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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo

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Volume 15, Issue 1

# On θ-commutators and the corresponding non-commuting graphs

S. Shalchi
/ A. Erfanian
• Corresponding author
• Department of Pure Mathematics and the Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box 1159, 91775 Mashhad, Iran
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• Other articles by this author:
/ M. Farrokhi DG
• Department of Pure Mathematics and the Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box 1159, 91775 Mashhad, Iran
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Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/math-2017-0126

## Abstract

The θ-commutators of elements of a group with respect to an automorphism are introduced and their properties are investigated. Also, corresponding to θ-commutators, we define the θ-non-commuting graphs of groups and study their correlations with other notions. Furthermore, we study independent sets in θ-non-commuting graphs, which enable us to evaluate the chromatic number of such graphs.

MSC 2010: 05C25; 05C15; 05C69; 20D45; 20F12

## 1 Introduction

The commutator of two elements x and y of a group G is defined usually as [x,y]: = x−1y−1xy. 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(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−1Iy(x), Iy being the inner automorphism associate to y, one may generalize them in a natural way to [x,θ] = x−1θ(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−1y−1(y). One observes that [x,y]θ = 1 if and only if θ(y) = yx. 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)−1θ(x)y, which is simply the image of y−1−1(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−1θ(y)−1θ(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 vV(Γ), the neighbor of v in Γ is denoted by NΓ(v) and the degree of v is given by degΓ(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−1θ(y)−1θ(x)y. Also, the autocommutator of x and θ is known to be [θ,x]−1 = [x,θ]: = x−1θ(x). We begin with the following lemma, which gives a θ-commutator analogue of some well-known commutator identities.

#### Lemma 2.1

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

1. [x,θ y]−1 = [y,θ x];

2. θ(x)θ(y) = x[x,θ y][y,θ];

3. [x,θ yz] = [x,θ z][θ, x]z[x,θ y]z;

4. [xy,θ z] = [x,θ z]y[z,θ]y[y,θ z]; and

5. [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:

#### Definition 2.2

Let G be a group and θ be an automorphism of G. The θ-centralizer of an element xG, denoted by $\begin{array}{}{C}_{G}^{\theta }\end{array}$(x), is defined as $CGθ(x)={y∈G∣[x,θy]=1}.$

Utilizing θ-centralizers, the θ-center of G is defined simply as $Zθ(G)=⋂x∈GCGθ(x)={y∈G∣[x,θy]=1,x∈G}.$

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〉≅ C3 and θ is the nontrivial automorphism of G, then Zθ(G) = ∅ and $\begin{array}{}{C}_{G}^{\theta }\end{array}$(x) = {x}. In what follows, we discuss the situations that θ-centralizers and the θ-center of a group turn into subgroups.

#### Theorem 2.3

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

1. $\begin{array}{}{C}_{G}^{\theta }\end{array}$(1) = Fix(θ);

2. x−1 $\begin{array}{}{C}_{G}^{\theta }\end{array}$(x) is a subgroup of G for all xG;

3. $\begin{array}{}{C}_{G}^{\theta }\end{array}$(x) is a subgroup of G if and only if x2 ∈ Fix(θ); and

4. |$\begin{array}{}{C}_{G}^{\theta }\end{array}$(x)| divides |G|.

#### Proof

1. It is obvious.

2. Let y,z$\begin{array}{}{C}_{G}^{\theta }\end{array}$(x). Then θ(x−1y) = (x−1y)x−1 and θ(x−1z) = (x−1z)x−1 so that θ(x−1yx−1z) = (x−1yx−1z)x−1. Hence yx−1z$\begin{array}{}{C}_{G}^{\theta }\end{array}$(x), that is, (x−1y)(x−1z) ∈ x−1 $\begin{array}{}{C}_{G}^{\theta }\end{array}$(x). On the other hand, xy−1x$\begin{array}{}{C}_{G}^{\theta }\end{array}$(x), from which it follows that (x−1y)−1 = x−1xy−1xx−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.

