup $\left(t-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p\right)$ if the support set $A_t^{*}$ is a Sylow's $p-$ subgroup of $\mathbb{G}$.

Definition 22

Let $A_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group $\mathbb{G}$. A $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ $A_t$ is called t-intuitionistic fuzzy Sylow's $p-$ subgroup$(t-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p)$ of group $\mathbb{G}$, if one of the level subgroups of $A_t$ is Sylow's $p-$ subgroup of $\mathbb{G}$.

There may be two or more level subgroups ${\mathrm{A}}_t$ which are Sylow $p-$ subgroup, but the support of $A_t$ may not be Sylow $p-$ subgroup of $\mathbb{G}$. However, definition (20) implies definition (21).

The following example illustrates the existence of the Sylow $p-$ subgroup of a $t-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p\left(\mathbb{G}\right)$.

Example 10

Consider the direct product of the groups: $\mathbb{G}=<r:r^5=1>$ and $\mathbb{H}=<s:s^2=1>$ as $\mathbb{G}\times \mathbb{H}=\{(r,s):r\in \mathbb{G},s\in \mathbb{H}\}$.

Consider the $0.70-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of $\mathbb{G}\times \mathbb{H}$ is given by:

$${\mu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.70\\ 0.50\\ 0.40\end{array}& \begin{array}{c}if{z}_1=\left(\mathrm{1,1}\right)\\ if{z}_1\in <(r,1)>-\left\{\left(\mathrm{1,1}\right)\right\}\\ if{z}_1\in \mathbb{G}\times \mathbb{H}-<(r,1)>\end{array}\end{array}$$

and

$${\nu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.30\\ 0.40\\ 0.60\end{array}& \begin{array}{c}if{z}_1=\left(\mathrm{1,1}\right)\\ if{z}_1\in <(r,1)>-\left\{\left(\mathrm{1,1}\right)\right\}\\ if{z}_1\in \mathbb{G}\times \mathbb{H}-<(r,1)>\end{array}\end{array}$$

In view of definition (3), we have

$${\mathrm{Ĉ}}_{\left(\mathrm{0.5,0.4}\right)}\left(A_{0.70}\right)=⟨\left(r,1\right)⟩\text{,}$$

which is a Sylow 5-subgroup of $\mathbb{G}\times \mathbb{H}$.

Thus, $A_{0.70}$ is a $t-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_5$ of $\mathbb{G}\times \mathbb{H}$.

The following result describes the t-intuitionistic fuzzy version of Sylow's First Theorem.

Theorem 26

(t-Intuitionistic Fuzzification of Sylow's First Theorem). Let $A_t$ be a $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$ such that $O\left(\mathbb{G}\right)=p^{k}m$ where $p$ is a prime and $k,m$ are positive integers with $(p,m)=1$. Assume that p divides $A_t^{*}=\mathbb{H}\text{.}$ Then there exists a $t-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p{B}_t$ of $\mathbb{G}$ such that $B_t\subseteq A_t$.

Proof. If $\mathbb{H}$ is a Sylow p-subgroup, then there is nothing to prove. Assume that $\mathbb{H}$ is not a Sylow p-subgroup of $\mathbb{G}$. Let $\rho =\mathit{Min}\{{\mu }_{A_t}\left(a_1\right):{\mu }_{A_t}\left(a_1\right)>0,a_1\in \mathbb{G}\}$ and $\eta =\mathit{Max}\{{\nu }_{A_t}\left(a_1\right):{\nu }_{A_t}\left(a_1\right)>0,a_1\in \mathbb{G}\}\text{.}$ Clearly $\rho \ne 0$ and $\mathrm{\eta }\ne 1\text{.}$ Since $\mathbb{G}$ is finite, therefore, ${\mathfrak{C}}_{(\rho ,\eta )}\left(A_t\right)=A_t^{*}=\mathbb{H}\text{.}$ Since $p$ divides the order of $\mathbb{H}$ so by Sylow's first theorem, there exists a Sylow $p-$ subgroup $\mathbb{L}$ of $\mathbb{H}$. By our assumption, $O\left(\mathbb{L}\right)=p^l$ where $1\le l\le k\text{.}$ Further, $\mathbb{L}$ will be contained in a subgroup ${\mathbb{L}}^{\prime }$ of $\mathbb{G}$. Consider a $t-\mathbb{I}\mathbb{F}\mathbb{S}{B}_t$ of $\mathbb{G}$ as follows:

