(α1, 2, β1, 2)-complex intuitionistic fuzzy subgroups and its algebraic structure

1.   Introduction

Intuitionistic Fuzzy Sets (IFS) ^[1] is a generalization of Fuzzy Sets (FS) ^[2]. Many problems have been successfully solved using IFS compared to FS. IFS is characterized by representing information that depends on human subjectivity with answers of "yes", "No", and "I am not sure, I do not know, … etc. i.e. IFS has membership, non-membership, and hesitant functions, respectively, which is help to represent human information in medical application, multi-attribute decision making, renewable energy, manufacturing industry, and other fields ^[3,4,5,6,7].

Two approaches in the literature have been generated from the idea of combining FS and the mathematical field of algebra. In 1971, the first approach was given by Rosenfeld ^[8] under the concept of fuzzy subgroup. Many researchers studied, generalized, and discussed Rosenfeld approach ^{[9,10,11,12,13,14,15]}. In 1989, Biswas introduced intuitionistic fuzzy subgroup (IFSG) and its properties ^[16]. The notion of (α, β)-cut of IFSG was presented by Sharma ^[17]. Sharma and other researcher have been widely studied IFSG and its properties ^{[18,19,20,21,22,23]}. In 1994, the second approach was introduced by Dip ^[24], named fuzzy group based on fuzzy space. Before started his new approach, Dib and Youssef established a new structure of fuzzy cartesian product, relations and functions ^[25]. Dib approach can be summarized in replacing fuzzy space and fuzzy binary operation instead of the universal set and binary operation, respectively, at traditional algebra. The fuzzy normal subgroup is defined using fuzzy space by Dib and Hassan ^[26]. Marashdeh and Salleh generalized the concept of fuzzy space to intuitionistic fuzzy space to create the theory of Intuitionistic Fuzzy Group (IFG) ^[27] and then Intuitionistic Fuzzy Normal Subgroup (IFNSG) ^[28].

Moreover, complicated information involving periodicity and changeable meaning for the same data gives a new type of uncertainty. This type of complicated information comes from the rapid development of our daily life and modern technology. Therefore, we need a proper mathematical tool that has the ability to represent the uncertainty and periodicity semantics of information at the same time. So, the expected tool should be helpful and easily used by decision-makers to decrease the difficulties of giving a proper solution to the decision-making problems. To beat this obstacle, Ramot et al. ^[29,30] introduced the concept of Complex Fuzzy Sets (CFS) and logic. Also, Alkouri and Salleh ^[31,32,33] introduced Complex Intuitionistic Fuzzy Set (CIFS), relations and its operations. Both CFS and CIFS generated an extra range which lies within the unit disk in the complex plane. This extension helps to represent complicated information that carries time-periodic problems and two-dimension data simultaneously in a single set. Differences between CFS and CIFS appear in the ability of CIFS to represent information in more detail by using an extra complex non-membership function. Besides CIFS is considered a generalization of CFS that can solve a problem that seems too much roughs in CF ^[30,31]. Several enhancement and development applications using CIFS and its generalizations have increased rapidly appeared in different fields ^{[34,35,36,37,38,39,40]}, for example, decision-making process, information security management, graph and group, cellular network and etc.

Thus, it is greatly necessary to create additional notions of IFS and CIFS relating to complex set members. In 2016, Alhusbann and Salleh ^[41] introduced the notion of complex fuzzy group based on Dib approach. One year Later, Alsarahead and Ahmed ^[42,43,44] produced different notions named complex fuzzy subgroup, complex fuzzy subring and complex fuzzy soft subgroups from Rosenfeld and Liu approach ^[8,45]. In 2021, (α, β)-complex fuzzy sets, subgroups, and their properties were introduced by Alolaiyan et al. ^[46]. Furthermore, they introduced the notion of (α, β)-complex fuzzy cosets to formulate (α, β)-complex fuzzy normal subgroup. Besides, (α, β)-complex fuzzy quotient ring induced by (α, β)-CFNSG and (α, β)-complex fuzzification of Lagrange's Theorem were also derived in ^[46]. Concurrently, with developing the fuzzy subgroup to the complex fuzzy subgroup. IFS has been clearly developed into IFG and IFNSG ^[27,28]. Also, in the realm of complex numbers, the notion of Complex Intuitionistic Fuzzy Group (CIFG) and Complex Intuitionistic Fuzzy normal Subgroup were produced by Alhusban et al. in 2016 and 2017 ^[47,48].

Recently, Xiao and other researchers studied, generalized and applied the Complex evidence theory in the field of quantum mechanics and decisions. Also, they designed several algorithms, model and methods using quantum information in complex plane to predicting and describing human decision-making behaviors to be applied in pattern classification ^{[49,50,51,52]}.

In this study, our motivations are, 1- to introduce more generalized notion under CIFS, i.e. $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-Complex Intuitionistic Fuzzy Subgroups (CIFSG). 2- to define the reduced relations between our concept and each of (α, β)-CFS, and CIFS. 3- to study the algebraic structure of $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-complex intuitionistic fuzzy normal subgroup (CIFNSG) and some algebraic notions as coset, quotient group, and Lagrange theorem under $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$ CIFSG. Therefore, the main purpose of this study is to introduce a powerful extension of CIFS and (α, β)-complex fuzzy set and subgroup. Besides, we show the relation between each of $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFNSG and CIF cosets. Also, we prove some theorems connected the $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFSG of the classical quotient group and the set of all $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIF cosets. Finally, we expand the index and Lagrange's theorem to be suitable under $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFSG.

