## 1 Introduction

The *commutator* of two elements *x* and *y* of a group *G* is defined usually as [*x*,*y*]: = *x*^{−1}*y*^{−1}*xy*. The influence of commutators in the theory of groups is inevitable and the analogy of computations encouraged some authors to define and study modifications of the ordinary commutators to include automorphisms or more generally endormorphisms of the underlying groups. The first of those is due to Ree [1] who generalizes the conjugation of *x* by *y* with respect to an endomorphism *θ* of *G* as *y*^{−1}*xθ*(*y*) and uses it to make relationships between the corresponding conjugacy classes with special ordinary conjugacy classes and irreducible characters of the group. Later, Acher [2] invokes a very similar generalization of conjugation as to that of Ree and studies the corresponding generalized conjugacy classes, centralizers and the center of groups in a more abstract way. Writing the commutators as [*x*,*y*] = *x*^{−1}*I _{y}*(

*x*),

*I*being the inner automorphism associate to

_{y}*y*, one may generalize them in a natural way to [

*x*,

*θ*] =

*x*

^{−1}

*θ*(

*x*), in which

*θ*is an endomorphism of the underlying group. The element [

*x*,

*θ*], called the

*autocommutator*of the element

*x*and automorphism

*θ*when

*θ*is an automorphism, seems to appear first in Gorenstein’s book [3, p. 33] while it first appears in practice in the pioneering papers [4, 5] of Hegarty.

According to Ree’s definition of conjugation, the commutator of two elements *x* and *y* of a group *G* with respect to an endomorphism *θ* will be [*x*,*y*]_{θ}: = *x*^{−1}*y*^{−1}*xθ*(*y*). One observes that [*x*,*y*]_{θ} = 1 if and only if *θ*(*y*) = *y ^{x}*. Hence [

*x*,

*y*]

_{θ}= 1 does not guarantee in general that [

*y*,

*x*]

_{θ}= 1. The aim of this paper is to introduce a new generalization of commutators, as a minor modification to that of Ree, in order to obtain a new commutator behaving more like the ordinary commutators. Indeed, we define the conjugation of

*x*by

*y*via

*θ*as

*θ*(

*y*)

^{−1}

*θ*(

*x*)

*y*, which is simply the image of

*y*

^{−1}

*xθ*

^{−1}(

*y*), the conjugate of

*x*by

*y*via

*θ*

^{−1}in the sense of Ree’s, under

*θ*. Hence the corresponding commutators, we call them the

*θ*-

*commutators*, will be [

*x*,

_{θ}

*y*]: =

*x*

^{−1}

*θ*(

*y*)

^{−1}

*θ*(

*x*)

*y*and we observe that [

*x*,

_{θ}

*y*] = 1 if and only if [

*y*,

_{θ}

*x*] = 1. This property of

*θ*-commutators, as we will see later, remains unchanged modulo a shift of elements by left multiplication corresponding to automorphisms which are congruent modulo the group of inner automorphisms. Therefore, all inner automorphisms give rise to same

*θ*-commutators modulo a shift of elements by left multiplication.

The paper is organized as follows: Section 2 initiates the study of *θ*-commutators by generalizing the ordinary commutator identities as well as centralizers and the center of a group, and determines the structure of *θ*-centralizers and *θ*-center of the groups under investigation. In section 3, we shall define the *θ*-non-commuting graph associated to *θ*-commutators of a group and describe some of its basic properties and its correlations with other notions, namely fixed-point-free and class preserving automorphisms. Sections 4 and 5 are devoted to the study of independent subsets of *θ*-non-commuting graphs where we give an explicit structural theorem for them and apply them to see under which conditions the *θ*-non-commuting graphs are union of particular independent sets.

Throughout this paper, we use the following notations: given a graph *Γ*, the set of its vertices and edges are denoted by *V*(*Γ*) and *E*(*Γ*), respectively. For every vertex *v* ∈ *V*(*Γ*), the neighbor of *v* in *Γ* is denoted by *N _{Γ}*(

*v*) and the degree of

*v*is given by deg

_{Γ}(

*v*). For convenience, we usually drop the index

*Γ*and write

*N*(

*v*) and deg(

*v*) for the neighbor and degree of the vertex

*v*, respectively. A subset of

*V*(

*Γ*) with no edges among its vertices is an

*independent set*. The maximum size of an independent set in

*Γ*is denoted by

*α*(

*Γ*) and called the

*independence number*of

*Γ*. Also, the minimum number of independent sets required to cover all vertices of

*Γ*is the

*chromatic number*of

*Γ*and it is denoted by

*χ*(

*Γ*). All other notations regarding groups, their subgraphs and automorphisms are standard and follow that of [6].

