Soft Groups and Characteristic Soft Subgroups

Abstract

Group theory is the part of mathematics which addresses the study of symmetry. This paper extends the investigation of the soft group theory which Aktaş and Çağman have defined. We study new concepts in the soft group theory such as the center of the soft group, the kernel of soft homomorphism and soft automorphisms with their basic properties. Furthermore, the concept of soft point groups is introduced and the properties of these soft groups are studied. We state the concept of characteristic soft subgroups of a given soft group. Also, some theorems related to this concept are investigated. We study the characteristic soft subgroups of a given soft point group. The characteristic soft subgroups play a significant part in the study of the soft group theory and are useful for understanding the structure of a soft group and its soft automorphisms. As an application of characteristic soft subgroups, they allow us to identify and study important soft subgroups that are preserved under soft automorphisms. Also, practical applications for our theory can be conducted in future work such as the relation with other disciplines in sciences.

1. Introduction

In 1999, Molodtsov [1] proposed the notion of soft set theory to solve complex problems and difficulties which are related to uncertainties in probability theory, fuzzy set theory [2], rough sets [3] and other mathematical tools. In fuzzy sets, every element is assigned a grade of membership function between 0 and 1. However, the selection of an appropriate membership function can be difficult in each particular case, especially when the underlying data are complex or uncertain. The rough sets are based on the idea of approximating a set by two subsets: the lower approximation and the upper approximation. This can be a useful way to handle incomplete or missing information, and it may become very complex when addressing large sets of data or complex relationships between elements and sets. While fuzzy sets and rough sets are beneficial frameworks for handling uncertainty, the concept of soft sets provides a more flexible and powerful approach to handling uncertainty by allowing each element of a set to be associated with a set of parameters that represent a different characterization or attributes. This can provide a more structured and complete representation of uncertainty than fuzzy sets or rough sets. Every fuzzy set and rough set can be represented as a particular case of a soft set, wherein the soft set parameters are defined by the degree of membership or the rough set approximations, respectively. Therefore, the soft sets have rapidly developed into a powerful and versatile framing for addressing uncertainty and have numerous applications in different fields. In 2002, Maji et al. [4] discussed the soft sets in decision-making problems and proposed a method for aggregating soft set parameters to make decisions. In a decision-making problem, a soft set can be used to represent each option or alternative being considered. The soft set parameters can represent the various factors that influence the decision, such as cost, risk or benefit. In 2003, Maji et al. [5] presented several basic notions of the soft set theory. In 2008, Yang [6] worked to correct some properties of the soft set theory that were introduced in [5]. In 2009, the authors of [7] improved some notions and introduced new operations in the soft sets theory. In 2012, Zorlutuna et al. [8] presented the concept of soft mappings between two collections of soft sets with some properties.

Rosenfeld [9] proposed fuzzy subgroups as an extension of the classical notion of subgroups in group theory. Biswas and Nanda [10] defined the rough groups as an extension of the rough set theory. The algebraic structure of the soft set was originated by Aktaş and Çağman in 2007. They [11] defined the soft group as a soft set ${\tilde{F}}_A$ over a group G such that $F\left(a\right)$ is subgroup of the group G for all $a\in A.$ In 2011, Sezgin and Atagün [12] did a correction for some properties of soft groups that were introduced in [11]. They also presented the concept of normalistic soft groups. Aslam et al. [13] introduced several concepts in soft groups with their properties such as cyclic and abelian soft groups, factor soft groups and others. In 2014, Aktaş and Özlü [14] proposed the order of the soft group and its basic properties were studied. In 2019, Nazmal [15] studied the homomorphic image and preimage of soft groups under the soft mapping that was defined in [8]. In 2023, Barazegar et al. [16] defined the commutator of soft groups and nilpotent soft groups. They also presented the second type nilpotent soft subgroup with its properties.

The relationship between the set theory and group theory is rather strong and the group theory is very rich with many applications in various fields of science. This motivates us to use its richness in the soft set theory. In the group theory, the concept of characteristic subgroups has been studied by many scholars. We say that a subgroup of a given group G is a characteristic subgroup of G if it is invariant under all automorphism of G. In [17,18], several researchers studied the concept of characteristic subgroups and the results related to this concept in fuzzy group theory. The objective of this paper is to further examine soft group theory and extend the concept of characteristic subgroups and their related properties and results to include soft groups. We will study some properties of the soft abelian group and cyclic soft group. We will study new concepts in the soft group such as the center of the soft group and the kernel of soft homomorphism with their basic properties. We will prove that the collection of all soft automorphisms of a given soft group forms a group and the collection of all its inner automorphisms forms a normal subgroup. We will state a new kind of soft group which is called a soft point group. Also, we will study its properties. We will define the characteristic soft subgroups as follows: Suppose that ${\tilde{F}}_A$ is a soft group over a given group G and ${\tilde{H}}_A$ is a soft subgroup of ${\tilde{F}}_A$. Then, ${\tilde{H}}_A$ is called a characteristic soft subgroup of ${\tilde{F}}_A$ if it is invariant under all soft automorphism of ${\tilde{F}}_A,$ and will be denoted by ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A.$ We will study equivalent definitions to the definition of characteristic soft subgroups. We will prove that every characteristic soft subgroup of a normalistic soft subgroup is a soft normal subgroup. We will show that if ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{K}}_A$ and ${\tilde{K}}_A$ $cha{r}_s$ ${\tilde{F}}_A$, then ${\tilde{K}}_A$ $cha{r}_s$ ${\tilde{F}}_A.$ We will prove that the restricted intersection of a collection of characteristic soft subgroups of ${\tilde{F}}_A$ is a characteristic soft subgroup of ${\tilde{F}}_A$. We will study the characteristic soft subgroups of a soft point group. We will prove that every characteristic soft subgroup of a soft point group is a soft normal subgroup. This allows us to identify soft normal subgroups of a given soft point group and study their attributes using the characteristic soft subgroups. We will show that the center of a soft point group and every cyclic soft subgroup of a soft point group is a characteristic soft subgroup. The characteristic soft subgroups play a significant part in the study of soft group theory and are useful for understanding the structure of a soft group and its soft automorphisms. As an application of characteristic soft subgroups, they allow us to identify and study important soft subgroups that are preserved under soft automorphisms. Also, practical applications for our theory can be conducted in future work such as the relation with other disciplines in sciences.

The structure of this paper is as follows: following the introduction, Section 2 provides a review of the notions and theorems in soft theory and soft groups. In Section 3, we study new notions of soft groups, soft homomorphism, soft isomorphism and soft automorphisms. Then, we deduce their properties. After that, we present the concept of soft point groups and examine some of their properties. In Section 4, the definition of the characteristic soft subgroups of a given soft group is presented. Then, we investigate their properties.

2. Preliminaries

This section is devoted to recalling the main concepts and results in soft sets theory and soft groups which will be used in the following sections.

2.1. Soft Sets Theory

Throughout this section, X denotes an initial universe set and $P\left(X\right)$ denotes the power set of X. We indicate the empty set by the symbol ∅.

Definition 1

([1]). Suppose that A is a set of parameters and F is a mapping from A into $P\left(X\right).$ Then, a soft set over X is presented as

$${\tilde{F}}_A=\{(a,F\left(a\right)):a\in AandF\left(a\right)\subseteq X\}.$$

The collection of all soft sets over X with respect to the set of parameters A will be symbolized by $SS{\left(X\right)}_A$.

Definition 2

([5]). Suppose that ${\tilde{F}}_A$ is a soft set over X. If $F\left(a\right)=\varnothing$ for each $a\in A,$ then ${\tilde{F}}_A$ is said to be a null soft set over X and is symbolized by ${\tilde{\mathsf{\Phi }}}_A$.

Definition 3

([5]). Suppose that ${\tilde{F}}_A$ is a soft set over X. If $F\left(a\right)=X$ for each $a\in A,$ then ${\tilde{F}}_A$ is said to be an absolute soft over X and is symbolized by ${\tilde{X}}_A$.

Definition 4

([7]). Consider ${\tilde{F}}_A$ is a soft set over X. Then, the relative complement of ${\tilde{F}}_A,$ symbolized by ${\tilde{F}}_A^c$, is presented as

$${\tilde{F}}_A^c=\{(a,F{\left(a\right)}^c):a\in Aand{F}^c:A\to P\left(X\right)suchthat{F}^c\left(a\right)=F{\left(a\right)}^c\}.$$

Definition 5

([6]). Suppose that ${\tilde{F}}_A$ and ${\tilde{H}}_B$ are soft sets over X. Then, ${\tilde{F}}_A$ is called a soft subset of ${\tilde{H}}_B$, symbolized by ${\tilde{F}}_A$ $\tilde{\subseteq }$ ${\tilde{H}}_B$, if $A\subseteq B$ and $F\left(a\right)\subseteq H\left(a\right)$ for each $a\in A$. The soft sets ${\tilde{F}}_A$ and ${\tilde{H}}_B$ are soft equal if ${\tilde{F}}_A$ $\tilde{\subseteq }$ ${\tilde{H}}_B$ and ${\tilde{H}}_B$ $\tilde{\subseteq }$ ${\tilde{F}}_A$.

Definition 6

([7]). Suppose ${\tilde{F}}_A$ and ${\tilde{H}}_B$ are soft sets over X such that $A\cap B\ne \varnothing$. Then, the restricted intersection of ${\tilde{F}}_A$ and ${\tilde{H}}_B,$ symbolized by ${\tilde{F}}_A$ $\tilde{\cap }$ ${\tilde{H}}_B,$ is the soft set

$${\tilde{T}}_C=\{(c,T\left(c\right)):c\in C,T:C\to P\left(X\right)\},$$

where $C=A\cap B$ and $T\left(c\right)=F\left(c\right)\cap H\left(c\right)$ for each $c\in C.$

Definition 7

([5]). Suppose that ${\tilde{F}}_A$ and ${\tilde{H}}_B$ are soft sets over X. Then, the union (extended union) of ${\tilde{F}}_A$ and ${\tilde{H}}_B,$ symbolized by ${\tilde{F}}_A$ $\tilde{\cup }$ ${\tilde{H}}_B,$ is the soft set ${\tilde{T}}_C=\{(c,T\left(c\right)):$ $c\in C,$ $T:C\to P\left(X\right)\}$, where $C=A\cup B$ and

$$\begin{array}{c}\hfill T\left(c\right)=\left\{\begin{array}{cc}F\left(c\right)\hfill & ifc\in A-B,\hfill \\ H\left(c\right)\hfill & ifc\in B-A,\hfill \\ F\left(c\right)\cup H\left(c\right)\hfill & ifc\in A\cap B.\hfill \end{array}\right.\end{array}$$

Definition 8

([8]). Suppose that $SS{\left(X\right)}_A$ is a collection of soft sets over X and $SS{\left(Y\right)}_B$ is a collections of soft sets over $Y.$ Let $f:X\to Y$ and $\psi :A\to B$ be two mappings. Then, ${\mathsf{\Psi }}_{f,\psi }:SS{\left(X\right)}_A\to SS{\left(Y\right)}_B$ is a map defined as

• If ${\tilde{F}}_A\in SS{\left(X\right)}_A,$ then the image of ${\tilde{F}}_A$ is a soft set in $SS{\left(Y\right)}_B$ such that for all $b\in B$,

  $$\begin{array}{c}\hfill {\mathsf{\Psi }}_{f,\psi }\left(\tilde{F}\right)\left(b\right)=\left\{\begin{array}{cc}\underset{a\in {\psi }^{-1}\left(b\right)}{\cup }f\left(F\left(a\right)\right)\hfill & if{\psi }^{-1}\left(b\right)\ne \varnothing ,\hfill \\ \varnothing \hfill & if{\psi }^{-1}\left(b\right)=\varnothing .\hfill \end{array}\right.\end{array}$$
• If ${\tilde{H}}_B\in SS{\left(Y\right)}_B,$ then the inverse image of a soft set ${\tilde{H}}_B$ is
  $${\mathsf{\Psi }}_{f,\psi }^{-1}\left({\tilde{H}}_B\right)\left(a\right)=f^{-1}\left(H\left(\psi \left(a\right)\right)\right)foralla\in A.$$

Remark 1.