2.   Preliminaries

(Alpha 1, 2, Beta 1, 2) CIF- complex intuitionistic fuzzy cosets is a cornerstone in the structure of Lagrange's theorem under CIFSG. Also, Cosets is a particular type of CIF subgroup named by a CIF normal subgroup. Its can be used as the elements of another group called a quotient group or factor group under CIFS. As a future research, Cosets OF CIFS also may appear in other areas of mathematics such as vector spaces and error-correcting codes.

In this section, we recall some useful definitions to produce our work successfully.

Definition 2.1. ^[2] A complex intuitionistic fuzzy set A, defined by $A = \{\left\langle{x, {\mu }_{A}\left(x\right), {\gamma }_{A}\left(x\right)}\right\rangle:xϵX\}$, where ${\mu }_{A}\left(x\right):X\to \left\{a|a\in \complement , \left|a\right.\le 1\right\}$, ${\gamma }_{A}\left(x\right):X\to \left\{\stackrel{´}{a}|\stackrel{´}{a}\in \complement , \left|\stackrel{´}{a}\right.\le 1\right\}$, and $\left|{\mu }_{A}\left(x\right)\right.\left.{+\gamma }_{A}\left(x\right)\right|\le 1,$ and $i = \sqrt{-1}$, each of ${r}_{A}\left(x\right)$, ${k}_{A}\left(x\right)$ belong to the interval [0, 1] such that $0\le {r}_{A}\left(x\right)+{k}_{A}\left(x\right)\le 1$, also ${{w}^{r}}_{A}\left(x\right)$ and ${{w}^{k}}_{A}\left(x\right)$ are real-valued.

Definition 2.2 ^[2]. Let $A = \{\left\langle{x, {\mu }_{A}\left(o\right), {\gamma }_{A}\left(o\right)}\right\rangle:oϵO\}$ be a complex intuitionistic fuzzy set. Define the complement of $A, c\left(A\right)$, as $c\left(A\right) = \ddot{A} = \left\{\left\langle{x, {\gamma }_{A}\left(o\right), {\mu }_{A}\left(o\right)}\right\rangle:xϵX\right\} = \left\{\left\langle{x, {k}_{A}\left(o\right).{e}^{i\left({{w}^{k}}_{A}\left(o\right)\right)}, {r}_{A}\left(o\right){e}^{i\left({{w}^{r}}_{A}\left(o\right)\right)}}\right\rangle:oϵO\right\}$, where ${{w}^{k}}_{A}\left(o\right) = {w}_{A}\left(o\right), 2\pi -{w}_{A}\left(o\right)$ or ${w}_{A}\left(o\right)+\pi$.

Definition 2.3. ^[2] Let ${\mu _A}(o) = {r_A}(o)e{{\text{ }}^{i{\text{ }}{\omega ^r}_{{\text{ }}A}(o)}}, \, \, and\, \, {\nu _A}(o) = {k_A}(o)e{{\text{ }}^{i{\text{ }}{\omega ^k}_{{\text{ }}A}(o)}},$ and ${\mu _B}(o) = {r_B}(o)e{{\text{ }}^{i{\text{ }}{\omega _{{\text{ }}B}}(o)}},$$and\, \, {\nu _B}(o) = {k_B}(o)e{{\text{ }}^{i{\text{ }}{\omega ^k}_B(o)}},$ Be two membership and non-membership functions of complex intuitionistic fuzzy sets A and B respectively, on O.

a. B is subset of A, "$A \supseteq B$ or $B \subseteq A$", if for any $o \in O,$ ${r_A}(o) \leqslant {r_B}(o), \, \, {k_A}(o) \geqslant {k_B}(o),$ ${\omega ^r}_A(o) \leqslant {\omega ^r}_B(o), \, \, and\, \, {\omega ^k}_A(o) \geqslant {\omega ^k}_B(o).$

b. A union B, $A \cup B$, as

c. A intersection B, denoted by $A \cap B$, as

Some definitions related to ($\alpha , \beta$)-CFS and ($\alpha , \beta$)-CFSG of a group G are selected from the reference of Alolaiyan et al. ^[46], as follow:

Definition 2.4. ^[46] $\mathrm{L}\mathrm{e}\mathrm{t}\;\mathrm{A} = \left\{ < a, S\left(a\right){e}^{i\alpha {w}_{A}^{S\left(a\right)}} > :a\mathrm{ϵ}G\right\}$, be CFS of group G, for any $\alpha ϵ\left[\mathrm{0, 1}\right], and\;\beta ϵ\left[\mathrm{0, 2}\pi \right],$ such that $S\left(a\right)\le \alpha ,$ ${w}_{A}^{S}\le \beta ,$ or ($S\left(a\right)\ge \alpha , {w}_{A}^{s}\ge \beta$). Then, the set ${A}_{(\alpha , \beta )}$ is called an ($\alpha , \beta$)-CFS and is defined as:

where, ${S}_{{A}_{\alpha }}{e}^{i{w}_{{A}_{\beta }}^{S\left(a\right)}}$ is called a complex membership function of ($\alpha , \beta$)-complex fuzzy sets.