## 2 Some basic results

Recall that *θ*-commutator of two elements *x* and *y* of a group *G* with respect to an automorphism *θ* of *G* is defined as [*x*,_{θ} *y*]: = *x*^{−1}*θ*(*y*)^{−1}*θ*(*x*)*y*. Also, the *autocommutator* of *x* and *θ* is known to be [*θ*,*x*]^{−1} = [*x*,*θ*]: = *x*^{−1}*θ*(*x*). We begin with the following lemma, which gives a *θ*-commutator analogue of some well-known commutator identities.

*Let G* *be a group*, *x*,*y*,*z be elements of G and θ be an automorphism of G*. *Then*

- [
*x*,_{θ}*y*]^{−1}= [*y*,_{θ}*x*]; *θ*(*x*)^{θ(y)}=*x*[*x*,_{θ}*y*][*y*,*θ*];- [
*x*,_{θ}*yz*] = [*x*,_{θ}*z*][*θ*,*x*]^{z}[*x*,_{θ}*y*]^{z}; - [
*xy*,_{θ}*z*] = [*x*,_{θ}*z*]^{y}[*z*,*θ*]^{y}[*y*,_{θ}*z*];*and* - [
*x*,_{θ}*y*^{−1}] = [*x*,*θ*][*yx*,_{θ}*x*]^{(yx)−1}.

The *θ*-centralizer of elements as well as the *θ*-center of a group can be defined analogously as follows:

*Let G be a group and θ be an automorphism of G*. *The θ*-*centralizer of an element x* ∈ *G*, *denoted by*
*x*), *is defined as*

*Utilizing θ*-*centralizers*, *the θ*-*center of G is defined simply as*

In contrast to natural centralizers and the center of a group, *θ*-centralizers and the *θ*-center of a group *G* need not be subgroups of *G*. For example, if *G* = 〈*x*〉≅ *C*_{3} and *θ* is the nontrivial automorphism of *G*, then *Z _{θ}*(

*G*) = ∅ and

*x*) = {

*x*}. In what follows, we discuss the situations that

*θ*-centralizers and the

*θ*-center of a group turn into subgroups.

*Let* *G* *be a group and θ be an automorphism of G*. *Then*

(1) = Fix($\begin{array}{}{C}_{G}^{\theta}\end{array}$ *θ*);*x*^{−1} ($\begin{array}{}{C}_{G}^{\theta}\end{array}$ *x*)*is a subgroup of G for all x*∈*G*; ($\begin{array}{}{C}_{G}^{\theta}\end{array}$ *x*)*is a subgroup of G if and only if x*^{2}∈ Fix(*θ*);*and*- |
($\begin{array}{}{C}_{G}^{\theta}\end{array}$ *x*)|*divides*|*G*|.

- It is obvious.
- Let
*y*,*z*∈ ($\begin{array}{}{C}_{G}^{\theta}\end{array}$ *x*). Then*θ*(*x*^{−1}*y*) = (*x*^{−1}*y*)^{x−1}and*θ*(*x*^{−1}*z*) = (*x*^{−1}*z*)^{x−1}so that*θ*(*x*^{−1}*yx*^{−1}*z*) = (*x*^{−1}*yx*^{−1}*z*)^{x−1}. Hence*yx*^{−1}*z*∈ ($\begin{array}{}{C}_{G}^{\theta}\end{array}$ *x*), that is, (*x*^{−1}*y*)(*x*^{−1}*z*) ∈*x*^{−1} ($\begin{array}{}{C}_{G}^{\theta}\end{array}$ *x*). On the other hand,*xy*^{−1}*x*∈ ($\begin{array}{}{C}_{G}^{\theta}\end{array}$ *x*), from which it follows that (*x*^{−1}*y*)^{−1}=*x*^{−1}*xy*^{−1}*x*∈*x*^{−1} ($\begin{array}{}{C}_{G}^{\theta}\end{array}$ *x*). Therefore,*x*^{−1} ($\begin{array}{}{C}_{G}^{\theta}\end{array}$ *x*) is a subgroup of*G*. - From (2) it follows that
($\begin{array}{}{C}_{G}^{\theta}\end{array}$ *x*) is a subgroup of*G*if and only if*x*^{−1}∈ ($\begin{array}{}{C}_{G}^{\theta}\end{array}$ *x*) and this holds if and only if*x*^{2}∈ Fix(*θ*). - It follows from (2). □

*Let* *G* *be a group and* *θ* *be an automorphism of* *G*. *Then*

*Z*(_{θ}*G*)≠∅*if and only if**θ*∈ Inn(*G*);*and**Z*(_{θ}*G*) =*Z*(*G*)*g*^{−1}*whenever**θ*=*I*∈ Inn(_{g}*G*).