3. 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 x2 ∈ Fix(θ).

4. It follows from (2). □

#### Lemma 2.4

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

1. Zθ(G)≠∅ if and only if θ ∈ Inn(G); and

2. Zθ(G) = Z(G)g−1 whenever θ = Ig ∈ Inn(G).

As a result, Zθ(G) is a subgroup of G if and only if θ is the identity automorphism.

#### Proof

1. If xZθ(G), then [x,θ x−1y] = 1 for all yG, from which it follows that θ(y) = xyx−1 for all yG. Hence θ = Ix−1 ∈ Inn(G). Conversely, if θ = Ix−1 for some xG, then θ(y) = xyx−1 so that [x,θ y] = 1 for all yG. Thus xZθ(G).

2. Assume xZθ(G). We are going to show that xZ(G)g−1 or equivalently gxZ(G). We first observe that θ(x) = x and hence xg = gx. Now, for yG we have $[x,θy]=1⇔x−1θ(y)−1θ(x)y=1⇔x−1(g−1yg)−1xy=1⇔gxy=ygx.$

Hence gxZ(G) and consequently Zθ(G) ⊆ Z(G)g−1. Conversely, if xZ(G)g−1, then gxZ(G) and the above argument shows that [x,θ y] = 1 for all yG. Therefore, Z(G)g−1Zθ(G) and the result follows. □

The above lemma states that Zθ(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.

#### Definition 3.1

Let G be a group and θ be an automorphism of G. The θ-non-commuting graph of G, denoted by $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$, is a simple undirected graph whose vertices are elements of GZθ(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 $\begin{array}{}\mathrm{\Theta }:V\left({\mathit{\Gamma }}_{G}^{\theta }\right)⟶V\left({\mathit{\Gamma }}_{G}^{{I}_{g}\theta }\right)\end{array}$ defined by Θ(x) = g−1x, for all xV( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$), presents an isomorphism between $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\text{\hspace{0.17em}and\hspace{0.17em}}{\mathit{\Gamma }}_{G}^{{I}_{g}\theta }.\end{array}$ Hence, every two automorphisms in the same cosets of Inn(G) in Aut(G) give rise to the same graphs.

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

#### Lemma 3.2

Let X be a subset of G with |X| ≤ |G|/2. If there exists a vertex x in $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ such that [x,θ y] = 1 for all yGX, then |X| = |G|/2.

#### Proof

Assume that [x,θ y] = 1 for all yGX. We claim that 〈x−1(GX)〉 is a proper subgroup of G. Suppose on the contrary that 〈x−1(GX)〉 = G. One can easily see that θ(x−1y) = (x−1y)x−1 for all yGX. Hence θ = Ix−1, which implies that Zθ(G) = Z(G)x by Lemma 2.4. But then xZθ(G), which is a contradiction. Thus, |GX| ≤ |〈x−1(GX)〉| ≤ |G|/2 and consequently |X| = |G|/2, as required. □

#### Corollary 3.3

For every xG we have deg(x) ≥ |G|/2.

#### Theorem 3.4

We have diam( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) ≤ 2.

#### Proof

If diam( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) > 2, then there exist two vertices 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 yN(x), that is, x and y are adjacent, a contradiction. □

#### Theorem 3.5

We have girth( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) ≤ 4 and equality holds if and only if G is an abelian group, [G, Fix(θ)] = 2 and [G,θ]2 = 1.