If f and ψ in Definition 8 are injective (resp. surjective, bijective) maps, then the map ${\mathsf{\Psi }}_{f,\psi }$ is injective (resp. surjective, bijective).

Theorem 1

([8]). Consider $SS{\left(X\right)}_A$ is a collections of soft sets over X and $SS{\left(Y\right)}_B$ is a collection of soft sets over $Y.$ We assume that ${\mathsf{\Psi }}_{f,\psi }:SS{\left(X\right)}_A\to SS{\left(Y\right)}_B,$ where $f:X\to Y$ and $\psi :A\to B$.

1. 

${\mathsf{\Psi }}_{f,\psi }\circ {\mathsf{\Psi }}_{f,\psi }^{-1}\left({\tilde{H}}_B\right)$$\tilde{\subseteq }$${\tilde{H}}_B$ for any ${\tilde{H}}_B\in SS{\left(Y\right)}_B.$ If ${\mathsf{\Psi }}_{f,\psi }$ is surjective, then ${\mathsf{\Psi }}_{f,\psi }\circ {\mathsf{\Psi }}_{f,\psi }^{-1}\left({\tilde{H}}_B\right)=$ ${\tilde{H}}_B.$

2. 

${\tilde{F}}_A$$\tilde{\subseteq }$${\mathsf{\Psi }}_{f,\psi }^{-1}\circ {\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)$ for any ${\tilde{F}}_A\in SS{\left(X\right)}_A.$ If ${\mathsf{\Psi }}_{f,\psi }$ is injective, then ${\tilde{F}}_A$ = ${\mathsf{\Psi }}_{f,\psi }^{-1}\circ {\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right).$

2.2. Soft Group

Within this section, G denotes a group with identity element $e_G.$ We call $(a,F(a\left)\right)$ an element of the soft set ${\tilde{F}}_A$, where $a\in A.$

Definition 9

([11]). Suppose that ${\tilde{F}}_A$ is a soft set over $G.$ If $F\left(a\right)$ is a subgroup of G for each $a\in A,$ then ${\tilde{F}}_A$ is called a soft group over $G.$

Definition 10

([11]). Suppose that ${\tilde{F}}_A$ is a soft group over G.

1. 

If $F\left(a\right)=\left\{e_G\right\}$ for each $a\in A,$ then ${\tilde{F}}_A$ is said to be an identity soft group and is denoted by ${\tilde{e}}_G.$

2. 

If $F\left(a\right)=G$ for each $a\in A,$ then ${\tilde{F}}_A$ is said to be an absolute soft group and is denoted by ${\tilde{G}}_A.$

Definition 11

([11]). Suppose that ${\tilde{F}}_A$ and ${\tilde{H}}_B$ are two soft groups over G. Then, ${\tilde{F}}_A$ is a soft subgroup of ${\tilde{H}}_B$, symbolized by ${\tilde{F}}_A$ $\tilde{\le }$ ${\tilde{H}}_B$, if $A\subseteq B$ and $F\left(a\right)$ is a subgroup of $H\left(a\right)$ for each $a\in A$. If $F\left(a\right)$ is a normal subgroup of $H\left(a\right)$ for each $a\in A$, then ${\tilde{F}}_A$ is a normal soft subgroup of ${\tilde{H}}_B$ and symbolized by ${\tilde{F}}_A$ $\widetilde{⊴}$ ${\tilde{H}}_B.$

Definition 12

([12]). Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ If $F\left(a\right)$ is a normal subgroup of G for each $a\in A,$ then ${\tilde{F}}_A$ is called a normalistic soft group over $G.$

Let us now recollect the definitions of soft homomorphism and soft isomorphism between two soft groups.

Definition 13

([11]). Suppose that ${\tilde{F}}_A$ is a soft group over G and ${\tilde{H}}_B$ is a soft group over the group $K.$ Suppose that $f:G\to K$ and $\psi :A\to B$ are two mappings. Then, a pair $(f,\psi )$ is considered a soft homomorphism from ${\tilde{F}}_A$ to ${\tilde{H}}_B$ if the following conditions are met:

• f is a group homomorphism from G onto K;
• ψ is a map from A onto B;
• $f\left(F\right(a\left)\right)=H\left(\psi \right(a\left)\right)$ for each $a\in A.$

Definition 14

([11]). Suppose that ${\tilde{F}}_A$ is a soft group over G and ${\tilde{H}}_B$ is a soft group over the group $K.$ Let $(f,\psi )$ be a soft homomorphism from ${\tilde{F}}_A$ to ${\tilde{H}}_B.$ If f is a group isomorphism from G to K and ψ is a bijective mapping from A to B, then the pair $(f,\psi )$ is said to be a soft isomorphism from ${\tilde{F}}_A$ to ${\tilde{H}}_B.$ Also, we call ${\tilde{F}}_A$ a soft isomorphic to ${\tilde{H}}_B$ and is symbolized by ${\tilde{F}}_A\simeq {\tilde{H}}_B.$

The following theorem clarifies the soft homomorphic image and preimage of soft groups under the soft mappings mentioned in Definition 8.

Theorem 2

([15]). Consider $SS{\left(G\right)}_A$ a collections of soft sets over G and $SS{\left(K\right)}_B$ is a collection of soft sets over the group $K.$ Suppose that ${\mathsf{\Psi }}_{f,\psi }:SS{\left(G\right)}_A\to SS{\left(K\right)}_B,$ where $(f,\psi )$ is a soft homomorphism.

1. 

${\mathsf{\Psi }}_{f,\psi }\left({\tilde{e}}_G\right)={\tilde{e}}_K$ and ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{G}}_A\right)={\tilde{K}}_B$.

2. 

If ${\tilde{H}}_B$ is a soft group over $K,$ then ${\mathsf{\Psi }}_{f,\psi }^{-1}\left({\tilde{H}}_B\right)$ is a soft group over G.

3. 

If ${\tilde{F}}_A$ is a soft group over G and ψ is an injective map, then ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)$ is a soft group over K.

4. 

If ${\tilde{F}}_A$ and ${\tilde{U}}_A$ are soft groups over G such that ${\tilde{F}}_A$ $\tilde{\le }$ ${\tilde{U}}_A$ and ψ is an injective map, then ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)$ $\tilde{\le }$ ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{U}}_A\right).$ Also, if ${\tilde{F}}_A$ $\widetilde{⊴}$ ${\tilde{U}}_A$ and ψ is an injective map, then ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)$ $\widetilde{⊴}$ ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{U}}_A\right).$

5. 

If ${\tilde{H}}_B$ and ${\tilde{N}}_B$ are soft groups over K such that ${\tilde{H}}_B$ $\tilde{\le }$ ${\tilde{N}}_B$, then ${\mathsf{\Psi }}_{f,\psi }^{-1}\left({\tilde{H}}_B\right)$ $\tilde{\le }$ ${\mathsf{\Psi }}_{f,\psi }^{-1}\left({\tilde{N}}_B\right).$ Also, if ${\tilde{H}}_B$ $\widetilde{⊴}$ ${\tilde{N}}_B$, then ${\mathsf{\Psi }}_{f,\psi }^{-1}\left({\tilde{H}}_B\right)$ $\widetilde{⊴}$ ${\mathsf{\Psi }}_{f,\psi }^{-1}\left({\tilde{N}}_B\right).$

In what follows, definitions and some theorems of the order of soft groups and cyclic soft groups are presented.

Definition 15

([14]). Suppose that ${\tilde{F}}_A$ is a soft group over G and $(a,F(a\left)\right)$ is an element of ${\tilde{F}}_A.$ Then, the order of $(a,F(a\left)\right),$ symbolized by $|F(a)|,$ is the smallest positive integer n such that $F{\left(a\right)}^n=\left\{e_G\right\}$, where $F{\left(a\right)}^n=\{g^n:g\in F\left(a\right)\}.$ The element $(a,F(a\left)\right)$ of ${\tilde{F}}_A$ has infinite order if n does not exist.

Theorem 3

([14]). Suppose that ${\tilde{F}}_A$ is a soft group over a finite group $G.$ Then, each element of ${\tilde{F}}_A$ has a finite order.

Theorem 4

([14]). Suppose that ${\tilde{F}}_A$ is a soft group over a finite group G and $(a,F\left(a\right))\in {\tilde{F}}_A.$ Then, $|F(a)|$ is the exponent of the subgroup $F\left(a\right).$

Definition 16

([14]). Suppose that ${\tilde{F}}_A$ is a soft group over G.

1. 

For a finite group G, the order of ${\tilde{F}}_A$ is the exponent of ${\tilde{F}}_A$, where the exponent of ${\tilde{F}}_A$ means the least common multiple of the orders of its elements.

2. 

For an infinite group G, the order of ${\tilde{F}}_A$ is determined by the number of its elements.

We denote the order of ${\tilde{F}}_A$ by $|{\tilde{F}}_A|.$

Example 1.

Consider $G=S_4$ and $A=\{a_1,a_2,a_3\}$. Then,

$${\tilde{F}}_A=\{(a_1,S_3),(a_2,\langle \left(1234\right),\left(13\right)\rangle \cong D_8),(a_3,\langle \left(1234\right)\rangle \cong C_4)\}$$

is a soft group over G. Thus, $|F(a_1)|=6$, $|F(a_2)|=4$ and $|F(a_3)|=4$. Therefore, $|{\tilde{F}}_A|=12.$

Example 2.

Consider $G=\mathbb{R}$ an additive group of real numbers and $A=\{a_1,a_2,a_3\}$. Then,

$${\tilde{F}}_A=\{(a_1,\mathbb{Z}),(a_2,\mathbb{Q}),(a_3,\mathbb{R})\}$$

is a soft group over G. Therefore, $|{\tilde{F}}_A|=3.$

Definition 17

([14]). Assuming that ${\tilde{F}}_A$ is a soft group defined over G and X is a subset of G. If ${\tilde{F}}_A=\{(a,\langle x\rangle ):$ for each $a\in A$ and $x\in X\subset G\},$ then ${\tilde{F}}_A$ is the soft group generated by the set $X.$ If ${\tilde{F}}_A=\langle X\rangle ,$ then we say that ${\tilde{F}}_A$ is a cyclic soft group over $G.$

Remark 2.

Every soft group over a cyclic group is a cyclic soft group but the converse may fail.