${\mu }_{B_t}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}1\\ \rho \end{array}& \begin{array}{c}if{z}_1=e\\ if{z}_1\in \mathbb{L}-\left\{e\right\}\end{array}\\ \begin{array}{c}{\rho }^{\prime }\\ 0\end{array}& \begin{array}{c}if{z}_1\in {\mathbb{L}}^{\prime }-\mathbb{L}\\ if{z}_1\in \mathbb{G}-{\mathbb{L}}^{\prime }\end{array}\end{array}$ and ${\nu }_{B_t}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0\\ \eta \end{array}& \begin{array}{c}if{z}_1=e\\ if{z}_1\in \mathbb{L}-\left\{e\right\}\end{array}\\ \begin{array}{c}{\eta }^{\prime }\\ 1\end{array}& \begin{array}{c}if{z}_1\in {\mathbb{L}}^{\prime }-\mathbb{L}\\ if{z}_1\in \mathbb{G}-{\mathbb{L}}^{\prime }\end{array}\end{array}$.where $0<{\rho }^{\prime },{\eta }^{\prime }<1\text{.}$ Note that $B_t$ is $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$ such that $B_t\subseteq A_t$. Moreover, in view of definition (21), we have $B_t$ is a $t-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ of $\mathbb{G}$ as ${\mathrm{Ĉ}}_{\left(\rho ,\eta \right)}\left(B_t\right)=\mathbb{L}$.

Theorem 27

A $t-\mathbb{I}\mathbb{F}\mathbb{C}\mathbb{S}\mathbb{G}$ of $t-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ of a group $\mathbb{G}$ is a $t-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ of $\mathbb{G}$.

Proof. Let $A_t$ be a $t-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ of a group $\mathbb{G}$ and $B_t$ is $t-\mathbb{I}\mathbb{F}\mathbb{C}$ to $A_t$ of $\mathbb{G}$. In view of definition (19), we have ${\mathrm{Ć}}_{(\rho ,\eta )}\left(B_t\right)=a_1^{-1}{\mathrm{Ć}}_{(\rho ,\eta )}\left(A_t\right)a_1,a_1\in \mathbb{G}\forall \rho ,\eta$. This implies that $B_t^{*}=a_1^{-1}{A}_t^{*}{a}_1\text{.}$ Application of the Theorem (26) and using the fact that $A_t$ is $t-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ of $\mathbb{G}$, we have a Sylow p-subgroup $\mathbb{H}$ of $\mathbb{G}$ contained in $A_t^{*}\text{.}$ Moreover, in view of Sylow's Second Theorem, $a_1^{-1}\mathbb{H}{a}_1$ being a conjugate of $\mathbb{H}$ is itself a Sylow p-subgroup of $\mathbb{G}$. Further, $a_1^{-1}\mathbb{H}{a}_1$ is contained in $a_1^{-1}{A}_t^{*}{a}_1$ and so $B_t$ is a $t-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ of $\mathbb{G}$.

Remark 4

Two distinct $t-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ need not be $t-\mathbb{I}\mathbb{F}\mathbb{C}$ to each other.

We describe the above algebraic aspect in the subsequent example.