*As a result*, *Z _{θ}*(

*G*)

*is a subgroup of*

*G*

*if and only if*

*θ*

*is the identity automorphism*.

- If
*x*∈*Z*(_{θ}*G*), then [*x*,_{θ}*x*^{−1}*y*] = 1 for all*y*∈*G*, from which it follows that*θ*(*y*) =*xyx*^{−1}for all*y*∈*G*. Hence*θ*=*I*_{x−1}∈ Inn(*G*). Conversely, if*θ*=*I*_{x−1}for some*x*∈*G*, then*θ*(*y*) =*xyx*^{−1}so that [*x*,_{θ}*y*] = 1 for all*y*∈*G*. Thus*x*∈*Z*(_{θ}*G*). - Assume
*x*∈*Z*(_{θ}*G*). We are going to show that*x*∈*Z*(*G*)*g*^{−1}or equivalently*gx*∈*Z*(*G*). We first observe that*θ*(*x*) =*x*and hence*xg*=*gx*. Now, for*y*∈*G*we have$$\begin{array}{}[x{,}_{\theta}y]=1\iff {x}^{-1}\theta (y{)}^{-1}\theta (x)y=1\iff {x}^{-1}({g}^{-1}yg{)}^{-1}xy=1\iff gxy=ygx.\end{array}$$

Hence *gx* ∈ *Z*(*G*) and consequently *Z _{θ}*(

*G*) ⊆

*Z*(

*G*)

*g*

^{−1}. Conversely, if

*x*∈

*Z*(

*G*)

*g*

^{−1}, then

*gx*∈

*Z*(

*G*) and the above argument shows that [

*x*,

_{θ}

*y*] = 1 for all

*y*∈

*G*. Therefore,

*Z*(

*G*)

*g*

^{−1}⊆

*Z*(

_{θ}*G*) and the result follows. □

The above lemma states that *Z _{θ}*(

*G*) = ∅ if and only if

*θ*is a non-inner automorphism of

*G*. This fact will be used frequently in the sequel.

## 3 The *θ*-non-commuting graphs

Having defined the *θ*-commutators, we can now define and study the *θ*-non-commuting graph analog of the non-commuting graphs. In this section, some primary properties if such graphs and their relationship to other notions will be established.

*Let* *G* *be a group and θ be an automorphism of G*. *The* *θ*-*non*-*commuting graph of G*, *denoted by*
*is a simple undirected graph whose vertices are elements of G*∖ *Z _{θ}*(

*G*)

*and two distinct vertices x and y are adjacent if*[

*x*,

_{θ}

*y*]≠ 1.

Clearly, the *θ*-non-commuting graph of a group coincides with the ordinary non-commuting graph whenever *θ* is the identity automorphism. Indeed, the map
*Θ*(*x*) = *g*^{−1}*x*, for all *x* ∈ *V*(
*G*) in Aut(*G*) give rise to the same graphs.

The following two results will be used in order to prove Theorem 3.4.

*Let* *X be a subset of G with* |*X*| ≤ |*G*|/2. *If there exists a vertex* *x* *in*
*such that* [*x*,_{θ} *y*] = 1 *for all* *y* ∈ *G* ∖ *X*, *then* |*X*| = |*G*|/2.

Assume that [*x*,_{θ} *y*] = 1 for all *y* ∈ *G* ∖ *X*. We claim that 〈*x*^{−1}(*G* ∖ *X*)〉 is a proper subgroup of *G*. Suppose on the contrary that 〈*x*^{−1}(*G* ∖ *X*)〉 = *G*. One can easily see that *θ*(*x*^{−1}*y*) = (*x*^{−1}*y*)^{x−1} for all *y* ∈ *G* ∖ *X*. Hence *θ* = *I*_{x−1}, which implies that *Z _{θ}*(

*G*) =

*Z*(

*G*)

*x*by Lemma 2.4. But then

*x*∈

*Z*(

_{θ}*G*), which is a contradiction. Thus, |

*G*∖

*X*| ≤ |〈

*x*

^{−1}(

*G*∖

*X*)〉| ≤ |

*G*|/2 and consequently |

*X*| = |

*G*|/2, as required. □

*For every* *x* ∈ *G* *we have* deg(*x*) ≥ |*G*|/2.

