A New Type of Soft Group: Soft Symmetric Difference Group with Group Theory Applications

Abstract

In this paper, a new type of soft group called the soft symmetric difference group (SSD-group) is introduced and systematically developed. This structure is constructed by integrating soft set theory with group theory through the symmetric difference operation and set inclusion. Fundamental concepts such as characteristic soft symmetric difference groups, soft symmetric difference subgroups, normal soft symmetric difference subgroups, soft normalizers, and soft cosets are defined, and their essential algebraic properties are investigated. Several characterizations of soft normality are also established through these concepts. Various axiomatic results are obtained, providing necessary and sufficient conditions for a soft set to form an SSD-group. Furthermore, soft quotient (factor) groups of SSD-groups are introduced and their structural properties are examined in detail. The relationship between SSD-group theory and classical group theory is also established through several corresponding concepts. Illustrative examples are provided to demonstrate the applicability and internal consistency of the proposed framework. Overall, the results obtained in this study extend existing soft group structures and contribute to the development of algebraic theory within the context of soft sets, while also providing a foundation for further generalizations to other algebraic frameworks such as semigroups, rings, and fields.

1. Introduction

A wide range of mathematically rigorous frameworks has been developed to model and analyze systems affected by uncertainty, vagueness, and indeterminacy—features that are intrinsic to diverse domains such as engineering, economics, social sciences, and medical diagnostics. Nevertheless, classical approaches—most notably fuzzy set theory introduced by Zadeh [1] and traditional probabilistic models—possess inherent epistemological and algebraic limitations. Specifically, fuzzy sets depend on subjectively defined membership functions, while probabilistic systems presuppose well-defined distributions and reproducible conditions, both of which are often unattainable in real-world contexts.

To address these shortcomings, Molodtsov [2] introduced soft set theory as a parameter-based framework that circumvents the constraints of probabilistic and fuzzy methodologies, providing an algebraically neutral means for handling uncertainty. The foundational operations were first proposed by Maji et al. [3] and later refined by Pei and Miao [4] from an information-theoretic perspective. Ali et al. [5] further broadened the operational framework by introducing restricted and extended versions of classical operations. Vital contributions by Yang [6], Feng et al. [7], Jiang et al. [8], Ali et al. [9], Sezgin and Atagün [10], Neog and Sut [11], Fu [12], Ge and Yang [13], Park et al. [14], Singh and Onyeozili [15,16,17,18], Zhu and Wen [19], Jayanta [20], Onyeozili and Gwary [21], and Husain and Shivani [22] have collectively deepened the algebraic foundations of soft set theory by clarifying definitional ambiguities, formulating new operations, and generalizing soft equality. Fundamental operations in soft set theory—namely, restricted and extended intersection, union, and difference—were systematically redefined and thoroughly investigated in [23,24,25]. In these works, earlier results were unified, previously omitted cases were addressed, and several incorrect theorems were rigorously corrected. Collectively, studies [23,24,25] represent a major advancement in the development of soft set theory by filling a crucial theoretical gap, providing a comprehensive reference for researchers new to the area, and opening promising directions for further investigation. Consequently, research on various extensions of soft sets has grown significantly in recent years [26,27,28,29,30,31,32,33,34,35,36,37,38,39] and has been successfully applied to a wide range of decision-making problems [40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55].

Parallel progress in generalized soft subsethood and equality has further consolidated this theoretical framework. Building upon early contributions by Pei and Miao [4] and Qin and Hong [56], subsequent efforts by Jun and Yang [57] and Liu et al. [58] introduced J-soft and L-soft equalities. Feng and Li [59] extended this line of work by classifying soft subsets under L-equality and revealing semigroup structures within the corresponding quotient classes. Broader generalizations—such as g-soft, gf-soft, and T-soft equalities—proposed by Abbas et al. [60,61], Al-shami [62], and Al-shami and El-Shafei [63], incorporated congruence and lattice-theoretic structures into the algebraic study of soft sets. In particular, Ali et al. [64] developed generalized finite relaxed soft equality together with the corresponding union and intersection operations, thereby forming a robust generalization framework. From a different perspective, Sezgin et al. [65] and Orbay et al. [66] conducted rigorous theoretical investigations of the AND-product and OR-product, respectively, and thoroughly analyzed their algebraic properties in connection with soft F-subsets and soft M-equality.

A major milestone in the formal development of soft set theory was achieved through the axiomatic reformulation proposed by Çağman and Enginoğlu [67], which resolved internal inconsistencies and established a mathematically rigorous foundation for algebraic analysis. This advancement facilitated the introduction of various soft products on classical algebraic structures. For instance, the soft intersection–union product has been generalized to ring [68], semigroup [69], and group [70] contexts, resulting in the notions of soft intersection rings, soft intersection semigroups, and soft intersection groups, respectively. Its dual—the soft union–intersection product—has been examined within group-theoretic [71], semigroup-theoretic [72], and ring-theoretic [73] frameworks, leading to the corresponding concepts of soft union groups, soft union semigroups, and soft union rings.

Aktaş and Çağman [74,75] introduced the notion of a soft group as a parametrized family of subgroups of a given group, which initiated the study of the algebraic structure of soft sets. Within this framework, Sezgin and Atagün [76] examined normalistic soft groups and corrected several properties of soft groups originally defined in [74]. Aslam et al. [77] further developed the theory by introducing normal and abelian soft groups together with their fundamental properties. Their study also addressed cyclic soft groups and soft subgroups, soft cosets, the order and index of a soft group, partitions of soft sets over groups, as well as simple and factor soft groups. Aktaş and Özlü [78] extended these investigations by defining the order and power of soft groups and soft sets, and by analyzing cyclic soft groups in relation to classical group theory. Nazmal [79] studied homomorphic images and preimages of soft groups under soft mappings. In 2023, Barazegar et al. [80] introduced the notion of the commutator of soft groups and investigated nilpotent soft groups. They also defined second-type nilpotent soft subgroups and studied their fundamental properties. Alajlan and Alghamdi [81] examined the center of soft groups, kernels of soft homomorphisms, and characteristic soft subgroups. In 2025, Atagün et al. [82] explored applications of Sylow’s theorems within the framework of soft groups. It should be noted that all these contributions rely on the original soft group structure proposed by Aktaş and Çağman [74].

A significant advancement in the field was made by Çağman et al. [83], who introduced soft intersection groups (soft int-groups). This framework integrates inclusion relations and set intersections with group-theoretic operations, thus providing a unifying structure that connects set theory, group theory, and soft set theory. In Ref. [83], the authors defined soft int-subgroups and normal soft int-subgroups and introduced related concepts. Subsequent studies expanded this theory by establishing additional properties of soft int-groups and investigating relationships among the soft union–intersection product, soft inclusions, and group-theoretic structures [71]. Normal soft int-groups were analyzed in detail in [84], including results on α-inclusion, normalizers, quotient groups, and related constructions. Homomorphism and isomorphism theorems for soft int-groups, as well as the notions of soft quotient subgroups and quotient dual soft subgroups, were studied in [85,86,87], together with soft analogues of the fundamental isomorphism theorems. In 2024, Sezgin et al. [88] studied soft cosets for soft intersection groups in detail. For related concepts concerning soft union groups, the reader is referred to [70].

Cosets, normal subgroups, and quotient groups are central concepts in abstract algebra and group theory. Lagrange’s theorem, derived from counting cosets and their cardinalities, plays a fundamental role in understanding group structure. Normal subgroups are essential for constructing quotient groups, which often exhibit simpler structures than the original group. Consequently, the study of quotient groups provides an effective approach for gaining deeper insight into the structural properties of groups. The relationship between set theory and group theory is particularly strong, and group theory possesses a rich structure with numerous applications across diverse scientific fields. This richness provides strong motivation for its incorporation into soft set theory. In the current literature, two types of soft groups based on inclusion structures have been introduced: the soft intersection group (soft int-group) defined in [83] and the soft union group (soft uni-group) defined in [89].

Motivated by ongoing developments in soft algebraic structures and the limitations of existing soft group frameworks, this paper introduces a new type of soft group called the soft symmetric difference group (SSD-group). It is important to emphasize that, despite the terminology, SSD-groups are not groups in the classical algebraic sense. The SSD-condition establishes a structural compatibility between a group and a set-valued mapping, but the collection of SSD-functions does not, in general, satisfy the group axioms under the induced operation. Thus, SSD-groups should be understood as generalized algebraic structures derived from group-theoretic principles rather than as groups themselves. By employing the symmetric difference operation together with set inclusion, the proposed structure extends classical soft group concepts in a systematic manner and offers a fresh perspective on the interaction between soft set operations and traditional algebraic properties. The study establishes fundamental notions such as characteristic soft symmetric difference groups, soft symmetric difference subgroups, normal soft symmetric difference subgroups, soft normalizers, and soft cosets of soft symmetric difference groups, and investigates their essential algebraic properties. Several characterization results are obtained, providing necessary and sufficient conditions for the existence of an SSD-group structure. In addition, soft quotient (factor) groups of SSD-groups are introduced and their algebraic properties are examined in detail. Key relationships between SSD-group theory and classical group theory are also established through concepts such as lower alpha inclusion and the soft characteristic function. Illustrative examples are presented throughout the paper to demonstrate the applicability and internal consistency of the proposed framework. The findings establish a link between group-based mathematical structures and soft algebraic processes, opening the door to further algebraic investigation.

The structure of the manuscript is as follows. Essential definitions and algebraic preliminaries are presented in Section 2. The soft symmetric difference group is introduced, and its theoretical foundations and basic concepts are thoroughly developed in Section 3. In addition, key relationships between SSD-group theory and classical group theory are established through corresponding concepts. The notions of e-sets, soft cosets of soft symmetric difference complement groups, soft quotient groups, p-SSD-groups, and the order of soft cosets are further examined in Section 4. The main conclusions are summarized in Section 5, together with future directions for research in soft algebra and its applicability to uncertainty-driven analysis.

2. Preliminaries

This section presents a rigorous and methodical re-evaluation of the foundational definitions of soft sets.

Definition 1

([67]). Let $E$ be a parameter set, $U$ be a universal set, $P\left(U\right)$ be the power set of $U$, and $\mathcal{H}\subseteq E$. The soft set ${\mathcal{f}}_{\mathcal{H}}$ over $U$ is a function such that ${\mathrm{ƭ}}_{\mathcal{H}}:E\to P\left(U\right)$, where for all $w\notin \mathcal{H}$, ${\mathrm{ƭ}}_{\mathcal{H}}\left(w\right)=\varnothing$.

Throughout this paper, a soft set over $U$ is abbreviated by $\mathcal{S}\mathcal{S}$, and universal set by $\mathcal{U}\mathcal{S}$.

Definition 2

([67]). Let ${\mathrm{ƭ}}_{\mathcal{H}}$ be an $\mathcal{S}\mathcal{S}$. If ${\mathrm{ƭ}}_{\mathcal{H}}\left(w\right)=\varnothing$ for all $w\in E$, then ${\mathrm{ƭ}}_{\mathcal{H}}$ is called the null $\mathcal{S}\mathcal{S}$ and denoted by ${\varnothing }_E$, and if ${\mathrm{ƭ}}_{\mathcal{H}}\left(w\right)=U$, for all $w\in E$, then ${\mathrm{ƭ}}_{\mathcal{H}}$ is called the absolute $\mathcal{S}\mathcal{S}$ denoted by $U_E$.

Definition 3

([67]). Let ${\mathrm{ƭ}}_{\mathcal{H}}$ and ${\mathcal{g}}_{\aleph }$ be two $\mathcal{S}\mathcal{S}$s. If ${\mathrm{ƭ}}_{\mathcal{H}}\left(w\right)\subseteq {\mathcal{g}}_{\aleph }\left(w\right)$, for all $w\in E$, then ${\mathrm{ƭ}}_{\mathcal{H}}$ is said to be a soft subset of ${\mathcal{g}}_{\aleph }$ and indicated by ${\mathrm{ƭ}}_{\mathcal{H}}\tilde{\subseteq }{\mathcal{g}}_{\aleph }$. If ${\mathrm{ƭ}}_{\mathcal{H}}\left(w\right)={\mathcal{g}}_{\aleph }\left(w\right)$, for all $w\in E$, then ${\mathrm{ƭ}}_{\mathcal{H}}$ is called soft equal to ${\mathcal{g}}_{\aleph }$, and denoted by ${\mathrm{ƭ}}_{\mathcal{H}}={\mathcal{g}}_{\aleph }$.

Definition 4

([67]). Let ${\mathrm{ƭ}}_{\mathcal{H}}$ be an $\mathcal{S}\mathcal{S}$. The complement of ${\mathrm{ƭ}}_{\mathcal{H}}$ denoted by ${{\mathrm{ƭ}}_{\mathcal{H}}}^c$, is defined by the soft set ${{\mathrm{ƭ}}_{\mathcal{H}}}^c: E\to P\left(U\right)$ such that ${{\mathrm{ƭ}}_{\mathcal{H}}}^c\left(e\right)=U\backslash {\mathrm{ƭ}}_{\mathcal{H}}\left(e\right)={\left({\mathrm{ƭ}}_{\mathcal{H}}\left(e\right)\right)}^{\prime }$, for all $e\in E$, where ${\left({\mathrm{ƭ}}_{\mathcal{H}}\left(e\right)\right)}^{\prime }$ is the complement of ${\mathrm{ƭ}}_{\mathcal{H}}\left(e\right)$.

From now on, $∆$ stands for the symmetric difference operation on sets, $G$ is a group with identity element $e$, $S_G\left(U\right)$ denotes the collection of all $\mathcal{S}\mathcal{S}$s over $U$ parameterized by $G$.

Definition 5

([90]). Let ${Ɉ}_G$ and ${\wp }_G$ be $\mathcal{S}\mathcal{S}$s. The soft symmetric difference of ${Ɉ}_G$ and ${\wp }_G$ is the $\mathcal{S}\mathcal{S}$ ${Ɉ}_G\widetilde{∆}{\wp }_G$, where $\left({Ɉ}_G\widetilde{∆}{\wp }_G\right)\left(w\right)={Ɉ}_G\left(w\right)∆{\wp }_G\left(w\right)$, for all $w\in G$.

Definition 6

([91]). Let ${Ɉ}_G$ and ${\wp }_G$ be $\mathcal{S}\mathcal{S}$s. The soft symmetric difference complement of ${Ɉ}_G$ and ${\wp }_G$ is the $\mathcal{S}\mathcal{S}$ ${Ɉ}_G{\displaystyle \coprod ^{~}}{\wp }_G$, defined by $\left({Ɉ}_G{\displaystyle \coprod ^{~}}{\wp }_G\right)\left(w\right)={\left({Ɉ}_G\widetilde{∆}{\wp }_G\right)}^c\left(w\right){=\left({Ɉ}_G\left(w\right)∆{\wp }_G\left(w\right)\right)}^{\prime }$, for all $w\in G$, where ${\left({Ɉ}_G\left(w\right)∆{\wp }_G\left(w\right)\right)}^{\prime }$ is the complement of ${Ɉ}_G\left(w\right)∆{\wp }_G\left(w\right)$.

Definition 7

([83]). Let ${\mathrm{ƭ}}_G\in S_G\left(U\right)$. The $e$-set of ${\mathrm{ƭ}}_G$, denoted by $G_{{\mathrm{ƭ}}_G}$, is defined as $G_{{\mathrm{ƭ}}_G}=\left\{ⱴ\in G:{ \mathrm{ƭ}}_G\left(ⱴ\right)={\mathrm{ƭ}}_G\left(e\right)\right\}$.

Definition 8

([92]). Let ${\mathrm{ƭ}}_G\in S_G\left(U\right)$. Then, ${\mathrm{ƭ}}_G$ is called a soft symmetric difference complement group (SSDC-group) if ${\mathrm{ƭ}}_G\left(\varkappa \mathcal{y}\right)\supseteq {\left({\mathrm{ƭ}}_G\left(\varkappa \right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)\right)}^{\prime }$ and ${\mathrm{ƭ}}_G\left({\varkappa }^{-1}\right)={\mathrm{ƭ}}_G\left(\varkappa \right)$, for all $\varkappa ,\mathcal{y}\in G$.

3. Soft Symmetric Difference Groups with Their Basic Concepts

In this section, we first introduce the concept of soft symmetric difference groups. We then define the related notions of soft symmetric difference subgroups, normal soft symmetric difference subgroups, characteristic soft symmetric difference groups, and the normalizer of a soft symmetric difference group, and examine their fundamental properties. Some relationships between soft symmetric difference group theory and classical group theory are also established through corresponding concepts.

Definition 9.

Let $G$ be a group and ${\mathrm{ƭ}}_G\in S_G\left(U\right)$. Then, ${\mathrm{ƭ}}_G$ is called a soft symmetric difference group over $U$, if

$${\mathrm{ƭ}}_G\left(\varkappa \mathcal{y}\right)\subseteq {\mathrm{ƭ}}_G\left(\varkappa \right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right) \mathit{and} {\mathrm{ƭ}}_G\left({\varkappa }^{-1}\right)={\mathrm{ƭ}}_G\left(\varkappa \right)$$

for all $\varkappa ,\mathcal{y}\in G$.

For the sake of brevity, the soft symmetric difference group over $U$ is abbreviated by SSD-group and all the $\mathcal{S}\mathcal{S}$s are the elements of ${\mathrm{S}}_{\mathrm{G}}\left(\mathrm{U}\right)$ unless otherwise stated throughout this paper,

Example 1.

Let $U$ = $D_2$ = $\left\{<\varkappa ,\mathcal{y}>:{ \varkappa }^2={\mathcal{y}}^2=e,\varkappa \mathcal{y}=\mathcal{y}\varkappa \right\}=\left\{e,\varkappa , \mathcal{y},\mathcal{y}\varkappa \right\}$ be the $\mathcal{U}\mathcal{S}$ and $G=\left\{e,\mathcal{o},\mathrm{Ҍ},\mathrm{ƶ}\right\}$ be the set of parameters, which forms a group isomorphic to ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$, that is, the Klein four group. In particular, $G$ is a non-cyclic abelian group in which every non-identity element has order two. The group operation on $G$ is given by the Table 1.

Table 1. Cayley table of the group in Example 1.

$.$	$e$	$\mathcal{o}$	$\mathrm{Ҍ}$	$\mathrm{ƶ}$
$e$	$e$	$\mathcal{o}$	$\mathrm{Ҍ}$	$\mathrm{ƶ}$
$\mathcal{o}$	$\mathcal{o}$	$e$	$\mathrm{ƶ}$	$\mathrm{Ҍ}$
$\mathrm{Ҍ}$	$\mathrm{Ҍ}$	$\mathrm{ƶ}$	$e$	$\mathcal{o}$
$Ƶ$	$\mathrm{ƶ}$	$\mathrm{Ҍ}$	$\mathcal{o}$	$e$

Let ${\mathrm{ƭ}}_G$ and ${ʈ}_G$ be $\mathcal{S}\mathcal{S}$s as follows:

$${\mathrm{ƭ}}_G=\left\{\left(e,\varnothing \right),\left(\mathcal{o},\left\{e,\varkappa \right\}\right),\left(\mathrm{Ҍ},\left\{e,\varkappa ,\mathcal{y}\right\}\right),\left(\mathrm{ƶ},\left\{\mathcal{y}\right\}\right)\right\},{ʈ}_G=\left\{\left(e,\varnothing \right)\left(\mathcal{o},\left\{e\right\}\right),\left(\mathrm{Ҍ},\left\{e,\mathcal{y}\varkappa \right\}\right),\left(\mathrm{ƶ},U\right)\right\}$$

One can easily show that ${\mathrm{ƭ}}_G$ is an SSD-group. For example, for a few elements of $G$, we have

$${\mathrm{ƭ}}_G\left(\mathcal{o}\mathrm{Ҍ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ƶ}\right)=\left\{\mathcal{y}\right\}\subseteq {\mathrm{ƭ}}_G\left(\mathcal{o}\right)∆{\mathrm{ƭ}}_G\left(\mathrm{Ҍ}\right)=\left\{\mathcal{y}\right\},$$

$${\mathrm{ƭ}}_G\left(\mathrm{Ҍ}\mathrm{ƶ}\right)={\mathrm{ƭ}}_G\left(\mathcal{o}\right)=\left\{e,\varkappa \right\}\subseteq {\mathrm{ƭ}}_G\left(\mathrm{Ҍ}\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ƶ}\right)=\left\{e,\varkappa \right\},$$

$${\mathrm{ƭ}}_G\left(\mathrm{ƶ}\mathcal{o}\right)={\mathrm{ƭ}}_G\left(\mathrm{Ҍ}\right)=\left\{e,\varkappa ,\mathcal{y}\right\}\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ƶ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{o}\right)=\left\{e,\varkappa ,\mathcal{y}\right\}.$$

It can be shown that the conditions are satisfied for the remaining elements of $G$ as well. Therefore, ${\mathrm{ƭ}}_G$ is an SSD-group. However, since

$${ʈ}_G\left(\mathrm{Ҍ}\mathrm{ƶ}\right)={ʈ}_G\left(\mathcal{o}\right)=\left\{e\right\}⊈{ʈ}_G\left(\mathrm{Ҍ}\right)∆{ʈ}_G\left(\mathrm{ƶ}\right)=\left\{\varkappa ,\mathcal{y}\right\},$$

${ʈ}_G$ is not an SSD-group.

Example 2.

Let  $G=(\mathbb{Z},+)$ be the infinite cyclic group of integers under addition and let $U=\{a,b\}$. Define an $\mathcal{S}\mathcal{S}$ ${\mathrm{ƭ}}_G:\mathbb{Z}⟶P\left(U\right)$ by

$${\mathrm{ƭ}}_G\left(k\right)=\left\{\begin{array}{c}\varnothing , if k even, \\ \left\{a\right\}, if k odd.\end{array}\right.$$

Then, ${\mathrm{ƭ}}_G$ is an SSD-group. To verify that ${\mathrm{ƭ}}_G\left(k+l\right)\subseteq {\mathrm{ƭ}}_G\left(k)∆{\mathrm{ƭ}}_G(l\right)$, for any $k,l\in \mathbb{Z}$, we examine the possible parity cases:

Case 1: If both $k$ and $l$ are even, then $k+l$ is even. Thus, ${\mathrm{ƭ}}_G\left(k+l\right)=\varnothing =\varnothing ∆\varnothing ={\mathrm{ƭ}}_G\left(k\right)∆{\mathrm{ƭ}}_G\left(l\right)$.