Example 11

Consider the Sylow $2-$ subgroup ${\mathbb{H}}_1$ and ${\mathbb{H}}_2$ of $S_5$ as follows: ${\mathbb{H}}_1=<\left(12\right)\left(35\right),\left(1325\right)>$ and ${\mathbb{H}}_2=<\left(24\right)\left(35\right),\left(2345\right)>$.

The $0.70-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}{A}_{0.70}$ and $B_{0.7}$ of $S_5$ as follows:

$${\mu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.70\\ 0.65\\ 0.60\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1\in ⟨\left(24\right)\left(35\right)⟩-\left\{1\right\}\\ if{z}_1\in ⟨\left(24\right),\left(35\right)⟩-⟨\left(24\right)\left(35\right)⟩\end{array}\\ \begin{array}{c}0.50\\ 0.30\end{array}& \begin{array}{c}if{z}_1\in ⟨\left(25\right)\left(34\right),\left(2345\right)⟩-⟨\left(24\right),\left(35\right)⟩\\ otherwise\end{array}\end{array}$$

and

$${\nu }_{A_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.30\\ 0.35\\ 0.40\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1\in ⟨\left(24\right)\left(35\right)⟩-\left\{1\right\}\\ if{z}_1\in ⟨\left(24\right),\left(35\right)⟩-⟨\left(24\right)\left(35\right)⟩\end{array}\\ \begin{array}{c}0.50\\ 0.60\end{array}& \begin{array}{c}iif{z}_1\in ⟨\left(25\right)\left(34\right),\left(2345\right)⟩-⟨\left(24\right),\left(35\right)⟩\\ otherwise\end{array}\end{array}$$

and

$${\mu }_{B_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.70\\ 0.65\\ 0.60\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1\in ⟨\left(12\right)\left(35\right)⟩-\left\{1\right\}\\ if{z}_1\in ⟨\left(1325\right)⟩-⟨\left(12\right)\left(35\right)⟩\end{array}\\ \begin{array}{c}0.50\\ 0.30\end{array}& \begin{array}{c}if{z}_1\in ⟨\left(12\right)\left(35\right),\left(1325\right)⟩-⟨\left(1325\right)⟩\\ otherwise\end{array}\end{array}$$

and

$${\nu }_{B_{0.70}}\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0.30\\ 0.35\\ 0.40\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1\in ⟨\left(12\right)\left(35\right)⟩-\left\{1\right\}\\ if{z}_1\in ⟨\left(1325\right)⟩-⟨\left(12\right)\left(35\right)⟩\end{array}\\ \begin{array}{c}0.50\\ 0.60\end{array}& \begin{array}{c}if{z}_1\in ⟨\left(12\right)\left(35\right),\left(1325\right)⟩-⟨\left(1325\right)⟩\\ otherwise\end{array}\end{array}$$

Clearly $A_{0.70}$ and $B_{0.70}$ are $0.70-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_2$ of $S_5$.

Moreover, $A_{0.70}$ is not $0.7-\mathbb{I}\mathbb{F}\mathbb{C}$ to $B_{0.70}$ because the level subgroups: ${\mathrm{Ć}}_{(0.6,0.4)}\left(A_{0.70}\right)=<\left(24\right),\left(35\right)>$ is noncyclic and ${\mathrm{Ć}}_{(0.6,0.4)}\left({\mathrm{B}}_{0.70}\right)=<\left(1325\right)>$ is cyclic.

The following Theorem describes a t-intuitionistic fuzzy version of Sylow’s Second Theorem.

Theorem 28

(t-Intuitionistic Fuzzification of Sylow Second Theorem). For any two $t-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$$A_t$ and $B_t$ having the same level subgroups such that ${\mathrm{Ć}}_{(\rho ,\eta )}\left(B_t\right)=a_1^{-1}{\mathrm{Ć}}_{(\rho ,\eta )}\left(A_t\right)a_1\text{,}$ for some fixed $a_1\in \mathbb{G}\text{,}$ then $A_t$ and $B_t$ are $t-\mathbb{I}\mathbb{F}\mathbb{C}$ to each other.