#### Proof

Suppose girth( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) > 3. We show that girth( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) = 4. Since girth( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) > 3 N(x1)∩ N(x2) = ∅ for every edge {x1,x2} ∈ E( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$). Moreover, |N(x1)|,|N(x2)| ≥ |G|/2 by Corollary 3.3, from which it follows that |N(x1)| = |N(x2)| = |G|/2, hence G = N(x1)∪̇ N(x2). Since for yN(xi) (i = 1,2) we have G = N(xi)∪̇ N(y) as well, it follows that N(y) = N(x3−i). Therefore, $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ is a complete bipartite graph with the bipartition (N(x1),N(x2)), which yields girth( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) = 4. To prove the second part, assume girth( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) = 4. Then G = N(x)∪̇ N(y) is an equally partition for each {x,y} ∈ E( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$). Suppose 1 ∈ N(x). Then $g∈N(x)⇔[g,θ1]=1⇔θ(g)=g⇔g∈Fix(θ),$

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,bN(x). Now let gGN(x). Clearly, N(y) = N(x)g. Since g,agN(y) for all aN(x), it follows that [g,θ ag] = 1 or equivalently (g) = θ(g)a. Therefore, G = 〈N(x),θ(g)〉 is abelian. As g2N(x) we have θ(g2) = g2 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.

#### Theorem 3.6

The graph $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ is complete if and only if θ is a fixed-point-free automorphism of G.

#### Proof

Assume $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ is a complete graph. Then θ is non-inner and [x,θ y]≠1 for all vertices x and y in $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$. If θ is not fixed-point-free, then there exists an element xG 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)∉ gG for all gG ∖{1}. Hence θ(x−1y)≠ (x−1y)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(gG) = gG for every conjugacy class gG of G.

#### Theorem 3.7

Let k(G) denote the number of conjugacy classes of G. Then $|E(ΓGθ)|≥12|G|(|G|−k(G))$

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

#### Proof

If θ is an inner automorphism, then $\begin{array}{}|E\left({\mathit{\Gamma }}_{G}^{\theta }\right)|=\frac{1}{2}|G|\left(|G|-k\left(G\right)\right)\end{array}$ and we are done. Hence, assume that θ is a non-inner automorphism. By Lemma 2.4, we observe that V( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) = G. Then $E(ΓGθ)c=12∑x∈G|CGθ(x)|−12|G|=12∑x∈G|{y∈G∣θ(x−1y)=(x−1y)x−1}|−12|G|=12∑x∈G|{y∈G∣θ(y)=yx−1}|−12|G|=12∑y∈G|{x∈G∣θ(y)=yx−1}|−12|G|≤12∑y∈G|CG(y)|−12|G|=12|G|k(G)−12|G|,$

from which, in conjunction with the fact that $\begin{array}{}\left|E\left(\left({\mathit{\Gamma }}_{G}^{\theta }{\right)}^{c}\right)\right|=\frac{1}{2}|G|\left(|G|-1\right)-|E\left({\mathit{\Gamma }}_{G}^{\theta }\right)|,\end{array}$ the result follows. □

## 4 Independent sets

In the section, we give a description of independent subsets of the graph $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$, which enables us to compute the independence number of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$. We begin with the easier case of abelian groups. Indeed, utilizing the following lemma, we can determine the structure of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ precisely when G is an abelian group.

#### Lemma 4.1

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

1. If x,y are in the same coset of Fix(θ), then xy if and only if xyyx.

2. If x,y are in different cosets of Fix(θ), then xy if xy = yx.

#### Proof

1. By assumption y−1x ∈ Fix(θ). Thus $xy/=yx⇔y−1x/=xy−1⇔θ(y−1x)/=xy−1⇔[x,θy]/=1⇔x∼y.$

2. We have y−1x∉Fix(θ) and consequently $xy=yx⇒y−1x=xy−1⇒θ(y−1x)/=xy−1⇒[x,θy]/=1⇒x∼y,$

as required. □

#### Corollary 4.2

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

#### Corollary 4.3

For every abelian group G, we have $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$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 $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$. The following key lemma illustrates this relationship in a much suitable form.

#### Lemma 4.4

Let I be an independent subset of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$. Then

1. I−1I is an abelian set; and

2. if I is non-abelian, then I is a product-free set.

#### Proof

1. Let x,y,z,wI. Then $θ(z−1w)=θ(x−1z)−1θ(x−1w)=(x−1z)−x−1(x−1w)x−1=(z−1w)x−1.$

Similarly, we have θ(z−1w) = (z−1w)y−1, from which the result follows.