Case 2: If both $k$ and $l$ are odd, then $k+l$ is even. Hence, ${\mathrm{ƭ}}_G\left(k+l\right)=\varnothing \subseteq \left\{a\right\}∆\left\{a\right\}={\mathrm{ƭ}}_G\left(k\right)∆{\mathrm{ƭ}}_G\left(l\right)$.

Case 3: If exactly one of $k, l$ is odd, then $k+l$ is odd. Thereby, ${\mathrm{ƭ}}_G\left(k+l\right)=\left\{a\right\}\subseteq \varnothing ∆\left\{a\right\}={\mathrm{ƭ}}_G\left(k\right)∆{\mathrm{ƭ}}_G\left(l\right)$.

Moreover, since the integers $k$ and $-k$ have the same parity, where $k\in \mathbb{Z}$, ${\mathrm{ƭ}}_G\left(k\right)={\mathrm{ƭ}}_G(-k)$ holds, for all $k\in \mathbb{Z}$. Therefore, ${\mathrm{ƭ}}_G$ is an SSD-group.

This example shows that the SSD-framework naturally extends beyond finite groups and applies equally to infinite groups.

Proposition 1.

If  ${\mathrm{ƭ}}_G$ is an SSD-group, then ${\mathrm{ƭ}}_G\left(e\right)=\varnothing$.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group. Then, for all $\mathrm{ⱴ}\in G$, we have

$${\mathrm{ƭ}}_G\left(e\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathrm{ⱴ}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)=\mathrm{\varnothing }.$$

Note that the condition ${\mathrm{ƭ}}_G\left(e\right)=\varnothing$ alone does not guarantee that ${\mathrm{ƭ}}_G$ is an SSD-group. This is consistent with Example 1, which illustrates the structural distinction. In Example 1, although ${ʈ}_G\left(e\right)=\varnothing$, it has been shown that ${ʈ}_G$ is not an SSD-group. □

Theorem 1.

An $\mathcal{S}\mathcal{S}$  ${\mathrm{ƭ}}_G$ is an SSD-group if and only if ${\mathrm{ƭ}}_G\left(ⱴ{\mathcal{y}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(ⱴ\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right),$ for all $\mathcal{ⱴ},\mathcal{y}\in G$.

Proof. 

Suppose that $\mathcal{S}\mathcal{S}$ ${\mathrm{ƭ}}_G$ is an SSD-group. Then, by Definition 9, for all $\mathcal{y}\in G$, we have ${\mathrm{ƭ}}_G\left(\mathcal{y}\right)={\mathrm{ƭ}}_G\left({\mathcal{y}}^{-1}\right)$. Thus,

$${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left({\mathcal{y}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right),$$

for all $\mathcal{ⱴ},\mathcal{y}\in G$. Hence, ${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)$.

Conversely, suppose that ${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $\mathcal{ⱴ},\mathcal{y}\in G$. Firstly, choosing $\mathrm{ⱴ}=e$ yields

$${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)={\mathrm{ƭ}}_G\left(e{\mathcal{y}}^{-1}\right)={\mathrm{ƭ}}_G\left({\mathcal{y}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(e\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)=\mathrm{\varnothing }∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right).$$

Moreover,

$${\mathrm{ƭ}}_G\left(\mathcal{y}\right)={\mathrm{ƭ}}_G\left({\left({\mathcal{y}}^{-1}\right)}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left({\mathcal{y}}^{-1}\right)$$

holds. From these two results, we have

$${\mathrm{ƭ}}_G\left(\mathcal{y}\right)={\mathrm{ƭ}}_G\left({\mathcal{y}}^{-1}\right)$$

is obtained. Secondly,

$${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left({\mathrm{ⱴ}\left({\mathcal{y}}^{-1}\right)}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left({\mathcal{y}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right).$$

Thus, the $\mathcal{S}\mathcal{S}$ ${\mathrm{ƭ}}_G$ is an SSD-group. □

We would also like to emphasize that Theorem 1 provides a characterization equivalent to Definition 9. Since the result is stated as an “if and only if” condition, it establishes an alternative but logically equivalent formulation of the SSD-group concept. For this reason, the use of the defining property in the proof is essential to demonstrate this equivalence.

Definition 10

([93]). Let ${\mathrm{ƭ}}_G$ and ${ʈ}_G$ be $\mathcal{S}\mathcal{S}$s. Then, the soft intersection-symmetric difference product ${\mathrm{ƭ}}_G {⨂}_{i/s}{ʈ}_G$ is defined by

$$\left({\mathrm{ƭ}}_G{⨂}_{i/s}{ʈ}_G\right)\left(ⱴ\right)=\bigcap _{\mathcal{ⱴ}=\mathcal{y}\mathcal{z}}\left({\mathrm{ƭ}}_G\left(\mathcal{y}\right)∆{ʈ}_G\left(\mathcal{z}\right)\right), \begin{array}{c} \mathcal{y},\mathcal{z}\in G\end{array}$$

for all $ⱴ\in G$.

The inverse of ${\mathrm{ƭ}}_G$ is defined as ${{\mathrm{ƭ}}_G}^{-1}\left(ⱴ\right)={\mathrm{ƭ}}_G\left({ⱴ}^{-1}\right)$, for all $ⱴ\in G$.

Example 3.

Let  $U$ = $D_2$ $=\left\{e,\varkappa , \mathcal{y},\mathcal{y}\varkappa \right\}$ be the $\mathcal{U}\mathcal{S}$ and $G=\left\{\mathcal{o},\mathrm{Ҍ},\mathrm{ƶ}\right\}$ be the set of parameters, which forms a cyclic group of order three (isomorphic to $\mathbb{Z}/3\mathbb{Z}$). The group operation on $G$ is given by Table 2:

Table 2. Cayley table of the group in Example 3.

$·$	$\mathcal{o}$	$\mathrm{Ҍ}$	$\mathrm{ƶ}$
$\mathcal{o}$	$\mathcal{o}$	$\mathrm{Ҍ}$	$\mathrm{ƶ}$
$\mathrm{Ҍ}$	$\mathrm{Ҍ}$	$\mathrm{ƶ}$	$\mathcal{o}$
$\mathrm{ƶ}$	$\mathrm{ƶ}$	$\mathcal{o}$	$\mathrm{Ҍ}$

Let ${\mathrm{ƭ}}_G$ and ${ʈ}_G$ be $\mathcal{S}\mathcal{S}$s as follows:

$${\mathrm{ƭ}}_G=\left\{\left(\mathcal{o},\left\{\varkappa ,\mathcal{y},\mathcal{y}\varkappa \right\}\right),\left(\mathrm{Ҍ},\left\{e,\mathcal{y}\varkappa \right\}\right),\left(\mathrm{ƶ},\left\{\varkappa ,\mathcal{y}\varkappa \right\}\right)\right\}, {ʈ}_G=\left\{\left(\mathcal{o},\left\{e,\mathcal{y}\varkappa \right\}\right),\left(\mathrm{Ҍ},\left\{\varkappa ,\mathcal{y}\varkappa \right\}\right),\left(\mathrm{ƶ},\left\{e,\mathcal{y}\right\}\right)\right\}.$$

Since

$$\begin{array}{l}\mathcal{o}=\mathcal{o}\mathcal{o}=\mathrm{Ҍ}\mathrm{ƶ}=\mathrm{ƶ}\mathrm{Ҍ},\mathrm{Ҍ}=\mathcal{o}\mathrm{Ҍ}=\mathrm{Ҍ}\mathcal{o}=\mathrm{ƶ}\mathrm{ƶ} \mathit{and} \mathrm{ƶ}=\mathcal{o}\mathrm{ƶ}=\mathrm{Ҍ}\mathrm{Ҍ}=\mathrm{ƶ}\mathcal{o}\\ \left({\mathrm{ƭ}}_G{\otimes }_{i/s}{ʈ}_G\right)\left(\mathcal{o}\right)=\left({\mathrm{ƭ}}_G\left(\mathcal{o}\right)∆{ʈ}_G\left(\mathcal{o}\right)\right)\cap \left({\mathrm{ƭ}}_G\left(\mathrm{Ҍ}\right)∆{ʈ}_G\left(\mathrm{ƶ}\right)\right)\cap \left({\mathrm{ƭ}}_G\left(\mathrm{ƶ}\right)∆{ʈ}_G\left(\mathrm{Ҍ}\right)\right)=\left\{\varkappa ,\mathcal{y}\right\}\cap \left\{\mathcal{y},\mathcal{y}\varkappa \right\}\cap \varnothing =\varnothing ,\\ \left({\mathrm{ƭ}}_G{\otimes }_{i/s}{ʈ}_G\right)\left(\mathrm{Ҍ}\right)=\left({\mathrm{ƭ}}_G\left(\mathcal{o}\right)∆{ʈ}_G\left(\mathrm{Ҍ}\right)\right)\cap \left({\mathrm{ƭ}}_G\left(\mathrm{Ҍ}\right)∆{ʈ}_G\left(\mathcal{o}\right)\right)\cap \left({\mathrm{ƭ}}_G\left(\mathrm{ƶ}\right)∆{ʈ}_G\left(\mathrm{ƶ}\right)\right)=\left\{\mathcal{y}\right\}\cap \varnothing \cap \left\{\mathcal{y}\right\}=\varnothing ,\\ \left({\mathrm{ƭ}}_G{\otimes }_{i/s}{ʈ}_G\right)\left(\mathrm{ƶ}\right)=\left({\mathrm{ƭ}}_G\left(\mathcal{o}\right)∆{ʈ}_G\left(\mathrm{ƶ}\right)\right)\cap \left({\mathrm{ƭ}}_G\left(\mathrm{Ҍ}\right)∆{ʈ}_G\left(\mathrm{Ҍ}\right)\right)\cap \left({\mathrm{ƭ}}_G\left(\mathrm{ƶ}\right)∆{ʈ}_G\left(\mathcal{o}\right)\right)=\left\{e,\varkappa ,\mathcal{y}\varkappa \right\}\cap \left\{e,\varkappa \right\}\cap \left\{e, \varkappa \right\}=\left\{e,\varkappa \right\}.\end{array}$$

Hence, we have

$${\mathrm{ƭ}}_G{\otimes }_{i/s}{ʈ}_G=\left\{\left(\mathcal{o},\varnothing \right),\left(\mathrm{Ҍ},\varnothing \right),\left(\mathrm{ƶ},\left\{e, \varkappa \right\}\right)\right\}.$$

Moreover, since ${\mathcal{o}}^{-1}=\mathcal{o}$, ${\mathrm{Ҍ}}^{-1}=\mathrm{ƶ}$ and ${\mathrm{ƶ}}^{-1}=\mathrm{Ҍ}$, we have

$${{\mathrm{ƭ}}_G}^{-1}=\left\{\left(\mathcal{o},\left\{\varkappa ,\mathcal{y},\mathcal{y}\varkappa \right\}\right),\left(\mathrm{Ҍ},\left\{\varkappa ,\mathcal{y}\varkappa \right\}\right),\left(\mathrm{ƶ},\left\{e,\mathcal{y}\varkappa \right\}\right)\right\}.$$

Theorem 2.

${\mathrm{ƭ}}_G$ is an SSD-group over $U$ if and only if ${\mathrm{ƭ}}_G{⨂}_{i/s}{\mathrm{ƭ}}_G\tilde{\supseteq }{\mathrm{ƭ}}_G$ and ${{\mathrm{ƭ}}_G}^{-1}={\mathrm{ƭ}}_G$.

Proof. 

Let ${\mathrm{ƭ}}_{\mathrm{G}}$ be an SSD-group. Then, for all $\mathrm{ⱴ}\in \mathrm{G}$, we have

$$\left({\mathrm{ƭ}}_{\mathrm{G}}{⨂}_{\mathrm{i}/\mathrm{s}}{\mathrm{ƭ}}_{\mathrm{G}}\right)\left(\mathrm{ⱴ}\right)=\bigcap _{\mathcal{ⱴ}=\mathcal{y}\mathcal{z}}\left({\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{y}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{z}\right)\right)\supseteq \bigcap _{\mathcal{ⱴ}=\mathcal{y}\mathcal{z}}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{y}\mathcal{z}\right)=\bigcap _{\mathcal{ⱴ}=\mathcal{y}\mathcal{z}}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right).$$

Thus, ${\mathrm{ƭ}}_{\mathrm{G}}{⨂}_{\mathrm{i}/\mathrm{s}}{\mathrm{ƭ}}_{\mathrm{G}}\tilde{\supseteq }{\mathrm{ƭ}}_{\mathrm{G}}$. The second part follows directly from the definition of the soft SSD-group, since for all $\mathrm{ⱴ}\in \mathrm{G}$, we have

$${{\mathrm{ƭ}}_G}^{-1}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right),$$

implying that ${{\mathrm{ƭ}}_G}^{-1}={\mathrm{ƭ}}_G$.

Conversely, assume that ${\mathrm{ƭ}}_{\mathrm{G}}{⨂}_{\mathrm{i}/\mathrm{s}}{\mathrm{ƭ}}_{\mathrm{G}}\tilde{\supseteq }{\mathrm{ƭ}}_{\mathrm{G}}$ and ${{\mathrm{ƭ}}_{\mathrm{G}}}^{-1}={\mathrm{ƭ}}_{\mathrm{G}}$. Then, for all $\mathrm{ⱴ}\in \mathrm{G}$, we have

$${\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{y}\mathcal{z}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)\subseteq \left({\mathrm{ƭ}}_{\mathrm{G}}{⨂}_{\mathrm{i}/\mathrm{s}}{\mathrm{ƭ}}_{\mathrm{G}}\right)\left(\mathrm{ⱴ}\right)=\bigcap _{\mathcal{ⱴ}=\mathcal{y}\mathcal{z}}\left({\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{y}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{z}\right)\right)\subseteq {\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{y}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{z}\right),$$

where $\mathcal{y},\mathcal{z}\in \mathrm{G}$. Thus, ${\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{y}\mathcal{z}\right)\subseteq {\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{y}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{z}\right)$, for all $\mathcal{y},\mathcal{z}\in \mathrm{G}$.

Moreover, since ${{\mathrm{ƭ}}_{\mathrm{G}}}^{-1}={\mathrm{ƭ}}_{\mathrm{G}}$, then for all $\mathrm{ⱴ}\in \mathrm{G}$, we have ${{\mathrm{ƭ}}_{\mathrm{G}}}^{-1}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)$. Besides, by Definition 10, ${{\mathrm{ƭ}}_{\mathrm{G}}}^{-1}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ⱴ}}^{-1}\right)$, implying that ${\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ⱴ}}^{-1}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)$, for all $\mathrm{ⱴ}\in G$. □

Proposition 2.

If  ${\mathrm{ƭ}}_G$ is an SSD-group, then ${{\mathrm{ƭ}}_G}^{-1}$ is also an SSD-group.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group. Then, for all $\mathcal{ⱴ},\mathcal{y}\in G$, we have

$${{\mathrm{ƭ}}_G}^{-1}\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left({\left(\mathcal{ⱴ}\mathcal{y}\right)}^{-1}\right)={\mathrm{ƭ}}_G\left({\mathcal{y}}^{-1}{\mathrm{ⱴ}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left({\mathcal{y}}^{-1}\right)∆{\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}\right)={\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}\right)∆{\mathrm{ƭ}}_G\left({\mathcal{y}}^{-1}\right)={{\mathrm{ƭ}}_G}^{-1}\left(\mathrm{ⱴ}\right)∆{{\mathrm{ƭ}}_G}^{-1}\left(\mathcal{y}\right).$$

Thus, ${{\mathrm{ƭ}}_G}^{-1}\left(\mathcal{ⱴ}\mathcal{y}\right)\subseteq {{\mathrm{ƭ}}_G}^{-1}\left(\mathrm{ⱴ}\right)∆{{\mathrm{ƭ}}_G}^{-1}\left(\mathcal{y}\right)$. Moreover, for all $\mathrm{ⱴ}\in \mathrm{G}$, we have

$${{\mathrm{ƭ}}_G}^{-1}\left({\mathrm{ⱴ}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}\right)={{\mathrm{ƭ}}_G}^{-1}\left(\mathrm{ⱴ}\right).$$

Hence, ${{\mathrm{ƭ}}_G}^{-1}\left({\mathrm{ⱴ}}^{-1}\right)={{\mathrm{ƭ}}_G}^{-1}\left(\mathrm{ⱴ}\right)$. So, we are done. □

Proposition 3.

If  ${\mathrm{ƭ}}_G$ and ${ʈ}_G$ are two SSD-groups, then ${\left({\mathrm{ƭ}}_G{⨂}_{i/s}{ʈ}_G\right)}^{-1}={{ʈ}_G}^{-1}{⨂}_{i/s}{{\mathrm{ƭ}}_G}^{-1}.$

Proof. 

Let ${\mathrm{ƭ}}_G$ and ${\mathrm{ʈ}}_G$ be two SSD-groups. Then, for all $\mathrm{ⱴ}\in \mathrm{G}$, we have

$$\begin{array}{c}{\left({\mathrm{ƭ}}_G{⨂}_{i/s}{\mathrm{ʈ}}_G\right)}^{-1}\left(\mathrm{ⱴ}\right)=\left({\mathrm{ʈ}}_G{⨂}_{i/s}{\mathrm{ƭ}}_G\right)\left({\mathrm{ⱴ}}^{-1}\right)\hfill \\ ={\displaystyle \bigcap _{{\mathrm{ⱴ}}^{-1}=\mathcal{y}\mathcal{z}}}\left({\mathrm{ƭ}}_G\left(\mathcal{y}\right)∆{\mathrm{ʈ}}_G\left(\mathcal{z}\right)\right)\hfill \\ ={\displaystyle \bigcap _{\mathrm{ⱴ}={\mathcal{z}}^{-1}{\mathcal{y}}^{-1}}}\left({\mathrm{ʈ}}_G\left({\mathcal{z}}^{-1}\right)∆{\mathrm{ƭ}}_G\left({\mathcal{y}}^{-1}\right)\right)\hfill \\ ={\displaystyle \bigcap _{\mathrm{ⱴ}={\mathcal{z}}^{-1}{\mathcal{y}}^{-1}}}\left({{\mathrm{ʈ}}_G}^{-1}\left({\mathcal{z}}^{-1}\right)∆{{\mathrm{ƭ}}_G}^{-1}\left({\mathcal{y}}^{-1}\right)\right)\hfill \\ \hspace{1em} =\left({{\mathrm{ʈ}}_G}^{-1}{⨂}_{i/s}{{\mathrm{ƭ}}_G}^{-1}\right)\left(\mathrm{ⱴ}\right).\hfill \end{array}$$

Note here that, since ${\mathrm{ʈ}}_G$ is an SSD-group, ${{\mathrm{ʈ}}_G}^{-1}$ is an SSD-group, by Proposition 2. Thus, ${\mathrm{ʈ}}_G\left({\mathcal{z}}^{-1}\right)={{\mathrm{ʈ}}_G}^{-1}\left(\mathcal{z}\right)={{\mathrm{ʈ}}_G}^{-1}\left({\mathcal{z}}^{-1}\right)$, for all $\mathcal{z}\in G$. Hence, ${\left({\mathrm{ƭ}}_G{⨂}_{i/s}{\mathrm{ʈ}}_G\right)}^{-1}={{\mathrm{ʈ}}_G}^{-1}{⨂}_{i/s}{{\mathrm{ƭ}}_G}^{-1}$. □

Definition 11

([84]). Let $A$ be a subset of $G$. We denote by ${\delta }_A$ the soft characteristic function of $A$, and define as

$${\delta }_A\left(x\right)=\left\{\begin{array}{c}U, if x\in A, \\ \varnothing , if x\in G\backslash A. \end{array}\right.$$

The soft characteristic function of $A$ is an $\mathcal{S}\mathcal{S}$ over $U$.

Proposition 4.

If  $X$ and $Y$ are two subsets of $G$, then ${ \delta }_X\widetilde{∆}{\delta }_Y={\delta }_{X∆Y}$.

Proof. 

Let $\mathrm{ⱴ}\in \mathrm{G}$. Suppose $\mathrm{ⱴ}\in \mathrm{X}∆\mathrm{Y}$. Then, $\mathrm{ⱴ}\in \mathrm{X}\setminus \mathrm{Y}$ or $\mathrm{ⱴ}\in \mathrm{Y}\setminus \mathrm{X}$. Firstly, let $\mathrm{ⱴ}\in \mathrm{X}\setminus \mathrm{Y}$. Then, we have

$$\left({\mathsf{\delta }}_{\mathrm{X}}\widetilde{∆}{\mathsf{\delta }}_{\mathrm{Y}}\right)\left(\mathrm{ⱴ}\right)={\mathsf{\delta }}_{\mathrm{X}}\left(\mathrm{ⱴ}\right)∆{\mathsf{\delta }}_{\mathrm{Y}}\left(\mathrm{ⱴ}\right)=\mathrm{U}∆\mathrm{\varnothing }=\mathrm{U}=\left({\mathsf{\delta }}_{\mathrm{X}∆\mathrm{Y}}\right)\left(\mathrm{ⱴ}\right).$$

Secondly, let $\mathrm{ⱴ}\in \mathrm{Y}\setminus \mathrm{X}$. Then,

$$\left({\mathsf{\delta }}_{\mathrm{X}}\widetilde{∆}{\mathsf{\delta }}_{\mathrm{Y}}\right)\left(\mathrm{ⱴ}\right)={\mathsf{\delta }}_{\mathrm{X}}\left(\mathrm{ⱴ}\right)∆{\mathsf{\delta }}_{\mathrm{Y}}\left(\mathrm{ⱴ}\right)=\mathrm{\varnothing }∆\mathrm{U}=\mathrm{U}=\left({\mathsf{\delta }}_{\mathrm{X}∆\mathrm{Y}}\right)\left(\mathrm{ⱴ}\right),$$

implying that ${\mathsf{\delta }}_{\mathrm{X}}\widetilde{∆}{\mathsf{\delta }}_{\mathrm{Y}}={\mathsf{\delta }}_{\mathrm{X}∆\mathrm{Y}}$.