Theorem 5

([14]). Suppose that ${\tilde{F}}_A$ is a soft group over G.

1. 

${\tilde{e}}_G$ is a cyclic soft group over G generated by $\left\{e_G\right\}.$

2. 

${\tilde{G}}_A$ is a cyclic soft group over G if and only if G is a cyclic group.

3. 

If ${\tilde{F}}_A$ is a cyclic soft group, then every soft subgroup of ${\tilde{F}}_A$ is a cyclic soft group.

Theorem 6

([14]). Suppose that ${\tilde{F}}_A$ is soft group over G and ${\tilde{H}}_B$ is a soft group over the group $K.$ Assume that $(f,\psi )$ is a soft homomorphism that maps ${\tilde{F}}_A$ to ${\tilde{H}}_B.$ If ${\tilde{F}}_A$ is a cyclic soft group over $G,$ then $(f\left({\tilde{F}}_A\right),\psi \left(A\right))$ is a cyclic soft group over $K.$

Definition 18

([13]). Assuming that ${\tilde{F}}_A$ is a soft group over $G.$ If $F\left(a\right)$ is an abelian subgroup of G for each $a\in A,$ then ${\tilde{F}}_A$ is called an abelian soft group over $G.$

Theorem 7

([13]). Suppose that ${\tilde{F}}_A$ is an abelian soft group over G and ${\tilde{H}}_B$ $\widetilde{⩽}$ ${\tilde{F}}_A.$ Then, ${\tilde{H}}_B$ is a normal soft subgroup of ${\tilde{F}}_A$.

Theorem 8

([13]). Assuming that ${\tilde{F}}_A$ is a cyclic soft group over G. Then, ${\tilde{F}}_A$ is an abelian soft group over G.

But the reverse of the above theorem is generally not true. For instance, ${\tilde{V}}_4$ is an abelian soft group over $V_4$ but not cyclic soft group over $V_4$.

Now, we recall the definitions and theorems which are related to the commutator of soft groups and nilpotent soft groups.

Definition 19

([16]). Suppose that ${\tilde{F}}_A$ is a soft group over G and ${\tilde{H}}_B$ is a soft group over the group $K.$ Then, the commutator of ${\tilde{F}}_A$ and ${\tilde{H}}_A$ is the soft set $[{\tilde{F}}_A,{\tilde{H}}_A]:A\to P\left(G\right)$ such that

$$[{\tilde{F}}_A,{\tilde{H}}_A]\left(a\right)=[F\left(a\right),H\left(a\right)]$$

for each $a\in A.$ It is clear that $[{\tilde{F}}_A,{\tilde{H}}_A]$ is a soft group of $G.$

Remark 3

([16]). Let $\{{\tilde{F}}_A^1,{\tilde{F}}_A^2,\dots ..,{\tilde{F}}_A^n\}$ be a collection of soft groups over G. Then, the n-commutator $[{\tilde{F}}_A^1,{\tilde{F}}_A^2,\dots ..,{\tilde{F}}_A^n]:A\to P\left(G\right)$ is defined as

$$[{\tilde{F}}_A^1,{\tilde{F}}_A^2,\dots ..,{\tilde{F}}_A^n]\left(a\right)=[F^1\left(a\right),F^2\left(a\right),\dots ,F^n\left(a\right)]$$

for each $a\in A$.

Definition 20

([16]). Suppose that ${\tilde{F}}_A$ is soft group over $G.$ A chain of soft groups over G

$${\tilde{e}}_G={\tilde{F}}_A^0\tilde{\subseteq }{\tilde{F}}_A^1\tilde{\subseteq }\dots \dots ..\tilde{\subseteq }{\tilde{F}}_A^n={\tilde{F}}_A$$

is said to be a central series if $[{\tilde{F}}_A^i,{\tilde{F}}_A]$ $\tilde{\subseteq }$ ${\tilde{F}}_A^{i-1}$ for all $1⩽i⩽n$.

Definition 21

([16]). Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ Then, ${\tilde{F}}_A$ is said to be a nilpotent soft group if it possesses a central series. The nilpotency class of ${\tilde{F}}_A$ is defined as the length of its shorter central series and symbolized by $c\left({\tilde{F}}_A\right).$

3. Some More Properties of Soft Groups, Soft Homomorphisms and Soft Automorphisms

In this section, we introduce concepts and results which are related to soft groups, soft isomorphisms and soft automorphisms. Also, we point out the concept of soft point groups and deduce their basic properties. The notations are the same as in Section 2.2.

First, we introduce theorems of abelian soft groups.

Theorem 9.

Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ If $|{\tilde{F}}_A|=2,$ then ${\tilde{F}}_A$ is an abelian soft group over $G.$

Proof. 

It is straightforward. □

Theorem 10.

Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ Then, $[{\tilde{F}}_A,{\tilde{F}}_A]={\tilde{e}}_G$ if and only if ${\tilde{F}}_A$ is an abelian soft group over G.

Proof. 

Necessity. Suppose that $[{\tilde{F}}_A,{\tilde{F}}_A]={\tilde{e}}_G.$ Then, $[F\left(a\right),F\left(a\right)]=\left\{e_G\right\}$ for each $a\in A.$ Therefore, $F\left(a\right)$ is an abelian subgroup of G for each $a\in A.$ Hence, ${\tilde{F}}_A$ is an abelian soft group over G.

Sufficiency. We assume that ${\tilde{F}}_A$ is an abelian soft group over $G.$ Then, $F\left(a\right)$ is an abelian subgroup of G for each $a\in A.$ Therefore, $[F\left(a\right),F\left(a\right)]=\left\{e_G\right\}$ for each $a\in A.$ Hence, $[{\tilde{F}}_A,{\tilde{F}}_A]={\tilde{e}}_G.$ □

Theorem 11.

Suppose that ${\tilde{F}}_A$ is an abelian soft group over $G.$ Then, ${\tilde{F}}_A$ is a nilpotent soft group.

Proof. 

Suppose that ${\tilde{F}}_A$ is an abelian soft group over $G.$ By Definition 20, a chain

$${\tilde{e}}_G={\tilde{F}}_A^0\tilde{\subseteq }{\tilde{F}}_A^1\tilde{\subseteq }\dots \dots ..\tilde{\subseteq }{\tilde{F}}_A^n={\tilde{F}}_A$$

of soft groups over G is central series if and only if $[{\tilde{F}}_A^i,{\tilde{F}}_A]$ $\tilde{\subseteq }$ ${\tilde{F}}_A^{i-1}$ for all $1⩽i⩽n.$ Then, by Theorem 10, $[{\tilde{F}}_A^n,{\tilde{F}}_A]=[{\tilde{F}}_A,{\tilde{F}}_A]=\left\{e_G\right\}.$ Thus, $[{\tilde{F}}_A^n,{\tilde{F}}_A]$ $\tilde{\subseteq }$ ${\tilde{F}}_A^{n-1}$ for all $1⩽i⩽n.$ Hence, ${\tilde{F}}_A$ is a nilpotent soft group. □

The reverse of the above theorem is generally not true, as illustrated in the next example.

Example 3.

Let $G=D_8=\langle a,b|a^4=b^2={\left(ab\right)}^2=1\rangle .$ Let $A=\{a_1,a_2,a_3\}$ and ${\tilde{F}}_A=\{(a_1,D_8),(a_2,\langle a\rangle ),(a_2,\langle a^2\rangle )\}$ be a soft group over $G.$ Therefore, ${\tilde{F}}_A$ is a nilpotent soft group over G since it has a central series

$${\tilde{e}}_G={\tilde{F}}_A^0\tilde{\subseteq }{\tilde{F}}_A^1\tilde{\subseteq }{\tilde{F}}_A^n={\tilde{F}}_A,$$

where ${\tilde{F}}_A^1=\{(a_1,\langle a^2\rangle ),(a_2,\langle a^2\rangle ),(a_3,\langle a^2\rangle )\}.$ It is obvious that ${\tilde{F}}_A$ is not an abelian soft group over $G.$

Now, we look at the image of abelian and cyclic soft groups under the soft mapping which is defined in Definition 8.

Theorem 12.

Suppose that ${\tilde{F}}_A$ is a soft group over G. We assume that ${\mathsf{\Psi }}_{f,\psi }:SS{\left(G\right)}_A\to SS{\left(K\right)}_B,$ where $f:G\to K$ is a group homomorphism and $\psi :A\to B$ is a bijection mapping.

1. 

If ${\tilde{F}}_A$ is an abelian soft group, then ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)$ is an abelian soft group over K.

2. 

If ${\tilde{F}}_A$ is a cyclic soft group, then ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)$ is a cyclic soft group over $K.$

Proof. 

• By Theorem 2, ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)$ is a soft group over $K.$ By the definition of ${\mathsf{\Psi }}_{f,\psi },$ ${\mathsf{\Psi }}_{f,\psi }\left(\tilde{F}\right)\left(b\right)=f\left(F\left({\psi }^{-1}\left(b\right)\right)\right)$ for each $b\in B.$ Since $F\left({\psi }^{-1}\left(b\right)\right)$ is an abelian subgroup of G for each $a\in A$ then $f\left(F\left({\psi }^{-1}\left(b\right)\right)\right)$ is an abelian subgroup of K for each $b\in B.$ Therefore, ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)$ is an abelian soft group over K.
• It is similar to the proof part 1.

□

Now, we present the notion of the center of a soft group.

Definition 22.

Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ Then, we define the center of ${\tilde{F}}_A$ as

$$Z\left({\tilde{F}}_A\right)=\{(a,Z\left(F\left(a\right)\right)):foreacha\in A\},$$

where $Z\left(F\right(a\left)\right)$ is the center of the subgroup $F\left(a\right)$. It is clear that $Z\left({\tilde{F}}_A\right)$ is a soft subgroup of ${\tilde{F}}_A$.

Example 4.

Consider $G=S_3$ and ${\tilde{F}}_A=\{(a_1,\langle \left(123\right)\rangle ),(a_2,S_3)\}$ is a soft group over $G.$ Then, $Z\left({\tilde{F}}_A\right)=\{(a_1,\langle \left(123\right)\rangle ),(a_2,\left\{e_G\right\})\}$.

Theorem 13.

Assuming that ${\tilde{F}}_A$ is a soft group over $G.$ Then, $Z\left({\tilde{F}}_A\right)$ $\widetilde{⊴}$ ${\tilde{F}}_A.$

Proof. 

Since $Z\left(F\right(a\left)\right)$ is a normal subgroup of $F\left(a\right)$ for each $a\in A$ then $Z\left({\tilde{F}}_A\right)$ $\widetilde{⊴}$ ${\tilde{F}}_A.$ □

Definition 23.

Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ If every element of ${\tilde{F}}_A$ has order a power of some fixed prime p, then ${\tilde{F}}_A$ is called a soft p-group.

Theorem 14.

Assuming that ${\tilde{F}}_A$ is a soft p-group over G. Then, $Z\left({\tilde{F}}_A\right)$ is a non-trivial soft group, meaning that $Z\left({\tilde{F}}_A\right)$ is not equal to the identity soft group.

Proof. 