2. Suppose on the contrary that I is not product-free so that abI for some a,bI. For xI we have $(x−1ab)x−1=θ(x−1ab)=θ(x−1a)θ(x)θ(x−1b)=(x−1a)x−1θ(x)(x−1b)x−1,$

from which we get θ(x) = x. Hence [x,y] = [x,θ y] = 1 for all x,yI. Therefore I is abelian, which is a contradiction. □

Now we can state our structural description of arbitrary independent sets in the graph $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$.

#### Theorem 4.5

Let G be a group and I be a subset of G. Then I is an independent (resp. a maximal independent) subset of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ if and only if IgA (resp. I = gA) for every gI, in which A is an abelian (resp. a maximal abelain) subgroup of Fix(Igθ).

#### Proof

First observe that if I is an independent subset of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$, then A = 〈I−1I〉 is an abelian subgroup of Fix(Igθ) and IgA for every gI by Lemma 4.4(1). Also, I = gA and A is a maximal abelian subgroup of Fix(Igθ) whenever I is a maximal independent subset of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$. Clearly, every subset of gA is an independent set in $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$. To complete the proof, we must show that any two independent sets xAyB in which A and B are maximal abelian subgroups of Fix(Ixθ) and Fix(Iyθ), respectively, coincide. First observe that Ax−1yB so that x−1y = b0B. Hence AB. Now, for every bB, we have $\begin{array}{}\theta \left(b\right)={b}^{{y}^{-1}}={b}^{{b}_{0}^{-1}{x}^{-1}}={b}^{{x}^{-1}},\end{array}$ which implies that B ⊆ Fix(Ixθ). The maximality of A yields A = B and consequently xA = yB, as required. □

#### Corollary 4.6

We have $α(ΓGθ)=max{|A|∣A≤Fix(Igθ) is abelian,g∈G}.$

#### Corollary 4.7

The graph $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ is empty if and only if G is abelian and θ is the identity automorphism, in which case $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ is the null graph.

#### Corollary 4.8

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( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) ≤ |G|/2 and the equality holds if and only if Fix(Igθ) is an abelian subgroup of G of index 2 for some element gG.

#### Proof

If α( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) > |G|/2, then Corollary 4.6 gives an element gG such that G = Fix(Igθ) is abelian. But then θ = Ig−1 = I, which is a contradiction. Hence α( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) ≤ |G|/2. Now if the equality holds, by using Corollary 4.6 once more, we observe that Fix(Igθ) is an abelian subgroup of G of index 2 for some gG. 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 $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ is a left coset to an abelian group, the evaluation of the chromatic number of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ relies on the theory of covering groups by left cosets of their proper subgroups. In this regard, the following result of Tomkinson plays an important role.

#### Theorem 5.1

(Tomkinson [7]). Let G be covered by some cosets giHi for i = 1,…,n. If the cover is irredundant, then $\begin{array}{}\left[G:\bigcap _{i=1}^{n}{H}_{i}\right]\le n!.\end{array}$

Tomkinson’s theorem has the following immediate result connecting the chromatic number of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ to the number of fixed points of θ.

#### Corollary 5.2

For any group G, we have $[G:Fix(θ)]≤χ(ΓGθ)!$

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

#### Proof

Let G = I1∪⋯∪ Iχ be the union of independent sets I1,…,Iχ in which IigiAi and Ai is an abelian subgroup of Fix(Igiθ), for i = 1,…,χ = χ( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$). If 1 ∈ Ik, then gkAk, which implies that Ak ⊆ Fix(θ). Thus $[G:Fix(θ)]≤[G:Ak]≤[G:⋂i=1nAi]≤χ!.$

Now assume the equality holds. Then Fix(θ) = AkAi, for all i = 1,…,χ. Since Ai are abelian, Fix(θ) commutes with all elements of A1,…,Aχ. On the other hand, as AkAi, we have a = θ(a) = $\begin{array}{}{a}^{{g}_{i}^{-1}}\end{array}$, for all aAk and i = 1,…,χ, which implies that Fix(θ) commutes with g1,…,gχ as well. Therefore Fix(θ) ⊆ Z(G), as required. □

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

#### Theorem 5.3

Let G be a finite group. Then χ( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) = 2 if and only if G is abelian and [G:Fix(θ)] = 2.