*We have* diam(

If diam(
*x* and *y* such that *d*(*x*,*y*) > 2. Thus *N*(*x*)∩ *N*(*y*) = ∅ so that |*N*(*x*)| = |*N*(*y*)| = |*G*|/2 by Corollary 3.3. Consequently, *G* = *N*(*x*)∪̇ *N*(*y*), which implies that *y* ∈ *N*(*x*), that is, *x* and *y* are adjacent, a contradiction. □

*We have* girth(
*and equality holds if and only if* *G* *is an abelian group*, [*G*, Fix(*θ*)] = 2 *and* [*G*,*θ*]^{2} = 1.

Suppose girth(
*N*(*x*_{1})∩ *N*(*x*_{2}) = ∅ for every edge {*x*_{1},*x*_{2}} ∈ *E*(
*N*(*x*_{1})|,|*N*(*x*_{2})| ≥ |*G*|/2 by Corollary 3.3, from which it follows that |*N*(*x*_{1})| = |*N*(*x*_{2})| = |*G*|/2, hence *G* = *N*(*x*_{1})∪̇ *N*(*x*_{2}). Since for *y* ∈ *N*(*x _{i}*) (

*i*= 1,2) we have

*G*=

*N*(

*x*)∪̇

_{i}*N*(

*y*) as well, it follows that

*N*(

*y*) =

*N*(

*x*

_{3−i}). Therefore,

*N*(

*x*

_{1}),

*N*(

*x*

_{2})), which yields girth(

*G*=

*N*(

*x*)∪̇

*N*(

*y*) is an equally partition for each {

*x*,

*y*} ∈

*E*(

*N*(

*x*). Then

that is, *N*(*x*) = Fix(*θ*) is a subgroup of *G*. Furthermore, *N*(*x*) is abelian as [*a*,_{θ} *b*] = 1 or equivalently *ab* = *ba* for all distinct elements *a*,*b* ∈ *N*(*x*). Now let *g* ∈ *G* ∖ *N*(*x*). Clearly, *N*(*y*) = *N*(*x*)*g*. Since *g*,*ag* ∈ *N*(*y*) for all *a* ∈ *N*(*x*), it follows that [*g*,_{θ} *ag*] = 1 or equivalently *aθ*(*g*) = *θ*(*g*)*a*. Therefore, *G* = 〈*N*(*x*),*θ*(*g*)〉 is abelian. As *g*^{2} ∈ *N*(*x*) we have *θ*(*g*^{2}) = *g*^{2} so that [*g*,*θ*]^{2} = 1. Hence [*G*,*θ*]^{2} = 1, as required. The converse is straightforward. □

In what follows, we obtain some criterion for an automorphism to be fixed-point-free (or regular) or class-preserving. Remind that an automorphism *θ* of *G* is *fixed-point-free* if the only fixed point of *θ* is the trivial element, that is, Fix(*θ*) = 〈1〉 is the trivial subgroup of *G*.

*The graph*
*is complete if and only if* *θ* *is a fixed-point-free automorphism of* *G*.

Assume
*θ* is non-inner and [*x*,_{θ} *y*]≠1 for all vertices *x* and *y* in
*θ* is not fixed-point-free, then there exists an element *x* ∈ *G* such that *θ*(*x*) = *x*. But then [*x*,_{θ} 1] = 1, which is impossible. Thus *θ* is fixed-point-free. Conversely, suppose that *θ* is a fixed-point-free automorphism. By [6, 10.5.1(iii)], *θ*(*g*)∉ *g ^{G}* for all

*g*∈

*G*∖{1}. Hence

*θ*(

*x*

^{−1}

*y*)≠ (

*x*

^{−1}

*y*)

^{x−1}for all distinct vertices

*x*and

*y*, which implies that [

*x*,

_{θ}

*y*]≠1, that is,

*x*and

*y*are adjacent. The proof is complete. □

An automorphism *θ* of *G* is called *class preserving* if *θ*(*g ^{G}*) =

*g*for every conjugacy class

^{G}*g*of

^{G}*G*.

*Let* *k*(*G*) *denote the number of conjugacy classes of* *G*. *Then*

*and the equality holds if and only if* *θ* *is a class preserving automorphism of* *G*.