Now, let $\mathrm{ⱴ}\notin \mathrm{X}∆\mathrm{Y}$. Then, $\mathrm{ⱴ}\in \mathrm{X}\cap \mathrm{Y}$ or $\mathrm{ⱴ}\in {\left(\mathrm{X}\cup \mathrm{Y}\right)}^{\prime }$. Firstly, let $\mathrm{ⱴ}\in \mathrm{X}\cap \mathrm{Y}$. Then, we have

$$\left({\delta }_X\widetilde{∆}{\delta }_Y\right)\left(\mathrm{ⱴ}\right)={\delta }_X\left(\mathrm{ⱴ}\right)∆{\delta }_Y\left(\mathrm{ⱴ}\right)=U∆U=\mathrm{\varnothing }=\left({\delta }_{X∆Y}\right)\left(\mathrm{ⱴ}\right).$$

Secondly, let $\mathrm{ⱴ}\in {\left(X\cup Y\right)}^{\prime }$. Then,

$$\left({\delta }_X\widetilde{∆}{\delta }_Y\right)\left(\mathrm{ⱴ}\right)={\delta }_X\left(\mathrm{ⱴ}\right)∆{\delta }_Y\left(\mathrm{ⱴ}\right)=\mathrm{\varnothing }∆\mathrm{\varnothing }=\mathrm{\varnothing }=\left({\delta }_{X∆Y}\right)\left(\mathrm{ⱴ}\right).$$

Hence, ${\delta }_X\widetilde{∆}{\delta }_Y={\delta }_{X∆Y}$. □

Proposition 5.

Let  $\varnothing \ne H$ be a subset of $G$. If ${\delta }_{H^c}$ is an SSD-group, then $H$ is a subgroup of $G$.

Proof. 

Let $\mathcal{ⱴ},\mathcal{y}\in \mathrm{H}$ and ${\mathsf{\delta }}_{{\mathrm{H}}^{\mathrm{c}}}$ be an SSD-group. We need to show that $\mathrm{ⱴ}{\mathcal{y}}^{-1}\in \mathrm{H}$. Then, we have

$${\delta }_{H^c}\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)\subseteq {\delta }_{H^c}\left(\mathrm{ⱴ}\right)∆{\delta }_{H^c}\left(\mathcal{y}\right)=\mathrm{\varnothing }∆\mathrm{\varnothing }=\mathrm{\varnothing }.$$

Hence,${\delta }_{H^c}\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)=\mathrm{\varnothing }$, implying that $\mathrm{ⱴ}{\mathcal{y}}^{-1}\in \mathrm{H}$. □

Note 1.

The converse of Proposition 5 need not be true. That is, if  $H$ is a subgroup of $G$, then ${\delta }_{H^c}$ need not be an SSD-group.

Example 4.

Let  $G={\mathbb{Z}}_6$ and $H=\left\{\overline{0},\overline{3}\right\}$ be the subgroup of ${\mathbb{Z}}_6$. Then, we have

$${\delta }_{H^c}=\{\left(\overline{0},\varnothing \right), \left(\overline{1},U\right), \left(\overline{2},U\right), \left(\overline{3},\varnothing \right), \left(\overline{4},U\right), \left(\overline{5},U\right)\}.$$

Since ${\delta }_{H^c}\left(\overline{5}\right)=U⊈{\delta }_{H^c}\left(\overline{1}\right)∆{\delta }_{H^c}\left(\overline{4}\right)=U∆U=\varnothing$, ${ \delta }_{H^c}$ is not an SSD-group.

Definition 12

([83]). Let ${\mathrm{ƭ}}_G$ be an $\mathcal{S}\mathcal{S}$ and $\alpha \subseteq U$. Then, lower $\alpha$-inclusion of ${\mathrm{ƭ}}_G$, denoted by $\mathcal{L}\left({\mathrm{ƭ}}_G;\alpha \right)$, is defined as $\mathcal{L}\left({\mathrm{ƭ}}_G;\alpha \right)=\left\{x\in G | {\mathrm{ƭ}}_G\left(x\right)\subseteq \alpha \right\}$.

Proposition 6.

Let  ${\mathrm{ƭ}}_G$ be an $\mathcal{S}\mathcal{S}$ and $\alpha$ be a subset of $U$ such that $\alpha \in Im\left({\mathrm{ƭ}}_G\right)$, where $Im\left({\mathrm{ƭ}}_G\right)=\left\{\alpha \subseteq U: {\mathrm{ƭ}}_G\left(ⱴ\right)=\alpha , for ⱴ\in G \right\}$. If ${\mathrm{ƭ}}_G$ is an SSD-group, then $\mathcal{L}\left({\mathrm{ƭ}}_G;\alpha \right)$ is a subgroup of $G$.

Proof. 

Since ${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)=\alpha$ for some $\mathrm{ⱴ}\in G$, then $\varnothing \ne \mathcal{L}\left({\mathrm{ƭ}}_G;\alpha \right)\subseteq G$. Let $\mathcal{ⱴ},\mathcal{y}\in \mathcal{L}\left({\mathrm{ƭ}}_G;\alpha \right)$, then ${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)\subseteq \alpha$ and ${\mathrm{ƭ}}_G\left(\mathcal{y}\right)\subseteq \alpha$. Now, we need to show that $\mathrm{ⱴ}{\mathcal{y}}^{-1}\in \mathcal{L}\left({\mathrm{ƭ}}_G;\alpha \right)$, for all $\mathcal{ⱴ},\mathcal{y}\in \mathcal{L}\left({\mathrm{ƭ}}_G;\alpha \right)$. Since ${\mathrm{ƭ}}_G$ is an SSD-group, it follows that ${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)\cup {\mathrm{ƭ}}_G\left(\mathcal{y}\right)\subseteq \alpha$ implying that $\mathrm{ⱴ}{\mathcal{y}}^{-1}\in \mathcal{L}\left({\mathrm{ƭ}}_G;\alpha \right)$. So, we are done. □

Note 2.

If each of the $\mathcal{L}\left({\mathrm{ƭ}}_G;\alpha \right)$ are a subgroup of $G$, then ${\mathrm{ƭ}}_G$ need not be an SSD-group.

Example 5.

Let  $G={\mathbb{Z}}_8$ and the $\mathcal{S}\mathcal{S}$ ${\mathrm{ƭ}}_G$ over $U=\mathbb{Q}$ be defined as follows:

$${\mathrm{ƭ}}_G=\{\left(\overline{0},\mathrm{\varnothing }\right),\left(\overline{1},U\right),\left(\overline{2},U\right),\left(\overline{3},U\right),\left(\overline{4},\mathrm{\varnothing }\right),\left(\overline{5},U\right),\left(\overline{6},U\right),\left(\overline{7},U\right)\}.$$

By considering $Im\left({\mathrm{ƭ}}_G\right)=\left\{\mathrm{\varnothing },U\right\}$, we have the following:

$$\mathcal{L}\left({\mathrm{ƭ}}_G;\alpha \right)=\left\{\begin{array}{c} {\mathbb{Z}}_8, \alpha =U, \\ \left\{\overline{0},\overline{4}\right\}, \alpha =\varnothing . \end{array}\right.$$

Here, $\left\{\overline{0},\overline{4}\right\}$ and ${\mathbb{Z}}_8$ are both subgroups of ${\mathbb{Z}}_8$; however, since

$${\mathrm{ƭ}}_G\left(\overline{7}\right)=U⊈{\mathrm{ƭ}}_G\left(\overline{2}\right)∆{\mathrm{ƭ}}_G\left(\overline{5}\right)=U∆U=\mathrm{\varnothing },$$

${\mathrm{ƭ}}_G$ is not an SSD-group.

Theorem 3.

${\mathrm{ƭ}}_G$ is an SSD-group if and only if ${{\mathrm{ƭ}}_G}^c$ is an SSDC-group.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group. Then, for all $\mathcal{ⱴ},\mathcal{y}\in G$, we have

$${{\mathrm{ƭ}}_G}^c\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)=U\setminus {\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)\supseteq U\setminus \left({\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)\right)={\left({\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)\right)}^{\prime }={\left({{\mathrm{ƭ}}_G}^c\left(\mathrm{ⱴ}\right)∆{{\mathrm{ƭ}}_G}^c\left(\mathcal{y}\right)\right)}^{\prime }.$$

Hence, ${{\mathrm{ƭ}}_G}^c\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)\supseteq {\left({{\mathrm{ƭ}}_G}^c\left(\mathrm{ⱴ}\right)∆{{\mathrm{ƭ}}_G}^c\left(\mathcal{y}\right)\right)}^{\prime }$, for all $\mathcal{ⱴ},\mathcal{y}\in G$. That is to say, ${{\mathrm{ƭ}}_G}^c$ is an SSDC-group.

Conversely, let ${{\mathrm{ƭ}}_G}^c$ be an SSDC-group. Then, for all $\mathcal{ⱴ},\mathcal{y}\in G$,

$${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)=U\setminus {{\mathrm{ƭ}}_G}^c\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)\subseteq U\setminus {\left({{\mathrm{ƭ}}_G}^c\left(\mathrm{ⱴ}\right)∆{{\mathrm{ƭ}}_G}^c\left(\mathcal{y}\right)\right)}^{\prime }={{\mathrm{ƭ}}_G}^c\left(\mathrm{ⱴ}\right)∆{{\mathrm{ƭ}}_G}^c\left(\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right).$$

Thus, ${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $\mathcal{ⱴ},\mathcal{y}\in G$. Hence, ${\mathrm{ƭ}}_G$ is an SSD-group. Note here that, in classical set theory, $A∆B=A^{\prime }∆B^{\prime }$, where $A$ and $B$ are fixed sets and $A^{\prime }, B^{\prime }$ are the complements of $A$ and $B$, respectively. □

Definition 13.

Let  $H$ be a subgroup of $G$, ${\mathrm{ƭ}}_G$ be an SSD-group and ${\mathrm{ƭ}}_H$ be a nonempty soft subset of ${\mathrm{ƭ}}_G$. If ${\mathrm{ƭ}}_H$ itself is an SSD-group, then ${\mathrm{ƭ}}_H$ is called an SSD-subgroup of ${\mathrm{ƭ}}_G$.

Example 6.

Consider the group  $G=\left\{e,\mathcal{o},\mathrm{Ҍ},\mathrm{ƶ}\right\}$ and the SSD-group ${\mathrm{ƭ}}_G=\left\{\left(e,\varnothing \right),\left(\mathcal{o},\left\{e,\varkappa \right\}\right),\left(\mathrm{Ҍ},\left\{e,\varkappa ,\mathcal{y}\right\}\right),\left(\mathrm{ƶ},\left\{\mathcal{y}\right\}\right)\right\}$ over $U$ $=\left\{e,\varkappa , \mathcal{y},\mathcal{y}\varkappa \right\}$ in Example 1. Let $H=\left\{e,\mathcal{o}\right\}$ be the subgroup of $G$, and ${\mathrm{ƭ}}_H=\left\{\left(e,\varnothing \right),\left(\mathcal{o},\left\{\varkappa \right\}\right)\right\}$ be an $\mathcal{S}\mathcal{S}$ over $U.$ It is obvious that ${\mathrm{ƭ}}_H$ is a nonempty soft subset of ${\mathrm{ƭ}}_G$ and

$$\begin{array}{c}{\mathrm{ƭ}}_H\left(\mathcal{o}e\right)={\mathrm{ƭ}}_H\left(\mathcal{o}\right)=\left\{\varkappa \right\}\subseteq {\mathrm{ƭ}}_H\left(\mathcal{o}\right)∆{\mathrm{ƭ}}_H\left(e\right)=\left\{\varkappa \right\},\\ {\mathrm{ƭ}}_H\left(\mathcal{o}\mathcal{o}\right)={\mathrm{ƭ}}_H\left(e\right)=\varnothing \subseteq {\mathrm{ƭ}}_H\left(\mathcal{o}\right)∆{\mathrm{ƭ}}_H\left(\mathcal{o}\right)=\varnothing ,\\ {\mathrm{ƭ}}_H\left(e\mathcal{o}\right)={\mathrm{ƭ}}_H\left(\mathcal{o}\right)=\left\{\varkappa \right\}\subseteq {\mathrm{ƭ}}_H\left(e\right)∆{\mathrm{ƭ}}_H\left(\mathcal{o}\right)=\left\{\varkappa \right\},\\ {\mathrm{ƭ}}_H\left(ee\right)={\mathrm{ƭ}}_H\left(e\right)=\varnothing \subseteq {\mathrm{ƭ}}_H\left(e\right)∆{\mathrm{ƭ}}_H\left(e\right)=\varnothing .\end{array}$$

Moreover, ${{\mathrm{ƭ}}_H}^{-1}\left(e\right)={\mathrm{ƭ}}_H\left(e^{-1}\right)$, ${{\mathrm{ƭ}}_H}^{-1}\left(\mathcal{o}\right)={\mathrm{ƭ}}_H\left({\mathcal{o}}^{-1}\right)$, ${\mathrm{ƭ}}_H$ is an SSD-subgroup of ${\mathrm{ƭ}}_G$.

Definition 14.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. Then, ${\mathrm{ƭ}}_G$ is called an abelian SSD-group if ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)$, for all $\mathcal{ⱴ},\mathcal{y}\in G$.

It is obvious that if $G$ is abelian, then ${\mathrm{ƭ}}_G$ is abelian.

Definition 15.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. Then, ${\mathrm{ƭ}}_G$ is called soft normal in $G$ (or normal SSD-group), if ${\mathrm{ƭ}}_G$ is an abelian SSD-group.

We present the equivalent conditions of the normality of an SSD-group ${\mathrm{ƭ}}_G$ with Theorem 4 in a similar manner as soft-int groups in [83]:

Theorem 4.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. Then, the following conditions are equivalent:

(1) 

${\mathrm{ƭ}}_G$ is a normal SSD-group, that is, ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)$, for all $ⱴ$, $\mathcal{y}\in G$,

(2) 

${\mathrm{ƭ}}_G$ is constant in the conjugate class of $G$, that is, ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}{ⱴ}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $ⱴ$, $\mathcal{y}\in G$,

(3) 

${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}{ⱴ}^{-1}\right)\supseteq {\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $ⱴ$, $\mathcal{y}\in G$,

(4) 

${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}{ⱴ}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $ⱴ$, $\mathcal{y}\in G$.

Proof. 

Let $\mathrm{ⱴ}$,$\mathcal{y}\in G$ and ${\mathrm{ƭ}}_G$ be an SSD-group.

(1$\Rightarrow 2)$ Suppose that ${\mathrm{ƭ}}_G$ is a normal SSD-group, that is, ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)$, for all $\mathrm{ⱴ}$,$\mathcal{y}\in G$. Then, ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}{\mathrm{ⱴ}}^{-1}\right)={\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $\mathrm{ⱴ}$,$\mathcal{y}\in G$.

(2$\Rightarrow 3)$ Let ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}{\mathrm{ⱴ}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $\mathrm{ⱴ}$,$\mathcal{y}\in G$. Then, it is clear that ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}{\mathrm{ⱴ}}^{-1}\right)\supseteq {\mathrm{ƭ}}_G\left(\mathcal{y}\right)$.

(3$\Rightarrow 4)$ Let ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}{\mathrm{ⱴ}}^{-1}\right)\supseteq {\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $\mathrm{ⱴ}$,$\mathcal{y}\in G$. Then, ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}{\mathrm{ⱴ}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}\left(\mathcal{ⱴ}\mathcal{y}{\mathrm{ⱴ}}^{-1}\right){\left({\mathrm{ⱴ}}^{-1}\right)}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right)$.

(4$\Rightarrow 1)$ Let ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}{\mathrm{ⱴ}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $\mathrm{ⱴ}$,$\mathcal{y}\in G$. Then, ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\left(\mathcal{ⱴ}\mathcal{y}\right)\mathrm{ⱴ}{\mathrm{ⱴ}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\left(\mathcal{y}\mathcal{ⱴ}\right){\mathrm{ⱴ}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)={\mathrm{ƭ}}_G\left(\left(\mathcal{y}\mathcal{ⱴ}\right)\mathcal{y}{\mathcal{y}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\left(\mathcal{ⱴ}\mathcal{y}\right){\mathcal{y}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)$. Hence, ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)$ is obtained. □

Definition 16.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group and ${\mathrm{ƭ}}_N$ be an SSD-subgroup of ${\mathrm{ƭ}}_G$. Then, ${\mathrm{ƭ}}_N$ is called a normal SSD-subgroup, if it is an abelian SSD-subgroup of ${\mathrm{ƭ}}_G$.

Note that normal SSD-group defined in Definition 15 and normal SSD-subgroup defined in Definition 16 are two distinct notions. The first concept concerns structural properties of the SSD-group defined on $G$, whereas the second refers to a subgroup $N$ of $G$ satisfying an additional compatibility condition, which is a soft substructure of an SSD-group.

Example 7.

Let  $U=\left\{\mathrm{1,2},\mathrm{3,4}\right\}$ be the $\mathcal{U}\mathcal{S}$ and consider the dihedral group $G=D_3$, whose Cayley table is as Table 3:

Table 3. Cayley table of the Dihedral group ${\mathrm{D}}_3$.

$.$	$e$	$\varkappa$	${\varkappa }^2$	$\mathcal{y}$	$\mathcal{y}\varkappa$	$\mathcal{y}{\varkappa }^2$
$e$	$e$	$\varkappa$	${\varkappa }^2$	$\mathcal{y}$	$\mathcal{y}\varkappa$	$\mathcal{y}{\varkappa }^2$
$\varkappa$	$\varkappa$	${\varkappa }^2$	$e$	$\mathcal{y}{\varkappa }^2$	$\mathcal{y}$	$\mathcal{y}\varkappa$
${\varkappa }^2$	${\varkappa }^2$	$e$	$\varkappa$	$\mathcal{y}\varkappa$	$\mathcal{y}{\varkappa }^2$	$\mathcal{y}$
$\mathcal{y}$	$\mathcal{y}$	$\mathcal{y}\varkappa$	$\mathcal{y}{\varkappa }^2$	$e$	$\varkappa$	${\varkappa }^2$
$\mathcal{y}\varkappa$	$\mathcal{y}\varkappa$	$\mathcal{y}{\varkappa }^2$	$\mathcal{y}$	${\varkappa }^2$	$e$	$\varkappa$
$\mathcal{y}{\varkappa }^2$	$\mathcal{y}{\varkappa }^2$	$\mathcal{y}$	$\mathcal{y}\varkappa$	$\varkappa$	${\varkappa }^2$	$e$

We define the $\mathcal{S}\mathcal{S}$ ${\mathrm{ƭ}}_G$ over $U$ by

$${\mathrm{ƭ}}_G=\left\{\left(e,\varnothing \right),\left(\varkappa ,\varnothing \right),\left({\varkappa }^2,\varnothing \right),\left(\mathcal{y},\left\{\mathrm{1,4}\right\}\right),\left(\mathcal{y}\varkappa ,\left\{\mathrm{1,4}\right\}\right),\left({\mathcal{y}\varkappa }^2,\left\{\mathrm{1,4}\right\}\right)\right\}.$$

Let the $\mathcal{S}\mathcal{S}$ ${\mathrm{ƭ}}_H$ over $U$ be as follows:

$${\mathrm{ƭ}}_H=\left\{\left(e,\varnothing \right),\left(\varkappa ,\varnothing \right),\left({\varkappa }^2,\varnothing \right)\right\},$$

where $H=\left\{e,\varkappa ,{\varkappa }^2\right\}$ is the subgroup of $G$. One can show that ${\mathrm{ƭ}}_H$ is a normal SSD-subgroup of ${\mathrm{ƭ}}_G$.

Note 3.

The soft intersection-symmetric difference product of groups is not commutative in general; however, we can state Proposition 7:

Proposition 7.

If  ${\mathrm{ƭ}}_G$ is an SSD-group and ${\mathrm{ƭ}}_N$ is a normal SSD-subgroup of ${\mathrm{ƭ}}_G$, then ${\mathrm{ƭ}}_N{\otimes }_{i/s}{\mathrm{ƭ}}_G={\mathrm{ƭ}}_G{\otimes }_{i/s}{\mathrm{ƭ}}_N$.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group and ${\mathrm{ƭ}}_N$ be a normal SSD-subgroup of ${\mathrm{ƭ}}_G$. Then, for all $\mathrm{ⱴ}\in G$,

$$\left({\mathrm{ƭ}}_H{\otimes }_{i/s}{\mathrm{ƭ}}_G\right)\left(\mathrm{ⱴ}\right)=\bigcap _{\mathcal{ⱴ}=\mathcal{y}\mathcal{z}}\left({\mathrm{ƭ}}_H\left(\mathcal{y}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{z}\right)\right)$$

where $\mathcal{y},$ $\mathcal{z}\in G$. Since ${\mathrm{ƭ}}_N$ is a normal SSD-subgroup of ${\mathrm{ƭ}}_G$ and $\mathcal{y}\mathcal{z}=\mathcal{ⱴ}$, we have $\mathcal{y}=\mathcal{ⱴ}{\mathcal{z}}^{-1}$ and

$$\begin{array}{cc}\left({\mathrm{ƭ}}_H{\otimes }_{i/s}{\mathrm{ƭ}}_G\right)\left(\mathrm{ⱴ}\right)& ={\displaystyle \bigcap _{\mathrm{ⱴ}=\left(\mathrm{ⱴ}{\mathcal{z}}^{-1}\right)\mathcal{z}}}\left({\mathrm{ƭ}}_H\left(\mathrm{ⱴ}{\mathcal{z}}^{-1}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{z}\right)\right)\hfill \\ & ={\displaystyle \bigcap _{\mathcal{ⱴ}=\mathcal{z}\left({\mathcal{z}}^{-1}\mathrm{ⱴ}\right)}}\left({\mathrm{ƭ}}_G\left(\mathcal{z}\right)∆{\mathrm{ƭ}}_H\left({\mathcal{z}}^{-1}\mathrm{ⱴ}\right)\right)\hfill \\ & =\left({\mathrm{ƭ}}_N{\otimes }_{i/s}{\mathrm{ƭ}}_H\right)\left(\mathrm{ⱴ}\right).\hfill \end{array}$$

Hence, ${\mathrm{ƭ}}_N{\otimes }_{i/s}{\mathrm{ƭ}}_G={\mathrm{ƭ}}_G{\otimes }_{i/s}{\mathrm{ƭ}}_N$. □

Definition 17.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group, $\psi$ be a map from $G$ into itself. We define a map

$${ \mathrm{ƭ}}_G^{\psi }:G⟶P\left(U\right)$$

by ${ \mathrm{ƭ}}_G^{\psi }\left(ⱴ\right)={\mathrm{ƭ}}_G\left(\psi \left(ⱴ\right)\right)$, for all $ⱴ\in G$. Then, ${\mathrm{ƭ}}_G$ is called a characteristic SSD-group, if ${ \mathrm{ƭ}}_G^{\psi }={ \mathrm{ƭ}}_G$ for every automorphism $\psi$ of $G$.

Proposition 8.

If  ${\mathrm{ƭ}}_G$ is an SSD-group and $\psi$ is a homomorphism of $G$, then ${ \mathrm{ƭ}}_G^{\psi }$ is an SSD-group.