Since ${\tilde{F}}_A$ is a soft p-group, every element $(a,F\left(a\right))\in {\tilde{F}}_A$ has order is a power of p. By a fact in the basic group theory, for each $a\in A,$ $F\left(a\right)$ has a non-trivial center. Then, $Z\left({\tilde{F}}_A\right)$ is a non-trivial soft group. □

We now introduce and study some concepts and properties which are related to soft homomorphisms, soft isomorphisms and soft automorphisms.

Definition 24.

Suppose that ${\tilde{F}}_A$ is soft group over G and ${\tilde{H}}_B$ is a soft group over the group $K.$ We assume that $(f,\psi )$ is a soft homomorphism that maps ${\tilde{F}}_A$ to ${\tilde{H}}_B.$ Then, we define the kernel of $(f,\psi )$ as

$$Ker(f,\psi )=\{(a,Ker{f}_a):foreacha\in A\},$$

where $Ker{f}_a$ is the kernel of the restriction of f to the subgroup $F\left(a\right).$

Theorem 15.

Suppose that ${\tilde{F}}_A$ is soft group over G and ${\tilde{H}}_B$ is a soft group over the group $K.$ We assume that $(f,\psi )$ is a soft homomorphism that maps ${\tilde{F}}_A$ to ${\tilde{H}}_B.$ Then, $Ker(f,\psi )$ is a normal soft subgroup of ${\tilde{F}}_A.$

Proof. 

By a fact in the basic group theory, we have $Ker{f}_a⊴F\left(a\right)$ for each $a\in A.$ Thus, $Ker(f,\psi )$ $\widetilde{⊴}$ ${\tilde{F}}_A.$ □

The following theorem explains the symmetry part between the kernel of a homomorphism between two groups G and K and the kernel of a soft homomorphism between the soft group ${\tilde{F}}_A$ over G and the soft group ${\tilde{H}}_B$ over $K.$

Theorem 16.

Suppose that ${\tilde{F}}_A$ is soft group over G and ${\tilde{H}}_B$ is a soft group over the group $K.$ We assume that $(f,\psi )$ is a soft homomorphism that maps ${\tilde{F}}_A$ to ${\tilde{H}}_B.$ If $Kerf=\left\{e_G\right\},$ then $Ker(f,\psi )={\tilde{e}}_G.$

Proof. 

Suppose that $Kerf=\left\{e_G\right\}.$ Let ${\tilde{U}}_A=Ker(f,\psi )$ then $f_a\left(U\left(a\right)\right)=\left\{e_K\right\}$ for each $a\in A.$ Therefore, $U\left(a\right)=\left\{e_G\right\}$ for each $a\in A$ since $Kerf=\left\{e_G\right\}.$ Hence, $Ker(f,\psi )={\tilde{e}}_G.$ □

The next example elaborates that the above theorem cannot be reversed.

Example 5.

Consider $G=(S_3,\circ )$ and $K=({\mathbb{Z}}_2,\oplus ).$ Consider that $A=\{a_1,a_2\}$ and $B=\{b_1,b_2\}$ are two sets of parameters. Then, ${\tilde{F}}_A=\{(a_1,\langle \left(12\right)\rangle ),(a_2,\langle \left(13\right)\rangle )\}$ is a soft group over G and ${\tilde{H}}_B=\{(b_1,{\mathbb{Z}}_2),(b_2,{\mathbb{Z}}_2)\}$ is a soft group over $K.$ Let $(f,\psi )$ be a soft homomorphism from ${\tilde{F}}_A$ to ${\tilde{H}}_B$, where f and ψ are defined as

$$\begin{gathered} f\left(\left(e\right)\right)=f\left(\left(123\right)\right)=f\left(\left(132\right)\right)=\overline{0}andf\left(\left(12\right)\right)=f\left(\left(13\right)\right)=f\left(\left(23\right)\right)=\overline{1}, \\ \psi \left(a_1\right)=b_1,\psi \left(a_2\right)=b_2. \end{gathered}$$

We note that $Kerf=A_3$ and $Ker(f,\psi )={\tilde{e}}_G.$

Corollary 1.

Suppose that ${\tilde{F}}_A$ is soft group over G and ${\tilde{H}}_B$ is a soft group over the group $K.$ We assume that $(f,\psi )$ is a soft homomorphism that maps ${\tilde{F}}_A$ to ${\tilde{H}}_B.$ If $(f,\psi )$ is a soft isomorphism, then $Ker(f,\psi )={\tilde{e}}_G.$

Proof. 

If $(f,\psi )$ is a soft isomorphism, then f is a monomorphism from G to $K.$ Then, by a fact in basic group theory, $Kerf=\left\{e_G\right\}.$ Therefore, by Theorem 16, $Ker(f,\psi )={\tilde{e}}_G.$ □

Theorem 17.

Suppose that ${\tilde{F}}_A$ is soft group over G and ${\tilde{H}}_B$ is a soft group over the group $K.$ We assume that $(f,\psi )$ is a soft isomorphism that maps ${\tilde{F}}_A$ to ${\tilde{H}}_B.$ Then, $(f^{-1},{\psi }^{-1})$ is a soft isomorphism from ${\tilde{H}}_B$ to ${\tilde{F}}_A.$

Proof. 

Since $f:G\to K$ is a group isomorphism, then $f^{-1}:K\to G$ is an isomorphism of groups. Also, since $\psi$ is a bijection mapping, there exists a bijection map ${\psi }^{-1}:B\to A$. Therefore,

$$\begin{array}{cc}\hfill f^{-1}\left(H\left(b\right)\right)& =f^{-1}\left(H\left(\psi \left(a\right)\right)\right),\sin \mathrm{ce}\psi \mathrm{is}\mathrm{bijection},\mathrm{let}\psi \left(a\right)=b\hfill \\ & =f^{-1}\left(f\left(F\left(a\right)\right)\right),\sin \mathrm{ce}(f,\psi )\mathrm{is}\mathrm{a}\mathrm{soft}\mathrm{isomorphism}\hfill \\ & =F\left(a\right),\sin \mathrm{ce}{f}^{-1}\circ f=i{d}_G:G\to G\hfill \\ & =F\left({\psi }^{-1}\left(b\right)\right),\sin \mathrm{ce}\psi \left(a\right)=b.\hfill \end{array}$$

for each $b\in B.$ Hence, $(f^{-1},{\psi }^{-1})$ is a soft isomorphism from ${\tilde{H}}_B$ to ${\tilde{F}}_A$. □

Remark 4.

A soft identity map is a pair $(i{d}_G,i{d}_A),$ where $i{d}_G:G\to G$ and $i{d}_A:A\to A$ such that $i{d}_G\left(g\right)=g$ for each $g\in G$ and $i{d}_A\left(a\right)=a$ for each $a\in A.$ Therefore, it is clear that $(i{d}_G,i{d}_A)$ is a soft isomorphism that maps ${\tilde{F}}_A$ to itself, where ${\tilde{F}}_A$ is a soft group over $G.$

Theorem 18.

Let Ω be the set of soft groups. Then, the relation ≃ is an equivalence relation on $\mathsf{\Omega }.$

Proof. 

• Suppose that ${\tilde{F}}_A$ is any soft group over $G.$ By Remark 4, $(i{d}_G,i{d}_A)$ is a soft isomorphism that maps ${\tilde{F}}_A$ to itself. Then, ${\tilde{F}}_A\simeq {\tilde{F}}_A.$ Therefore, the relation ≃ is a reflexive relation.
• Suppose that ${\tilde{F}}_A$ is a soft group over G and ${\tilde{H}}_B$ is a soft group over the group $K.$ We assume that ${\tilde{F}}_A\simeq {\tilde{H}}_B.$ Then, there exists a soft isomorphism $(f,\psi )$ from ${\tilde{F}}_A$ to ${\tilde{H}}_B$. By Theorem 17, $(f^{-1},{\psi }^{-1})$ is a soft isomorphism that maps ${\tilde{H}}_B$ to ${\tilde{F}}_A.$ Therefore, the relation ≃ is a symmetric relation.
• Suppose that ${\tilde{F}}_A,$ ${\tilde{H}}_B$ and ${\tilde{U}}_C$ are soft groups over the groups $G,$K and $S,$ respectively. We assume that ${\tilde{F}}_A\simeq {\tilde{H}}_B$ and ${\tilde{H}}_B\simeq {\tilde{U}}_C.$ Then, there exist a soft isomorphism $(f,\psi )$ from ${\tilde{F}}_A$ to ${\tilde{H}}_B$ and a soft isomorphism $(f_1,{\psi }_1)$ from ${\tilde{H}}_B$ to ${\tilde{U}}_C$. Since f and $f_1$ are isomorphisms of groups then $f_1\circ f$ is an isomorphism from G to S. Similarly, since $\psi$ and ${\psi }_1$ are bijection mappings then ${\psi }_1\circ \psi$ is a bijection mapping from A to $C.$ Moreover,

  $$\begin{array}{cc}\hfill f_1\circ f\left(F\left(a\right)\right)& =f_1\left(f\left(F\left(a\right)\right)\right)\hfill \\ & =f_1\left(H\left(\psi \left(a\right)\right)\right),\sin \mathrm{ce}(f,\psi )\mathrm{is}\mathrm{a}\mathrm{soft}\mathrm{isomorphism}\hfill \\ & =U\left({\psi }_1\left(\psi \left(a\right)\right)\right),\sin \mathrm{ce}(f_1,{\psi }_1)\mathrm{is}\mathrm{a}\mathrm{soft}\mathrm{isomorphism}\hfill \\ & =U\left(({\psi }_1\circ \psi )\left(a\right)\right)\hfill \end{array}$$

  for each $a\in A.$ Thus, $(f_1\circ f,{\psi }_1\circ \psi )$ is a soft isomorphism from ${\tilde{F}}_A$ to ${\tilde{U}}_C$. Therefore, the relation ≃ is a transitive relation.

Hence, the relation ≃ is an equivalence relation on $\mathsf{\Omega }.$ □

Theorem 19.

Suppose that ${\tilde{F}}_A$ is soft group over a finite group $G.$ Assuming that ${\mathsf{\Psi }}_{f,\psi }:SS{\left(G\right)}_A\to SS{\left(K\right)}_B,$ where $f:G\to K$ is a group isomorphism and $\psi :A\to B$ is a bijection mapping. Then, $|{\tilde{F}}_A|=|{\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)|.$

Proof. 

Suppose that $|{\tilde{F}}_A|=n,$ $F{\left(a\right)}^n=\left\{e_G\right\}$ for each $a\in A.$ By the definition of ${\mathsf{\Psi }}_{f,\psi },$ ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)\left(b\right)=f\left(F\left({\psi }^{-1}\left(b\right)\right)\right)$ for each $b\in B.$ Since f is an isomorphism of groups, $|f\left(F\left({\psi }^{-1}\left(b\right)\right)\right)|=|F\left({\psi }^{-1}\left(b\right)\right)|$ for each $b\in B.$ Thus, $|{\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)|=n.$ Now, if $|{\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)|=m,$ then $f{\left(F\left({\psi }^{-1}\left(b\right)\right)\right)}^m=\left\{e_K\right\}$ for each $b\in B.$ Since f is an isomorphism of groups, $F{\left({\psi }^{-1}\left(b\right)\right)}^m=\left\{e_G\right\}$ for each $a\in A.$ Then, $n\mid m$ since n is the exponent of ${\tilde{F}}_A$. This proves the theorem. □

Theorem 20.