Proof. Let ${\mathrm{Ć}}_{(\rho ,\eta )}\left(B_t\right)=a_1^{-1}{\mathrm{Ć}}_{(\rho ,\eta )}\left(A_t\right)a_1\text{.}$ Assume that $A_t$ and $B_t$ are not $t-\mathbb{I}\mathbb{F}\mathbb{C}$ to each other it follows that ${\mu }_{A_t}\left(b_1\right)>{\mu }_{B_t}\left(a_1^{-1}{b}_1{a}_1\right)$ and ${\nu }_{A_t}\left(b_1\right)<{\nu }_{B_t}\left(a_1^{-1}{b}_1{a}_1\right)\text{,}$ for $a_1,b_1\in \mathbb{G}$ (13)

Then $b_1\in {\mathrm{Ć}}_{(\rho ,\eta )}\left(B_t\right)$ and $a_1^{-1}{b}_1{a}_1\notin {\mathrm{Ć}}_{(\rho ,\eta )}\left(A_t\right)$. This means that $b_1\in a_1^{-1}{\mathrm{Ć}}_{(\rho ,\eta )}\left(A_t\right)a_1$ and $a_1^{-1}{b}_1{a}_1\notin {\mathrm{Ć}}_{(\rho ,\eta )}\left(A_t\right)$.

This implies that $a_1{b}_1{a}_1^{-1}\in {\mathrm{Ć}}_{(\rho ,\eta )}\left(A_t\right)$ and $a_1^{-1}{b}_1{a}_1\notin {\mathrm{Ć}}_{\left(\rho ,\eta \right)}\left(A_t\right)\text{.}$ This further implies that ${\left(a_1^{-1}{b}_1{a}_1\right)}^{-1}\in {\mathrm{Ć}}_{(\rho ,\eta )}\left(A_t\right)$ and $a_1^{-1}{b}_1{a}_1\notin {\mathrm{Ć}}_{\left(\rho ,\eta \right)}\left(A_t\right)$.

This means that $c^{-1}\in {\mathrm{Ć}}_{(\rho ,\eta )}\left(A_t\right)$ and $c\notin {\mathrm{Ć}}_{\left(\rho ,\eta \right)}\left(A_t\right)$.

This shows that ${\mathrm{Ć}}_{(\rho ,\eta )}\left(A_t\right)$ is not a subgroup of $\mathbb{G}$.

Which is a contradiction against relation (13).

Hence, ${\mu }_{A_t}\left(b_1\right)={\mu }_{B_t}\left(a_1^{-1}{b}_1{a}_1\right)$ and ${\nu }_{A_t}\left(b_1\right)={\nu }_{B_t}\left(a_1^{-1}{b}_1{a}_1\right)\text{,}$ $\forall a_1,b_1\in \mathbb{G}$.

We conclude that $A_t$ and $B_t$ are $t-\mathbb{I}\mathbb{F}\mathbb{C}$ to each other.

Remark 5

Let $A_t$ and $B_t$ be any two $t-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ of a group $\mathbb{G}$ and $\Lambda \left(A_t\right)=\Lambda \left(B_t\right)$ then $A_t$ and $B_t$ are $t-\mathbb{I}\mathbb{F}\mathbb{C}$ to each other.

The following Theorem describes the t-intuitionistic fuzzy version of Sylow's third Theorem.

Theorem 29

(t-Intuitionistic Fuzzification of Sylow Third Theorem). The number of distinct $t-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ of a finite group $\mathbb{G}$ is equal to the index of the normalizer of $t-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ $A_t$ in $\mathbb{G}$.