Proof. 

Suppose that ${\mathrm{ƭ}}_G$ is an SSD-group and $\psi$ be a homomorphism of $G$. Then, for all $\mathcal{ⱴ},\mathcal{y}\in G$,

$${\mathrm{ƭ}}_G^{\psi }\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\psi \left(\mathcal{ⱴ}\mathcal{y}\right)\right)={\mathrm{ƭ}}_G\left(\psi \left(\mathrm{ⱴ}\left)\psi \right(\mathcal{y}\right)\right)\subseteq {\mathrm{ƭ}}_G\left(\psi \left(\mathrm{ⱴ}\left)\right){∆\mathrm{ƭ}}_G\left(\psi \right(\mathcal{y}\right)\right)={\mathrm{ƭ}}_G^{\psi }\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G^{\psi }\left(\mathcal{y}\right).$$

Moreover, we have

$${\mathrm{ƭ}}_G^{\psi }\left({\mathrm{ⱴ}}^{-1}\right)={\mathrm{ƭ}}_G\left(\psi \left({\mathrm{ⱴ}}^{-1}\right)\right)={\mathrm{ƭ}}_G{\left(\psi \left(\mathrm{ⱴ}\right)\right)}^{-1}={\mathrm{ƭ}}_G\left(\psi \left(\mathrm{ⱴ}\right)\right)={\mathrm{ƭ}}_G^{\psi }\left(\mathrm{ⱴ}\right).$$

Thus, ${\mathrm{ƭ}}_G^{\psi }$ is an SSD-group. □

Proposition 9.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group, where ${\mathrm{ƭ}}_G$ is an injective mapping and $\psi$ be an epimorphism of $G$. Then, ${ \mathrm{ƭ}}_G^{\psi }$ is a normal SSD-group if and only if $G$ is an abelian group.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group, where ${\mathrm{ƭ}}_G$ is an injective mapping and $\psi$ is an epimorphism of $G$. Let $G$ be an abelian group. By Proposition 8, ${\mathrm{ƭ}}_G^{\psi }$ is an SSD-group. Thus, we only need to show the normality of ${\mathrm{ƭ}}_G^{\psi }$. Since $G$ is an abelian group, then for all $\mathcal{ⱴ},\mathcal{y}\in G$, ${\mathrm{ƭ}}_G^{\psi }\left(\mathcal{ⱴ}\mathcal{y}\right){=\mathrm{ƭ}}_G^{\psi }\left(\mathcal{y}\mathcal{ⱴ}\right)$, which shows the normality of ${\mathrm{ƭ}}_G^{\psi }$.

Conversely, let ${\mathrm{ƭ}}_G^{\psi }$ be a normal SSD-group, where $\psi$ be an epimorphism of $G$. Let $\mathcal{ⱴ},\mathcal{y}\in G$. Then,

$${\mathrm{ƭ}}_G^{\psi }\left(\mathcal{ⱴ}\mathcal{y}\right){=\mathrm{ƭ}}_G^{\psi }\left(\mathcal{y}\mathcal{ⱴ}\right)⟹{\mathrm{ƭ}}_G\left(\left(\psi (\mathrm{ⱴ}\right)\psi \left(\mathcal{y}\right)\right)={\mathrm{ƭ}}_G\left(\left(\psi (\mathcal{y}\right)\psi (\mathrm{ⱴ}\right)⟹\psi \left(\mathrm{ⱴ}\right)\psi \left(\mathcal{y}\right)=\psi \left(\mathcal{y}\right)\psi \left(\mathrm{ⱴ}\right).$$

Since $\psi$ is an epimorphism of $G$, it follows that $G$ is an abelian group, □

Proposition 10.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. If ${\mathrm{ƭ}}_G$ is a characteristic SSD-group, then ${\mathrm{ƭ}}_G$ is a normal SSD-group.

Proof: 

Let ${\mathrm{ƭ}}_{\mathrm{G}}$ be a characteristic SSD-group. We need to show that ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right){=\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)$, for all $\mathcal{ⱴ},\mathcal{y}\in G$. Let ${\psi }_{\mathrm{ⱴ}}$ be a homomorphism from $G$ to $G$ defined by

$${\psi }_{\mathrm{ⱴ}}\left(\mathfrak{g}\right)={\mathrm{ⱴ}}^{-1}\mathfrak{g}\mathfrak{ⱴ}$$

for all $\mathfrak{g}\in G$. One can easily show that ${\psi }_{\mathrm{ⱴ}}$ is an automorphism of $G$ induced by $\mathrm{ⱴ}$. Since ${\mathrm{ƭ}}_G$ is a characteristic SSD-group, then ${\mathrm{ƭ}}_G^{\varphi }={\mathrm{ƭ}}_G$ for every automorphism $\varphi$ of $G$. Thus, we have

$${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G^{{\psi }_{\mathrm{ⱴ}}}\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left({\psi }_{\mathrm{ⱴ}}\left(\mathcal{ⱴ}\mathcal{y}\right)\right)={\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}\left(\mathcal{ⱴ}\mathcal{y}\right)\mathrm{ⱴ}\right){=\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)$$

for all $\mathcal{ⱴ},\mathcal{y}\in \mathrm{G}$. Hence, ${\mathrm{ƭ}}_{\mathrm{G}}$ is a normal SSD-group. □

Definition 18.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group, ${\mathrm{ƭ}}_H$ and ${\mathrm{ƭ}}_K$ be SSD-subgroups of ${\mathrm{ƭ}}_G$. We say that ${\mathrm{ƭ}}_H$ is soft conjugate to ${\mathrm{ƭ}}_K$, if there exists $\mathcal{y}\in G$ such that

$${\mathrm{ƭ}}_H(ⱴ)={\mathrm{ƭ}}_K\left({\mathcal{y}}^{-1}\mathcal{ⱴ}\mathcal{y}\right)$$

for all $ⱴ\in G$.

Notation 1.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group, $\mathfrak{g}\in G$. We denote by

$${ \mathrm{ƭ}}_G^{\mathfrak{g}}:G⟶P\left(U\right)$$

the map defined by ${ \mathrm{ƭ}}_G^{\mathfrak{g}}\left(ⱴ\right)={\mathrm{ƭ}}_G{(\mathfrak{g}}^{-1}\mathfrak{ⱴ}\mathfrak{g})$, for all $ⱴ\in G$.

Example 8.

Consider the group  $G=D_3$ and the SSD-group

$${\mathrm{ƭ}}_G=\left\{\left(e,\varnothing \right),\left(\varkappa ,\varnothing \right),\left({\varkappa }^2,\varnothing \right),\left(\mathcal{y},\left\{\mathrm{1,4}\right\}\right),\left(\mathcal{y}\varkappa ,\left\{\mathrm{1,4}\right\}\right),\left({\mathcal{y}\varkappa }^2,\left\{\mathrm{1,4}\right\}\right)\right\}$$

over $U=\left\{\mathrm{1,2},\mathrm{3,4}\right\}$ in Example 7. Let $\mathfrak{g}=\mathcal{y}\varkappa \in D_3$. Then,

$$\begin{array}{c}{ \mathrm{ƭ}}_G^{\mathcal{y}\varkappa }=\left\{\left(e,{\mathrm{ƭ}}_G\left(e\right)\right),\left(\varkappa ,{\mathrm{ƭ}}_G\left({\varkappa }^2\right)\right),\left({\varkappa }^2,{\mathrm{ƭ}}_G(\varkappa \right)),\left(\mathcal{y},{\mathrm{ƭ}}_G\left(\mathcal{y}{\varkappa }^2\right)\right),\left(\mathcal{y}\varkappa ,{\mathrm{ƭ}}_G\left(\mathcal{y}\varkappa \right)\right),\left({\mathcal{y}\varkappa }^2,{\mathrm{ƭ}}_G\right.(\mathcal{y})\right\}=\\ \left\{\left(e,\varnothing \right),\left(\varkappa ,\varnothing \right),\left({\varkappa }^2,\varnothing \right),\left(\mathcal{y},\left\{\mathrm{1,4}\right\}\right),\left(\mathcal{y}\varkappa ,\left\{\mathrm{1,4}\right\}\right),\left({\mathcal{y}\varkappa }^2,\left\{\mathrm{1,4}\right\}\right)\right\}.\end{array}$$

Proposition 11.

If ${\mathrm{ƭ}}_G$ is an SSD-group, then ${ \mathrm{ƭ}}_G^{\mathfrak{g}}$ is an SSD-group, for all $\mathfrak{g}\in G$.

Proof. 

Suppose that ${\mathrm{ƭ}}_G$ is an SSD-group, $\mathfrak{g}\in G$. Then, for all $\mathcal{ⱴ},\mathcal{y}\in G$,

$$\begin{array}{c}{\mathrm{ƭ}}_G^{\mathfrak{g}}\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left({\mathfrak{g}}^{-1}\left(\mathcal{ⱴ}\mathcal{y}\right)\mathfrak{g}\right)={\mathrm{ƭ}}_G\left({\mathfrak{g}}^{-1}\mathrm{ⱴ}\left({\mathfrak{g}\mathfrak{g}}^{-1}\right)\mathcal{y}\mathfrak{g}\right)={\mathrm{ƭ}}_G\left(({\mathfrak{g}}^{-1}\mathrm{ⱴ}{\mathfrak{g}\left)\right(\mathfrak{g}}^{-1}\mathcal{y}\mathfrak{g}\right))\subseteq {\mathrm{ƭ}}_G({\mathfrak{g}}^{-1}\mathrm{ⱴ}{\mathfrak{g})∆{\mathrm{ƭ}}_G(\mathfrak{g}}^{-1}\mathcal{y}\mathfrak{g})=\\ {\mathrm{ƭ}}_G^{\mathfrak{g}}\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G^{\mathfrak{g}}\left(\mathcal{y}\right).\end{array}$$

Moreover, we have

$${\mathrm{ƭ}}_G^{\mathfrak{g}}\left({\mathrm{ⱴ}}^{-1}\right)={\mathrm{ƭ}}_G\left({\mathfrak{g}}^{-1}{\mathrm{ⱴ}}^{-1}\mathfrak{g}\right)={\mathrm{ƭ}}_G\left({\left({\mathfrak{g}}^{-1}\mathfrak{ⱴ}\mathfrak{g}\right)}^{-1}\right)={\mathrm{ƭ}}_G\left({\mathfrak{g}}^{-1}\mathfrak{ⱴ}\mathfrak{g}\right)={\mathrm{ƭ}}_G^{\mathfrak{g}}\left(\mathrm{ⱴ}\right).$$

Thus, ${\mathrm{ƭ}}_G^{\mathfrak{g}}$ is an SSD-group. □

Definition 19.

The SSD-group  ${ \mathrm{ƭ}}_G^{\mathfrak{g}}$ in Proposition 11 is called the conjugate SSD-group determined by ${\mathrm{ƭ}}_G$ and $\mathfrak{g}\in G$.

Definition 20.

Let  ${ \mathrm{ƭ}}_G$ be an SSD-group. Then, the normalizer of ${ \mathrm{ƭ}}_G$ is the set given by

$${ N(\mathrm{ƭ}}_G)=\left\{\mathfrak{g}\in G, { \mathrm{ƭ}}_G\left(\mathfrak{g}\mathcal{y}\right)={ \mathrm{ƭ}}_G\left(\mathcal{y}\mathfrak{g}\right), for all \mathcal{y}\in G\right\}.$$

Proposition 12.

If ${\mathrm{ƭ}}_G$ is an SSD-group and $A=\left\{\mathfrak{g}\in G, { \mathrm{ƭ}}_G^{\mathfrak{g}} ={ \mathrm{ƭ}}_G\right\}$, then $A={ N(\mathrm{ƭ}}_G)$.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group, $\mathfrak{g}\in {\mathrm{N}(\mathrm{ƭ}}_G)$. Then, ${\mathrm{ƭ}}_G\left(\mathfrak{g}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathfrak{g}\right)$, for all $\mathcal{y}\in G$. Thus, ${\mathrm{ƭ}}_G\left({\mathfrak{g}}^{-1}\mathcal{y}\mathfrak{g}\right)={\mathrm{ƭ}}_G\left({\mathfrak{g}}^{-1}\mathfrak{g}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $\mathcal{y}\in G$. Thus, $\mathfrak{g}\in A$, implying that ${\mathrm{N}(\mathrm{ƭ}}_G)$ $\subseteq A$. Similarly, let $\mathfrak{g}\in A$. Then, ${\mathrm{ƭ}}_G^{\mathfrak{g}}\left(\mathrm{ɍ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)$, for all $\mathrm{ɍ}\in G$. Thus, ${\mathrm{ƭ}}_G\left({\mathfrak{g}}^{-1}\mathfrak{ɍ}\mathfrak{g}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)$, for all $\mathrm{ɍ}\in G$. Since $\mathfrak{g}\mathcal{y}\in G$, ${\mathrm{ƭ}}_G\left({\mathfrak{g}}^{-1}\left(\mathfrak{g}\mathcal{y}\right)\mathfrak{g}\right)={\mathrm{ƭ}}_G\left(\mathfrak{g}\mathcal{y}\right)$. Hence, ${\mathrm{ƭ}}_G\left(\mathcal{y}\mathfrak{g}\right)={\mathrm{ƭ}}_G\left(\mathfrak{g}\mathcal{y}\right)$, for all $\mathcal{y}\in G$. Thus, $\mathfrak{g}\in {\mathrm{N}(\mathrm{ƭ}}_G)$, implying that $A\subseteq {\mathrm{N}(\mathrm{ƭ}}_G)$. Then, $A={\mathrm{N}(\mathrm{ƭ}}_G)$. □

Proposition 13.

If ${\mathrm{ƭ}}_G$ is an SSD-group, then ${N(\mathrm{ƭ}}_G)$ is a subgroup of $G$.

Proof. 

Suppose that ${\mathrm{ƭ}}_{\mathrm{G}}$ is an SSD-group. Since ${\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{e}}^{-1}\mathrm{ⱴ}\mathrm{e}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)$, for all $\mathrm{ⱴ}\in \mathrm{G}$, we have ${\mathrm{ƭ}}_{\mathrm{G}}^{\mathrm{e}}={\mathrm{ƭ}}_{\mathrm{G}}$. Thus, $\mathrm{e}\in {\mathrm{N}(\mathrm{ƭ}}_{\mathrm{G}})$ and ${\mathrm{N}(\mathrm{ƭ}}_{\mathrm{G}})\ne \mathrm{\varnothing }. {\mathrm{N}(\mathrm{ƭ}}_{\mathrm{G}})\subseteq \mathrm{G}$ is obvious. Let $\mathrm{ɍ},\mathrm{ɗ}$ $\in {\mathrm{N}(\mathrm{ƭ}}_{\mathrm{G}})$. Then, we have ${\mathrm{ƭ}}_{\mathrm{G}}^{\mathrm{ɍ}}={\mathrm{ƭ}}_{\mathrm{G}}$, ${\mathrm{ƭ}}_{\mathrm{G}}^{\mathrm{ɗ}}={\mathrm{ƭ}}_{\mathrm{G}}$. We, firstly, need to show that $\mathrm{ɍ}\mathrm{ɗ}$ $\in {\mathrm{N}(\mathrm{ƭ}}_{\mathrm{G}})$. Then, for all $\mathrm{ⱴ}\in \mathrm{G}$,

$${\mathrm{ƭ}}_{\mathrm{G}}^{\mathrm{ɍ}\mathrm{ɗ}}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\left(\mathrm{ɍ}\mathrm{ɗ}\right)}^{-1}\mathrm{ⱴ}\left(\mathrm{ɍ}\mathrm{ɗ}\right)\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɗ}}^{-1}{\mathrm{ɍ}}^{-1}\mathrm{ⱴ}\mathrm{ɍ}\mathrm{ɗ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɗ}}^{-1}{(\mathrm{ɍ}}^{-1}\mathrm{ⱴ}\mathrm{ɍ})\mathrm{ɗ}\right)={\mathrm{ƭ}}_{\mathrm{G}}^{\mathrm{ɗ}}\left({\mathrm{ɍ}}^{-1}\mathrm{ⱴ}\mathrm{ɍ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɍ}}^{-1}\mathrm{ⱴ}\mathrm{ɍ}\right)={\mathrm{ƭ}}_{\mathrm{G}}^{\mathrm{ɍ}}\left(\mathrm{ⱴ}\right){=\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right).$$

Thus, we have $\mathrm{ɍ}\mathrm{ɗ}$ $\in {\mathrm{N}(\mathrm{ƭ}}_{\mathrm{G}})$.

Let $\mathrm{ɍ}\in {\mathrm{N}(\mathrm{ƭ}}_G)$. We need to show that ${{\mathrm{ɍ}}^{-1}\in \mathrm{N}(\mathrm{ƭ}}_G)$. Since $\mathrm{ɍ}\in {\mathrm{N}(\mathrm{ƭ}}_G),{\mathrm{ƭ}}_G\left(\mathrm{ɍ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathrm{ɍ}\right),$ for all $\mathcal{y}\in G$. Thus, ${\mathrm{ƭ}}_G\left({\left(\mathrm{ɍ}\mathcal{y}\right)}^{-1}\right)={\mathrm{ƭ}}_G\left({\left(\mathcal{y}\mathrm{ɍ}\right)}^{-1}\right)$, for all $\mathcal{y}\in G$. Hence, ${\mathrm{ƭ}}_G\left({\mathcal{y}}^{-1}{\mathrm{ɍ}}^{-1}\right)={\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}{\mathcal{y}}^{-1}\right)$, for all ${\mathcal{y}}^{-1}\in G$. Since ${\mathcal{y}}^{-1}$ is arbitrary, ${{\mathrm{ɍ}}^{-1}\in \mathrm{N}(\mathrm{ƭ}}_G)$. Thus, ${\mathrm{N}(\mathrm{ƭ}}_G)$ is a subgroup of $G$. □

Proposition 14.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. Then, ${\mathrm{ƭ}}_G$ is a normal SSD-group if and only if ${N(\mathrm{ƭ}}_G)=G$.

Proof. 

Let ${\mathrm{ƭ}}_G$ be a normal SSD-group. ${\mathrm{N}(\mathrm{ƭ}}_G)\subseteq G$ is obvious. Let $\mathfrak{g}\in G$. For any $\mathcal{v}\in G$,

$${\mathrm{ƭ}}_G^{\mathfrak{g}}\left(\mathcal{v}\right)={\mathrm{ƭ}}_G\left({\mathfrak{g}}^{-1}\mathcal{v}\mathfrak{g}\right)={\mathrm{ƭ}}_G\left({\mathfrak{g}}^{-1}\mathfrak{g}\mathcal{v}\right)={\mathrm{ƭ}}_G\left(\mathcal{v}\right).$$

Thus, ${\mathrm{ƭ}}_G^{\mathfrak{g}}={\mathrm{ƭ}}_G$. Hence, $\mathfrak{g}\in {\mathrm{N}(\mathrm{ƭ}}_G)$ and $G\subseteq {\mathrm{N}(\mathrm{ƭ}}_G)$, so ${\mathrm{N}(\mathrm{ƭ}}_G)=G$.

Conversely, let ${\mathrm{N}(\mathrm{ƭ}}_G)=G$ and $\mathcal{ⱴ},\mathcal{y}\in G$. We need to show that ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)$, for all $\mathcal{ⱴ},\mathcal{y}\in G$. We have

$${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}(\mathcal{ⱴ}{\mathrm{ⱴ}}^{-1}\right))={\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\left(\mathcal{y}\mathcal{ⱴ}\right){\mathrm{ⱴ}}^{-1}\right){=\mathrm{ƭ}}_G^{{\mathrm{ⱴ}}^{-1}}\left(\mathcal{y}\mathcal{ⱴ}\right){=\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right).$$

Thus, ${\mathrm{ƭ}}_G$ is a normal SSD-group. □

We now give an alternative formulation of SSD-group in terms of “commutators” of a group. First, we recall the following definition:

Definition 21.

Let  $G$ be any group and $ⱴ,\mathcal{y}\in G$. The element ${ ⱴ}^{-1}{\mathcal{y}}^{-1} ⱴ\mathcal{y}\in G$ is usually denoted $\left[ⱴ,\mathcal{y}\right]$ and called the commutator of $ⱴ$ and $\mathcal{y}$.

It is well known that $G$ is abelian if and only if $\left[\mathrm{ⱴ},\mathcal{y}\right]=e$, for all $\mathrm{ⱴ},\mathcal{y}\in G$. This motivates the following proposition:

Proposition 15.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. Then, we have the following:

i. 

If ${\mathrm{ƭ}}_G$ is a normal SSD-group, then ${\mathrm{ƭ}}_G\left[ⱴ,\mathcal{y}\right]={\mathrm{ƭ}}_G(e$), for all $ⱴ,\mathcal{y}\in G$,

ii. 

${\mathrm{ƭ}}_G\left[ⱴ,\mathcal{y}\right]={\mathrm{ƭ}}_G\left(e\right)$ for all $ⱴ,\mathcal{y}\in G$ if and only is ${\mathrm{ƭ}}_G$ is a normal SSD-group, where ${\mathrm{ƭ}}_G$ is bijective.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group and $\mathrm{ⱴ},\mathcal{y}\in G$.

i.

Suppose that ${\mathrm{ƭ}}_G$ is a normal SSD-group. Then, ${\mathrm{ƭ}}_G\left[\mathrm{ⱴ},\mathcal{y}\right]={\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}{\mathcal{y}}^{-1}\mathrm{ⱴ}\mathcal{y}{)={\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}\mathrm{ⱴ}{\mathcal{y}}^{-1}\mathcal{y}\right)=\mathfrak{k}}_G\right(e$), for all $\mathrm{ⱴ},\mathcal{y}\in G$.

ii.

Let $\mathrm{ⱴ},\mathcal{y}\in G$. Then,

$${\mathrm{ƭ}}_G\left[\mathrm{ⱴ},\mathcal{y}\right]={\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}{\mathcal{y}}^{-1}\mathrm{ⱴ}\mathcal{y}{)=\mathrm{ƭ}}_G\right(e)⟺{\mathrm{ⱴ}}^{-1}{\mathcal{y}}^{-1}\mathrm{ⱴ}\mathcal{y}=e⟺\mathcal{y}\mathrm{ⱴ}=\mathrm{ⱴ}\mathcal{y}⟺{\mathrm{ƭ}}_G\left(\mathcal{y}\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathrm{ⱴ}\right).\square$$

4. On E-Set, Soft Cosets, and Soft Quotient Groups of Soft Symmetric Difference Groups

In this section, we introduce the concept of e-sets and construct and study soft cosets and soft quotient groups of soft symmetric difference groups. Moreover, we characterize these notions in detail in terms of the normality of soft symmetric difference groups.