If *θ* is an inner automorphism, then
*θ* is a non-inner automorphism. By Lemma 2.4, we observe that *V*(
*G*. Then

from which, in conjunction with the fact that

## 4 Independent sets

In the section, we give a description of independent subsets of the graph
*G* is an abelian group.

*Let* *G* *be a group and* *θ be an automorphism of* *G*. *Then*

*If**x*,*y are in the same coset of*Fix(*θ*),*then**x*∼*y if and only if xy*≠*yx*.*If**x*,*y**are in different cosets of*Fix(*θ*),*then**x*∼*y if**xy*=*yx*.

- By assumption
*y*^{−1}*x*∈ Fix(*θ*). Thus$$\begin{array}{}xy/=yx\iff {y}^{-1}x/=x{y}^{-1}\iff \theta ({y}^{-1}x)/=x{y}^{-1}\iff [x{,}_{\theta}y]/=1\iff x\sim y.\end{array}$$ - We have
*y*^{−1}*x*∉Fix(*θ*) and consequentlyas required. □$$\begin{array}{}xy=yx\Rightarrow {y}^{-1}x=x{y}^{-1}\Rightarrow \theta ({y}^{-1}x)/=x{y}^{-1}\Rightarrow [x{,}_{\theta}y]/=1\Rightarrow x\sim y,\end{array}$$

*A coset of* Fix(*θ*) *is an independent set if and only if it is an abelian set*.

*For every abelian group* *G*, *we have*
*K*_{|F|,…,|F|} *is a complete m*-*partite graph in which F* = Fix(*θ*) *and m* = [*G*:Fix(*θ*)].

As we have seen in Lemma 4.1, there is a close relationship between independence and commutativity of vertices in the graph

*Let* *I* *be an independent subset of*
*Then*

*I*^{−1}*I is an abelian set*;*and**if**I**is non-abelian*,*then**I**is a product-free set*.

- Let
*x*,*y*,*z*,*w*∈*I*. ThenSimilarly, we have$$\begin{array}{}\theta ({z}^{-1}w)=\theta ({x}^{-1}z{)}^{-1}\theta ({x}^{-1}w)=({x}^{-1}z{)}^{-{x}^{-1}}({x}^{-1}w{)}^{{x}^{-1}}=({z}^{-1}w{)}^{{x}^{-1}}.\end{array}$$ *θ*(*z*^{−1}*w*) = (*z*^{−1}*w*)^{y−1}, from which the result follows. - Suppose on the contrary that
*I*is not product-free so that*ab*∈*I*for some*a*,*b*∈*I*. For*x*∈*I*we havefrom which we get$$\begin{array}{}({x}^{-1}ab{)}^{{x}^{-1}}=\theta ({x}^{-1}ab)=\theta ({x}^{-1}a)\theta (x)\theta ({x}^{-1}b)=({x}^{-1}a{)}^{{x}^{-1}}\theta (x)({x}^{-1}b{)}^{{x}^{-1}},\end{array}$$ *θ*(*x*) =*x*. Hence [*x*,*y*] = [*x*,_{θ}*y*] = 1 for all*x*,*y*∈*I*. Therefore*I*is abelian, which is a contradiction. □

Now we can state our structural description of arbitrary independent sets in the graph

*Let* *G* *be a group and I be a subset of* *G*. *Then* *I* *is an independent (resp*. *a maximal independent) subset of*
*if and only if I* ⊆ *gA (resp*. *I* = *gA) for every* *g* ∈ *I*, *in which* *A is an abelian (resp*. *a maximal abelain) subgroup of* Fix(*I _{g}θ*).

First observe that if *I* is an independent subset of
*A* = 〈*I*^{−1}*I*〉 is an abelian subgroup of Fix(*I _{g}θ*) and

*I*⊆

*gA*for every

*g*∈

*I*by Lemma 4.4(1). Also,

*I*=

*gA*and

*A*is a maximal abelian subgroup of Fix(

*I*) whenever

_{g}θ*I*is a maximal independent subset of

*gA*is an independent set in

*xA*⊆

*yB*in which

*A*and

*B*are maximal abelian subgroups of Fix(

*I*) and Fix(

_{x}θ*I*), respectively, coincide. First observe that

_{y}θ*A*⊆

*x*

^{−1}

*yB*so that

*x*

^{−1}

*y*=

*b*

_{0}∈

*B*. Hence

*A*⊆

*B*. Now, for every

*b*∈

*B*, we have

*B*⊆ Fix(

*I*). The maximality of

_{x}θ*A*yields

*A*=

*B*and consequently

*xA*=

*yB*, as required. □

*We have*

*The graph*
*is empty if and only if G is abelian and θ is the identity automorphism*, *in which case*
*is the null graph*.