Proof. In view of corollary (7) and using the fact that $A_t$ is a $t-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$, we have

$$O\left[Cl\left(A_t\right)\right)]=\frac{O\left(\mathbb{G}\right)}{O\left[N\left(A_t\right)\right]}$$

Consider the collection of all $t-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ conjugate to $A_t$ has

$$Cl\mathit{(}{Ć}_{\left(\mathit{\rho }\mathit{,}\mathit{\eta }\right)}\left({\mathit{A}}_{\mathit{t}}\right)\mathit{=}\left\{{\mathit{Ć}}_{\left(\mathit{\rho }\mathit{,}\mathit{\eta }\right)}\left({\mathit{B}}_{\mathit{t}}\right)\mathit{:}{\mathit{a}}_{\mathit{1}}^{\mathit{-}\mathit{1}}{\mathit{Ć}}_{\left(\mathit{\rho }\mathit{,}\mathit{\eta }\right)}\left({\mathit{A}}_{\mathit{t}}\right){\mathit{a}}_{\mathit{1}}\mathit{=}{\mathit{Ć}}_{\left(\mathit{\rho }\mathit{,}\mathit{\eta }\right)}\left({\mathit{B}}_{\mathit{t}}\right)\mathit{,}{\mathit{a}}_{\mathit{1}}\mathit{\in }\mathbb{G}\right\}$$

Consequently, the number of $t-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ of $\mathbb{G}$ is $\frac{O\left(\mathbb{G}\right)}{O\left[N\left(A_t\right)\right]}$.

We illustrate the above algebraic fact in the following example.

Example 12

Consider the finite presentation of the dihedral group $D_5$ of order 10 as follows:

$$D_5=<r,s:r^5=s^2=1,sr=r^{4}s>$$

The $\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of $D_5$ is defined as follows: ${\mu }_A\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}1\\ 0.60\\ 0.40\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1\in <r>-\left\{1\right\}\\ if{z}_1\in D_5-<r>\end{array}\end{array}$ and ${\nu }_A\left(z_1\right)=\{\begin{array}{cc}\begin{array}{c}0\\ 0.40\\ 0.50\end{array}& \begin{array}{c}if{z}_1=1\\ if{z}_1\in <r>-\left\{1\right\}\\ if{z}_1\in D_5-<r>\end{array}\end{array}$.

The $\mathbf{t}-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ ${\mathbf{A}}_{\mathbf{t}}$ of ${\mathbf{D}}_{\mathbf{5}}$ corresponding to the value $\mathbf{t}=\mathbf{0.70}$ is given by: ${\mu }_{{\mathit{A}}_{\mathit{0}\mathit{.}\mathit{70}}}\left({\mathit{z}}_{\mathit{1}}\right)\mathit{=}\mathit{\{}\begin{array}{cc}\begin{array}{c}\mathit{0}\mathit{.}\mathit{70}\\ \mathit{0}\mathit{.}\mathit{60}\\ \mathit{0}\mathit{.}\mathit{40}\end{array}& \begin{array}{c}\mathit{i}\mathit{f}{\mathit{z}}_{\mathit{1}}\mathit{=}\mathit{1}\\ \mathit{i}\mathit{f}{\mathit{z}}_{\mathit{1}}\mathit{\in }\mathit{<}\mathit{r}\mathit{>}\mathit{-}\left\{\mathit{1}\right\}\\ \mathit{i}\mathit{f}{\mathit{z}}_{\mathit{1}}\mathit{\in }{\mathit{D}}_{\mathit{5}}\mathit{-}\mathit{<}\mathit{r}\mathit{>}\end{array}\end{array}$ and ${\nu }_{{\mathit{A}}_{\mathit{0}\mathit{.}\mathit{70}}}\left({\mathit{z}}_{\mathit{1}}\right)\mathit{=}\mathit{\{}\begin{array}{cc}\begin{array}{c}\mathit{0}\mathit{.}\mathit{30}\\ \mathit{0}\mathit{.}\mathit{40}\\ \mathit{0}\mathit{.}\mathit{50}\end{array}& \begin{array}{c}\mathit{i}\mathit{f}{\mathit{z}}_{\mathit{1}}\mathit{=}\mathit{1}\\ \mathit{i}\mathit{f}{\mathit{z}}_{\mathit{1}}\mathit{\in }\mathit{<}\mathit{r}\mathit{>}\mathit{-}\left\{\mathit{1}\right\}\\ \mathit{i}\mathit{f}{\mathit{z}}_{\mathit{1}}\mathit{\in }{\mathit{D}}_{\mathit{5}}\mathit{-}\mathit{<}\mathit{r}\mathit{>}\end{array}\end{array}$.