Proposition 16.

If  ${\mathrm{ƭ}}_G$ is an SSD-group, then $G_{{\mathrm{ƭ}}_G}$ is a subgroup of $G$, where

$$G_{{\mathrm{ƭ}}_G}=\left\{ⱴ\in G:{\mathrm{ƭ}}_G\left(ⱴ\right)={\mathrm{ƭ}}_G\left(e\right)\right\}.$$

Proof. 

It is clear that $e\in G_{{\mathrm{ƭ}}_G}$, thus, $G_{{\mathrm{ƭ}}_G}\ne \mathrm{\varnothing }$ and $G_{{\mathrm{ƭ}}_G}\subseteq G$. Let $\mathcal{ⱴ},\mathcal{y}\in G_{{\mathrm{ƭ}}_G}$. Then, ${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right)={\mathrm{ƭ}}_G\left(e\right)$. Moreover, for all $\mathcal{ⱴ},\mathcal{y}\in G$, we have

$${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left({\mathcal{y}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)={\mathrm{ƭ}}_G\left(e\right)∆{\mathrm{ƭ}}_G\left(e\right)=\mathrm{\varnothing }={\mathrm{ƭ}}_G\left(e\right).$$

Since ${\mathrm{ƭ}}_G\left(e\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)$, for all $\mathcal{ⱴ},\mathcal{y}\in G$, it follows that ${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathcal{y}}^{-1}\right)={\mathrm{ƭ}}_G\left(e\right)$. Thus, $\mathrm{ⱴ}{\mathcal{y}}^{-1}\in G_{{\mathrm{ƭ}}_G}$, for all $\mathcal{ⱴ},\mathcal{y}\in G$. Thus, $G_{{\mathrm{ƭ}}_G}$ is a subgroup of $G$. □

Proposition 17.

If  ${\mathrm{ƭ}}_G$ is a normal SSD-group, then $G_{{\mathrm{ƭ}}_G}$ is a normal subgroup of $G$.

Proof. 

Let ${\mathrm{ƭ}}_{\mathrm{G}}$ be a normal SSD-group. By Proposition 16, it is clear that ${\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}$ is a subgroup of $\mathrm{G}$. Let $\mathrm{m}\in \mathrm{G}$ and $\mathrm{ⱴ}\in {\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}$. Then, we have

$${\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{m}\mathrm{ⱴ}{\mathrm{m}}^{-1}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{m}{\mathrm{m}}^{-1}\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{e}\right).$$

Hence, $\mathrm{m}\mathrm{ⱴ}{\mathrm{m}}^{-1}\in {\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}$, therefore ${\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}$ is a normal subgroup of $\mathrm{G}$. □

Proposition 18.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. Then, for all $\mathcal{y}\in G$, ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right)$ if and only if $ⱴ\in G_{{\mathrm{ƭ}}_G}$.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group and ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $\mathcal{y}\in G$. By setting $\mathcal{y}=e$,

$${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}e\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\left(e\right).$$

Thus, $\mathrm{ⱴ}\in G_{{\mathrm{ƭ}}_G}.$ Conversely, assume that $\mathrm{ⱴ}\in G_{{\mathrm{ƭ}}_G}.$ Then, ${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\left(e\right)$. Thus, for all $\mathcal{y}\in G$, we have

$${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)={\mathrm{ƭ}}_G\left(e\right)∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)=\mathrm{\varnothing }∆{\mathrm{ƭ}}_G\left(\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right).$$

Therefore, for all $\mathcal{y}\in G$, ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathcal{y}\right)$ holds. Moreover, for all $\mathcal{y}\in G$,

$${\mathrm{ƭ}}_G\left(\mathcal{y}\right)={\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}\mathcal{ⱴ}\mathcal{y}\right)\subseteq {\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)=\mathrm{\varnothing }∆{\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right).$$

Thus, ${\mathrm{ƭ}}_G\left(\mathcal{y}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)$, for all $\mathcal{y}\in G$. From these results, it follows that ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $\mathcal{y}\in G$. □

Proposition 19.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. If $ⱴ\in G_{{\mathrm{ƭ}}_G}$, then ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)$, for all $\mathcal{y}\in G$.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group. Suppose that $\mathrm{ⱴ}\in G_{{\mathrm{ƭ}}_G}$. Then, ${\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\left(e\right)$. By Proposition 18, ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right)$, for all $\mathcal{y}\in G$. Then, for all $\mathcal{y}\in G$,

$${\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathcal{y}\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right)∆{\mathrm{ƭ}}_G\left(e\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\right)∆\mathrm{\varnothing }={\mathrm{ƭ}}_G\left(\mathcal{y}\right).$$

Thus, ${\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathcal{y}\right)$. Moreover, we have

$${\mathrm{ƭ}}_G\left(\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}{\mathrm{ⱴ}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)∆{\mathrm{ƭ}}_G\left({\mathrm{ⱴ}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)∆{\mathrm{ƭ}}_G\left(e\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)∆={\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right).$$

Hence, ${\mathrm{ƭ}}_G\left(\mathcal{y}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)$, for all $\mathcal{y}\in G$. From these results, it follows that ${\mathrm{ƭ}}_G\left(\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)$, for all $\mathcal{y}\in G$. Hence, ${\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)$, for all $\mathcal{y}\in G$. □

Proposition 20.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group and $ɍ,ɗ \in G$. We have the following statements:

(i) 

If $ɍɗ \in G_{{\mathrm{ƭ}}_G},$ then ${\mathrm{ƭ}}_G\left(ɍ\right)={\mathrm{ƭ}}_G\left(ɗ\right)$,

(ii) 

If $ɍ{ɗ}^{-1}\in G_{{\mathrm{ƭ}}_G},$ then ${\mathrm{ƭ}}_G\left(ɍ\right)={\mathrm{ƭ}}_G\left(ɗ\right)$,

(iii) 

If ${ɍ}^{-1}ɗ \in G_{{\mathrm{ƭ}}_G},$ then ${\mathrm{ƭ}}_G\left(ɍ\right)={\mathrm{ƭ}}_G\left(ɗ\right)$,

(iv) 

If ${ɍ}^{-1}{ɗ}^{-1\hspace{0.33em}}\in G_{{\mathrm{ƭ}}_G},$ then ${\mathrm{ƭ}}_G\left(ɍ\right)={\mathrm{ƭ}}_G\left(ɗ\right)$.

Proof: 

Let ${\mathrm{ƭ}}_G$ be an SSD-group and $\mathrm{ɍ},\mathrm{ɗ}\in \mathrm{G}$.

(i)

If $\mathrm{ɍ}\mathrm{ɗ}\hspace{0.33em}\in G_{{\mathrm{ƭ}}_G},$ then ${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(e\right)$. Thus,

$${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}{\mathrm{ɗ}\mathrm{ɗ}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ɍ}\mathrm{ɗ}\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(e\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right).$$

Therefore, ${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right).$ Similarly,

$${\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\mathrm{ɍ}\mathrm{ɗ}\right)\subseteq {\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ɍ}\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)∆{\mathrm{ƭ}}_G\left(e\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right).$$

Hence, ${\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)$. Thus, ${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)$.

(ii)

If $\mathrm{ɍ}{\mathrm{ɗ}}^{-1}\in G_{{\mathrm{ƭ}}_G},$ then ${\mathrm{ƭ}}_G\left(\mathrm{ɍ}{\mathrm{ɗ}}^{-1}\right)={\mathrm{ƭ}}_G\left(e\right)$. Thus,

$${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}{\mathrm{ɗ}}^{-1}\mathrm{ɗ}\right)\subseteq \hspace{0.33em}{\mathrm{ƭ}}_G\left(\mathrm{ɍ}{\mathrm{ɗ}}^{-1}\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(e\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right).$$

Therefore, ${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right).$ Similarly,

$${\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left({\mathrm{ɗ}}^{-1}\right)={\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\mathrm{ɍ}{\mathrm{ɗ}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ɍ}{\mathrm{ɗ}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)∆{\mathrm{ƭ}}_G\left(e\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right).$$

Hence, ${\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)$. Thus, ${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)$.

(iii)

If ${\mathrm{ɍ}}^{-1}\mathrm{ɗ}\in G_{{\mathrm{ƭ}}_G},$ then ${\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(e\right)$. Thus,

$${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)={\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\right)={\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\mathrm{ɗ}{\mathrm{ɗ}}^{-1}\right)\subseteq \hspace{0.33em}{\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\mathrm{ɗ}\right)\hspace{0.33em}∆{\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(e\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right).$$

Therefore, ${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right).$ Similarly,

$${\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}{\mathrm{ɍ}}^{-1}\mathrm{ɗ}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)∆{\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)∆{\mathrm{ƭ}}_G\left(e\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right).$$

Hence, ${\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)$. Thus, ${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)$.

(iv)

If ${\mathrm{ɍ}}^{-1}{\mathrm{ɗ}}^{-1}\in G_{{\mathrm{ƭ}}_G},$ then ${\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}{\mathrm{ɗ}}^{-1}\right)={\mathrm{ƭ}}_G\left(e\right)$. Thus,

$${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)={\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\right)={\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}{\mathrm{ɗ}}^{-1}\mathrm{ɗ}\right)\subseteq \hspace{0.33em}{\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}{\mathrm{ɗ}}^{-1}\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(e\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right).$$

Therefore, ${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right).$ Similarly,

$${\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left({\mathrm{ɗ}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}{\mathrm{ɍ}}^{-1}{\mathrm{ɗ}}^{-1}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)∆{\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}{\mathrm{ɗ}}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)∆{\mathrm{ƭ}}_G\left(e\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right).$$

Hence, ${\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)\subseteq {\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)$. Thus, ${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)$. □

In Refs. [71,83,84,85,88], soft cosets within the framework of soft int-groups were investigated. In parallel, our study extends this notion to SSD-groups as follows:

Definition 22.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group and $\mathcal{o}\in G$. The soft left coset of ${\mathrm{ƭ}}_G$ by $\mathcal{o}$, denoted by $\mathcal{o}{\mathrm{ƭ}}_G$, is defined by

$$\mathcal{o}{\mathrm{ƭ}}_G=\left\{\left(ⱴ,{\mathrm{ƭ}}_G\left({\mathcal{o}}^{-1}ⱴ\right)\right),ⱴ\in G \right\}$$

and the soft right coset of ${\mathrm{ƭ}}_G$ by $\mathcal{o}$, denoted by ${\mathrm{ƭ}}_G\mathcal{o}$, is defined by

$${\mathrm{ƭ}}_G\mathcal{o}=\left\{\left(ⱴ,{\mathrm{ƭ}}_G\left(ⱴ{\mathcal{o}}^{-1}\right)\right),ⱴ\in G \right\}$$

for all $ⱴ\in G$.

Note 4.

If  $G$ is an abelian group, then ${ɍ\mathrm{ƭ}}_G={\mathrm{ƭ}}_Gɍ$, for all $ɍ\in G$, since ${\mathrm{ƭ}}_G({ɍ}^{-1}ⱴ)={\mathrm{ƭ}}_G(ⱴ{ɍ}^{-1})$, for all $ⱴ, ɍ\in G$.

The converse of Note 4 holds when the SSD-group is injective, as shown in Proposition 21:

Proposition 21.

If  ${\mathrm{ƭ}}_G$ is an injective SSD-group and ${ɍ\mathrm{ƭ}}_G={\mathrm{ƭ}}_Gɍ$ for all $ɍ\in G$, then $G$ is an abelian group.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an injective SSD-group and ${\mathrm{ɍ}\mathrm{ƭ}}_G={\mathrm{ƭ}}_G\mathrm{ɍ}$, for all $\mathrm{ɍ}\in G$. Then, for all $\mathrm{ⱴ}$,$\mathcal{y}\in G$, we have

$${\mathrm{ƭ}}_G(\mathcal{ⱴ}\mathcal{y})=({\mathrm{ⱴ}}^{-1}{\mathrm{ƭ}}_G\left)\right(\mathcal{y})=({\mathrm{ƭ}}_G{\mathrm{ⱴ}}^{-1}\left)\right(\mathcal{y})={\mathrm{ƭ}}_G\left(\mathrm{y}{\left({\mathrm{ⱴ}}^{-1}\right)}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right).$$

Since ${\mathrm{ƭ}}_G$ is injective, $\mathcal{ⱴ}\mathcal{y}$ $=\mathcal{y}\mathcal{ⱴ}$ for all $\mathrm{ⱴ}$,$\mathcal{y}\in G$. Thus, $G$ is abelian. □

Theorem 5.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. Then, ${\mathrm{ƭ}}_G$ is a normal SSD-group if and only if ${ɍ\mathrm{ƭ}}_G={\mathrm{ƭ}}_Gɍ$, for all $ɍ$ $\in G$.

Proof. 

Let $\mathrm{ɍ}$ $\in G$ and ${\mathrm{ƭ}}_{\mathrm{G}}$ be a normal SSD-group. Then, ${\mathrm{ɍ}\mathrm{ƭ}}_G{\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathrm{ɍ}}^{-1}\right)=\mathrm{ƭ}}_G\mathrm{ɍ}\left(\mathrm{ⱴ}\right)$, for all $\mathrm{ⱴ}\in G$. Thus, ${\mathrm{ɍ}\mathrm{ƭ}}_G={\mathrm{ƭ}}_G\mathrm{ɍ}$, for all $\mathrm{ɍ}$ $\in G$.

Conversely, let ${\mathrm{ɍ}\mathrm{ƭ}}_G={\mathrm{ƭ}}_G\mathrm{ɍ}$, for all $\mathrm{ɍ}\in G$. Then, ${\mathrm{ɍ}\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\mathrm{ɍ}\left(\mathrm{ⱴ}\right)$, for all $\mathrm{ⱴ}\in G$. Thus, ${\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\mathrm{ⱴ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ⱴ}{\mathrm{ɍ}}^{-1}\right)$, for all $\mathrm{ⱴ}\in G$. Since ${\mathrm{ɍ}}^{-1}\in G$ is arbitrary, ${\mathrm{ƭ}}_G$ is a normal SSD-group. □

Example 9.

Consider the group  $G=\left\{e,\mathcal{o},\mathrm{Ҍ},\mathrm{ƶ}\right\}$ and the SSD-group ${\mathrm{ƭ}}_G=\left\{\left(e,\varnothing \right),\left(\mathcal{o},\left\{e,\varkappa \right\}\right),\left(\mathrm{Ҍ},\left\{e,\varkappa ,\mathcal{y}\right\}\right),\left(\mathrm{ƶ},\left\{\mathcal{y}\right\}\right)\right\}$over $U =D_2$ in Example 1. Then, the soft left cosets of ${\mathrm{ƭ}}_G$ are as follows:

$$e{\mathrm{ƭ}}_G=\left\{\left(e,\varnothing \right),\left(\mathcal{o},\left\{e,\varkappa \right\}\right),\left(\mathrm{Ҍ},\left\{e,\varkappa ,\mathcal{y}\right\}\right),\left(\mathrm{ƶ},\left\{\mathcal{y}\right\}\right)\right\},$$

$${\mathcal{o}\mathrm{ƭ}}_G=\left\{\left(e,\left\{e,\varkappa \right\}\right),\left(\mathcal{o},\varnothing \right),\left(\mathrm{Ҍ},\left\{\mathcal{y}\right\}\right),\left(\mathrm{ƶ},\left\{e,\varkappa ,\mathcal{y}\right\}\right)\right\},$$

$${\mathrm{Ҍ}\mathrm{ƭ}}_G=\left\{\left(e,\left\{e,\varkappa ,\mathcal{y}\right\}\right),\left(\mathcal{o},\left\{\mathcal{y}\right\}\right),\left(\mathrm{Ҍ},\varnothing \right),\left(\mathrm{ƶ},\left\{e,\varkappa \right\}\right)\right\},$$

$$\mathrm{ƶ}{\mathrm{ƭ}}_G=\left\{\left(e,\left\{\mathcal{y}\right\}\right),\left(\mathcal{o},\left\{e,\varkappa ,\mathcal{y}\right\}\right),\left(\mathrm{Ҍ},\left\{e,\varkappa \right\}\right),\left(\mathrm{ƶ},\varnothing \right)\right\}.$$

Since $G$ is an abelian group, the soft right cosets of ${\mathrm{ƭ}}_G$ are as follows:

$${\mathrm{ƭ}}_{G}e=\left\{\left(e,\varnothing \right),\left(\mathcal{o},\left\{e,\varkappa \right\}\right),\left(\mathrm{Ҍ},\left\{e,\varkappa ,\mathcal{y}\right\}\right),\left(\mathrm{ƶ},\left\{\mathcal{y}\right\}\right)\right\},$$

$${\mathrm{ƭ}}_G\mathcal{o}=\left\{\left(e,\left\{e,\varkappa \right\}\right),\left(\mathcal{o},\varnothing \right),\left(\mathrm{Ҍ},\left\{\mathcal{y}\right\}\right),\left(\mathrm{ƶ},\left\{e,\varkappa ,\mathcal{y}\right\}\right)\right\},$$

$${\mathrm{ƭ}}_G\mathrm{Ҍ}=\left\{\left(e,\left\{e,\varkappa ,\mathcal{y}\right\}\right),\left(\mathcal{o},\left\{\mathcal{y}\right\}\right),\left(\mathrm{Ҍ},\varnothing \right),\left(\mathrm{ƶ},\left\{e,\varkappa \right\}\right)\right\},$$

$${\mathrm{ƭ}}_G\mathrm{ƶ}=\left\{\left(e,\left\{\mathcal{y}\right\}\right),\left(\mathcal{o},\left\{e,\varkappa ,\mathcal{y}\right\}\right),\left(\mathrm{Ҍ},\left\{e,\varkappa \right\}\right),\left(\mathrm{ƶ},\varnothing \right)\right\}.$$

Note 5.

If $G$ is an abelian group, then ${\mathrm{ƭ}}_G$ is a normal SSD-group. Thus, ${ɍ\mathrm{ƭ}}_G={\mathrm{ƭ}}_Gɍ$, for all $ɍ\in G$. However, ${\mathrm{ƭ}}_G$ may be also a normal SSD-group, that is, it is possible that ${ɍ\mathrm{ƭ}}_G={\mathrm{ƭ}}_Gɍ$ for all $ɍ\in G$, even when $G$ is not abelian. Now, we give the following example:

Example 10.

Consider the group $G=\{e,\varkappa ,{\varkappa }^2,\mathcal{y},\mathcal{y}\varkappa ,\mathcal{y}{\varkappa }^2\}$ and the SSD-group

$${\mathrm{ƭ}}_G=\left\{\left(e,\varnothing \right),\left(\varkappa ,\varnothing \right),\left({\varkappa }^2,\varnothing \right),\left(\mathcal{y},\left\{\mathrm{1,4}\right\}\right),\left(\mathcal{y}\varkappa ,\left\{\mathrm{1,4}\right\}\right),\left({\mathcal{y}\varkappa }^2,\left\{\mathrm{1,4}\right\}\right)\right\}$$

over $U=\left\{\mathrm{1,2},\mathrm{3,4}\right\}$ in Example 7. Then, the soft left cosets of ${\mathrm{ƭ}}_G$ are as follows:

$$e{\mathrm{ƭ}}_G=\varkappa {\mathrm{ƭ}}_G={\varkappa }^2{\mathrm{ƭ}}_G=\left\{\left(e,\varnothing \right),\left(\varkappa ,\varnothing \right),\left({\varkappa }^2,\varnothing \right),\left(\mathcal{y},\left\{\mathrm{1,4}\right\}\right),\left(\mathcal{y}\varkappa ,\left\{\mathrm{1,4}\right\}\right),\left({\mathcal{y}\varkappa }^2,\left\{\mathrm{1,4}\right\}\right)\right\}={\mathrm{ƭ}}_G,$$

$${\mathcal{y}\mathrm{ƭ}}_G={\mathcal{y}\varkappa \mathrm{ƭ}}_G=\mathcal{y}{\varkappa }^2{\mathrm{ƭ}}_G=\{\left(e,\left\{\mathrm{1,4}\right\}\right),\left(\varkappa ,\left\{\mathrm{1,4}\right\}\right),\left({\varkappa }^2,\left\{\mathrm{1,4}\right\}\right),\left(\mathcal{y},\varnothing \right),\left(\mathcal{y}\varkappa ,\varnothing \right),\left(\mathcal{y}{\varkappa }^2,\varnothing \right)\}.$$

The soft right cosets of ${\mathrm{ƭ}}_G$ are as follows:

$${\mathrm{ƭ}}_{G}e={\mathrm{ƭ}}_G\varkappa ={\mathrm{ƭ}}_G{\varkappa }^2=\left\{\left(e,\varnothing \right),\left(\varkappa ,\varnothing \right),\left({\varkappa }^2,\varnothing \right),\left(\mathcal{y},\left\{\mathrm{1,4}\right\}\right),\left(\mathcal{y}\varkappa ,\left\{\mathrm{1,4}\right\}\right),\left({\mathcal{y}\varkappa }^2,\left\{\mathrm{1,4}\right\}\right)\right\}={\mathrm{ƭ}}_G,$$

$${\mathrm{ƭ}}_G\mathcal{y}={\mathrm{ƭ}}_G\mathcal{y}\varkappa ={\mathrm{ƭ}}_G\mathcal{y}{\varkappa }^2=\{\left(e,\left\{\mathrm{1,4}\right\}\right),\left(\varkappa ,\left\{\mathrm{1,4}\right\}\right),\left({\varkappa }^2,\left\{\mathrm{1,4}\right\}\right),\left(\mathcal{y},\varnothing \right),\left(\mathcal{y}\varkappa ,\varnothing \right),\left(\mathcal{y}{\varkappa }^2,\varnothing \right)\}.$$

Hence, $e{\mathrm{ƭ}}_G={\mathrm{ƭ}}_{G}e$, $\varkappa {\mathrm{ƭ}}_G={\mathrm{ƭ}}_G\varkappa , {\varkappa }^2{\mathrm{ƭ}}_G={\mathrm{ƭ}}_G{\varkappa }^2$, $\mathcal{y}{\mathrm{ƭ}}_G={\mathrm{ƭ}}_G\mathcal{y}$, $\mathcal{y}x{\mathrm{ƭ}}_G$ $={\mathrm{ƭ}}_G\mathcal{y}\varkappa$, $\mathcal{y}{\varkappa }^2{\mathrm{ƭ}}_G={\mathrm{ƭ}}_G\mathcal{y}{\varkappa }^2$.