Definition 2.5. ^[46]. Let ${A}_{(\alpha , \beta )}$ be an ($\alpha , \beta$)-CFS of group G for $\alpha , \beta ϵ\left[\mathrm{0, 1}\right]$. Then, ${A}_{(\alpha , \beta )}$ is called

($\alpha , \beta$)-CFSG of group G if it satisfies the following axioms:

Definition 2.6. ^[46]. Let ${A}_{(\alpha , \beta )}$ be an $(\alpha , \beta )-$ CFSG of group $G$, where $\alpha \in \left[\mathrm{0, 1}\right]$ and $\beta \in \left[\mathrm{0, 2}\mathrm{\pi }\right]$. Then the $\left(\alpha , \beta \right)-$ CFS $g{A}_{(\alpha , \beta )}\left(a\right) = \left\{\left(a, {S}_{g{A}_{\alpha }}\left(a\right){e}^{i{w}_{{gA}_{\beta }}^{{S}_{g{A}_{\alpha }}}\left(a\right)}\right), a\in G\right\}$ of $G$ is called a $(\alpha , \beta )\text{-}$ complex fuzzy left coset of $G$ determined by ${A}_{(\alpha , \beta )}$ and $g$ and is described as:

Similarly, they defined $(\alpha , \beta )$-complex fuzzy right coset ^[45].

Definition 2.7. ^[46]. Let ${A}_{(\alpha , \beta )}$ be an $(\alpha , \beta )-$ CFSG of group $G$, where $\alpha \in \left[\mathrm{0, 1}\right]$ and $\beta \in \left[\mathrm{0, 2}\mathrm{\pi }\right]$. Then, ${A}_{(\alpha , \beta )}$ is called a $(\alpha , \beta )$-CFNSG if ${A}_{(\alpha , \beta )}\left(gh\right) = {A}_{(\alpha , \beta )}\left(hg\right)$. Equivalently $(\alpha , \beta )$-CFSG ${A}_{(\alpha , \beta )}$ is $(\alpha , \beta )$-CFNSG of group $G$ if: ${A}_{(\alpha , \beta )}g\left(h\right) = g{A}_{(\alpha , \beta )}\left(h\right)$, for all $g, h\in G$.

Definition 2.8. ^[45]. Let ${A}_{\left(\alpha , \beta \right)}$ be an $\left(\alpha , \beta \right)$-CFSG of finite group $G$. Then, the cardinality of the set $G/{A}_{\left(\alpha , \beta \right)}$ of all $\left(\alpha , \beta \right)$-complex fuzzy left cosets of $G$ by ${A}_{\left(\alpha , \beta \right)}$ is called the index of $\left(\alpha , \beta \right)$-CFSG and is denoted by $\left[G:{A}_{\left(\alpha , \beta \right)}\right]$.

3.   (${\mathit{\alpha }}_{1, 2}, {\mathit{\beta }}_{1, 2}$)-complex intuitionistic fuzzy subgroups

In this section, we define the hybrid models of (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSs and (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSGs. We prove that every CIFSG is also (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSG but the converse may not be true generally, and we discuss some basic characterization of this phenomenon.

Definition 3.1. $\mathrm{L}\mathrm{e}\mathrm{t}\;\mathrm{A} = \left\{ < a, S\left(a\right){e}^{i\alpha {w}_{A}^{S\left(a\right)}}, L\left(a\right){e}^{i\alpha {w}_{A}^{L\left(a\right)}} > :\mathrm{a}\mathrm{ϵ}\mathrm{G}\right\}$, be CIFS of group G, for any ${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}ϵ\left[\mathrm{0, 1}\right], S, L, {w}_{A}^{S}, {w}_{A}^{L}\in \left[\mathrm{0, 1}\right]$ such that $S\le {\alpha }_{1}, L\ge {\alpha }_{2}$ ${w}_{A}^{S}\le {\beta }_{1}, {w}_{A}^{L}\ge {\beta }_{2}$ or ($S\ge {\alpha }_{1}, L\le {\alpha }_{2}, {w}_{A}^{s}\ge {\beta }_{1}, {w}_{A}^{l}\le {\beta }_{2}$), Then, the set ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is called an (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFS and is defined as:

where

In this paper we shall use $\mathrm{S}{A}_{{\alpha }_{1}}{e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{\mathrm{S}{A}_{{\alpha }_{1}}\left(a\right)}}, \mathrm{L}{A}_{{\alpha }_{2}}{e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{\mathrm{L}{A}_{{\alpha }_{2}}\left(a\right)}}$ as a membership function and non-membership function of (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSs, ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}and{B}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ respectively.

Definition Let ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$, ${B}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ be a two (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSs of G. Then

(1) A (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)–CIFS ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is homogeneous (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)–CIFS if, for all $p, qϵG$, we have ${\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right)\le {\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(q\right), {L}_{{A}_{{\alpha }_{2}}}\left(p\right)\le {L}_{{A}_{{\alpha }_{2}}}\left(q\right)$ if and only if ${w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(a\right)}\left(p\right)\le {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(a\right)}\left(q\right)$, ${w}_{{A}_{{\beta }_{2}}}^{{L}_{{A}_{{\alpha }_{2}}}\left(a\right)}\left(p\right)\le {w}_{{A}_{{\beta }_{2}}}^{{L}_{{A}_{{\alpha }_{2}}}\left(a\right)}\left(q\right)$.

(2) A (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)–CIFS ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is homogeneous (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)–CIFS with ${B}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ for all $p, qϵG$, we have ${\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right)\le {\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(q\right), {\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(p\right)\le {L}_{{A}_{{\alpha }_{2}}}\left(q\right)$ if and only if ${w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(a\right)}\left(p\right)\le {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(a\right)}\left(q\right)$, ${w}_{{A}_{{\beta }_{2}}}^{{L}_{{A}_{{\alpha }_{2}}}\left(a\right)}\left(p\right)\le {w}_{{A}_{{\beta }_{2}}}^{{L}_{{A}_{{\alpha }_{2}}}\left(a\right)}\left(q\right)$.

In this paper, we take (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFS as homogeneous (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFS.