Suppose that ${\tilde{F}}_A$ is a soft group over an infinite group G. Assuming that ${\mathsf{\Psi }}_{f,\psi }:SS{\left(G\right)}_A\to SS{\left(K\right)}_B,$ where $f:G\to K$ is a group isomorphism and $\psi :A\to B$ is a bijection mapping. Then, $|{\tilde{F}}_A|=|{\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)|.$

Proof. 

Since the order of ${\tilde{F}}_A$ is equal to the number of its elements and $(f,\psi )$ is a soft isomorphism then it is clear that $|{\tilde{F}}_A|=|{\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)|.$ □

Theorem 21.

Every cyclic soft group over a finite cyclic group of order n is a soft isomorphic to a soft group over additive group ${\mathbb{Z}}_n.$

Proof. 

We assume that $G=\langle g\rangle$ is a cyclic group of order $n.$ Then, by a fact in the basic group theory, there exists a group isomorphism, $f:{\mathbb{Z}}_n\to G$ defined by $f\left(\overline{k}\right)=g^k.$ Let $\psi :B\to A$ be any bijection map between two sets of parameters. Therefore, for any soft group ${\tilde{F}}_A=\{(\psi \left(b\right),\langle g^k\rangle ):k<n$ for all $b\in B\}$ over G, there exists a soft group ${\tilde{H}}_B=\{(b,\langle \overline{k}\rangle ):k<n$ for all $b\in B\}$ over ${\mathbb{Z}}_n.$ This yields that f is a group isomorphism, $\psi$ is a bijection map and

$$\begin{array}{cc}\hfill f\left(H\right(b\left)\right)& =f\left(\langle \overline{k}\rangle \right)\hfill \\ & =\langle g^k\rangle \hfill \\ & =F\left(\psi \right(b\left)\right)\hfill \end{array}$$

for each $b\in B.$ Hence, ${\tilde{H}}_B$ is a soft isomorphic to ${\tilde{F}}_A.$ □

Theorem 22.

Every cyclic soft group over an infinite cyclic group is a soft isomorphic to a soft group over additive group $\mathbb{Z}.$

Proof. 

It is similar to the proof Theorem 21. □

Theorem 23.

Suppose that ${\tilde{F}}_A$ is soft group over G and ${\tilde{H}}_B$ is a soft group over the group $K.$ We assume that ${\mathsf{\Psi }}_{f,\psi }:SS{\left(G\right)}_A\to SS{\left(K\right)}_B,$ where $f:G\to K$ and $\psi :A\to B$ are mappings. If $(f,\psi )$ is a soft isomorphism, then ${\mathsf{\Psi }}_{f,\psi }\circ {\mathsf{\Psi }}_{f,\psi }^{-1}\left({\tilde{H}}_B\right)={\tilde{H}}_B$ and ${\mathsf{\Psi }}_{f,\psi }^{-1}\circ {\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)={\tilde{F}}_A,$ where ${\mathsf{\Psi }}_{f,\psi }^{-1}={\mathsf{\Psi }}_{f^{-1},{\psi }^{-1}}.$

Proof. 

Since $(f,\psi )$ is a soft isomorphism, f is a group isomorphism and $\psi$ is a bijection mapping. Then, by Theorem 1, ${\mathsf{\Psi }}_{f,\psi }\circ {\mathsf{\Psi }}_{f,\psi }^{-1}\left({\tilde{H}}_B\right)={\tilde{H}}_B$ and ${\mathsf{\Psi }}_{f,\psi }^{-1}\circ {\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)={\tilde{F}}_A.$ □

Definition 25.

Suppose that ${\tilde{F}}_A$ is a soft group over G. Let $f:G\to G$ and $\psi :A\to A$ be two mappings.

• A pair $(f,\psi )$ is said to be a soft endomorphism if $(f,\psi )$ is a soft homomorphism.
• A pair $(f,\psi )$ is said to be a soft automorphism if $(f,\psi )$ is a soft isomorphism.

Definition 26.

Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ Let $f:G\to G$ and $\psi :A\to A$ be two mappings. Then, a pair $(f,\psi )$ is said to be a soft inner automorphism if the pair $(f,\psi )$ is a soft automorphism and $f\in Inn\left(G\right).$ The collection of all soft inner automorphisms is symbolized by $Inn\left({\tilde{F}}_A\right).$

The next results demonstrate that the collection of all soft automorphism of any soft group over a group G forms a group and the collection of all soft inner automorphism forms a normal subgroup of it.

Theorem 24.

Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ Let

$$Aut\left({\tilde{F}}_A\right)=\{(f,\psi ):(f,\psi )\mathit{is\; a\; soft\; automorphism\; from}{\tilde{F}}_A\mathit{to\; itself}\}.$$

Then, $(Aut\left(\tilde{F}\right),\circ )$ is a group, where the operation ∘ is defined as

$$(f,\psi )\circ (f_1,{\psi }_1)=(f\circ f_1,\psi \circ {\psi }_1).$$

Proof. 

Let $(f,\psi ),(f_1,{\psi }_1)\in Aut\left({\tilde{F}}_A\right).$ We show that $(f,\psi )\circ (f_1,{\psi }_1)\in Aut\left(\tilde{F}\right).$ Since f and $f_1$ are isomorphisms of groups, $f\circ f_1$ is an isomorphism of groups. Also, since $\psi$ and ${\psi }_1$ are bijection, $\psi \circ {\psi }_1$ is a bijection mapping. We need to show that $f\circ f_1\left(F\left(a\right)\right)=F\left((\psi \circ {\psi }_1)\left(a\right)\right)$ for each $a\in A.$ Then,

$$\begin{array}{cc}\hfill f\circ f_1\left(F\left(a\right)\right)& =f\left(f_1\left(F\left(a\right)\right)\right)\hfill \\ & =f\left(F\left({\psi }_1\left(a\right)\right)\right),\sin \mathrm{ce}(f_1,{\psi }_1)\mathrm{is}\mathrm{a}\mathrm{soft}\mathrm{isomorphism}\hfill \\ & =F\left(\psi \left({\psi }_1\left(a\right)\right)\right),\sin \mathrm{ce}(f,\psi )\mathrm{is}\mathrm{a}\mathrm{soft}\mathrm{isomorphism}\hfill \\ & =F\left((\psi \circ {\psi }_1)\left(a\right)\right)\hfill \end{array}$$

for each $a\in A.$ Hence, $(f,\psi )\circ (f_1,{\psi }_1)\in Aut\left(\tilde{F}\right).$ Since the function compositions always associative, $Aut\left(\tilde{F}\right)$ has the associative property. By Remark 4, $(i{d}_G,i{d}_A)\in Aut\left({\tilde{F}}_A\right).$ It is clear that $(f,\psi )\circ (i{d}_G,i{d}_A)=(i{d}_G,i{d}_A)\circ (f,\psi )=(f,\psi ).$ Hence, $(i{d}_G,i{d}_A)$ is an identity element in $Aut\left(\tilde{F}\right).$ By Theorem 17, if $(f,\psi )\in Aut\left({\tilde{F}}_A\right),$ then $(f^{-1},{\psi }^{-1})\in Aut\left({\tilde{F}}_A\right).$ It is clear that $(f,\psi )\circ (f^{-1},{\psi }^{-1})=(f^{-1},{\psi }^{-1})\circ (f,\psi )=(i{d}_G,i{d}_A).$ Hence, $(Aut\left({\tilde{F}}_A\right),\circ )$ is a group. □

Theorem 25.

Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ Then, $Inn\left({\tilde{F}}_A\right)$ is a normal subgroup of $Aut\left({\tilde{F}}_A\right).$

Proof. 

It is clear that $(i{d}_G,i{d}_A)\in Inn\left({\tilde{F}}_A\right)$ then $Inn\left({\tilde{F}}_A\right)\ne \varnothing .$ Let $(f_x,{\psi }_1),(f_y,{\psi }_2)\in Inn\left({\tilde{F}}_A\right),$ where $x,y\in G.$ By a fact in the basic group theory, $f_x\circ f_y^{-1}\in Inn\left(G\right)$ and ${\psi }_1\circ {\psi }_2^{-1}$ is a bijection map since ${\psi }_1$ and ${\psi }_2$ are bijection mappings. It remains to show that $f_x\circ f_y^{-1}\left(F\left(a\right)\right)=F\left(({\psi }_1\circ {\psi }_2^{-1})\left(a\right)\right).$ Then,

$$\begin{array}{cc}\hfill f_x\circ f_y^{-1}\left(F\left(a\right)\right)& =f_x\left(f_y^{-1}\left(F\left(a\right)\right)\right)\hfill \\ & =f_x\left(F\left({\psi }_2^{-1}\left(a\right)\right)\right),\sin \mathrm{ce}(f_y^{-1},{\psi }_2^{-1})\mathrm{is}\mathrm{a}\mathrm{soft}\mathrm{automorphism}\hfill \\ & =F\left({\psi }_1\left({\psi }_2^{-1}\left(a\right)\right)\right),\sin \mathrm{ce}(f_x,{\psi }_1)\mathrm{is}\mathrm{a}\mathrm{soft}\mathrm{automorphism}\hfill \\ & =F\left(({\psi }_1\circ {\psi }_2^{-1})\left(a\right)\right)\hfill \end{array}$$

for each $a\in A.$ Therefore, $(f_x,{\psi }_1)\circ (f_y^{-1},{\psi }_2^{-1})\in Inn\left({\tilde{F}}_A\right).$ Hence, $Inn\left({\tilde{F}}_A\right)$ is a subgroup of $Aut\left({\tilde{F}}_A\right).$ Now, let $(f,\psi )\in Aut\left(\tilde{F}\right)$ and $(f_x,{\psi }_1)\in Inn\left(\tilde{F}\right).$ Then, $f\circ f_x\circ f^{-1}\in Inn\left(G\right)$ and $\psi \circ {\psi }_1\circ {\psi }^{-1}$ is a bijection mapping. It is clear that $f\circ f_x\circ f^{-1}\left(F\left(a\right)\right)=F\left((\psi \circ {\psi }_1\circ \psi )\left(a\right)\right)$ for all $a\in A.$ Hence, $Inn\left({\tilde{F}}_A\right)$ is a normal subgroup of $Aut\left({\tilde{F}}_A\right).$ □

Theorem 26.

Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ If $(f,\psi )\in Aut\left({\tilde{F}}_A\right),$ where $f\left(g\right)=g^{-1}$ for each $g\in G,$ then ${\tilde{F}}_A$ is an abelian soft group over G.

Proof. 

We assume that $(f,\psi )\in Aut\left({\tilde{F}}_A\right),$ where $f\left(g\right)=g^{-1}$ for each $g\in G.$ Then, $f\in Aut\left(G\right).$ By a fact in the basic group theory, G is an abelian group. Hence, ${\tilde{F}}_A$ is an abelian soft group over $G.$ □

Here is an example to point out that the reverse of the above theorem does not hold in general.

Example 6.