We see that, although $D_3=G$ is not abelian, ${\mathrm{ɍ}\mathrm{ƭ}}_G={\mathrm{ƭ}}_G\mathrm{ɍ}$, for all $\mathrm{ɍ}\in G$; that is, ${\mathrm{ƭ}}_G$ is a normal SSD-group by Theorem 5.

Proposition 22.

If  ${\mathrm{ƭ}}_G$ is an SSD-group, then $\mathrm{e}{\mathrm{ƭ}}_{\mathrm{G}}$= ${\mathrm{ƭ}}_{\mathrm{G}}\mathrm{e}={\mathrm{ƭ}}_G$.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group. Then, for all $ⱴ\in G$, we have

$$\left(\mathrm{e}{\mathrm{ƭ}}_{\mathrm{G}}\right)(ⱴ)={\mathrm{ƭ}}_G({\mathrm{e}}^{-1}ⱴ)={\mathrm{ƭ}}_G(ⱴ)={\mathrm{ƭ}}_G(ⱴ{\mathrm{e}}^{-1})=\left({\mathrm{ƭ}}_{\mathrm{G}}\mathrm{e}\right)(ⱴ).$$

Thus, ${\mathrm{ƭ}}_G$ itself is a left as well as right coset. □

Note 6.

Soft left (right) cosets are not necessarily SSD-groups.

Example 11.

Consider the left coset

$$\mathcal{y}{\mathrm{ƭ}}_{\mathrm{G}}=\{\left(\mathrm{e},\left\{\mathrm{1,4}\right\}\right),\left(\mathrm{\varkappa },\left\{\mathrm{1,4}\right\}\right),\left({\mathrm{\varkappa }}^2,\left\{\mathrm{1,4}\right\}\right),\left(\mathcal{y},\varnothing \right),\left(\mathcal{y}\mathrm{\varkappa },\mathrm{\varnothing }\right),\left(\mathcal{y}{\mathrm{\varkappa }}^2,\mathrm{\varnothing }\right)\}$$

given in Example 10. Since

$$\mathcal{y}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{\varkappa }{\mathrm{\varkappa }}^2\right)=\mathcal{y}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{e}\right)=\left\{\mathrm{1,4}\right\}⊈\mathcal{y}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{\varkappa }\right)∆\mathcal{y}{\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{\varkappa }}^2\right)=\mathrm{\varnothing },$$

$\mathcal{y}{\mathrm{ƭ}}_{\mathrm{G}}$ is not an SSD-group.

Theorem 6.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group and $\mathcal{o}\in G$. Then, $\mathcal{o}\in G_{{\mathrm{ƭ}}_G}$ if and only if $\mathcal{o}{\mathrm{ƭ}}_G={\mathrm{ƭ}}_G={\mathrm{ƭ}}_G\mathcal{o}$, where $G_{{\mathrm{ƭ}}_G}$ $=\{ⱴ\in G$ $| {\mathrm{ƭ}}_G(ⱴ)$= ${\mathrm{ƭ}}_G\left(e\right)\}$.

Proof. 

Let ${\mathrm{ƭ}}_{\mathrm{G}}$ be an SSD-group and $\mathcal{o}\in {\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}$. Then, ${\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{o}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{e}\right)=\mathrm{\varnothing }$. Thus, we have

$$\mathcal{o}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathcal{o}}^{-1}\mathrm{ⱴ}\right)\subseteq {\mathrm{ƭ}}_{\mathrm{G}}\left({\mathcal{o}}^{-1}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{o}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{e}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right).$$

Thus, $\mathcal{o}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)\subseteq {\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)$. Besides, for all $\mathrm{ⱴ}\in \mathrm{G}$,

$${\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{o}{\mathcal{o}}^{-1}\mathrm{ⱴ}\right)\subseteq {\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{o}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left({\mathcal{o}}^{-1}\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{e}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left({\mathcal{o}}^{-1}\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathcal{o}}^{-1}\mathrm{ⱴ}\right)=\left(\mathcal{o}{\mathrm{ƭ}}_{\mathrm{G}}\right)\left(\mathrm{ⱴ}\right).$$

Hence, ${\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)\subseteq \mathcal{o}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)$. Then, $\mathcal{o}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)$ for all $\mathrm{ⱴ}\in \mathrm{G}$. Similarly, for all $\mathrm{ⱴ}\in \mathrm{G}$, we have

$$\left({\mathrm{ƭ}}_{\mathrm{G}}\mathcal{o}\right)\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}{\mathcal{o}}^{-1}\right)\subseteq {\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left({\mathcal{o}}^{-1}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{o}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{e}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right).$$

Thus, $\left({\mathrm{ƭ}}_{\mathrm{G}}\mathcal{o}\right)\left(\mathrm{ⱴ}\right)\subseteq {\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)$. Besides, for all $\mathrm{ⱴ}\in \mathrm{G}$,

$${\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}{\mathcal{o}}^{-1}\mathcal{o}\right)\subseteq {\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}{\mathcal{o}}^{-1}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{o}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}{\mathcal{o}}^{-1}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{e}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}{\mathcal{o}}^{-1}\right)=\left({\mathrm{ƭ}}_{\mathrm{G}}\mathcal{o}\right)\left(\mathrm{ⱴ}\right).$$

Thereby, ${\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)\subseteq \left({\mathrm{ƭ}}_{\mathrm{G}}\mathcal{o}\right)\left(\mathrm{ⱴ}\right)$. Then, $\left({\mathrm{ƭ}}_{\mathrm{G}}\mathcal{o}\right)\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)$, for all $\mathrm{ⱴ}\in \mathrm{G}$. In conclusion, $\mathcal{o}{\mathrm{ƭ}}_{\mathrm{G}}={\mathrm{ƭ}}_{\mathrm{G}}={\mathrm{ƭ}}_{\mathrm{G}}\mathcal{o}$, for all $\mathrm{ɍ}\in \mathrm{G}$.

Conversely, let us assume that $\mathcal{o}{\mathrm{ƭ}}_{\mathrm{G}}={\mathrm{ƭ}}_{\mathrm{G}}={\mathrm{ƭ}}_{\mathrm{G}}\mathcal{o}$. Then, for all $\mathrm{ⱴ}\in \mathrm{G}$, we have

$$\left(\mathcal{o}{\mathrm{ƭ}}_{\mathrm{G}}\right)\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)=\left({\mathrm{ƭ}}_{\mathrm{G}}\mathcal{o}\right)\left(\mathrm{ⱴ}\right).$$

Moreover, if we choose $\mathcal{ⱴ}=\mathcal{o}$, then

$$\mathcal{o}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{o}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{o}\right)={\mathrm{ƭ}}_{\mathrm{G}}\mathcal{o}\left(\mathcal{o}\right)\Rightarrow {\mathrm{ƭ}}_{\mathrm{G}}\left({\mathcal{o}}^{-1}\mathcal{o}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{o}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{o}{\mathcal{o}}^{-1}\right)\Rightarrow {\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{e}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathcal{o}\right)\Rightarrow \mathcal{o}\in {\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}.$$

This completes the proof. □

Note 7.

Theorem 6 plays a central role because it determines which of these cosets coincide even when  $G$ is non-abelian and ${\mathrm{ƭ}}_G$ is not a normal SSD-group. Moreover, Theorem 6 allows us to conclude that certain soft left (or right) cosets coincide whenever the order of $G_{{\mathrm{ƭ}}_G}$ exceeds 1.

Proposition 23.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group and $ɍ,ɗ \in G$. Then,

$$ɍ\beta ɗ\iff {\mathrm{ƭ}}_G(ɍ{ɗ}^{-1})={\mathrm{ƭ}}_G\left(e\right)$$

is an equivalence relation on $G$.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group and $\mathrm{ɍ},\mathrm{ɗ},c\in G$. Since for all $\mathrm{ɍ}\in G;{\mathrm{ƭ}}_G\left(\mathrm{ɍ}{\mathrm{ɍ}}^{-1}\right)={\mathrm{ƭ}}_G\left(e\right)$, $\beta$ is reflexive.

Suppose that $\mathrm{ɍ}\beta \mathrm{ɗ}$. Then,$\hspace{0.33em}{\mathrm{ƭ}}_G\left(\mathrm{ɍ}{\mathrm{ɗ}}^{-1}\right)={\mathrm{ƭ}}_G\left(e\right).$ Thus, ${\mathrm{ƭ}}_G\left(\mathrm{ɗ}{\mathrm{ɍ}}^{-1}\right)={\mathrm{ƭ}}_G\left(e\right)$, implying that $\mathrm{ɗ}\beta \mathrm{ɍ}$. Thus, $\beta$ is symmetric.

Let $\mathrm{ɍ}\beta \mathrm{ɗ}$ and $\mathrm{ɗ}\beta c$. Then, $\left(\mathrm{ɍ}{\mathrm{ɗ}}^{-1}\right)={\mathrm{ƭ}}_G\left(e\right)\hspace{0.33em}$ and ${\mathrm{ƭ}}_G\left(\mathrm{ɗ}{c}^{-1}\right)={\mathrm{ƭ}}_G\left(e\right)$. Hence,

$${\mathrm{ƭ}}_G\left(\mathrm{ɍ}{c}^{-1}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɍ}{\mathrm{ɗ}}^{-1}\mathrm{ɗ}{c}^{-1}\right)\supseteq {\mathrm{ƭ}}_G\left(\mathrm{ɍ}{\mathrm{ɗ}}^{-1}\right)\cap {\mathrm{ƭ}}_G\left(\mathrm{ɗ}{c}^{-1}\right)={\mathrm{ƭ}}_G\left(e\right)\cap \hspace{0.33em}{\mathrm{ƭ}}_G\left(e\right)={\mathrm{ƭ}}_G\left(e\right).$$

This implies that $\hspace{0.33em}\mathrm{ɍ}\beta c$. Thus, $\beta$ is transitive.

Since $\beta$ is an equivalence relation on $G$, we can mention the equivalence classes for all $\mathrm{ɍ}\in G$:

$$\stackrel{-}{\mathrm{ɍ}}=\left\{\mathrm{ɗ}|\hspace{0.33em}{\mathrm{ƭ}}_G\left(\mathrm{ɍ}{\mathrm{ɗ}}^{-1}\right)={\mathrm{ƭ}}_G\left(e\right)\right\}=\left\{\mathrm{ɗ}|\hspace{0.33em}\mathrm{ɍ}{\mathrm{ɗ}}^{-1}\in G_{{\mathrm{ƭ}}_G}\right\}=\left\{\mathrm{ɗ}|\hspace{0.33em}\mathrm{ɗ}{\mathrm{ɍ}}^{-1}\in G_{{\mathrm{ƭ}}_G}\right\}=\left\{\mathrm{ɗ}|\hspace{0.33em}\mathrm{ɗ}\in G_{{\mathrm{ƭ}}_G}\mathrm{ɍ}\right\}=G_{{\mathrm{ƭ}}_G}\mathrm{ɍ}.$$

That is $\stackrel{-}{\mathrm{ɍ}}=G_{{\mathrm{ƭ}}_G}\mathrm{ɍ}$. Note here that if the relation is defined as follows, $\mathrm{ɍ}\beta \mathrm{ɗ}\iff {\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(e\right)$, then $\stackrel{-}{\mathrm{ɍ}}=\mathrm{ɍ}{G}_{{\mathrm{ƭ}}_G}.$ □

Theorem 7.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. Then,

$$ɍ{\mathrm{ƭ}}_G=ɗ{\mathrm{ƭ}}_G\iff ɍG_{{\mathrm{ƭ}}_G}=ɗG_{{\mathrm{ƭ}}_G}\iff {ɍ}^{-1}ɗ \in G_{{\mathrm{ƭ}}_G}$$

for all $ɍ,ɗ \in$ G.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group. Suppose that $ɍ{\mathrm{ƭ}}_G$ = $ɗ{\mathrm{ƭ}}_G$, for all $ɍ,ɗ\in G$. Thus, $ɍ{\mathrm{ƭ}}_G(ⱴ)$ = $ɗ{\mathrm{ƭ}}_G(ⱴ)$, for all $ⱴ\in G$, which means that ${\mathrm{ƭ}}_G\left({ɍ}^{-1}ⱴ\right)={\mathrm{ƭ}}_G\left({ ɗ}^{-1}ⱴ\right)$, for all $ⱴ\in G$. Choosing $ⱴ=ɗ$ yields ${\mathrm{ƭ}}_G\left({ɍ}^{-1}ɗ\right)={\mathrm{ƭ}}_G\left({ ɗ}^{-1}ɗ\right)={\mathrm{ƭ}}_G\left(e\right)$. Thus, ${ɍ}^{-1}ɗ\in G_{{\mathrm{ƭ}}_G}$, so $ɍG_{{\mathrm{ƭ}}_G}$ = $ɗG_{{\mathrm{ƭ}}_G}$.

Conversely, suppose that ${ɍ}^{-1}ɗ\in G_{{\mathrm{ƭ}}_G}.$ Since $G_{{\mathrm{ƭ}}_G}$ is a subgroup of $G$, then ${ɗ}^{-1} ɍ\in G_{{\mathrm{ƭ}}_G}$. It follows that

$$ɍ{\mathrm{ƭ}}_G\left(ⱴ\right)={\mathrm{ƭ}}_G\left({ɍ}^{-1}ⱴ\right)={\mathrm{ƭ}}_G\left({ɍ}^{-1}ɗ{ ɗ}^{-1}ⱴ\right)\subseteq {\mathrm{ƭ}}_G\left({ɍ}^{-1}ɗ)∆{ {\mathrm{ƭ}}_G(ɗ}^{-1}ⱴ\right)={\mathrm{ƭ}}_G\left(e)∆{ {\mathrm{ƭ}}_G(ɗ}^{-1}ⱴ\right)={ {\mathrm{ƭ}}_G(ɗ}^{-1}ⱴ)={ɗ\mathrm{ƭ}}_G\left(ⱴ\right).$$

Thus, $ɍ{\mathrm{ƭ}}_G\left(ⱴ\right)\subseteq {ɗ\mathrm{ƭ}}_G\left(ⱴ\right)$, for all $ⱴ\in G$. Similarly,

$$ɗ{\mathrm{ƭ}}_G\left(ⱴ\right)={\mathrm{ƭ}}_G\left({ɗ}^{-1}ⱴ\right)={\mathrm{ƭ}}_G\left({ɗ}^{-1}ɍ{ ɍ}^{-1}ⱴ\right)\subseteq {\mathrm{ƭ}}_G\left({ɗ}^{-1}ɍ)∆{ {\mathrm{ƭ}}_G(ɍ}^{-1}ⱴ\right)={\mathrm{ƭ}}_G\left(e)∆{ {\mathrm{ƭ}}_G(ɍ}^{-1}ⱴ\right)={ {\mathrm{ƭ}}_G(ɍ}^{-1}ⱴ)={ɍ\mathrm{ƭ}}_G\left(ⱴ\right).$$

Hence, $ɗ{\mathrm{ƭ}}_G\left(ⱴ\right)\subseteq {ɍ\mathrm{ƭ}}_G\left(ⱴ\right)$ for all $ⱴ\in G$, implying that $ɍ{\mathrm{ƭ}}_G={ɗ\mathrm{ƭ}}_G$. □

Theorem 8.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. Then,

$${\mathrm{ƭ}}_Gɍ={\mathrm{ƭ}}_Gɗ\iff G_{{\mathrm{ƭ}}_G}ɍ=G_{{\mathrm{ƭ}}_G}ɗ\iff ɍ{ɗ}^{-1}\in G_{{\mathrm{ƭ}}_G}$$

for all $ɍ,ɗ\in G$.

Proof. 

The proof is similar to the proof of Theorem 7. □

Note 8.

Theorems 7 and 8 are highly significant, as they identify which soft left (resp., right) cosets are equal to one another.

Example 12.

Consider the group  $G=\{e,\varkappa ,{\varkappa }^2,\mathcal{y},\mathcal{y}\varkappa ,\mathcal{y}{\varkappa }^2\}$ and the SSD-group

$${\mathrm{ƭ}}_G=\left\{\left(e,\varnothing \right),\left(\varkappa ,\varnothing \right),\left({\varkappa }^2,\varnothing \right),\left(\mathcal{y},\left\{\mathrm{1,4}\right\}\right),\left(\mathcal{y}\varkappa ,\left\{\mathrm{1,4}\right\}\right),\left({\mathcal{y}\varkappa }^2,\left\{\mathrm{1,4}\right\}\right)\right\}$$

over $U=\left\{\mathrm{1,2},\mathrm{3,4}\right\}$ in Example 7, where $G_{{\mathrm{ƭ}}_G}=\left\{e,\varkappa ,{\varkappa }^2\right\}$.

Then, ${eG}_{{\mathrm{ƭ}}_G}=e\left\{e,\varkappa ,{\varkappa }^2\right\}=\{e,\varkappa ,{\varkappa }^2\}$, ${\varkappa G}_{{\mathrm{ƭ}}_G}=\varkappa \left\{e,\varkappa ,{\varkappa }^2\right\}=\{\varkappa ,{\varkappa }^2,e\}$, ${{\varkappa }^{2}G}_{{\mathrm{ƭ}}_G}={\varkappa }^2\left\{e,\varkappa ,{\varkappa }^2\right\}=\{{\varkappa }^2, e,\varkappa \}$.

Similarly, ${\mathcal{y}G}_{{\mathrm{ƭ}}_G}=\mathcal{y}\left\{e,\varkappa ,{\varkappa }^2\right\}=\{\mathcal{y},\mathcal{y}\varkappa ,\mathcal{y}{\varkappa }^2\}$, ${\mathcal{y}\varkappa G}_{{\mathrm{ƭ}}_G}=\mathcal{y}\varkappa \left\{e,\varkappa ,{\varkappa }^2\right\}=\{\mathcal{y}\varkappa ,\mathcal{y}{\varkappa }^2,e\}$, ${\mathcal{y}{\varkappa }^{2}G}_{{\mathrm{ƭ}}_G}=\mathcal{y}{\varkappa }^2\left\{e,\varkappa ,{\varkappa }^2\right\}=\{\mathcal{y}{\varkappa }^2,e,\mathcal{y}\varkappa \}$.

It is seen that $e{G}_{{\mathrm{ƭ}}_G}=\varkappa G_{{\mathrm{ƭ}}_G}={\varkappa }^2{G}_{{\mathrm{ƭ}}_G}$ and $\mathcal{y}{G}_{{\mathrm{ƭ}}_G}=\mathcal{y}\varkappa G_{{\mathrm{ƭ}}_G}=\mathcal{y}{\varkappa }^2{G}_{{\mathrm{ƭ}}_G}$. In Example 10, it is also shown that $e{\mathrm{ƭ}}_G=\varkappa {\mathrm{ƭ}}_G={\varkappa }^2{\mathrm{ƭ}}_G$ and ${\mathcal{y}\mathrm{ƭ}}_G={\mathcal{y}\varkappa \mathrm{ƭ}}_G=\mathcal{y}{\varkappa }^2{\mathrm{ƭ}}_G$.

Proposition 24.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group and $ɍ,ɗ \in G$. If $ɍ{\mathrm{ƭ}}_G= ɗ{\mathrm{ƭ}}_G,$ then ${\mathrm{ƭ}}_G(ɍ)={\mathrm{ƭ}}_G(ɗ)$.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group, $\mathrm{ɍ},\mathrm{ɗ}\in \mathrm{G}$ and $\mathrm{ɍ}{\mathrm{ƭ}}_G=\mathrm{ɗ}{\mathrm{ƭ}}_G$. Then, □

$$\mathrm{ɍ}{\mathrm{ƭ}}_G=\mathrm{ɗ}{\mathrm{ƭ}}_G\Rightarrow \hspace{0.33em}\mathrm{ɍ}{G}_{{\mathrm{ƭ}}_G}=\mathrm{ɗ}{G}_{{\mathrm{ƭ}}_G}\Rightarrow {\mathrm{ɍ}}^{-1}\mathrm{ɗ}\in G_{{\mathrm{ƭ}}_G}\Rightarrow {\mathrm{ƭ}}_G\left(\mathrm{ɍ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right).$$

Proposition 25.

Let ${\mathrm{ƭ}}_G$ be an SSD-group and $ɍ,ɗ \in G$. If ${\mathrm{ƭ}}_Gɍ={\mathrm{ƭ}}_Gɗ,$ then ${\mathrm{ƭ}}_G(ɍ)={\mathrm{ƭ}}_G(ɗ)$.

Proof. 

The proof is similar to the proof of Proposition 24. □

Proposition 26.

Let ${\mathrm{ƭ}}_G$ be an SSD-group and $ɍ,ɗ \in G$. Then, $ɍ{\mathrm{ƭ}}_G= ɗ{\mathrm{ƭ}}_G\hspace{0.33em}$ if and only if ${\mathrm{ƭ}}_G({ɍ}^{-1}ɗ)={\mathrm{ƭ}}_G({ɗ}^{-1}ɍ) ={\mathrm{ƭ}}_G\left(e\right)$.

Proof. 

Let ${\mathrm{ƭ}}_{\mathrm{G}}$ be an SSD-group, $\mathrm{ɍ},\mathrm{ɗ}\in \mathrm{G}$ and $\hspace{0.33em}\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}=\mathrm{ɗ}{\mathrm{ƭ}}_{\mathrm{G}}$. Then,

$$\hspace{0.33em}\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}=\mathrm{ɗ}{\mathrm{ƭ}}_{\mathrm{G}}\Rightarrow \hspace{0.33em}\mathrm{ɍ}{\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}=\mathrm{ɗ}{\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}\Rightarrow {\mathrm{ɍ}}^{-1}\mathrm{ɗ}\in {\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}},\hspace{0.33em}{\mathrm{ɗ}}^{-1}\mathrm{ɍ}\in {\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}\Rightarrow {\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɍ}}^{-1}\mathrm{ɗ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɗ}}^{-1}\mathrm{ɍ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{e}\right).$$

Conversely, let ${\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɍ}}^{-1}\mathrm{ɗ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɗ}}^{-1}\mathrm{ɍ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{e}\right)$. Then, for all $\mathrm{ⱴ}\in \mathrm{G}$, we have

$$\begin{array}{c}\hspace{0.33em}\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɍ}}^{-1}\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɍ}}^{-1}\mathrm{ɗ}{\mathrm{ɗ}}^{-1}\mathrm{ⱴ}\right)\subseteq {\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɍ}}^{-1}\mathrm{ɗ}\right)∆{\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɗ}}^{-1}\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɗ}}^{-1}\mathrm{ⱴ}\right)=\mathrm{ɗ}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right)\hspace{0.33em}={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɗ}}^{-1}\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɗ}}^{-1}\mathrm{ɍ}{\mathrm{ɍ}}^{-1}\mathrm{ⱴ}\right)\\ \subseteq {\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɗ}}^{-1}\mathrm{ɍ}\right)∆\hspace{0.33em}{\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɍ}}^{-1}\mathrm{ⱴ}\right)={\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɍ}}^{-1}\mathrm{ⱴ}\right)=\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{ⱴ}\right).\end{array}$$

Thus, $\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}=\mathrm{ɗ}{\mathrm{ƭ}}_{\mathrm{G}}$. □

Proposition 27.