Remark 1. Let, ${M}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ and ${N}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ be two (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSs of G. Then $({M\cap N)}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = {M}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\cap {N}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$

Example 3.1. Consider a group ${\mathrm{Z}}_{4}$ = {0, 1, 2, 3} is a group. Let ${\alpha }_{1} = 0.7, {\alpha }_{2} = 0.2, {\beta }_{1} = 0.4, and{\beta }_{2} = 0.5$, and a CIFS $\mathrm{A} = \left\{ < 0, 0.8{e}^{i\alpha 0.4}, 0.1{e}^{i\alpha 0.3} > , < 1, 0.4{e}^{i\alpha 0.1}, 0.5{e}^{i\alpha 0.6} > , \; < 2, 0.3{e}^{i0.6}, 0.7{e}^{i\alpha 0.3} > , \; < 3, 0.7{e}^{i\alpha 0.4}, 0.1{e}^{i\alpha 0.5} > \right\}\mathrm{o}\mathrm{f}\;\mathrm{a}\;\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}{\mathrm{Z}}_{4}$, Then, the set ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is called an (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFS and is defined as:

Definition 3.2. Let ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ be an (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFS of group G for ${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}ϵ\left[\mathrm{0, 1}\right]$. Then, ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is called (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-Complex Intuitionistic Fuzzy subgroupoid of group G if the following axioms hold:

Example 3.2. Recalling that ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})} = \left\{ < 0, 0.7{e}^{i\alpha 0.4}, 0.2{e}^{i\alpha 0.5} > , < 1, 0.4{e}^{i\alpha 0.1}, 0.5{e}^{i\alpha 0.6} > , \; < 2, 0.3{e}^{i0.4}, 0.7{e}^{i\alpha 0.5} > \right.,$ $\left. < 3, 0.7{e}^{i\alpha 0.4}, 0.2{e}^{i\alpha 0.5} > \right\}$ from Example 3.1. satisfies all axioms of Definition 3.3. for all elements in the group ${\mathrm{Z}}_{4}$. For example (Let p = 1, q = 2), so pq = 3, ${p}^{-1} = 3$ in ${\mathrm{Z}}_{4}$. Then:

Axiom 1:

Axiom 2:

Axiom 3:

Axiom 4:

Definition 3.3. Let ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ be an (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFS of group G for ${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}ϵ\left[\mathrm{0, 1}\right]$. Then, ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is named.

(${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSG of group G if the following axioms hold:

$(1) \;{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(pq\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(pq\right)}}\ge \mathrm{m}\mathrm{i}\mathrm{n}\{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{S\left(p\right)}}, {\mathrm{S}}_{{A}_{{\alpha }_{1}}}(q\left){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(q\right)}}\right\} .$

$(2) \;{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left({p}^{-1}\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left({p}^{-1}\right)}}\ge {\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right)}}.$

$(3) \;{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(pq\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{L}_{{A}_{{\alpha }_{2}}}\left(pq\right)}}\le \mathrm{max}\{{L}_{{A}_{{\alpha }_{2}}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{L}_{{A}_{{\alpha }_{2}}}\left(p\right)}}, {L}_{{A}_{{\alpha }_{2}}}(q\left){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{L}_{{A}_{{\alpha }_{2}}}\left(q\right)}}\right\} .$

$(4)\; {L}_{{A}_{{\alpha }_{2}}}\left({p}^{-1}\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{L}_{{A}_{{\alpha }_{2}}}\left({p}^{-1}\right)}}\le {L}_{{A}_{{\alpha }_{2}}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{L}_{{A}_{{\alpha }_{2}}}\left(p\right)}}, \mathrm{f}\mathrm{o}\mathrm{r}\;\mathrm{a}\mathrm{l}\mathrm{l}p, qϵG .$

Remark 2. ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ be an (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFS of group G, for ${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}ϵ\left[\mathrm{0, 1}\right]$. Then,

$(1)\; {\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left({p}^{-1}q\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left({p}^{-1}q\right)}}\ge \mathrm{m}\mathrm{i}\mathrm{n}\{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right)}}, {\mathrm{S}}_{{A}_{{\alpha }_{1}}}(q\left){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(q\right)}}\right\} .$

$(2) \;{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left({p}^{-1}q\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left({p}^{-1}q\right)}}\le \mathrm{max}\{\mathrm{L}{A}_{{\alpha }_{2}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(p\right)}}, \mathrm{L}{A}_{{\alpha }_{2}}(q\left){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(q\right)}}\right\} .$

Theorem 3.1. If ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ be an (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFS of group G, for all $p, qϵG$. Then,

(1) ${\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(q\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(q\right)}}\le {\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(e\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(e\right)}}$, (2) ${\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left({p}^{-1}q\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left({p}^{-1}q\right)}} = {\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(e\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(e\right)}}$,

(3) ${\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(q\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(q\right)}}\ge {\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(e\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(e\right)}}$, (4) ${\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left({p}^{-1}q\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left({p}^{-1}q\right)}} = {\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(e\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(e\right)}}$,

which implies that

and

Theorem 3.2. Let ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ be an (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$) complex intuitionistic fuzzy subgroupoid of a finite G, and then ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSG of finite G.

Proof. Let $pϵG$ and G is a finite group; therefore, pⁿ = e, where p has finite order n., where e is the natural element of group G. Then, we have ${p}^{-1} = {p}^{n-1},$ now, by using the Definition 3.3 twice, we get

Theorem3.3. Let ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ be an (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSG of a group G, Let $pϵG$ and

then

and

Proof.