Consider $G=S_3$ and ${\tilde{F}}_A=\{(a_1,\langle \left(123\right)\rangle ),(a_2,\langle \left(12\right)\rangle ),(a_3,\langle \left(13\right)\rangle )\}$ is an abelian soft group over G. Let $f\in Aut\left(G\right)$ such that $f\left(g\right)=g^{-1}$ for all $g\in G$. We define the map $\psi :A\to A$ as

$$\psi \left(a_1\right)=a_1,\psi \left(a_2\right)=a_{3}and\psi \left(a_3\right)=a_2,$$

where $A=\{a_1,a_2,a_3\}.$ Then, $f\left(F\left(a_2\right)\right)=F\left(a_2\right)$ and $F\left(a_2\right)\ne F\left(\psi \left(a_2\right)\right)$. Hence, $(f,\psi )\notin Aut\left({\tilde{F}}_A\right)$.

Theorem 27.

Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ Then, ${\tilde{F}}_A$ is a normalistic soft group over G if and only if $(f_g,i{d}_A)\in Inn\left({\tilde{F}}_A\right)$ for each $g\in G.$

Proof. 

Necessity. Suppose that ${\tilde{F}}_A$ is a normalistic soft group over $G.$ For each $g\in G$, we have $f_g\in Inn\left(G\right)$ and $i{d}_A$ is a bijection map. Since ${\tilde{F}}_A$ is normalistic soft group over G, then $f_g\left(F\left(a\right)\right)=F\left(a\right)=F\left(\psi \left(a\right)\right)$ for each $g\in G$ and for each $a\in A$. Hence, $(f_g,i{d}_A)\in Inn\left({\tilde{F}}_A\right)$ for all $g\in G.$

Sufficiency. It is clear. □

Theorem 28.

Suppose that ${\tilde{F}}_A$ is soft group over G and ${\tilde{H}}_B$ is a soft group over the group $K.$ Let ${\mathsf{\Psi }}_{f,\psi }:SS{\left(G\right)}_A\to SS{\left(K\right)}_B,$ where $f:G\to K$ and $\psi :A\to B$ are mappings. If $(f,\psi )$ is a soft isomorphism then $(f\left({\tilde{F}}_A\right),\psi \left(A\right))={\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right),$ where $(f\left({\tilde{F}}_A\right),\psi \left(A\right))=\left\{\right(\psi \left(a\right),f\left(F\left(a\right)\right):$ for each $a\in A\}.$

Proof. 

By the definition of ${\mathsf{\Psi }}_{f,\psi },$ ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)\left(b\right)=f\left(F\left({\psi }^{-1}\left(b\right)\right)\right)$ for each $b\in B.$ Then, ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)=\left\{\right(b,f\left(F\left({\psi }^{-1}\left(b\right)\right)\right):$ for each $b\in B\}.$ Since $\psi$ is a bijection map, we take ${\psi }^{-1}\left(b\right)=a.$ Therefore, ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)=\{(\psi \left(a\right),f\left(F\left(a\right)\right)):$ for each $a\in A\}.$ This proves the theorem. □

Now, we study a new type of soft group which is called soft point groups.

Definition 27.

Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ Then, ${\tilde{F}}_A$ is said to be a soft point group over $G,$ symbolized by ${\tilde{F}}_a,$ if $F\left(a\right)=H⩽G$ and $F\left(b\right)=\left\{e_G\right\}$ for each $b\in A-\left\{a\right\}.$

Example 7.

Consider $G=S_3$ and $A=\left\{\right(e),(12\left)\right\}$. We define $F:A\to P\left(G\right)$ as

$$F\left(x\right)=\{y\in S_3:xRy\iff y\in \langle x\rangle \},$$

where R is a relation that connects an element from the set A to an element of $G.$ Then,

$$F\left(e_G\right)=\left\{e_G\right\}andF\left(\right(12\left)\right)=\langle \left(12\right)\rangle .$$

Thus, ${\tilde{F}}_{\left(12\right)}$ is a soft point group over $G.$

From now, ${\tilde{F}}_a$ and ${\tilde{H}}_b$ are referred to as soft point groups with respect to the set of parameters A and B, respectively.

Theorem 29.

Suppose that ${\tilde{F}}_a$ and ${\tilde{H}}_b$ are two soft point groups over $G.$ If $A\cap B\ne \varnothing$, then the restricted intersection of ${\tilde{F}}_a$ and ${\tilde{H}}_b$ is either a soft point group over G or equal to the identity soft group.

Proof. 

Since ${\tilde{F}}_a$ and ${\tilde{H}}_b$ are soft point groups over G, then

$$\begin{array}{c}\hfill {\tilde{F}}_a\tilde{\cap }{\tilde{H}}_b=\left\{\begin{array}{cc}{\tilde{e}}_G\hfill & \mathrm{if}a\ne b,\hfill \\ {\tilde{L}}_a\hfill & \mathrm{if}a=b,\hfill \end{array}\right.\end{array}$$

where $L\left(a\right)=F\left(a\right)\cap H\left(a\right).$ Therefore, if $a\ne b$, then ${\tilde{F}}_a$ $\tilde{\cap }$ ${\tilde{H}}_b={\tilde{e}}_G$ and if $a=b$, then ${\tilde{F}}_a$ $\tilde{\cap }$ ${\tilde{H}}_b$ is a soft point group over G since the intersection of $F\left(a\right)$ and $H\left(b\right)$ is a subgroup of G. □

Theorem 30.

Suppose that ${\tilde{F}}_a$ and ${\tilde{H}}_b$ are two soft point groups over $G.$ If $a\ne b$, then the union (extended union) of ${\tilde{F}}_a$ and ${\tilde{H}}_b$ is a soft group over $G.$

Proof. 

Suppose that $a\ne b.$ We assume that ${\tilde{F}}_a$ $\tilde{\cup }$ ${\tilde{H}}_b=\tilde{L}$. Then,

$$\begin{array}{c}\hfill L\left(x\right)=\left\{\begin{array}{cc}\left\{e_G\right\}\hfill & \mathrm{if}x\ne a\ne b,\hfill \\ F\left(x\right)\hfill & \mathrm{if}x=a,\hfill \\ H\left(x\right)\hfill & \mathrm{if}x=b.\hfill \end{array}\right.\end{array}$$

Therefore, ${\tilde{F}}_a$ $\tilde{\cup }$ ${\tilde{H}}_b$ is a soft group over G since $F\left(a\right)$ and $H\left(b\right)$ are subgroups of $G.$ □

The next example demonstrates that if $a=b$, then the union (extended union) of two soft point groups ${\tilde{F}}_a$ and ${\tilde{H}}_a$ over G may not be a soft group.

Example 8.

Consider $G=S_3$. Let ${\tilde{F}}_a=\{(a,\langle \left(12\right)\rangle ),(d,\left\{e_G\right\})\}$ and ${\tilde{H}}_a=\{(a,\langle \left(13\right)\rangle ),(c,\left\{e_G\right\})\}$ be two soft point groups over $G.$ It is clear that ${\tilde{F}}_a$ $\tilde{\cup }$ ${\tilde{H}}_b$ is not a soft group over G since $F\left(a\right)\cup H\left(a\right)$ is not a subgroup of $G.$

Remark 5.

From the above example, we note that if $a=b,$ then the union (extended union) of two soft point groups ${\tilde{F}}_a$ and ${\tilde{H}}_a$ over G is a soft point group over G if and only if $F\left(a\right)\subseteq H\left(a\right)$ or $H\left(a\right)\subseteq F\left(a\right).$

Theorem 31.

Suppose that ${\tilde{F}}_a$ is a soft point group over G. We assume that f is a monomorphism from G to the group $K.$ Then, $(f\left({\tilde{F}}_a\right),\psi \left(A\right))$ is a soft point group over $K,$ where ψ is any mapping from A to any set of parameters.

Proof. 

Since f is a monomorphism from G to K, $f\left(\left\{e_G\right\}\right)=\left\{e_K\right\}$ and $f\left(F\right(a\left)\right)$ is a subgroup of $K.$ Thus, $(f\left({\tilde{F}}_a\right),\psi \left(A\right))$ is a soft point group over the group K. □

Theorem 32.

Suppose that ${\tilde{F}}_a$ is a soft point group over $G.$

1. 

If $H\left(a\right)\le F\left(a\right)$, then ${\tilde{H}}_a$ $\tilde{\le }$ ${\tilde{F}}_a$ and if $H\left(a\right)⊴F\left(a\right),$ then ${\tilde{H}}_a$ $\widetilde{⊴}$ ${\tilde{F}}_a.$

2. 

If $F\left(a\right)$ is a normal subgroup of $G,$ then ${\tilde{F}}_a$ is a normalistic soft group over $G.$

3. 

If $F\left(a\right)$ is an abelian subgroup of $G,$ then ${\tilde{F}}_a$ is an abelian soft group over $G.$

4. 

If $F\left(a\right)$ is a cyclic subgroup of $G,$ then ${\tilde{F}}_a$ is a cyclic soft group over $G.$

5. 

$|{\tilde{F}}_a|=|F\left(a\right)|.$

6. 

$[{\tilde{F}}_a,{\tilde{F}}_a]$ is a soft point group over $G.$ If $F\left(a\right)$ is an abelian subgroup of G, then $[{\tilde{F}}_a,{\tilde{F}}_a]={\tilde{e}}_G.$

7. 

$Z\left({\tilde{F}}_a\right)$ is either a soft point group over G or equal to the identity soft group.

Proof. 

It is straightforward. □

Theorem 33.

Suppose that ${\tilde{F}}_a$ is a soft point group over G and ${\tilde{H}}_b$ is a soft point group over the group $K.$ If $(f,\psi )$ is a soft isomorphism from ${\tilde{F}}_a$ to ${\tilde{H}}_b,$ then $\psi \left(a\right)=b$.

Proof. 

Suppose that $(f,\psi )$ is a soft isomorphism from ${\tilde{F}}_a$ to ${\tilde{H}}_b.$ Then, f is an isomorphism from G to $K.$ This implies that

$$f\left(F\right(a\left)\right)=N\le K\mathrm{and}f\left(F\left(a_1\right)\right)=\left\{e_K\right\}\mathrm{for\; all}{a}_1\ne a.$$

Therefore, since $(f,\psi )$ is a soft isomorphism from ${\tilde{F}}_a$ to ${\tilde{H}}_b,$ $f\left(F\right(a\left)\right)=H\left(\psi \right(a\left)\right)$ and $f\left(F\left(a_1\right)\right)=H\left(\psi \left(a_1\right)\right)$ for all $a_1\ne a.$ Hence, if $H\left(b\right)\ne \left\{e_K\right\}$, then we must have $\psi \left(a\right)=b$. □

Theorem 34.

Suppose that ${\tilde{F}}_a$ is a soft point group over G and ${\tilde{H}}_b$ is a soft point group over the group $K.$ Then, $(f,\psi )$ is a soft isomorphism maps ${\tilde{F}}_a$ to ${\tilde{H}}_b$ if and only if the following conditions are met:

• f is a group isomorphism from G to K;
• ψ is a bijection mapping from A to B;
• $\psi \left(a\right)=b$ and $f\left(F\right(a\left)\right)=H\left(b\right).$

Proof. 

It is straightforward. □

Theorem 35.

Suppose that ${\tilde{F}}_a$ is a soft point group over $G.$ If ψ is a bijection mapping from A to itself such that $\psi \left(a\right)=a,$ then $(f_g,\psi )\in Inn\left({\tilde{F}}_a\right)$ for all $g\in F\left(a\right).$

Proof. 