*Let* *G* *be a finite group and* *θ* *be an automorphism of* *G*. *If either* *G* *is non-abelian or* *θ* *is non-identity*, *then α*(
*G*|/2 *and the equality holds if and only if* Fix(*I _{g}θ*)

*is an abelian subgroup of G of index*2

*for some element*

*g*∈

*G*.

If *α*(
*G*|/2, then Corollary 4.6 gives an element *g* ∈ *G* such that *G* = Fix(*I _{g}θ*) is abelian. But then

*θ*=

*I*

_{g−1}=

*I*, which is a contradiction. Hence

*α*(

*G*|/2. Now if the equality holds, by using Corollary 4.6 once more, we observe that Fix(

*I*) is an abelian subgroup of

_{g}θ*G*of index 2 for some

*g*∈

*G*. The converse is straightforward. □

## 5 Chromatic number

The results of section 4 on the independence number can be applied to study the chromatic number of *θ*-non-commuting graphs. Since every maximal independent set in

(Tomkinson [7]). *Let* *G* *be covered by some cosets* *g _{i}*

*H*

_{i}*for*

*i*= 1,…,

*n*.

*If the cover is irredundant*,

*then*

Tomkinson’s theorem has the following immediate result connecting the chromatic number of
*θ*.

*For any group G*, *we have*

*and the equality holds only if* Fix(*θ*) ⊆ *Z*(*G*).

Let *G* = *I*_{1}∪⋯∪ *I*_{χ} be the union of independent sets *I*_{1},…,*I _{χ}* in which

*I*⊆

_{i}*g*and

_{i}A_{i}*A*is an abelian subgroup of Fix(

_{i}*I*), for

_{gi}θ*i*= 1,…,

*χ*=

*χ*(

*I*, then

_{k}*g*∈

_{k}*A*, which implies that

_{k}*A*⊆ Fix(

_{k}*θ*). Thus

Now assume the equality holds. Then Fix(*θ*) = *A _{k}* ⊆

*A*, for all

_{i}*i*= 1,…,

*χ*. Since

*A*are abelian, Fix(

_{i}*θ*) commutes with all elements of

*A*

_{1},…,

*A*. On the other hand, as

_{χ}*A*⊆

_{k}*A*, we have

_{i}*a*=

*θ*(

*a*) =

*a*∈

*A*and

_{k}*i*= 1,…,

*χ*, which implies that Fix(

*θ*) commutes with

*g*

_{1},…,

*g*

_{χ}as well. Therefore Fix(

*θ*) ⊆

*Z*(

*G*), as required. □

In the sequel, we shall characterize those graphs having small chromatic numbers.

*Let* *G* *be a finite group*. *Then* *χ*(
*if and only if* *G* *is abelian and* [*G*:Fix(*θ*)] = 2.

Clearly, *α*(
*G*|/2. On the other hand, by Corollary 4.6, *α*(
*G*|/2, from which it follows that *α*(
*G*|/2. Hence *G* = *I*_{1}∪ *I*_{2}, where (*I*_{1},*I*_{2}) is a bipartition of
*I*_{1}| = |*I*_{2}| = |*G*|/2. Assume 1 ∈ *I*_{1}. Then, by Theorem 4.5, *I*_{1} = *g*_{1}*A*_{1} for some *g*_{1} ∈ *G* in which *A*_{1} = Fix(*I*_{g1}*θ*) is an abelian subgroup of *G*. Since *g*_{1} ∈ *A*_{1}, it follows that *A*_{1} = Fix(*θ*). Clearly, *A*_{2} = *A*_{1} and *g*_{2} ∈ *G* ∖ *A*_{1}. Now
*a* ∈ *A*_{1}, from which it follows that *g*_{2} commutes with *A*_{1}. Thus *G* is abelian. The converse is obvious by Corollary 4.3. □

*Let* *G* *be a finite group*. *Then* *χ*(
*if and only if either* *G* *is abelian and* [*G*:Fix(*θ*)] = 3 *or* *G* *is non-abelian and one of the following holds*:

*G*/*Z*(*G*)≅*C*_{2}×*C*_{2}*and**θ**is an inner automorphism*;- [
*G*:Fix(*θ*)] = [Fix(*θ*):*Z*(*G*)] = 2;*or* *G**has a characteristic subgroup**A**such that*[*G*:*A*] = [*A*:Fix(*θ*)] = 2, Fix(*θ*) =*Z*(*G*)*and there exist elements**x*∈*A*∖*Z*(*G*)*and**y*∈*G*∖*A such that**θ*(*x*) =*x*.^{y}

If *G* is abelian, then we are done by Corollary 4.3. Hence, we assume that *G* is non-abelian. Let *G* = *I*_{1}∪ *I*_{2}∪ *I*_{3} be a tripartition of *G* in which |*I*_{1}| ≥ |*I*_{2}| ≥ |*I*_{3}| and *I _{i}* ⊆

*g*(

_{i}A_{i}*i*= 1,2,3) for some elements

*g*∈

_{i}*G*and abelian subgroups

*A*of Fix(

_{i}*I*). Without loss of generality, we may assume that

_{gi}θ*I*

_{1}=

*g*

_{1}

*A*

_{1}. From Theorem 5.1, we know that [

*G*:

*A*

_{1}∩

*A*

_{2}∩

*A*

_{3}] ≤ 3! = 6. We distinguish two cases:

Case 1. *G* = 〈*A*_{1},*A*_{2},*A*_{3}〉. Then *A*_{1} ∩ *A*_{2} ∩ *A*_{3} ⊆ *Z*(*G*) and we must have *G*/*Z*(*G*)≅ *C*_{2}× *C*_{2} or *S*_{3}. Hence *A*_{1} ∩ *A*_{2} ∩ *A*_{3} = *Z*(*G*). One can verify that 2 = [*G*:*A*_{1}] ≥ [*G*:*A*_{2}] ≥ 3 and *A*_{1} ≠ *A*_{2}. Since *G* ≠ *A*_{1} ∪ *A*_{2} and every element of *G*/*Z*(*G*) has order 1, 2 or 3, one can always find an element *g* ∈ *G* ∖ *A*_{1} ∪ *A*_{2} such that *gA _{i}* =

*g*, for

_{i}A_{i}*i*= 1,2. Thus, for

*a*∈

*A*we have

_{i}*θ*acts by conjugation via

*g*

^{−1}on 〈

*A*

_{1},

*A*

_{2}〉 =

*G*. Hence

*θ*=

*I*

_{g−1}is an inner automorphism. If

*G*/

*Z*(

*G*)≅

*S*

_{3}, then

*Γ*has a subgraph isomorphic to

_{G}*Γ*

_{S3}≅

*K*

_{5}∖

*K*

_{2}with chromatic number 4, a contradiction. Therefore

*G*/

*Z*(

*G*)≅

*C*

_{2}×

*C*

_{2}, which gives us part (1).

Case 2. *G* ≠〈*A*_{1},*A*_{2},*A*_{3}〉. Clearly, *A*_{2},*A*_{3} ⊆ *A*_{1}. Then *A*_{2},*A*_{3} ⊂ *A*_{1} otherwise *A*_{1} = *A*_{2} and hence *θ*(*a*) = *a*^{g1} = *a*^{g2} for all *a* ∈ *A*_{1}. Since *g*_{1}*A*_{1} ≠ *g*_{2}*A*_{2}, it follows that *g*_{1}
*G* ∖ *A*_{1} commutes with *A*_{1} so that *G* = *A*_{1}〈*g*_{1}
*A*_{2},*A*_{3} ⊂ *A*_{1}, from which together with Tomkinson’s result we must have [*G*:*A*_{1}] = 2, [*A*_{1}:*A*_{2}] = 2 and *A*_{2} = *A*_{3}. Clearly, *g*_{2}*A*_{2} ∪ *g*_{3}*A*_{3} =
*A*_{1} where *G* = *g*_{1}*A*_{1} ∪
*A*_{1}. If *g*_{1} ∈ *A*_{1}, then *A*_{1} = Fix(*θ*). As *A*_{2} ⊆ *A*_{1} we have *a* = *θ*(*a*) =
*a* ∈ *A*_{2}, which implies that *A*_{2} = *Z*(*G*). Hence we obtain part (2). Next assume that *g*_{1} ∉ *A*_{1}. Then
*A*_{1} and consequently *g*_{2},*g*_{3} ∈
*A*_{1} = *A*_{1}. This implies that *A*_{2} ⊆ Fix(*θ*). As *A*_{2} ⊆ *A*_{1}, we have *a* = *θ*(*a*) =
*a* ∈ *A*_{2} showing that *A*_{2} = *Z*(*G*). Assuming *g*_{2} ∈ *A*_{1} ∖ *A*_{2}, we obtain *θ*(*g*_{2}) = *g*_{2}*z* for some *z* ∈ *A*_{2}. Furthermore,
*g*_{2} ∈ *A*_{1}. Thus [*g*_{2},*g*_{1}] = *z* and this yields part (3).