#### Proof

Clearly, α( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) ≥ |G|/2. On the other hand, by Corollary 4.6, α( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) ≤ |G|/2, from which it follows that α( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) = |G|/2. Hence G = I1I2, where (I1,I2) is a bipartition of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ satisfying |I1| = |I2| = |G|/2. Assume 1 ∈ I1. Then, by Theorem 4.5, I1 = g1A1 for some g1G in which A1 = Fix(Ig1θ) is an abelian subgroup of G. Since g1A1, it follows that A1 = Fix(θ). Clearly, A2 = A1 and g2GA1. Now $\begin{array}{}\theta \left(a\right)={a}^{{g}_{1}^{-1}}={a}^{{g}_{2}^{-1}},\end{array}$ for all aA1, from which it follows that g2 commutes with A1. Thus G is abelian. The converse is obvious by Corollary 4.3. □

#### Theorem 5.4

Let G be a finite group. Then χ( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) = 3 if and only if either G is abelian and [G:Fix(θ)] = 3 or G is non-abelian and one of the following holds:

1. G/Z(G)≅ C2× C2 and θ is an inner automorphism;

2. [G:Fix(θ)] = [Fix(θ):Z(G)] = 2; or

3. G has a characteristic subgroup A such that [G:A] = [A:Fix(θ)] = 2, Fix(θ) = Z(G) and there exist elements xAZ(G) and yGA such that θ(x) = xy.

#### Proof

If G is abelian, then we are done by Corollary 4.3. Hence, we assume that G is non-abelian. Let G = I1I2I3 be a tripartition of G in which |I1| ≥ |I2| ≥ |I3| and IigiAi (i = 1,2,3) for some elements giG and abelian subgroups Ai of Fix(Igiθ). Without loss of generality, we may assume that I1 = g1A1. From Theorem 5.1, we know that [G:A1A2A3] ≤ 3! = 6. We distinguish two cases:

Case 1. G = 〈A1,A2,A3〉. Then A1A2A3Z(G) and we must have G/Z(G)≅ C2× C2 or S3. Hence A1A2A3 = Z(G). One can verify that 2 = [G:A1] ≥ [G:A2] ≥ 3 and A1A2. Since GA1A2 and every element of G/Z(G) has order 1, 2 or 3, one can always find an element gGA1A2 such that gAi = giAi, for i = 1,2. Thus, for aAi we have $\begin{array}{}\theta \left(a\right)={a}^{{g}_{i}^{-1}}={a}^{{g}^{-1}}\end{array}$ which implies that θ acts by conjugation via g−1 on 〈A1,A2〉 = G. Hence θ = Ig−1 is an inner automorphism. If G/Z(G)≅ S3, then $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ΓG has a subgraph isomorphic to ΓS3K5K2 with chromatic number 4, a contradiction. Therefore G/Z(G)≅ C2× C2, which gives us part (1).