(1) Given that ${\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right)}}$ = ${\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(e\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(e\right)}}$. Then form Theorem we have that

Let

Now, assume that

Again, from Theorem 3.1, we have

Therefore, we obtain

From Eqs (1) and (2), we have

(2) Assume that ${\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(p\right)}} = {\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(e\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(e\right)}}$. Then from Theorem 3.3 we have that

Let

Now, assume that

Again, from Theorem 3.1, we have

Therefore, we obtain

From Eqs (3) and (4), we have

Theorem 3.4 Every CIFSG of group G is also (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSG of G.

Proof. Assume A be CIFSG of group G, and p, q $ϵ$ G. then

Further, we assume that

Consider

Further, we assume that

Theorem 3.5. Suppose A is a CIFS of group G where

such that ${\alpha }_{1}\le {r}_{1}, {\beta }_{1}\le {w}_{1}$, ${\alpha }_{2}\le {r}_{2}, {\beta }_{2}\le {w}_{2}$.

Where

Then, ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is an (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSG of G.

Proof. Let ${\alpha }_{1}{e}^{i{\beta }_{1}}\le {r}_{1}{e}^{i{w}_{1}}$, implies that $\mathrm{min}\{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right)}}:p\in G\ge {\alpha }_{1}{e}^{i{\beta }_{1}}$, which implies that $\mathrm{min}\{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right)}},$ ${\alpha }_{1}{e}^{i{\beta }_{1}}\} = {\alpha }_{1}{e}^{i{\beta }_{1}}, \forall p\in G$, which implies that ${\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{\mathrm{S}}_{{A}_{{\alpha }_{1}}}\left(p\right)}} = {\alpha }_{1}{e}^{i{\beta }_{1}}$.

Moreover,

This implies that

Let ${\alpha }_{2}{e}^{i{\beta }_{2}}\le {r}_{2}{e}^{i{w}_{2}}$, implies that $\mathrm{max}\{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(p\right)}}:p\in G\le {\alpha }_{2}{e}^{i{\beta }_{2}},$} which implies that $\mathrm{max}\{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(p\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{\mathrm{L}}_{{A}_{{\alpha }_{2}}}\left(p\right)}}:p\in G, {\alpha }_{2}{e}^{i{\beta }_{2}}\} = {\alpha }_{2}{e}^{i{\beta }_{2}}$, $\forall p\in G$, which implies that

Therefore, ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is an (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSG of G.

Theorem 3.6. If ${M}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ and ${N}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ are two (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSGs of G, then ${M}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}\cap {N}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is also (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)- CIFSG of G.

Proof.

Further,

And

Further,

Consequently, ${M}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}\cap {N}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is also (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)- CIFSG of G.

Remark 3. The union of two $\left({\mathrm{\alpha }}_{\mathrm{1, 2}}, {\mathrm{\beta }}_{\mathrm{1, 2}}\right)-$ CIFSGs may not be $\left({\mathrm{\alpha }}_{\mathrm{1, 2}}, {\mathrm{\beta }}_{\mathrm{1, 2}}\right)-$ CIFSGs.

Example 3.3. Let ${S}_{3}$ be a symmetric group of all permutation of 3 elements. Suppose ${M}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ and ${N}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ are two sets of $\left({\mathrm{\alpha }}_{\mathrm{1, 2}}, {\mathrm{\beta }}_{\mathrm{1, 2}}\right)-$ CIFSGs of ${S}_{4}$, and ${\alpha }_{1} = 0.6, {\alpha }_{2} = 0.3, {\beta }_{1} = 0.4, and{\beta }_{2} = 0.5$, given as:

So,

Take ${M}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}{\cup N}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\left({\rho }_{1}\right) = < 0.6{e}^{i\alpha 0.2}, 0.3{e}^{i\alpha 0.6} >$ and ${M}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}{\cup N}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\left({\rho }_{2}\right) = < 0.5{e}^{i\alpha 0.1}, 0.5{e}^{i\alpha 0.5} >$, and ${M}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}{\cup N}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\left({\rho }_{1}{\rho }_{2} = {\rho }_{0}\right) = < 0.3{e}^{i\alpha 0.1}, 0.5{e}^{i\alpha 0.6} >$.

Clearly, Axiom 1 from definition 3.3 does not hold

4.   CIFNSG and lagrange's theorem under ${\text{(}}{{\rm{ \mathsf{ α} }}_{{\text{1, 2}}}}{\text{, }}{{\rm{ \mathsf{ β} }}_{{\text{1, 2}}}}{\text{)}}$-CIFSG

We get started by introducing the notation of $({\mathrm{\alpha }}_{\mathrm{1, 2}}, {\mathrm{\beta }}_{\mathrm{1, 2}})$-complex intuitionistic fuzzy cosets of $({\mathrm{\alpha }}_{\mathrm{1, 2}}, {\mathrm{\beta }}_{\mathrm{1, 2}})$-CIFSG. Hence, a quotient group induced by CIDNSGs generalized. After that we study Lagrange's theorem under (${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}$)-CIFSG.