We assume that ${\tilde{F}}_a$ is a soft point group over $G.$ Let $f_g\in Inn\left(G\right),$ $g\in F\left(a\right).$ Then, $f_g\left(F\left(a\right)\right)=F\left(a\right)$ for each $g\in F\left(a\right).$ Since $\psi \left(a\right)=a$, $f_g\left(F\left(a\right)\right)=F\left(\psi \left(a\right)\right)$. Hence, $(f_g,\psi )\in Inn\left({\tilde{F}}_a\right)$ for each $g\in F\left(a\right).$ □

Theorem 36.

Suppose that ${\tilde{F}}_a$ is a soft point group over $G.$ Let ψ be a bijection mapping from A to itself such that $\psi \left(a\right)=a$. Then, $F\left(a\right)$ is a normal subgroup of G if and only if $(f_g,\psi )\in Inn\left({\tilde{F}}_a\right)$ for each $g\in G.$

Proof. 

Necessity. We assume that $F\left(a\right)$ is a normal subgroup of $G.$ Then, $f_g\left(F\left(a\right)\right)=F\left(a\right)$ for each $g\in G,$ where $f_g\in Inn\left(G\right).$ Thus, $f_g\left(F\left(a\right)\right)=F\left(\psi \left(a\right)\right)$ since $\psi \left(a\right)=a$. Therefore, $(f_g,\psi )\in Inn\left({\tilde{F}}_a\right)$ for each $g\in G.$

Sufficiency. Let $(f_g,\psi )\in Inn\left({\tilde{F}}_a\right)$ for each $g\in G.$ Then, $f_g\left(F\left(a\right)\right)=gF\left(a\right)g^{-1}=F\left(a\right)=F\left(\psi \left(a\right)\right)$ for each $g\in G.$ Hence, $F\left(a\right)$ is a normal subgroup of $G.$ □

4. Characteristic Soft Subgroup

This section introduces the concept of characteristic soft subgroups of a given soft group over the group G. Also, we study the features of these soft subgroups. The notations are the same as in Section 2.2.

Definition 28.

Suppose that ${\tilde{F}}_A$ is a soft group over G and ${\tilde{H}}_A$ $\widetilde{⩽}$ ${\tilde{F}}_A.$ Then, ${\tilde{H}}_A$ is said to be a characteristic soft subgroup of ${\tilde{F}}_A,$ symbolized by ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A,$ if for any soft automorphism $(f,\psi )\in Aut\left({\tilde{F}}_A\right),$ we have $(f\left({\tilde{H}}_A\right),\psi \left(A\right))={\tilde{H}}_A$.

Remark 6.

By Theorem 28, we have ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{H}}_A\right)=(f\left({\tilde{H}}_A\right),\psi \left(A\right)).$ Then, ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A$ if ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{H}}_A\right)={\tilde{H}}_A.$

Example 9.

1. 

${\tilde{e}}_G$ $cha{r}_s$ ${\tilde{F}}_A,$ for each soft group ${\tilde{F}}_A$ over $G.$

2. 

${\tilde{F}}_A$ $cha{r}_s$ ${\tilde{F}}_A,$ for each soft group ${\tilde{F}}_A$ over $G.$

3. 

$Z\left({\tilde{G}}_A\right)$ $cha{r}_s$ ${\tilde{G}}_A$.

Theorem 37.

Suppose that ${\tilde{F}}_A$ is a soft group over G and ${\tilde{H}}_A$ $\widetilde{⩽}$ ${\tilde{F}}_A.$ If ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{H}}_A\right)$ $\tilde{\subseteq }$ ${\tilde{H}}_A$ for every soft automorphism $(f,\psi )\in Aut\left({\tilde{F}}_A\right),$ then ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A.$

Proof. 

Let $(f,\psi )\in Aut\left({\tilde{F}}_A\right).$ By Theorem 17, $(f^{-1},{\psi }^{-1})\in Aut\left({\tilde{F}}_A\right)$. Therefore, ${\mathsf{\Psi }}_{f,\psi }^{-1}\left({\tilde{H}}_A\right)$ $\tilde{\subseteq }$ ${\tilde{H}}_A$ by the assumption. By applying ${\mathsf{\Psi }}_{f,\psi }$, we have

$${\mathsf{\Psi }}_{f,\psi }\circ {\mathsf{\Psi }}_{f,\psi }^{-1}\left({\tilde{H}}_A\right)\tilde{\subseteq }{\mathsf{\Psi }}_{f,\psi }\left({\tilde{H}}_A\right).$$

By Theorem 23, we obtain ${\tilde{H}}_A$ $\tilde{\subseteq }$ ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{H}}_A\right).$ Then, we obtain ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{H}}_A\right)={\tilde{H}}_A.$ Hence, ${\tilde{H}}_A$ is a characteristic soft subgroup of ${\tilde{F}}_A.$ □

Theorem 38.

Suppose that ${\tilde{F}}_A$ is a soft group over G and ${\tilde{H}}_A$ $\widetilde{⩽}$ ${\tilde{F}}_A.$ Then, ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A$ if and only if for each $(f,\psi )\in Aut\left({\tilde{F}}_A\right),$ $f\left(H\right(a\left)\right)=H\left(\psi \right(a\left)\right)$ for each $a\in A.$

Proof. 

Necessity. We assume that ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A.$ Then, $(f\left({\tilde{H}}_A\right),\psi \left(A\right))={\tilde{H}}_A$ for each $(f,\psi )\in Aut\left({\tilde{F}}_A\right).$ Therefore, for each $a\in A,$ we have $\left(f\right(H\left(a\right)),\psi (a\left)\right)=\left(H\right(\psi \left(a\right)),\psi (a\left)\right).$ Hence, $f\left(H\right(a\left)\right)=H\left(\psi \right(a\left)\right)$ for each $a\in A.$

Sufficiency. We assume that $f\left(H\right(a\left)\right)=H\left(\psi \right(a\left)\right)$ for each $a\in A$ and for each $(f,\psi )\in Aut\left({\tilde{F}}_A\right).$ Therefore,

$$\begin{array}{ccc}\hfill (f\left({\tilde{H}}_A\right),\psi \left(A\right))& =& \left\{\right(f\left(H\right(a\left)\right),\psi \left(a\right)):\mathrm{for}\mathrm{all}a\in A\}\hfill \\ & =& \left\{\right(H\left(\psi \right(a\left)\right),\psi \left(a\right)):\mathrm{for}\mathrm{all}a\in A\},\sin \mathrm{ce}f\left(H\right(a\left)\right)=H\left(\psi \right(a\left)\right)\mathrm{for}\mathrm{all}a\in A\hfill \\ & =& {\tilde{H}}_A.\hfill \end{array}$$

Thus, ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A.$ □

Theorem 39.

Suppose that ${\tilde{F}}_A$ is a normalistic soft group over G and ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A.$ Then, ${\tilde{H}}_A$ is a normalistic soft group over G and ${\tilde{H}}_A$ $\widetilde{⊴}$ ${\tilde{F}}_A.$

Proof. 

We assume that ${\tilde{F}}_A$ is a normalistic soft group over $G.$ By Theorem 27, $(f_g,i{d}_A)\in Inn\left({\tilde{F}}_A\right)$ for each $g\in G$. By Theorem 38, for each $a\in A,$ $f_g\left(H\left(a\right)\right)=H\left(i{d}_A\left(a\right)\right)$ for each $g\in G.$ Therefore, we have for each $a\in A,$ $gH\left(a\right)g^{-1}=H\left(a\right)$ for each $g\in G.$ Thus, ${\tilde{H}}_A$ is a normalistic soft group over $G.$ Since $H\left(a\right)\le F\left(a\right)$ for each $a\in A,$ ${\tilde{H}}_A$ $\widetilde{⊴}$ ${\tilde{F}}_A.$ □

If ${\tilde{F}}_A$ is a normalistic soft group over G and ${\tilde{H}}_A$ $\widetilde{⊴}$ ${\tilde{F}}_A,$ then ${\tilde{H}}_A$ is not necessarily a characteristic soft subgroup of ${\tilde{F}}_A$ as the next example shows.

Example 10.

Consider $G=\mathbb{Q}$ is an additive group of rational numbers. Then, ${\tilde{F}}_A=\{(a,\mathbb{Q}),(b,\mathbb{Q})\}$ is a normalistic soft group over G and ${\tilde{H}}_A=\{(a,\mathbb{Z}),(b,\mathbb{Z})\}$ is a soft normal subgroup of ${\tilde{F}}_A.$ We have $(f,i{d}_A)\in Aut\left({\tilde{F}}_A\right),$ where $f:\mathbb{Q}\to \mathbb{Q}$ is defined as $f\left(q\right)={\displaystyle \frac{q}{2}}$ for all $q\in \mathbb{Q}.$ It is clear that $(f\left({\tilde{H}}_A\right),i{d}_A)\ne {\tilde{H}}_A$. Thus, ${\tilde{H}}_A$ is not a characteristic soft subgroup of ${\tilde{F}}_A.$

Theorem 40.

Suppose that ${\tilde{F}}_A$ is a soft group over G and ${\tilde{F}}_A$ $cha{r}_s$ ${\tilde{G}}_A.$ If ${\tilde{H}}_A$ is a unique soft subgroup of ${\tilde{F}}_A,$ then ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A.$

Proof. 

Let $(f,\psi )\in Aut\left({\tilde{G}}_A\right).$ Then, $(f\left({\tilde{F}}_A\right),\psi \left(A\right))={\tilde{F}}_A={\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)$ since ${\tilde{F}}_A$ $cha{r}_s$ ${\tilde{G}}_A.$ Therefore, by Theorem 38, $f\left(F\right(a\left)\right)=F\left(\psi \right(a\left)\right)$ for each $a\in A.$ So, $(f,\psi )\in Aut\left({\tilde{F}}_A\right).$ By Theorem 2, ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{H}}_A\right)$ $\tilde{\le }$ ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{F}}_A\right)={\tilde{F}}_A$. Then, ${\mathsf{\Psi }}_{f,\psi }\left({\tilde{H}}_A\right)={\tilde{H}}_A$ since ${\tilde{H}}_A$ is a unique soft subgroup of ${\tilde{F}}_A$ and $|{\tilde{H}}_A|=|{\mathsf{\Psi }}_{f,\psi }\left({\tilde{H}}_A\right)|.$ Hence, ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A.$ □

Theorem 41.

Suppose that ${\tilde{F}}_A,{\tilde{H}}_A$ and ${\tilde{K}}_A$ are soft groups over G. If ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{K}}_A$ and ${\tilde{K}}_A$ $cha{r}_s$ ${\tilde{F}}_A,$ then ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A.$

Proof. 

Since ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{K}}_A$, $(f\left({\tilde{H}}_A\right),\psi \left(A\right))={\tilde{H}}_A$ for each $(f,\psi )\in Aut\left({\tilde{K}}_A\right).$ Similarly, $(f_1\left({\tilde{K}}_A\right),{\psi }_1\left(A\right))={\tilde{K}}_A$ for each $(f_1,{\psi }_1)\in Aut\left({\tilde{F}}_A\right)$ since ${\tilde{K}}_A$ $cha{r}_s$ ${\tilde{F}}_A.$ By Theorem 38, $f_1\left(K\left(a\right)\right)=K\left({\psi }_1\left(a\right)\right)$ for each $a\in A.$ Therefore, $(f_1,{\psi }_1)\in Aut\left({\tilde{K}}_A\right)$ and $(f_1\left({\tilde{H}}_A\right),{\psi }_1\left(A\right))={\tilde{H}}_A.$ Hence, ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A.$ □

Theorem 42.

Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ If ${\tilde{H}}_A$ $cha{r}_s$ ${\tilde{F}}_A$ and ${\tilde{K}}_A$ $cha{r}_s$ ${\tilde{F}}_A,$ then $[{\tilde{H}}_A,{\tilde{K}}_A]$ $cha{r}_s$ ${\tilde{F}}_A.$

Proof. 

Let ${\tilde{L}}_A=[{\tilde{H}}_A,{\tilde{K}}_A]=\{(a,[H\left(a\right),K\left(a\right)]):$ for each $a\in A\}.$ Let $(f,\psi )\in Aut\left({\tilde{F}}_A\right).$ Then,

$$\begin{array}{ccc}\hfill f\left(L\right(a\left)\right)& =& f\left(\right[H\left(a\right),K\left(a\right)\left]\right)\hfill \\ & =& \left[f\right(H\left(a\right)),f(K\left(a\right)\left)\right],\mathrm{by}\mathrm{a}\mathrm{fact}\mathrm{in}\mathrm{basic}\mathrm{group}\mathrm{theory}\hfill \\ & =& \left[H\right(\psi \left(a\right)),K(\psi \left(a\right)\left)\right],\sin \mathrm{ce}{\tilde{H}}_{A}cha{r}_s{\tilde{F}}_A\mathrm{and}{\tilde{K}}_{A}cha{r}_s{\tilde{F}}_A\hfill \\ & =& L\left(\psi \right(a\left)\right)\hfill \end{array}$$

for each $a\in A.$ Hence, by Theorem 38, ${\tilde{L}}_A$ $cha{r}_s$ ${\tilde{F}}_A.$ □

Corollary 2.

Suppose that ${\tilde{F}}_A$ is a soft group over $G.$ Then, $[{\tilde{F}}_A,{\tilde{F}}_A]$ $cha{r}_s$ ${\tilde{F}}_A.$

Theorem 43.

Suppose that ${\tilde{F}}_A$ is a soft group over G. Then, for any collection of characteristic soft subgroups of ${\tilde{F}}_A$, their restricted intersection is a characteristic soft subgroup of ${\tilde{F}}_A.$

Proof. 

Suppose that $\mathsf{\Delta }$ is a collection of characteristic soft subgroups of ${\tilde{F}}_A.$ Let $\tilde{N}=\underset{{\tilde{H}}_A\in \mathsf{\Delta }}{\tilde{\cap }}$ ${\tilde{H}}_A.$ Let $(f,\psi )\in Aut\left({\tilde{F}}_A\right).$ Then,

$$\begin{array}{ccc}\hfill {\mathsf{\Psi }}_{f,\psi }\left({\tilde{N}}_A\right)& \tilde{\subseteq }& {\mathsf{\Psi }}_{f,\psi }\left({\tilde{H}}_A\right),\mathrm{for}\mathrm{each}{\tilde{H}}_A\in \mathsf{\Delta }\hfill \\ & =& {\tilde{H}}_A,\mathrm{for}\mathrm{each}{\tilde{H}}_A\in \mathsf{\Delta }\sin \mathrm{ce}{\tilde{H}}_{A}cha{r}_s{\tilde{F}}_A.\hfill \end{array}$$

Therefore, ${\mathsf{\Psi }}_{f,\psi }\left(\tilde{N}\right)$ $\tilde{\subseteq }$ $\underset{{\tilde{H}}_A\in \mathsf{\Delta }}{\tilde{\cap }}$ ${\tilde{H}}_A.$ Thus, ${\mathsf{\Psi }}_{f,\psi }\left(\tilde{N}\right)$ $\tilde{\subseteq }$ $\tilde{N}.$ By Theorem 37, $\tilde{N}$ $cha{r}_s$ ${\tilde{F}}_A.$ □

It can be seen in the next results the properties of the characteristic soft subgroups of a given soft point group over G. We note that these properties may not hold in the soft group.

Theorem 44.

Suppose that ${\tilde{F}}_a$ is a soft point group over G and ${\tilde{H}}_a$ $cha{r}_s$ ${\tilde{F}}_a.$ Then, ${\tilde{H}}_a$ is a soft normal subgroup of ${\tilde{F}}_a.$

Proof. 

Since ${\tilde{H}}_a$ $cha{r}_s$ ${\tilde{F}}_a,$ $(f\left({\tilde{H}}_a\right),\psi \left(A\right))={\tilde{H}}_a$ for each $(f,\psi )\in Aut\left({\tilde{F}}_a\right).$ By Theorem 33, we have $\psi \left(a\right)=a.$ From Theorem 35, we have $(f_g,\psi )\in Inn\left({\tilde{F}}_a\right)\le Aut\left({\tilde{F}}_a\right)$ for each $g\in F\left(a\right).$ Then, $(f_g\left({\tilde{H}}_a\right),\psi \left(A\right))={\tilde{H}}_a$ for each $g\in F\left(a\right).$ Therefore, $f_g\left(H\left(a\right)\right)=gH\left(a\right)g^{-1}=H\left(\psi \left(a\right)\right)$ for all $g\in F\left(a\right).$ Thus, $H\left(a\right)⊴F\left(a\right).$ Hence, by Theorem 32, ${\tilde{H}}_a$ is a soft normal subgroup of ${\tilde{F}}_a.$ □

The reverse of the above theorem is generally not true, as illustrated in the next example.

Example 11.

Consider $G=\mathbb{Q}$ is an additive group of rational numbers. Then, ${\tilde{F}}_A=\{(a,\mathbb{Q}),(b,\left\{e_G\right\})\}$ is a soft point group over $G.$ Then, ${\tilde{H}}_A=\{(a,\mathbb{Z}),(b,\left\{e_G\right\})\}$ is a soft normal subgroup of ${\tilde{F}}_A$ but not a characteristic soft subgroup of ${\tilde{F}}_a$ for the same reason as in Example 10.

Theorem 45.

Assuming that ${\tilde{F}}_A$ is a soft group over $G.$ Then, $Z\left({\tilde{F}}_a\right)$ $cha{r}_s$ ${\tilde{F}}_a.$

Proof. 

Let ${\tilde{H}}_a=Z\left({\tilde{F}}_a\right)=\{(a,Z\left(F\left(a\right)\right)):$ for each $a\in A\}.$ Let $(f,\psi )\in Aut\left({\tilde{F}}_a\right)$. By Theorem 33, $\psi \left(a\right)=a.$ Therefore, $f\left(F\right(a\left)\right)=F\left(a\right).$ Thus, $f\left(H\right(a\left)\right)=H\left(a\right)$ since $H\left(a\right)=Z\left(F\right(a\left)\right)$ and $Z\left(F\right(a\left)\right)$ is a characteristic subgroup of $F\left(a\right).$ Hence, $f\left(H\right(a\left)\right)=H\left(\psi \right(a\left)\right)$ for each $a\in A.$ By Theorem 38, ${\tilde{H}}_a$ $cha{r}_s$ ${\tilde{F}}_a.$ □

Theorem 46.

Suppose that ${\tilde{F}}_a$ is a soft point group over $G.$ If ${\tilde{H}}_a$ is a cyclic soft subgroup of ${\tilde{F}}_a,$ then ${\tilde{H}}_a$ $cha{r}_s$ ${\tilde{F}}_a.$

Proof. 

It is similar to the proof of Theorem 45. □

Theorem 47.

Suppose that ${\tilde{F}}_a$ is a soft point group over $G.$ If ${\tilde{H}}_a$ is a unique soft subgroup of ${\tilde{F}}_a,$ then ${\tilde{H}}_a$ $cha{r}_s$ ${\tilde{F}}_a.$

Proof. 

We assume that ${\tilde{H}}_a$ is a unique soft subgroup of ${\tilde{F}}_a$; then, $H\left(a\right)$ is a unique subgroup of $F\left(a\right)$. Let $(f,\psi )\in Aut\left({\tilde{F}}_a\right)$. By Theorem 33, $\psi \left(a\right)=a.$ Therefore, $f\left(F\right(a\left)\right)=F\left(a\right)$. By a fact in basic group theory, $f\left(H\right(a\left)\right)=H\left(a\right)$ since $H\left(a\right)$ is a unique subgroup of $F\left(a\right)$. Hence, $f\left(H\right(a\left)\right)=H\left(\psi \right(a\left)\right)$ for each $a\in A.$ By Theorem 38, ${\tilde{H}}_a$ $cha{r}_s$ ${\tilde{F}}_a.$ □

Theorem 48.

Suppose that ${\tilde{F}}_a$ is a soft point group over $G.$ Let ψ be any bijection mapping from A to itself such that $\psi \left(a\right)=a.$ If ${\tilde{H}}_a$ is a soft normal subgroup of ${\tilde{F}}_a$ and ${\tilde{K}}_a$ $cha{r}_s$ ${\tilde{H}}_a,$ then ${\tilde{K}}_a$ is a soft normal subgroup of ${\tilde{F}}_a$

Proof. 

Let $g\in F\left(a\right).$ By Theorem 35, $(f_g,\psi )\in Aut\left({\tilde{F}}_a\right).$ Since ${\tilde{H}}_a$ $\widetilde{⊴}$ ${\tilde{F}}_a$, $gH\left(a\right)g^{-1}=H\left(a\right)=H\left(\psi \left(a\right)\right).$ Therefore, $(f_g,\psi )\in Aut\left({\tilde{H}}_a\right).$ Therefore, $(f_g\left({\tilde{K}}_a\right),\psi \left(A\right))={\tilde{K}}_a$ since ${\tilde{K}}_a$ $cha{r}_s$ ${\tilde{H}}_a.$ Then, $f_g\left(K\left(a\right)\right)=gK\left(a\right)g^{-1}=K\left(\psi \left(a\right)\right)=K\left(a\right).$ Since g is an arbitrary element of $F\left(a\right)$ and $gK\left(a\right)g^{-1}=K\left(a\right),$ $K\left(a\right)⊴F\left(a\right).$ By Theorem 32, ${\tilde{K}}_a$ $\widetilde{⊴}$ ${\tilde{F}}_a.$ □

5. Conclusions

In this paper, we have studied new concepts in the soft group theory with their results such as the center of the soft group, the kernel of soft homomorphism and soft automorphism. The soft point group as a type of soft group has been introduced with its basic properties. We have extended the concept of characteristic subgroups to include soft groups. We have studied the equivalent definitions to the definition of characteristic soft subgroups. We have studied the properties of characteristic soft subgroups. In addition, we have investigated the characteristic soft subgroups of a soft point group. In future work, more concepts and results related to characteristic subgroups in group theory could be analogously presented in the soft group theory. In addition, we plan to study many applications of characteristic soft subgroups in actions of soft groups, soft representation of soft groups and soft topological groups. Also, we plan to build more theory and to investigate some crucial relationships with advanced group theory in this setting.