Let  ${\mathrm{ƭ}}_{ G}$ be an SSD-group and $ɍ\in G$. Then, $ɍ{\mathrm{ƭ}}_G={ɍ}^{-1}{\mathrm{ƭ}}_G$ if and only if ${ɍ}^2\in G_{{\mathrm{ƭ}}_G}$.

Proof. 

Let ${\mathrm{ƭ}}_{\mathrm{G}}$ be an SSD-group and $\mathrm{ɍ}\in \mathrm{G}$. Then, □

$$\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}={\mathrm{ɍ}}^{-1}{\mathrm{ƭ}}_{\mathrm{G}}\iff \mathrm{ɍ}{\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}={\mathrm{ɍ}}^{-1}{\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}\iff {\mathrm{ɍ}}^{-1}{\mathrm{ɍ}}^{-1}\in {\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}\iff ({\mathrm{ɍ}}^{-1}{)}^2\in {\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}\iff \hspace{0.33em}({\mathrm{ɍ}}^2{)}^{-1}\in {\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}\iff {\mathrm{ɍ}}^2\in {\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}.$$

Example 13.

In Example 7 and Example 10, ${\varkappa }^{-1}={\varkappa }^2$, $\varkappa {\mathrm{ƭ}}_G={\varkappa }^2{\mathrm{ƭ}}_G$ and ${\varkappa }^2\in G_{{\mathrm{ƭ}}_G}$.

Proposition 28.

Let ${\mathrm{ƭ}}_{ G}$ be an SSD-group and $ɍ\in G$. If ${ɍ}^2\in G_{{ \mathrm{ƭ}}_G}$, for all $ɍ\in G$, then ${\mathrm{ƭ}}_G$ is a normal SSD-group.

Proof. 

Let ${\mathrm{ƭ}}_G$ be an SSD-group and for all $\mathrm{ɍ}\in G,{\mathrm{ɍ}}^2\in G_{{\mathrm{ƭ}}_G}$. Thus, ${\mathrm{ƭ}}_G\left(e\right)={\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^2\right)$. Let $\mathrm{ɍ},\mathrm{ɗ}\in G.$ Then, $\mathrm{ɍ}\mathrm{ɗ}\in G$. Thus,

$${\mathrm{ƭ}}_G\left(e\right)={\mathrm{ƭ}}_G\left(\right(\mathrm{ɍ}\mathrm{ɗ}{)}^2)={\mathrm{ƭ}}_G(\mathrm{ɍ}\mathrm{ɗ}\mathrm{ɍ}\mathrm{ɗ})={\mathrm{ƭ}}_G(\mathrm{ɍ}\mathrm{ɗ}\left({\mathrm{ɍ}}^{-1}{\mathrm{ɍ}}^2\right)\mathrm{ɗ})={\mathrm{ƭ}}_G(\left(\mathrm{ɍ}\mathrm{ɗ}{\mathrm{ɍ}}^{-1}\right)\left({\mathrm{ɍ}}^2\mathrm{ɗ}\right)).$$

We obtain that ${\mathrm{ƭ}}_G\left(\right(\mathrm{ɍ}\mathrm{ɗ}{\mathrm{ɍ}}^{-1}\left)\right({\mathrm{ɍ}}^2\mathrm{ɗ}\left)\right)={\mathrm{ƭ}}_G\left(e\right)$. By Proposition 20 (i), ${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\mathrm{ɗ}{\mathrm{ɍ}}^{-1}\right)=$ ${\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^2\mathrm{ɗ}\right)$. Therefore, we have

$${\mathrm{ƭ}}_G\left(\mathrm{ɍ}\mathrm{ɗ}{\mathrm{ɍ}}^{-1}\right)={\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^2\mathrm{ɗ}\right)\subseteq {\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^2\right)\hspace{0.33em}∆{\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(e\right)∆{\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɗ}\right).$$

for all $\mathrm{ɍ},\mathrm{ɗ}\in G.$ By Theorem 4, ${\mathrm{ƭ}}_G$ is a normal SSD-group. □

Theorem 9.

Let  ${\mathrm{ƭ}}_{ G}$ be an SSD-group and $ɍ\in G$. If $ɍ{\mathrm{ƭ}}_G={ɍ}^{-1}{\mathrm{ƭ}}_G$, for all $ɍ\in G,$ then ${\mathrm{ƭ}}_G$ is a normal SSD-group.

Proof. 

It follows by Proposition 27 and Proposition 28. □

Theorem 10.

The number of different (not identical/not equal) soft left cosets is equal to the number of different (not identical/not equal) soft right cosets of an SSD-group.

Proof. 

Let ${\mathrm{ƭ}}_G$ an SSD-group and $A=\{{\mathrm{ƭ}}_G\mathrm{ɍ}$: $\mathrm{ɍ}$ ∈ $G\}$ and $B=\{\mathrm{ɍ}{\mathrm{ƭ}}_G$: $\mathrm{ɍ}$ ∈ $G\}$ be the set of soft right and soft left cosets of ${\mathrm{ƭ}}_G$, respectively. Define a map from $A$ to $B$ as follows:

$$\phi : A⟶B$$

$${\mathrm{ƭ}}_G\mathrm{ɍ}⟶\phi ({\mathrm{ƭ}}_G\mathrm{ɍ})={\mathrm{ɍ}}^{-1}{\mathrm{ƭ}}_G$$

First, let us show $\phi$ is a well-defined function. Let $\mathrm{ɍ},\mathrm{ɗ}\in G$ and ${\mathrm{ƭ}}_G\mathrm{ɍ}$ = ${\mathrm{ƭ}}_G\mathrm{ɗ}$. We need to show that ${\mathrm{ɍ}}^{-1}{\mathrm{ƭ}}_G={\mathrm{ɗ}}^{-1}{\mathrm{ƭ}}_G$.

$${\mathrm{ƭ}}_G\mathrm{ɍ}={\mathrm{ƭ}}_G\mathrm{ɗ}\Rightarrow G_{{\mathrm{ƭ}}_G}\mathrm{ɍ}=G_{{\mathrm{ƭ}}_G}\mathrm{ɗ}\Rightarrow \mathrm{ɍ}{\mathrm{ɗ}}^{-1}\in G_{{\mathrm{ƭ}}_G}\Rightarrow {{(\mathrm{ɍ}}^{-1})}^{-1}{\mathrm{ɗ}}^{-1}\in G_{{\mathrm{ƭ}}_G}\Rightarrow {\mathrm{ɍ}}^{-1}{G}_{{\mathrm{ƭ}}_G}{=\mathrm{ɗ}}^{-1}{G}_{{\mathrm{ƭ}}_G}\Rightarrow {\mathrm{ɍ}}^{-1}{\mathrm{ƭ}}_G={\mathrm{ɗ}}^{-1}{\mathrm{ƭ}}_G.$$

Hence, $\phi$ is a well-defined function. Now, we show that $\phi$ is an injective function. Let $\mathrm{ɍ},\mathrm{ɗ}\in G$ and $\phi \left({\mathrm{ƭ}}_G\mathrm{ɍ}\right)$ = $\phi \left({\mathrm{ƭ}}_G\mathrm{ɗ}\right)$. We need to show that ${\mathrm{ƭ}}_G\mathrm{ɍ}$ = ${\mathrm{ƭ}}_G\mathrm{ɗ}$.

$$\phi ({\mathrm{ƭ}}_G\mathrm{ɍ})=\phi ({\mathrm{ƭ}}_G\mathrm{ɗ}) \Rightarrow {\mathrm{ɍ}}^{-1}{\mathrm{ƭ}}_G{=\mathrm{ɍ}}^{-1}{\mathrm{ƭ}}_G\Rightarrow {\mathrm{ɍ}}^{-1}{G}_{{\mathrm{ƭ}}_G}={\mathrm{ɗ}}^{-1}{G}_{{\mathrm{ƭ}}_G}\Rightarrow \mathrm{ɍ}{\mathrm{ɗ}}^{-1}\in G_{{\mathrm{ƭ}}_G}\Rightarrow G_{{\mathrm{ƭ}}_G}\mathrm{ɍ}=G_{{\mathrm{ƭ}}_G}\mathrm{ɗ}\Rightarrow {\mathrm{ƭ}}_G\mathrm{ɍ}={\mathrm{ƭ}}_G\mathrm{ɗ}.$$

Hence, $\phi$ is an injective function. Since, for all $\mathrm{ɍ}{\mathrm{ƭ}}_G\in \mathrm{B}$, there exists ${\mathrm{ƭ}}_G{\mathrm{ɍ}}^{-1}\in \mathrm{A}$ such that $\phi ({\mathrm{ƭ}}_G{\mathrm{ɍ}}^{-1}$) = $\mathrm{ɍ}{\mathrm{ƭ}}_G$, $\phi$ is a surjective function. Therefore, $\phi$ is bijective. So, we are done. □

Definition 23.

The number of different (not identical/not equal) soft right cosets of  ${\mathrm{ƭ}}_G$ is equal to the number of different soft left cosets, and is known as the soft index of ${\mathrm{ƭ}}_G$ in $G$, denoted by $[G$: ${\mathrm{ƭ}}_G]$.

Example 14.

In Example 1, different soft left cosets of  ${\mathrm{ƭ}}_G$ are $e{\mathrm{ƭ}}_G,\mathcal{o}{\mathrm{ƭ}}_G,\mathrm{Ҍ}{\mathrm{ƭ}}_G$ and $\mathrm{ƶ}{\mathrm{ƭ}}_G$, and so $[G$: ${\mathrm{ƭ}}_G]=4$. In Example 10, different soft left cosets of ${\mathrm{ƭ}}_G$ are $e{\mathrm{ƭ}}_G and$ $\mathcal{y}{\mathrm{ƭ}}_G$, and so $[G$: ${\mathrm{ƭ}}_G]=2$.

Corollary 1.

By the proof of Theorem 10, we can conclude that  ${\mathrm{ƭ}}_Gɍ={\mathrm{ƭ}}_Gɗ\iff {{ɍ}^{-1}\mathrm{ƭ}}_G={{ɗ}^{-1}\mathrm{ƭ}}_G$$(ɍ{\mathrm{ƭ}}_{ G}=ɗ{\mathrm{ƭ}}_G\iff {\mathrm{ƭ}}_G{ɍ}^{-1}={\mathrm{ƭ}}_G{ɗ}^{-1}$) for all $ɍ$, $ɗ \in G$, where ${\mathrm{ƭ}}_G$ is an SSD-group. In fact, in Example 10, ${\mathrm{ƭ}}_G\varkappa ={\mathrm{ƭ}}_G{\varkappa }^2$, ${\varkappa }^{-1}={\varkappa }^2$ and ${{\varkappa }^2\mathrm{ƭ}}_G=\varkappa {\mathrm{ƭ}}_G$.

Theorem 11.

Let  $ɍ$, $ɗ\in G$ and ${\mathrm{ƭ}}_G$ be a normal SSD-group. Then, ${\mathrm{ƭ}}_Gɍ={\mathrm{ƭ}}_Gɗ$ if and only if ${\mathrm{ƭ}}_G{ɍ}^{-1}={\mathrm{ƭ}}_G{ɗ}^{-1}$, for all $ɍ$, $ɗ\in$ $G$.

Proof. 

Let $\mathrm{ɍ}$, $\mathrm{ɗ}$ ∈ $G$ and ${\mathrm{ƭ}}_G\mathrm{ɍ}$ = ${\mathrm{ƭ}}_G\mathrm{ɗ}$. We need to show that ${\mathrm{ƭ}}_G{\mathrm{ɍ}}^{-1}={\mathrm{ƭ}}_G{\mathrm{ɗ}}^{-1}$. Since ${\mathrm{ƭ}}_G$ is a normal SSD-group, then

$${\mathrm{ƭ}}_G\mathrm{ɍ}={\mathrm{ƭ}}_G\mathrm{ɗ}\Rightarrow G_{{\mathrm{ƭ}}_G}\mathrm{ɍ}=G_{{\mathrm{ƭ}}_G}\mathrm{ɗ}\Rightarrow \mathrm{ɍ}{\mathrm{ɗ}}^{-1}\in G_{{\mathrm{ƭ}}_G}\Rightarrow \mathrm{ɗ}{\mathrm{ɍ}}^{-1}\in G_{{\mathrm{ƭ}}_G}\Rightarrow {\mathrm{ɍ}}^{-1}\mathrm{ɗ}\in G_{{\mathrm{ƭ}}_G}\Rightarrow G_{{\mathrm{ƭ}}_G}{\mathrm{ɍ}}^{-1}=G_{{\mathrm{ƭ}}_G}{\mathrm{ɗ}}^{-1}\Rightarrow {\mathrm{ƭ}}_G{\mathrm{ɍ}}^{-1}={\mathrm{ƭ}}_G{\mathrm{ɗ}}^{-1}.$$

Here note that since ${\mathrm{ƭ}}_G$ is a normal SSD-group, then $\mathrm{ɗ}{\mathrm{ɍ}}^{-1}\in G_{{\mathrm{ƭ}}_G}\Rightarrow {\mathrm{ƭ}}_G\left(\mathrm{ɗ}{\mathrm{ɍ}}^{-1}\right)={\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\mathrm{ɗ}\right)$= ${\mathrm{ƭ}}_G\left(e\right)$, thus ${\mathrm{ɍ}}^{-1}\mathrm{ɗ}$ $\in G_{{\mathrm{ƭ}}_G}$.

Now, let ${\mathrm{ƭ}}_G{\mathrm{ɍ}}^{-1}={\mathrm{ƭ}}_G{\mathrm{ɗ}}^{-1}$ for all $\mathrm{ɍ}$, $\mathrm{ɗ}$ ∈$G$. We need to show that ${\mathrm{ƭ}}_G\mathrm{ɍ}$ = ${\mathrm{ƭ}}_G\mathrm{ɗ}$.

$${\mathrm{ƭ}}_G{\mathrm{ɍ}}^{-1}={\mathrm{ƭ}}_G{\mathrm{ɗ}}^{-1}\Rightarrow G_{{\mathrm{ƭ}}_G}{\mathrm{ɍ}}^{-1}=G_{{\mathrm{ƭ}}_G}{\mathrm{ɗ}}^{-1}\Rightarrow {\mathrm{ɍ}}^{-1}\mathrm{ɗ}\in G_{{\mathrm{ƭ}}_G}\Rightarrow \mathrm{ɗ}{\mathrm{ɍ}}^{-1}\in G_{{\mathrm{ƭ}}_G}\Rightarrow \mathrm{ɍ}{\mathrm{ɗ}}^{-1}\in G_{{\mathrm{ƭ}}_G}\Rightarrow G_{{\mathrm{ƭ}}_G}\mathrm{ɍ}=G_{{\mathrm{ƭ}}_G}\mathrm{ɗ}\Rightarrow {\mathrm{ƭ}}_G\mathrm{ɍ}={\mathrm{ƭ}}_G\mathrm{ɗ}.$$

Here again note that since ${\mathrm{ƭ}}_G$ is a normal SSD-group, then ${\mathrm{ɍ}}^{-1}\mathrm{ɗ}\in G_{{\mathrm{ƭ}}_G}\Rightarrow {\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{-1}\mathrm{ɗ}\right)={\mathrm{ƭ}}_G\left(\mathrm{ɗ}{\mathrm{ɍ}}^{-1}\right)$= ${\mathrm{ƭ}}_G\left(e\right)$, thus $\mathrm{ɗ}{\mathrm{ɍ}}^{-1}\in G_{{\mathrm{ƭ}}_G}.$ In fact, in Example 10, ${\mathrm{ƭ}}_G\varkappa ={\mathrm{ƭ}}_G{\varkappa }^2$, ${\varkappa }^{-1}={\varkappa }^2$ and ${\mathrm{ƭ}}_G\varkappa ={\mathrm{ƭ}}_G{\varkappa }^2$. □

Theorem 12.

Let  $ɍ$, $ɗ\in G$ and ${\mathrm{ƭ}}_G$ be a normal SSD-group. Then, $ɍ{\mathrm{ƭ}}_{ G}= ɗ{\mathrm{ƭ}}_G$ if and only if ${{ɍ}^{-1}\mathrm{ƭ}}_G={ɗ}^{-1}{\mathrm{ƭ}}_G$ for all $ɍ$, $ɗ \in G$.

Proof. 

The proof is similar to the proof of Theorem 11. □

In Refs. [84,85,88], soft quotient (or factor) group within the framework of soft int-group was investigated. In parallel, our study extends this notion to SSD-groups via normal SSD-groups.

Definition 24.

Let  ${\mathrm{ƭ}}_G$ be a normal SSD-group. Consider the set ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$, where ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}=\left\{ ⱴ{\mathrm{ƭ}}_G: ⱴ\in G \right\}$ is the set of all the soft left (right) cosets of ${\mathrm{ƭ}}_G$. Define an operation of composition on ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ as follows:

$$\odot :{\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.} x{\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}⟶{\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$$

$$(ⱴ{\mathrm{ƭ}}_G,\mathcal{y}{\mathrm{ƭ}}_G)=\left(ⱴ{\mathrm{ƭ}}_G\right)\odot \left(\mathcal{y}{\mathrm{ƭ}}_G\right)=ⱴ{\mathcal{y}\mathrm{ƭ}}_G$$

for all $\mathcal{ⱴ},\mathcal{y}\in G$. Then, $\left({\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.},\odot \right)$ is called the soft quotient (or factor) group of $G$ relative to the normal SSD-group ${\mathrm{ƭ}}_G$.

Theorem 13.

If  ${\mathrm{ƭ}}_G$ is a normal SSD-group, then ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}=\left\{ⱴ{\mathrm{ƭ}}_G: ⱴ\in G\right\}$ forms a group under the composition $\odot$ on ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$, where $ⱴ{\mathrm{ƭ}}_G\odot \mathcal{y}{\mathrm{ƭ}}_G=ⱴ{\mathcal{y}\mathrm{ƭ}}_G$, for all $\mathcal{ⱴ},\mathcal{y}\in G$.

Proof. 

Let ${\mathrm{ƭ}}_G$ be a normal SSD-group and consider the set ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}=\left\{\mathrm{ⱴ}{\mathrm{ƭ}}_G:\mathrm{ⱴ}\in G\right\}$. Define the composition on ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ as $\mathrm{ⱴ}{\mathrm{ƭ}}_G\odot \mathcal{y}{\mathrm{ƭ}}_G=\mathrm{ⱴ}{\mathcal{y}\mathrm{ƭ}}_G$. To show that the above composition is well-defined, let ${\mathrm{ɍ}}_1,{\mathrm{ɍ}}_2,{\mathrm{ɗ}}_1,{\mathrm{ɗ}}_2\in G$ and $({\mathrm{ɍ}}_1{\mathrm{ƭ}}_G,{{\mathrm{ɗ}}_1\mathrm{ƭ}}_G)=({\mathrm{ɍ}}_2{\mathrm{ƭ}}_G,{\mathrm{ɗ}}_2{\mathrm{ƭ}}_G)$. We need to show that

$${\mathrm{ɍ}}_1{\mathrm{ƭ}}_G\odot {{\mathrm{ɗ}}_1\mathrm{ƭ}}_G={\mathrm{ɍ}}_2{\mathrm{ƭ}}_G\odot {\mathrm{ɗ}}_2{\mathrm{ƭ}}_G.$$

Since $({\mathrm{ɍ}}_1{\mathrm{ƭ}}_G,{{\mathrm{ɗ}}_1\mathrm{ƭ}}_G)=({\mathrm{ɍ}}_2{\mathrm{ƭ}}_G,{\mathrm{ɗ}}_2{\mathrm{ƭ}}_G)$, we have ${\mathrm{ɍ}}_1{\mathrm{ƭ}}_G={{\mathrm{ɍ}}_2\mathrm{ƭ}}_G$ and ${\mathrm{ɗ}}_1{\mathrm{ƭ}}_G={{\mathrm{ɗ}}_2\mathrm{ƭ}}_G$. Let $\mathrm{ⱴ}\in G$. Then, we have

$$\begin{array}{c}\left({\mathrm{ɍ}}_1{{\mathrm{ɗ}}_1)\mathrm{ƭ}}_G\right(\mathrm{ⱴ})={\mathrm{ƭ}}_G({\left({\mathrm{ɍ}}_1{\mathrm{ɗ}}_1\right)}^{-1}\mathrm{ⱴ})={\mathrm{ƭ}}_G({({\mathrm{ɗ}}_1}^{-1}{{\mathrm{ɍ}}_1}^{-1})\mathrm{ⱴ})=({{\mathrm{ɗ}}_1\mathrm{ƭ}}_G\left)\right({{\mathrm{ɍ}}_1}^{-1}\mathrm{ⱴ})=\\ ({{\mathrm{ɗ}}_2\mathrm{ƭ}}_G\left)\right({{\mathrm{ɍ}}_1}^{-1}\mathrm{ⱴ})=({\mathrm{ƭ}}_G{\mathrm{ɗ}}_2\left)\right({{\mathrm{ɍ}}_1}^{-1}\mathrm{ⱴ})=({\mathrm{ƭ}}_G\left)\right({{\mathrm{ɍ}}_1}^{-1}\mathrm{ⱴ}{{\mathrm{ɗ}}_2}^{-1})=({{\mathrm{ɍ}}_1\mathrm{ƭ}}_G\left)\right(\mathrm{ⱴ}{{\mathrm{ɗ}}_2}^{-1})=({{\mathrm{ɍ}}_2\mathrm{ƭ}}_G\left)\right(\mathrm{ⱴ}{{\mathrm{ɗ}}_2}^{-1})=({\mathrm{ƭ}}_G{\mathrm{ɍ}}_2\left)\right(\mathrm{ⱴ}{{\mathrm{ɗ}}_2}^{-1})={\mathrm{ƭ}}_G{(\mathrm{ⱴ}{\mathrm{ɗ}}_2}^{-1}{{\mathrm{ɍ}}_2}^{-1})=\\ {\mathrm{ƭ}}_G(\mathrm{ⱴ}{\left({\mathrm{ɍ}}_2{\mathrm{ɗ}}_2\right)}^{-1})=({\mathrm{ɍ}}_2{{\mathrm{ɗ}}_2)\mathrm{ƭ}}_G\left(\mathrm{ⱴ}\right).\end{array}$$

Thus, ${\mathrm{ɍ}}_1{{\mathrm{ɗ}}_1\mathrm{ƭ}}_G={\mathrm{ɍ}}_2{\mathrm{ɗ}}_2{\mathrm{ƭ}}_G$ and so ${\mathrm{ɍ}}_1{\mathrm{ƭ}}_G\odot {{\mathrm{ɗ}}_1\mathrm{ƭ}}_G={\mathrm{ɍ}}_2{\mathrm{ƭ}}_G\odot {\mathrm{ɗ}}_2{\mathrm{ƭ}}_G$. So, we are done.