By using definition (3), we have

$${\mathfrak{C}}_{\left(\mathit{0}\mathit{.}\mathit{6}\mathit{,}\mathit{0}\mathit{.}\mathit{4}\right)}\left({\mathit{A}}_{\mathit{0}\mathit{.}\mathit{70}}\right)\mathit{=}\mathit{<}r\mathit{>}\mathit{,}$$

which is a $0.70-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_5$ of ${\mathrm{D}}_5$.

The class of $0.70-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_5$ of ${\mathrm{D}}_5$ is

$$\mathit{Cl}\left(\mathit{<}\mathit{r}\mathit{>}\right)\mathit{=}\mathit{<}r\mathit{>}\mathit{.}$$

The $0.70-$ intuitionistic fuzzy normalizer of $0.70-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_5$ of ${\mathrm{D}}_5$ is

$$\mathcal{N}\left({\mathit{A}}_{\mathit{0}\mathit{.}\mathit{70}}\right)\mathit{=}{D}_{\mathit{5}}\mathit{.}$$

In view of Theorem (29), we have only one $0.70-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_5$ of ${\mathrm{A}}_{0.70}$ in ${\mathrm{D}}_5$.

The subsequent results show that a t-intuitionistic fuzzy normal Sylow $\mathrm{p}-$ subgroup of a $\mathbb{G}$ is unique.

Theorem 30

Let $A_t$ and $B_t$ be any two $t-\mathbb{l}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_p$ of a group $\mathbb{G}$. If $A_t$ is $t-\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a group $\mathbb{G}$, then $A_t\leftrightarrow B_t$. Moreover, if ${\Lambda (A}_t)=\Lambda \left(B_t\right)$ then $A_t=B_t\text{.}$

Proof. If O($\mathbb{G}$) is a power of p, then the case is obvious. Assume that O($\mathbb{G}$) is not a power of p.

Consider ${\mathrm{\mu }}_{{\mathrm{A}}_{\mathrm{t}}}\left(\mathrm{e}\right)=\mathrm{\rho },{\mathrm{\nu }}_{{\mathrm{A}}_{\mathrm{t}}}\left(\mathrm{e}\right)=\mathrm{\eta }$ and ${\mathrm{\mu }}_{{\mathrm{B}}_{\mathrm{t}}}\left(\mathrm{e}\right)={\mathrm{\rho }}^{\prime },{\mathrm{\nu }}_{{\mathrm{B}}_{\mathrm{t}}}\left(\mathrm{e}\right)={\mathrm{\eta }}^{\prime }$. Then ${\mathrm{Ć}}_{(\mathrm{\rho },\mathrm{\eta })}\left({\mathrm{A}}_{\mathrm{t}}\right)$ and ${\mathrm{Ć}}_{(\mathrm{\rho },\mathrm{\eta })}\left({\mathrm{B}}_{\mathrm{t}}\right)$ are $\mathrm{t}-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_{\mathrm{p}}$ of $\mathbb{G}$. Since ${\mathrm{A}}_{\mathrm{t}}$ is a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{N}\mathbb{S}\mathbb{G}$. By using Theorem (2), we have ${\mathrm{Ć}}_{(\mathrm{\rho },\mathrm{\eta })}\left({\mathrm{A}}_{\mathrm{t}}\right)$ is a normal subgroup of $\mathbb{G}$. Consequently, ${\mathrm{Ć}}_{(\mathrm{\rho },\mathrm{\eta })}\left({\mathrm{A}}_{\mathrm{t}}\right)$ =${\mathrm{Ć}}_{(\mathrm{\rho },\mathrm{\eta })}\left({\mathrm{B}}_{\mathrm{t}}\right)$. This shows that $\Lambda \left({\mathrm{A}}_{\mathrm{t}}\right)=\Lambda \left({\mathrm{B}}_{\mathrm{t}}\right)$). We obtain that ${\mathrm{A}}_{\mathrm{t}}\leftrightarrow {\mathrm{B}}_{\mathrm{t}}$. From Theorem (19), we get ${\mathrm{A}}_{\mathrm{t}}={\mathrm{B}}_{\mathrm{t}}$.