Definition 13. Let ${A}_{({\mathrm{\alpha }}_{\mathrm{1, 2}}, {\mathrm{\beta }}_{\mathrm{1, 2}})}$ be an $({\mathrm{\alpha }}_{\mathrm{1, 2}}, {\mathrm{\beta }}_{\mathrm{1, 2}})-$ CIFSG of group $G$, where ${\mathrm{\alpha }}_{\mathrm{1, 2}}, {\mathrm{\beta }}_{\mathrm{1, 2}}\in \left[\mathrm{0, 1}\right]$. Then the $\left({\mathrm{\alpha }}_{\mathrm{1, 2}}, {\mathrm{\beta }}_{\mathrm{1, 2}}\right)-$ CIFS $g{A}_{({\mathrm{\alpha }}_{\mathrm{1, 2}}, {\mathrm{\beta }}_{\mathrm{1, 2}})}\left(a\right) = \left\{\left(a, {S}_{g{A}_{\;{\alpha }_{1}}}\left(a\right){e}^{i\alpha {w}_{{gA}_{\;{\beta }_{1}}}^{{S}_{g{A}_{\;{\alpha }_{1}}}}\left(a\right)}, {L}_{g{A}_{{\alpha }_{2}}}\left(a\right){e}^{i\alpha {w}_{{gA}_{\;{\beta }_{2}}}^{{L}_{g{A}_{\;{\alpha }_{2}}}}\left(a\right)}\right), a\in G\right\}$ of $G$ is called a $({\mathrm{\alpha }}_{\mathrm{1, 2}}, {\mathrm{\beta }}_{\mathrm{1, 2}})\text{-}$ complex intuitionistic fuzzy left coset of $G$ determined by ${A}_{({\mathrm{\alpha }}_{\mathrm{1, 2}}, {\mathrm{\beta }}_{\mathrm{1, 2}})}$ and $g$ and is describe as:

and

for all $o, g\in G.$

Similarly we can define $({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})$ - complex intuitionistic fuzzy right coset is described as:

and

for all $o, g\in G.$

Definition 14. Let ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ be an $({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})$-CIFSG of group $G$, where ${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\in \left[\mathrm{0, 1}\right]$. Then, ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is called a $({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})$-CIFNSG if ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}\left(gh\right) = {A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}\left(hg\right)$. Equivalently $({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})$-CIFSG ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is $({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})$-CIFNSG of group $G$ if: ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}g\left(h\right) = g{A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}\left(h\right)$, for all $g, h\in G$.

Remark 4. Let ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ be an $({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})$-CIFNSG of group $G$. Then, ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}\left({h}^{-1}gh\right) = {A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}\left(g\right)$, for all $g, h\in G$.

Theorem 7. If $A$ is CIFNSG of group $G$, then ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is an $({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})$-CIFNSG of $G$.

Proof. Suppose $o, g$ are elements of $G$. Then, for the membership function we have

This implies that

which implies that ${S}_{g{A}_{\;{\alpha }_{1}}}\left(o\right){e}^{i\alpha {w}_{g{A}_{{\beta }_{1}}}^{{S}_{g{A}_{\;{\alpha }_{1}}}}\left(o\right)} = {S}_{{A}_{{\alpha }_{1}}g}\left(o\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}g}^{{S}_{{A}_{{\alpha }_{1}}g}}\left(o\right)}.$ This implies that $g{A}_{({\alpha }_{1}, {\beta }_{1})}\left(o\right) = {A}_{({\alpha }_{1}, {\beta }_{1})}g\left(o\right).$ Now, for the non-membership function we have:

This implies that

which implies that ${L}_{g{A}_{{\alpha }_{2}}}\left(o\right){e}^{i\alpha {w}_{g{A}_{{\beta }_{2}}}^{{L}_{g{A}_{\;{\alpha }_{2}}}}\left(o\right)} = {L}_{{A}_{{\alpha }_{2}}g}\left(o\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}g}^{{L}_{{A}_{{\alpha }_{2}}g}}\left(o\right)}.$ This implies that $g{A}_{({\alpha }_{2}, {\beta }_{2})}\left(o\right) = {A}_{({\alpha }_{2}, {\beta }_{2})}g\left(o\right).$

Therefore, ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is $({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})$-CIFNSG of $G$.

Theorem 8. Let ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ be $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFSG of group $G$ such that for membership we have ${\alpha }_{1}{e}^{i{\beta }_{1}} < r{e}^{i{w}^{r}}$ such that ${\alpha }_{1}\le r$ and ${\beta }_{1}\le {w}^{r}$, where ${re}^{i{w}^{r}} = \mathrm{m}\mathrm{i}\mathrm{n}\left\{{S}_{A}\left(o\right){e}^{i\mathrm{\alpha }{\mathrm{w}}_{A}^{S}\left(o\right)}, \forall o\in G\right\}$ and for non-membership we have ${\alpha }_{2}{e}^{i{\beta }_{2}} < k{e}^{i{w}^{k}}$ such that ${\alpha }_{2}\ge k$ and ${\beta }_{2}\ge {w}^{k}$, where ${ke}^{i{w}^{k}} = \mathrm{m}\mathrm{a}\mathrm{x}\left\{{L}_{A}\left(o\right){e}^{i\mathrm{\alpha }{\mathrm{w}}_{A}^{L}\left(o\right)}, \forall o\in G\right\}$ and ${\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}, r, k, {w}^{r}, {w}^{k}\in \left[\mathrm{0, 1}\right].$ with CIFS conditions: $0\le {\alpha }_{1}+{\alpha }_{2}\le 1$, $0\le {\beta }_{1}+{\beta }_{2}\le 1$, $0\le r+k\le 1$, and $0\le {w}^{r}+{w}^{k}\le 1.$ Then, ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ is a $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-complex intuitionistic fuzzy normal subgroup of group $G$.