The converse is straightforward. □

We conclude this section with a characterization of complete multipartite-ness of the graphs

*The graph*
*is a complete multipartite graph if and only if* Fix(*I _{g}θ*)

*is abelian for all g*∈

*G*.

First assume that
*I _{g}θ*) are disjoint so that Fix(

*I*) is abelian for all

_{g}θ*g*∈

*G*. Now assume that Fix(

*I*) is abelian for all

_{g}θ*g*∈

*G*. Let

*x*∈

*G*. By assumption,

*x*Fix(

*I*) is an independent subset of

_{x}θ*y*∉

*x*Fix(

*I*), we have

_{x}θ*x*

^{−1}

*y*∉Fix(

*I*) so that

_{x}θ*θ*(

*x*

^{−1}

*y*) ≠(

*x*

^{−1}

*y*)

^{x−1}. Hence

*y*is adjacent to

*x*. Since, by Theorem 4.5, the sets

*g*Fix(

*I*) are maximal independent subsets of

_{g}θ*g*Fix(

*I*) partition

_{g}θ*G*and hence

## 6 Conclusion/Open problems

In this paper, we have generalized commutators of a group, in a compatible way, to *θ*-commutators with respect to a given automorphism *θ*. Accordingly, commutator identities as well as the corresponding centralizers and center are studied.

One may define *θ*-nilpotent and *θ*-solvable groups by means of *θ*-commutators in a natural way. So, we may ask:

*How are the automorphism* *θ* *and the structure of* *θ*-*nilpotent and* *θ*-*solvable groups related*?

Next, we have defined the non-commuting graph
*θ*-commutators of a group *G* and established some connections between graph theoretical properties of
*θ*. For instance, it is proved, among other results, that
*θ* is fixed-point-free and that
*θ* is class preserving.

*Which other graph theoretical properties of*
*can be interpreted (simply) in terms of group theoretical properties of θ (and vice versa)?*

The rest of paper is devoted to the study of independent sets in the graph
*I _{g}θ*) is established, where

*I*denotes the inner automorphism of

_{g}*G*induced by the element

*g*∈

*G*. This result provided us with another partial answer to the above question: the graph

*G*is abelian and

*θ*is the identity automorphism. Secondly, a relationship between covers of

*G*by subgroups and the chromatic number

*χ*(

*χ*(

*θ*) is deduced. Also, the structure of

*G*or properties of

*θ*when

In contrast to our investigations on independent sets one may ask:

*How can cliques of*
*be described in terms of G and θ*?

Finally, a fundamental question to ask is:

*Suppose θ*_{1} *and* *θ*_{2} *are automorphisms of groups* *G*_{1} *and* *G*_{2}, *respectively*.

*Under which conditions on*(*G*_{1},*θ*_{1})*and*(*G*_{2},*θ*_{2})*are two graphs*$\begin{array}{}{\mathit{\Gamma}}_{{G}_{1}}^{{\theta}_{1}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}{\mathit{\Gamma}}_{{G}_{2}}^{{\theta}_{2}}\end{array}$ *isomorphic (in particular when G*_{1}=*G*_{2}*)?**How are the pairs*(*G*_{1},*θ*_{1})*and*(*G*_{2},*θ*_{2})*related*,*provided that*$\begin{array}{}{\mathit{\Gamma}}_{{G}_{1}}^{{\theta}_{1}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}{\mathit{\Gamma}}_{{G}_{2}}^{{\theta}_{2}}\end{array}$ *are isomorphic*?

The authors would like to thank the referees for their kind comments and suggestions.

## References

- [1]↑
Ree R., On generalized conjugate classes in a finite group, Illinois J. Math., 1959, 3, 440–444.

- [2]↑
Achar P.N., Generalized conjugacy classes, Rose-Hulman Mathematical Sciences Technical Report Series, no. 97-01, 1997.

- [6]↑
Robinson D.J.S., A Course in the Theory of Groups, Second Edition, Springer-Verlag, New York, 1996.

- [7]↑
Tomkinson M.J., Groups covered by finitely many cosets or subgroups, Comm. Algebra, 1987, 15, 845–859.