We can easily show that $\left({\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.},\odot \right)$ is a group with the identity $e{\mathrm{ƭ}}_G$. It is obvious that the order of ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ is the soft index of ${\mathrm{ƭ}}_G$ in $G$. □

Example 15.

Consider the normal SSD-group  ${\mathrm{ƭ}}_G$ in Example 1. By Example 9, ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}=\{e{\mathrm{ƭ}}_G,\mathcal{o}{\mathrm{ƭ}}_G,\mathrm{Ҍ}{\mathrm{ƭ}}_G,\mathrm{ƶ}{\mathrm{ƭ}}_G\}$ and the Cayley table of the soft quotient group ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ is as Table 4:

Table 4. Cayley table of the soft quotient group in Example 15.

$\odot$	$e{\mathrm{ƭ}}_G$	$\mathcal{o}{\mathrm{ƭ}}_G$	$\mathrm{Ҍ}{\mathrm{ƭ}}_G$	$\mathrm{ƶ}{\mathrm{ƭ}}_G$
$e{\mathrm{ƭ}}_G$	$e{\mathrm{ƭ}}_G$	$\mathcal{o}{\mathrm{ƭ}}_G$	$\mathrm{Ҍ}{\mathrm{ƭ}}_G$	$\mathrm{ƶ}{\mathrm{ƭ}}_G$
$\mathcal{o}{\mathrm{ƭ}}_G$	$\mathcal{o}{\mathrm{ƭ}}_G$	$e{\mathrm{ƭ}}_G$	$\mathrm{ƶ}{\mathrm{ƭ}}_G$	$\mathrm{Ҍ}{\mathrm{ƭ}}_G$
$\mathrm{Ҍ}{\mathrm{ƭ}}_G$	$\mathrm{Ҍ}{\mathrm{ƭ}}_G$	$\mathrm{ƶ}{\mathrm{ƭ}}_G$	$e{\mathrm{ƭ}}_G$	$\mathcal{o}{\mathrm{ƭ}}_G$
$\mathrm{ƶ}{\mathrm{ƭ}}_G$	$\mathrm{ƶ}{\mathrm{ƭ}}_G$	$\mathrm{Ҍ}{\mathrm{ƭ}}_G$	$\mathcal{o}{\mathrm{ƭ}}_G$	$e{\mathrm{ƭ}}_G$

One can easily show that $\left({\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.},\odot \right)$ is a group, even isomorphic to $D_2.$

Theorem 14.

If  ${\mathrm{ƭ}}_G$ is a normal SSD-group, then ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}\cong {\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$G_{{\mathrm{ƭ}}_G}$}\right.}.$

Proof. 

Since ${\mathrm{ƭ}}_G$ is a normal SSD-group, by Proposition 17, $G_{{\mathrm{ƭ}}_G}$ is a normal subgroup of G. Hence, ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$G_{{\mathrm{ƭ}}_G}$}\right.}$ is a quotient group. Let the operation defined on ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ be $“\odot ”$ and ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$G_{{\mathrm{ƭ}}_G}$}\right.}$ be $“⊡$”. By Theorem 13, $\left({\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.},\odot \right)$ is a group. Now, define a map by

$$\mathsf{\Phi }: \left({\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.},\odot \right) ⟶ \left({\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$G_{{\mathrm{ƭ}}_G}$}\right.},⊡\right)$$

$$\mathrm{ⱴ}{\mathrm{ƭ}}_G⟶\mathrm{ⱴ}{G}_{{\mathrm{ƭ}}_G}$$

Firstly, $\mathsf{\Phi }$ is a homomorphism, since

$$\mathsf{\Phi }\left(\right({\mathrm{ⱴ}\mathrm{ƭ}}_G)\odot ({\mathcal{y}\mathrm{ƭ}}_G\left)\right)=\mathsf{\Phi }(\mathrm{ⱴ}\mathrm{y}{\mathrm{ƭ}}_G)=\mathcal{ⱴ}\mathcal{y}{G}_{{\mathrm{ƭ}}_G}=(\mathrm{ⱴ}{G}_{{\mathrm{ƭ}}_G})⊡(\mathcal{y}{G}_{{\mathrm{ƭ}}_G})=\mathsf{\Phi }(\mathrm{ⱴ}{\mathrm{ƭ}}_G)⊡\mathsf{\Phi }(\mathcal{y}{\mathrm{ƭ}}_G).$$

for all $\mathrm{ⱴ}{\mathrm{ƭ}}_G$, ${\mathrm{y}\mathrm{ƭ}}_G\in {\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$. On the other hand, if $\mathrm{ⱴ}{G}_{{\mathrm{ƭ}}_G}=\mathcal{y}{G}_{{\mathrm{ƭ}}_G},$ then $\mathrm{ⱴ}{\mathrm{ƭ}}_G={\mathcal{y}\mathrm{ƭ}}_G$. Thus, ψ is injective. Moreover, for all $\mathrm{ⱴ}{G}_{{\mathrm{ƭ}}_G}\in {\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$G_{{\mathrm{ƭ}}_G}$}\right.},$ there exists at least one $\mathrm{ⱴ}{\mathrm{ƭ}}_G\in {\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ such that $\mathsf{\Phi }\left({\mathrm{ⱴ}\mathrm{ƭ}}_G\right)=\mathrm{ⱴ}{G}_{{\mathrm{ƭ}}_G}$. Hence, $\mathsf{\Phi }$ is bijective and so, ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}\cong {\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$G_{{\mathrm{ƭ}}_G}$}\right.}.$ □

Example 16.

In Example 9,$G_{{\mathrm{ƭ}}_G}=\{e$}. Thus, ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$G_{{\mathrm{ƭ}}_G}$}\right.}=$ $\{e{G}_{{\mathrm{ƭ}}_G},$ $ɍG_{{\mathrm{ƭ}}_G},ɗG_{{\mathrm{ƭ}}_G},\mathrm{ƶ}{G}_{{\mathrm{ƭ}}_G}\}$ and ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}=\{{e\mathrm{ƭ}}_G,$ $ɍ{\mathrm{ƭ}}_G,ɗ{\mathrm{ƭ}}_G,\mathrm{ƶ}{\mathrm{ƭ}}_G\}.$

Note 9.

By Theorems 8 and 14,  $[G$: ${ G}_{{\mathrm{ƭ}}_G}]=[G$: ${\mathrm{ƭ}}_G]$.

This equality indicates that once the set $G_{{\mathrm{ƭ}}_G}$ is known, we can determine the number of distinct soft left or right cosets of $G_{{\mathrm{ƭ}}_G}$ in $G$; that is, we can compute the index of $G_{{\mathrm{ƭ}}_G}$ in $G$, and consequently the soft index of ${\mathrm{ƭ}}_G$ in $G$.

Example 17.

In Example 9, [  $G$ : $G_{{\mathrm{ƭ}}_G}]=4$ and in Example 10, [$G$: $G_{{\mathrm{ƭ}}_G}]=2$. Thus, in Example 9, [$G$: ${\mathrm{ƭ}}_G]=4$ and in Example 10, [$G$: ${\mathrm{ƭ}}_G]=2$.

Proposition 29.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group and $G_{{\mathrm{ƭ}}_G}=\left\{e\right\}$. Then, $ɍ{\mathrm{ƭ}}_G=ɗ{\mathrm{ƭ}}_G$ if and only if $ɍ=ɗ$, where $ɍ,ɗ\in G$.

Proof. 

Let $G_{{\mathrm{ƭ}}_G}=\left\{e\right\}$ and $\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}=\mathrm{ɗ}{\mathrm{ƭ}}_{\mathrm{G}}$, where $ɍ,ɗ\in G$. Then, ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$G_{{\mathrm{ƭ}}_G}$}\right.}$ $\cong$ $G$, and so ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}\cong$ $G$. Since [$G$: $G_{{\mathrm{ƭ}}_G}]=$ [$G$: ${\mathrm{ƭ}}_G$], the number of different soft left (right) cosets is the order of $G$. Thus, $ɍ=ɗ$. The converse is obvious. □

Example 18.

In Example 9,  $G_{{\mathrm{ƭ}}_G}=\left\{e\right\}$ and there do not exist any soft left (right) coset such that $ɍ\ne ɗ$ and $ɍ{\mathrm{ƭ}}_G=$ $ɗ{\mathrm{ƭ}}_G$, where $ɍ,ɗ \in G.$

In fact in Example 9, $G_{{\mathrm{ƭ}}_G}=\left\{e\right\}$ and ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}=\{e{\mathrm{ƭ}}_G,$ $\mathrm{ɍ}{\mathrm{ƭ}}_G,\mathrm{ɗ}{\mathrm{ƭ}}_G,c{\mathrm{ƭ}}_G\}.$ That is, there are four different soft (left) right cosets, which equals the order of $G$.

Theorem 15.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. Then, ${\mathrm{ƭ}}_G$ is a normal SSD-group if and only if ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ is abelian.

Proof. 

Let ${\mathrm{ƭ}}_G$ be a normal SSD-group and the operation defined on ${\displaystyle \raisebox{1ex}{$\mathrm{G}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_{\mathrm{G}}$}\right.}$ be $\odot$. Since ${\mathrm{ƭ}}_{\mathrm{G}}$ is a normal SSD-group, we can construct the quotient group ${\displaystyle \raisebox{1ex}{$\mathrm{G}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_{\mathrm{G}}$}\right.}$. Let $\left(\mathcal{ⱴ}\mathcal{y}{\mathrm{ⱴ}}^{-1}{\mathcal{y}}^{-1}\right)={\mathrm{ƭ}}_G\left(e\right)$ for all $\mathcal{ⱴ},\mathcal{y}\in G$. Thus, we have

$$\begin{array}{c}{\mathrm{ƭ}}_G\left(\right(\mathcal{ⱴ}\mathcal{y}\left)\right(\mathcal{y}\mathcal{ⱴ}{)}^{-1})={\mathrm{ƭ}}_G(e)\Rightarrow {\mathrm{ƭ}}_G(\left(\mathcal{y}\mathcal{ⱴ}{)}^{-1}\right(\mathcal{ⱴ}\mathcal{y}\left)\right)={\mathrm{ƭ}}_G\left(e\right)\Rightarrow \left(\mathcal{y}\mathcal{ⱴ}{)}^{-1}\right(\mathcal{ⱴ}\mathcal{y})\in G_{{\mathrm{ƭ}}_G}\Rightarrow \hspace{0.33em}\mathcal{y}\mathcal{ⱴ}{G}_{{\mathrm{ƭ}}_G}=\mathcal{ⱴ}\mathcal{y}{G}_{{\mathrm{ƭ}}_G}\Rightarrow \hspace{0.33em}\mathcal{y}\mathcal{ⱴ}{\mathrm{ƭ}}_G=\\ \mathcal{ⱴ}\mathcal{y}{\mathrm{ƭ}}_G\Rightarrow \mathcal{y}{\mathrm{ƭ}}_G\odot \mathrm{ⱴ}{\mathrm{ƭ}}_G=\mathrm{ⱴ}{\mathrm{ƭ}}_G\odot \mathcal{y}{\mathrm{ƭ}}_G.\end{array}$$

Thus, ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ is abelian.

Conversely, let ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ be an abelian group. Then, we have

$$\begin{array}{c}\mathrm{ⱴ}{\mathrm{ƭ}}_G\odot \mathcal{y}{\mathrm{ƭ}}_G=\mathcal{y}{\mathrm{ƭ}}_G\odot \mathrm{ⱴ}{\mathrm{ƭ}}_G\Rightarrow \mathcal{ⱴ}\mathcal{y}{\mathrm{ƭ}}_G=\mathcal{y}\mathcal{ⱴ}{\mathrm{ƭ}}_G\hspace{0.33em}\Rightarrow \mathcal{ⱴ}\mathcal{y}{G}_{{\mathrm{ƭ}}_G}=\mathcal{y}\mathcal{ⱴ}{G}_{{\mathrm{ƭ}}_G}\Rightarrow {\left(\mathcal{ⱴ}\mathcal{y}\right)}^{-1}\left(\mathcal{y}\mathcal{ⱴ}\right)\in G_{{\mathrm{ƭ}}_G}\Rightarrow {\mathrm{ƭ}}_G\left(\right(\mathcal{ⱴ}\mathcal{y}{)}^{-1}\left(\mathcal{y}\mathcal{ⱴ}\right))={\mathrm{ƭ}}_G(e)\\ \Rightarrow {\mathrm{ƭ}}_G\left(\mathcal{ⱴ}\mathcal{y}\right)={\mathrm{ƭ}}_G\left(\mathcal{y}\mathcal{ⱴ}\right)\end{array}$$

for all $\mathcal{ⱴ},\mathcal{y}\in G$. Thus,$\hspace{0.33em}{\mathrm{ƭ}}_G$ is a normal SSD-group. □

Corollary 2.

If [$G$ : ${\mathrm{ƭ}}_G]=2$, $p$ or $p^2$, where $p$ is a prime number, then ${\mathrm{ƭ}}_G$ is a normal SSD-group.

In [88], the order of an element induced by the SSD-group and the order of soft cosets within the framework of soft int-groups were investigated. In parallel, our study extends this notion to SSD-groups for constructing $p$-SSD-group as follows:

Definition 25.

Let  ${\mathrm{ƭ}}_G$ be an SSD-group. If there exists a positive integer $n$ such that ${\mathrm{ƭ}}_G\left({ɍ}^n\right)={\mathrm{ƭ}}_G\left(e\right)$, then $ɍ$ is said to have finite order and the smallest such positive integer $n$ such that ${\mathrm{ƭ}}_G\left({ɍ}^n\right)={\mathrm{ƭ}}_G\left(e\right)$ is called the order of $ɍ$ with respect to ${\mathrm{ƭ}}_G$, and is denoted by ${SO\mathrm{ƭ}}_G(ɍ)$. If there does not exist such a positive integer, then $ɍ$ is said to have infinite order.

Definition 26.

If  ${SO\mathrm{ƭ}}_G(ɍ)$ is a power of a prime number $p$ for all $ɍ\in G$, ${\mathrm{ƭ}}_G$ is called $p$-SSD-group.

Example 19.

In Example 1,  ${SO\mathrm{ƭ}}_G\left(e\right)$ =1, ${SO\mathrm{ƭ}}_G\left(\mathcal{o}\right)$= ${SO\mathrm{ƭ}}_G\left(\mathrm{Ҍ}\right)=$ ${SO\mathrm{ƭ}}_G\left(\mathrm{ƶ}\right)=2$. Thus, ${\mathrm{ƭ}}_G$ is a $2$-SSD-group.

Definition 27.

Let  ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ be a quotient group and ${ɍ\mathrm{ƭ}}_G\in {\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$. If there exists a positive integer $n$ such that ${\left({ɍ\mathrm{ƭ}}_G\right)}^n={\mathrm{ƭ}}_G$ then ${ɍ\mathrm{ƭ}}_G$ is said to have finite order and the smallest such positive integer $n$ such that ${\left({ɍ\mathrm{ƭ}}_G\right)}^n={\mathrm{ƭ}}_G$ is called the order of ${ ɍ\mathrm{ƭ}}_G$ is denoted by $SO{(ɍ\mathrm{ƭ}}_G)$.

Definition 28.

Let  ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ be a quotient group. If there exist nonnegative integers $k$ and a prime $p$ such that $(ɍ{\mathrm{ƭ}}_G{)}^{p^k}=e{\mathrm{ƭ}}_G$, for any $ɍ\in G$, then ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ is called a $p$-group.

Theorem 16.

Let  ${\mathrm{ƭ}}_G$ be a normal SSD-group. Then, ${\mathrm{ƭ}}_G$ is a $p$-SSD-group if and only if $\hspace{0.33em}{\displaystyle \raisebox{1ex}{$G $}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ is a $p$-group.

Proof. 

Let ${\mathrm{ƭ}}_G\hspace{0.33em}$ be a normal $p$-SSD-group. Then, for any $\mathrm{ɍ}\in G$, there exist nonnegative integers $k$ and a prime $p$ such that ${\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{p^k}\right)={\mathrm{ƭ}}_G\left(e\right)$. Then, we have

$${\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{p^k}\right)={\mathrm{ƭ}}_G\left(e\right)⟹{\mathrm{ɍ}}^{p^k}\in G_{{\mathrm{ƭ}}_G}⟹{\mathrm{ɍ}}^{p^k}{\mathrm{ƭ}}_G=e{\mathrm{ƭ}}_G={\mathrm{ƭ}}_G⟹(\mathrm{ɍ}{\mathrm{ƭ}}_G{)}^{p^k}=e{\mathrm{ƭ}}_G.$$

Thus, ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ is a $p$-group. Conversely, let ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ be a $p$-group. Then, for any $\mathrm{ɍ}\in G$, there exist nonnegative integers $k$ and prime $p$ such that $(\mathrm{ɍ}{\mathrm{ƭ}}_G{)}^{p^k}=e{\mathrm{ƭ}}_G$. Thus,

$$(\mathrm{ɍ}{\mathrm{ƭ}}_G{)}^{p^k}=e{\mathrm{ƭ}}_G⟹{\mathrm{ɍ}}^{p^k}{\mathrm{ƭ}}_G=e{\mathrm{ƭ}}_G={\mathrm{ƭ}}_G\hspace{0.33em}⟹{\mathrm{ɍ}}^{p^k}\in G_{{\mathrm{ƭ}}_G}⟹{\mathrm{ƭ}}_G\left({\mathrm{ɍ}}^{p^k}\right)={\mathrm{ƭ}}_G\left(e\right).$$

Hence, ${\mathrm{ƭ}}_G$ is a $p$-SSD-group. □

Proposition 30.

Let  ${\mathrm{ƭ}}_{ G}$ be an SSD-group and $ɍ\in G$. If ${ɍ}^2\in G_{{ \mathrm{ƭ}}_G},$ for all $ɍ\in G,$ then ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ is an abelian $2$-group.

Proof. 

Let ${\mathrm{ƭ}}_{\mathrm{G}}$ be an SSD-group, ${ɍ}^2\in G_{{ \mathrm{ƭ}}_G},$ for all $\mathrm{ɍ}\in \mathrm{G}$, and the operation defined on ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ be $\odot$. By Proposition 28, ${\mathrm{ƭ}}_{\mathrm{G}}$ is a normal SSD-group. Then, by Theorem 15, ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ is an abelian group. Thus, we need to demonstrate that ${\displaystyle \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{${\mathrm{ƭ}}_G$}\right.}$ is a $2$-group. Let ${\mathrm{ɍ}}^2\in {\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}$ for all $\mathrm{ɍ}\in \mathrm{G}$. Then, ${\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɍ}}^2\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{e}\right)$, for all $\mathrm{ɍ}\in \mathrm{G}$. Thus, we have

$${\mathrm{ƭ}}_{\mathrm{G}}\left({\mathrm{ɍ}}^2\right)={\mathrm{ƭ}}_{\mathrm{G}}\left(\mathrm{e}\right)\Rightarrow {\mathrm{ɍ}}^2\in {\mathrm{G}}_{{\mathrm{ƭ}}_{\mathrm{G}}}\Rightarrow {\mathrm{ɍ}}^2{\mathrm{ƭ}}_{\mathrm{G}}={\mathrm{ƭ}}_{\mathrm{G}}\Rightarrow \mathrm{ɍ}\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}=\left(\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}\right)\odot \left(\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}\right)={\mathrm{ƭ}}_{\mathrm{G}}\Rightarrow {\left(\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}\right)}^2={\mathrm{ƭ}}_{\mathrm{G}}.$$

Thus, for all $\mathrm{ɍ}\in \mathrm{G}$, $ord\left(\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}\right)\le 2$, where $ord\left(\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}\right)$ is the order of $\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}$. Therefore, for all $\mathrm{ɍ}\in \mathrm{G}$, $ord\left(\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}\right)=$1 or $ord\left(\mathrm{ɍ}{\mathrm{ƭ}}_{\mathrm{G}}\right)=$ 2. So, we are done. □

5. Conclusions

In this study, the notion of a soft symmetric difference group (SSD-group) is introduced as a new type of soft group that integrates soft set theory with group theory through the symmetric difference operation and set inclusion. Fundamental concepts, including soft symmetric difference subgroups, normal and characteristic soft symmetric difference groups, soft normalizers, and soft cosets of soft symmetric difference groups, are rigorously defined, and their essential algebraic properties are systematically investigated. Several characterization results are established, providing necessary and sufficient conditions for a soft set to admit an SSD-group structure. Furthermore, soft quotient (factor) groups of SSD-groups are introduced, and their structural properties are examined within the proposed framework. Important connections between SSD-group theory and classical group theory are also established through related concepts. The inclusion of illustrative examples demonstrates the internal consistency of the theory and clarifies the behavior of the newly defined notions. The results presented in this paper extend existing soft group constructions and enrich the algebraic foundations of soft set theory. The proposed framework provides a solid basis for further investigations into homomorphisms, isomorphism theorems, and additional algebraic properties of soft symmetric difference groups. We note that no finiteness assumption is imposed on the group in the main results. Although the examples presented in this paper involve generally finite groups for computational transparency, the theory developed here applies equally to infinite groups. A systematic investigation of SSD-structures over infinite groups may lead to further structural classification results and will be considered in future. In particular, SSD-structures over infinite cyclic groups or infinite abelian groups provide natural testing grounds for further theoretical refinement. Moreover, it supports future research on uncertainty-based algebraic systems and the extension of soft operations to broader algebraic structures such as semigroups and rings. These developments may ultimately contribute to a deeper unification of soft algebra and classical algebraic theory, and enhance our understanding of how soft set operations interact with conventional algebraic structures.