Proof. Having ${\alpha }_{1}{e}^{i{\beta }_{1}} < r{e}^{i{w}^{r}}$ then we get $\mathrm{m}\mathrm{i}\mathrm{n}\left\{{S}_{A}\left(o\right){e}^{i\mathrm{\alpha }{\mathrm{w}}_{A}^{S}\left(o\right)}, \forall o\in G\right\}\ge {\alpha }_{1}{e}^{i{\beta }_{1}}$, which implies that ${S}_{A}\left(o\right){e}^{i\mathrm{\alpha }{\mathrm{w}}_{A}^{S}\left(o\right)}\ge {\alpha }_{1}{e}^{i{\beta }_{1}}$, for all $o\in G.$ And ${\alpha }_{2}{e}^{i{\beta }_{2}} > k{e}^{i{w}^{k}}$ implies that $\mathrm{m}\mathrm{a}\mathrm{x}\left\{{L}_{A}\left(o\right){e}^{i\mathrm{\alpha }{\mathrm{w}}_{A}^{L}\left(o\right)}, \forall o\in G\right\}\le {\alpha }_{2}{e}^{i{\beta }_{2}}$, which implies that ${L}_{A}\left(o\right){e}^{i\mathrm{\alpha }{\mathrm{w}}_{A}^{L}\left(o\right)}\le {\alpha }_{2}{e}^{i{\beta }_{2}}$, for all $o\in G.$

Thus,

and

${L}_{g{A}_{{\alpha }_{2}}}\left(o\right){e}^{i\alpha {w}_{g{A}_{{\beta }_{2}}}^{{L}_{g{A}_{\;{\alpha }_{2}}}}\left(o\right)} = \mathrm{m}\mathrm{a}\mathrm{x}\left\{{L}_{A}\left({g}^{-1}o\right){e}^{i\alpha {w}_{A}^{L}\left({g}^{-1}o\right)}, {\alpha }_{2}{e}^{i{\beta }_{2}}\right\} = {\alpha }_{2}{e}^{i{\beta }_{2}}$, for any $a\in G$.

Similarly,

and

This implies that

and

Theorem 9. Let ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ be an $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFSG of a group $G$, then ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ is an $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFNSG if and only if ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ is constant in the in the conjugacy class of group $G$.

Proof. Suppose that ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ is an $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFNSG. Then, so we get

and

Conversely, suppose that ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ is constant in all conjugate classes of group $G.$ Then,

and

Theorem 10. If ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ is an $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFSG of a group $G$, then ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ is an $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-complex intuitionistic fuzzy normal subgroup if and only if ${S}_{{A}_{{\alpha }_{1}}}\left(\right[g, h\left]\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{S}_{{A}_{\;{\alpha }_{1}}}}\;\left(\right[g, h\left]\right)}\ge {S}_{{A}_{{\alpha }_{1}}}\left(g\right){e}^{i\alpha {w}_{{A}_{{\beta }_{1}}}^{{S}_{{A}_{\;{\alpha }_{1}}}}\;\left(g\right)}$, and ${L}_{{A}_{{\alpha }_{2}}}\left(\left[g, h\right]\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{L}_{{A}_{\;{\alpha }_{2}}}}\;\left(\left[g, h\right]\right)}\le {L}_{{A}_{{\alpha }_{2}}}\left(g\right){e}^{i\alpha {w}_{{A}_{{\beta }_{2}}}^{{L}_{{A}_{\;{\alpha }_{2}}}}\;\left(g\right)}, \;\forall g, h\in G$.

Proof. Suppose that ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ is an $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$- CIFNSG. Let $x, y\in G$ be element of group. So,

and

Conversely, suppose that

and

Let $g, o\in G$ be an element. Consider

and,

Thus,

and,

Now,

And,

Here. Two cases have to be proved.

Case 1. If

Then, we obtain

And if

Then, we obtain

This implies that ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is a constant mapping.

Case 2. If

Then, from Eq (7) we have

and if

Then, from Eq (7) we have

In the view of Equations (6, 6*) and (8, 8*) we have

and

Hence, ${A}_{({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}})}$ is constant.

Theorem 11. Let ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ be $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFNSG of group $G$. Then, the set ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id} = \{x\in G:\left.{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\left({x}^{-1}\right) = {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\left(id\right)\right\}$ is a normal subgroup of group $G$.

Proof. Clearly that ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}\ne \varnothing$ because $id\in G$. Let $x, y\in {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ be any elements. Consider

And

This implies that

and

However,

and

Therefore,

This implies that

which implies that $xy\in {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$.

Further,

and

However,

and

Thus, ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ is subgroup of group $G$. Moreover, let $x\in {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ and $y\in G$. We have

and

This implies that ${y}^{-1}xy\in {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$. Hence, ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ is a normal subgroup.

Theorem 12. Let ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ be an $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFNSG of group $G$. Then,

Proof. (i) For any $g, h\in G$, we have $g{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = h{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$.Consider,

and

Therefore, ${g}^{-1}h\in {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$.

Conversely, let ${g}^{-1}h\in {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ implies that

and

Replace the position of $g$ and $h$, and we gain

and

Therefore,

and

(ⅱ) Similar to part (ⅰ).

Theorem 13. Let ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ be an $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFNSG of group $G$ and $g, h, o$, and $f$ be any elements in $G$. If $g{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = o{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ and $h{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = f{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$, then $gh{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = of{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$.

Proof. Given that $g{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = o{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ and $h{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = f{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$. This implies that ${g}^{-1}o, {h}^{-1}f\in {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$. Consider,

As ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ is normal subgroup of $G$. Therefor,

Consequently,

Theorem 14. Let $G/{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = \left\{g{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}:g\in G\right\}$ be the collection of all $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-complex intuitionistic fuzzy cosets of $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFNSG ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ of $G$. Then, $(\star )$ is a well-defined binary operation under $G/{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ and is defined as $g{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\star h{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = gh{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ for all $g, h\in G$.