Case 2. G ≠〈A1,A2,A3〉. Clearly, A2,A3A1. Then A2,A3A1 otherwise A1 = A2 and hence θ(a) = ag1 = ag2 for all aA1. Since g1A1g2A2, it follows that g1 $\begin{array}{}{g}_{2}^{-1}\end{array}$GA1 commutes with A1 so that G = A1g1 $\begin{array}{}{g}_{2}^{-1}\end{array}$〉 is abelian, a contradiction. Hence A2,A3A1, from which together with Tomkinson’s result we must have [G:A1] = 2, [A1:A2] = 2 and A2 = A3. Clearly, g2A2g3A3 = $\begin{array}{}{g}_{1}^{{}^{\prime }}\end{array}$A1 where G = g1A1$\begin{array}{}{g}_{1}^{{}^{\prime }}\end{array}$A1. If g1A1, then A1 = Fix(θ). As A2A1 we have a = θ(a) = $\begin{array}{}{a}^{{g}_{2}^{-1}}\end{array}$ for all aA2, which implies that A2 = Z(G). Hence we obtain part (2). Next assume that g1A1. Then $\begin{array}{}{g}_{1}^{{}^{\prime }}\end{array}$A1 and consequently g2,g3$\begin{array}{}{g}_{1}^{{}^{\prime }}\end{array}$A1 = A1. This implies that A2 ⊆ Fix(θ). As A2A1, we have a = θ(a) = $\begin{array}{}{a}^{{g}_{1}^{-1}}\end{array}$ for all aA2 showing that A2 = Z(G). Assuming g2A1A2, we obtain θ(g2) = g2z for some zA2. Furthermore, $\begin{array}{}{g}_{2}z=\theta \left({g}_{2}\right)={g}_{2}^{{g}_{1}^{-1}}={g}_{2}^{{g}_{1}}\end{array}$ as g2A1. Thus [g2,g1] = z and this yields part (3).

The converse is straightforward. □

We conclude this section with a characterization of complete multipartite-ness of the graphs $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$.

#### Theorem 5.5

The graph $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ is a complete multipartite graph if and only if Fix(Igθ) is abelian for all gG.

#### Proof

First assume that $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ is a complete multipartite graph. From Theorem 4.5, it follows that all maximal abelian subgroups of Fix(Igθ) are disjoint so that Fix(Igθ) is abelian for all gG. Now assume that Fix(Igθ) is abelian for all gG. Let xG. By assumption, xFix(Ixθ) is an independent subset of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$. On the other hand, for yxFix(Ixθ), we have x−1y∉Fix(Ixθ) so that θ(x−1y) ≠(x−1y)x−1. Hence y is adjacent to x. Since, by Theorem 4.5, the sets gFix(Igθ) are maximal independent subsets of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$, it follows the sets gFix(Igθ) partition G and hence $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ is a complete multipartite graph, as required. □

## 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:

#### Question

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

Next, we have defined the non-commuting graph $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ associated to θ-commutators of a group G and established some connections between graph theoretical properties of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ and group theoretical properties of the automorphism θ. For instance, it is proved, among other results, that $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ is complete as a graph if and only if θ is fixed-point-free and that $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ receives minimum number of edges if and only if θ is class preserving.

#### Question

Which other graph theoretical properties of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ 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 $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$. Firstly, a one-to-one correspondence between independent sets and abelian subgroups of Fix(Igθ) is established, where Ig denotes the inner automorphism of G induced by the element gG. This result provided us with another partial answer to the above question: the graph $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ is empty if and only if G is abelian and θ is the identity automorphism. Secondly, a relationship between covers of G by subgroups and the chromatic number χ( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$, i.e. the minimum number of independent sets to cover all vertices of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$, is revealed and a lower bound for χ( $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$) in terms of Fix(θ) is deduced. Also, the structure of G or properties of θ when $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ admits special colorings is obtained.

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

#### Question

How can cliques of $\begin{array}{}{\mathit{\Gamma }}_{G}^{\theta }\end{array}$ be described in terms of G and θ?

Finally, a fundamental question to ask is:

#### Question

Suppose θ1 and θ2 are automorphisms of groups G1 and G2, respectively.

1. Under which conditions on (G1,θ1) and (G2,θ2) 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 G1 = G2)?

2. How are the pairs (G1,θ1) and (G2,θ2) 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?

## Acknowledgement

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

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Accepted: 2017-09-27

Published Online: 2017-12-29

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1530–1538, ISSN (Online) 2391-5455,

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