In the following Theorem, we establish a unique $\mathrm{t}-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_{\mathrm{p}}$ of $\mathbb{G}$ is a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{N}\mathbb{S}\mathbb{G}$ of group $\mathbb{G}$.

Theorem 31

For any $\mathrm{t}-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_{\mathrm{p}}$ ${\mathrm{A}}_{\mathrm{t}}$ of a group $\mathbb{G}$. If ${\mathrm{A}}_{\mathrm{t}}\sim {\mathrm{B}}_{\mathrm{t}}$ $\forall {\mathrm{B}}_{\mathrm{t}}\in \mathrm{\Omega }$ where $\mathrm{\Omega }$ is the collection of all $\mathrm{t}-\mathbb{I}\mathbb{F}{\mathbb{S}\mathcal{y}\mathcal{l}}_{\mathrm{p}}$ of group $\mathbb{G}$ then ${\mathrm{A}}_{\mathrm{t}}$ is a $\mathrm{t}-\mathbb{I}\mathbb{F}\mathbb{N}\mathbb{S}\mathbb{G}$ of $\mathbb{G}$.

Proof. According to the Theorem (2), we obtain that ${\mathbf{Ć}}_{(\mathbf{\rho },\mathbf{\eta })}\left({\mathbf{A}}_{\mathbf{t}}\right)$ is a normal subgroup of $\mathbb{G}$ and the fact that $\Lambda \left({\mathrm{A}}_{\mathrm{t}}\right)=\Lambda \left({\mathrm{B}}_{\mathrm{t}}\right),\forall {\mathrm{B}}_{\mathrm{t}}\in \mathrm{\Omega }$.

7. Conclusion and future scopes

The t-intuitionistic fuzzy form of Sylow's theorem may be applied in real-world situations with subgroup analysis uncertainty, ambiguity, and imprecision. This approach improves decision-making, data analysis, optimization, and social network analysis, allowing for a complete knowledge of real-life scenarios with unpredictable group structures. In this article, we have introduced the concept of $\mathrm{t}-$ $\mathbb{I}\mathbb{F}\mathbb{C}$ element. After, we initiated the t-intuitionistic fuzzy conjugacy classes of t-$\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$. Moreover, the fundamental algebraic attributes of these phenomena were derived. Additionally, the class equation for t-$\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ was explained, along with an example. This paper has established the theory to formulate the t-intuitionistic fuzzy version of Sylow's theorems of t-$\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ of a finite group.

The theory presented in this article can be generalized to high-level uncertain environments, specifically t-picture fuzzy environments, t-spherical fuzzy environments, t-linear Diophantine fuzzy environments, and many more. Furthermore, we will apply the concept of $\mathrm{t}-$ $\mathbb{I}\mathbb{F}\mathbb{S}\mathbb{G}$ in the fields of cryptography and image processing in our future work.

Author contribution statement

Laila Latif: Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; wrote the paper.

Umer Shuaib: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data.

Additional information

No additional information is available for this paper.

Data availability statement

Data included in article/supplementary material/referenced in article.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.