Proof. We have $g{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = h{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ and $o{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = f{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$, for any $o, f, g, h\in G$. Let $v\in G$ be any element, then

Consider,

And,

Hence, we concluded that the axiom of associative and closure under the presented binary operation * are satisfied for the set $/{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$.Further,

and

an element of $G/{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$. So the inverse of every element of $G/{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ exist if

And

and then there exists an element,

And

such that

And

As a result, $G/{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ is a group. And is called the CIF quotient group of the G.

Lemma 1. Let $m:G\to G/{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ be natural homomorphism and defined by the rule, $m\left(g\right) = g{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ with the kernel $m = {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$.

Proof. Let $g, h$ be any elements of group $G$, and then

And

Thus, $m$ is homomorphism. Now,

Theorem 15. Let ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ be a normal subgroup of $G$. If

is $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFSG, then, the $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFS.

of $G/{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ is also a $(\alpha , \beta )$-CIFSG of $G/{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ where

And

Proof. First, we shall prove that

and

is well-defined. Let $g{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id} = h{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ then $h = go$, for some $o\in {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$.

Now,

And,

Therefore,

and

is will-defined.

Consider

And

And

Remark 5. If ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ is an $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFSG of a group $G$, let $g\in G$ and

and

for all $h\in G$ then

and

Definition 15. Let ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ be a $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-CIFSG. Then, the cardinality of the set $G/{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ of all $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-complex intuitionistic fuzzy left cosets of $G$ by ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ is called the index of $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-complex intuitionistic fuzzy subgroup and is represented by $\left[G:{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\right]$.

Theorem 16. ($\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-Complex intuitionistic Fuzzification of Lagrange's Theorem): Assume that a $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-complex intuitionistic fuzzy subgroup ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ of finite group $G$. Then, the index of $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-complex intuitionistic fuzzy subgroup of $G$ divides the order of $G$.

Proof. By Lemma 1, Define a subgroup $\stackrel{‾}{M} = \left\{x\in G:x{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = id{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\right\}$. Using the Definition 13 $x\in \stackrel{‾}{M}$ and $v\in G$, we have $x{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\left(v\right) = id{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\left(v\right)$. This implies that ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\left({x}^{-1}v\right) = {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\left(v\right)$, by Remark 5, which shows that $x\in {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$. Therefore, $\stackrel{‾}{M}$ is contained in ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}.$ Now, we take any element $x\in {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ and using the fact ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ is subgroup of $G$, we have ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\left({x}^{-1}\right) = {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\left(id\right)$. From Theorem 13, the elements ${x}^{-1}, v\in {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$, which means that $x{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = id{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$, which implies that $x\in \stackrel{‾}{M}$. Hence, ${A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ is contained in $\stackrel{‾}{M}$. From this discussion, we can say that $\stackrel{‾}{M} = {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$.

Now, we define the partition of the group $G$ into the disjoint union of right cosets, and this is defined as $G = {s}_{1}\stackrel{-}{M}$

where ${s}_{1}\stackrel{‾}{M} = \stackrel{‾}{M}$. Now, we prove that, to each coset ${s}_{j}\stackrel{‾}{M}$ in relation (i), there exists an $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-complex intuitionistic fuzzy coset ${s}_{j}{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$ in $G/{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$, and this corresponding is injective.

Consider any coset ${s}_{j}{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$. Let $x\in {A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$, then

Thus, $m$ maps each element of ${s}_{j}{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$ into the $\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)$-complex intuitionistic fuzzy coset ${s}_{j}{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$.

Between the set ${}^{G}\!\!\diagup\!\!{}_{A_{\left( {{\alpha }_{\text{1},\text{2}}},{{\beta }_{\text{1},\text{2}}} \right)}^{id}}\;$ and ${s}_{j}{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}, 1\le j\le k$ a natural correspondence $\stackrel{‾}{m}$ can be defined by

The correspondence $\stackrel{‾}{m}$ is injective.

For this, let ${s}_{i}{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = {s}_{l}{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$, then ${s}_{l}^{-1}{s}_{i}{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)} = id{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}$. By using $\left(A\right)$, we have ${s}_{l}^{-1}{s}_{i}\in \stackrel{‾}{M}$, which means that ${s}_{i}{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id} = {s}_{i}{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}$, and hence $\stackrel{‾}{m}$ is injective. It is quite clear from the above discussion that $\left[G:{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}\right]$ and $\left[G:{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}\right]$ are equal. Hence, $\left[G:{A}_{\left({\alpha }_{\mathrm{1, 2}}, {\beta }_{\mathrm{1, 2}}\right)}^{id}\right]$ divides $O\left(G\right)$.

5.   Conclusions

The concept of (α1, 2, β1, 2)-CIFSs was introduced as a generalization of (α1, 2, β1, 2)-CFS and classical CIFS. Then we defined (α1, 2, β1, 2)-CIF Subgroup and studied its algebraic structure. In order to establish Lagrange theorem under (α1, 2, β1, 2)-CIFSs, we introduced the notion of (α1, 2, β1, 2)−CIFS cosets. A special type of subgroup, named (α1, 2, β1, 2)-CIFNSGs, is created by using the notion of cosets under (α1, 2, β1, 2)-CIFSs. Moreover, we established an (α1, 2, β1, 2)-complex intuitionistic fuzzy quotient ring induced by (α1, 2, β1, 2)-CIFNSG. As future research, we may use our concepts to improve the assessment and prioritization method of key engineering characteristic for complex products ^[53]. Also can be employed current concepts in decision making problems and machine learning algorithm ^[54] to enhance the results in both references ^[53,54].

Conflict of interest

The authors declare there are no conflicts of interest.