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On θ-commutators and the corresponding non-commuting graphs

  • S. Shalchi , A. Erfanian EMAIL logo and M. Farrokhi DG
Published/Copyright: December 29, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

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.

Keywords: Commutator; Automorphism; Independence number; Chromatic number; Multipartite graph
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−1xθ(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−1xθ(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−1xθ−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 v ∈ V(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 x ∈ G, denoted by CGθ (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 CGθ (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. CGθ (1) = Fix(θ);

  2. x−1 CGθ (x) is a subgroup of G for all x ∈ G;

  3. CGθ (x) is a subgroup of G if and only if x2 ∈ Fix(θ); and

  4. | CGθ (x)| divides |G|.

Proof

  1. It is obvious.

  2. Let y,z ∈ CGθ (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 ∈ CGθ (x), that is, (x−1y)(x−1z) ∈ x−1 CGθ (x). On the other hand, xy−1x ∈ CGθ (x), from which it follows that (x−1y)−1 = x−1xy−1x ∈ x−1 CGθ (x). Therefore, x−1 CGθ (x) is a subgroup of G.

  3. From (2) it follows that CGθ (x) is a subgroup of G if and only if x−1 ∈ CGθ (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 x ∈ Zθ(G), then [x,θ x−1y] = 1 for all y ∈ G, from which it follows that θ(y) = xyx−1 for all y ∈ G. Hence θ = Ix−1 ∈ Inn(G). Conversely, if θ = Ix−1 for some x ∈ G, then θ(y) = xyx−1 so that [x,θ y] = 1 for all y ∈ G. Thus x ∈ Zθ(G).

  2. Assume x ∈ Zθ(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

    [x,θy]=1⇔x−1θ(y)−1θ(x)y=1⇔x−1(g−1yg)−1xy=1⇔gxy=ygx.

Hence gx ∈ Z(G) and consequently Zθ(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 ⊆ Zθ(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 ΓGθ , is a simple undirected graph whose vertices are elements of G∖ Zθ(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:V(ΓGθ)⟶V(ΓGIgθ) defined by Θ(x) = g−1x, for all x ∈ V( ΓGθ ), presents an isomorphism between ΓGθ and ΓGIgθ. 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 ΓGθ such that [x,θ y] = 1 for all y ∈ G ∖ X, then |X| = |G|/2.

Proof

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−1y) = (x−1y)x−1 for all y ∈ G ∖ X. Hence θ = Ix−1, which implies that Zθ(G) = Z(G)x by Lemma 2.4. But then x ∈ Zθ(G), which is a contradiction. Thus, |G ∖ X| ≤ |〈x−1(G ∖ X)〉| ≤ |G|/2 and consequently |X| = |G|/2, as required. □

Corollary 3.3

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

Theorem 3.4

We have diam( ΓGθ ) ≤ 2.

Proof

If diam( ΓGθ ) > 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 y ∈ N(x), that is, x and y are adjacent, a contradiction. □

Theorem 3.5

We have girth( ΓGθ ) ≤ 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( ΓGθ ) > 3. We show that girth( ΓGθ ) = 4. Since girth( ΓGθ ) > 3 N(x1)∩ N(x2) = ∅ for every edge {x1,x2} ∈ E( ΓGθ ). 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 y ∈ N(xi) (i = 1,2) we have G = N(xi)∪̇ N(y) as well, it follows that N(y) = N(x3−i). Therefore, ΓGθ is a complete bipartite graph with the bipartition (N(x1),N(x2)), which yields girth( ΓGθ ) = 4. To prove the second part, assume girth( ΓGθ ) = 4. Then G = N(x)∪̇ N(y) is an equally partition for each {x,y} ∈ E( ΓGθ ). 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,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 g2 ∈ N(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 ΓGθ is complete if and only if θ is a fixed-point-free automorphism of G.

Proof

Assume ΓGθ is a complete graph. Then θ is non-inner and [x,θ y]≠1 for all vertices x and y in ΓGθ . If 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)∉ gG for all g ∈ G ∖{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 |E(ΓGθ)|=12|G|(|G|−k(G)) and we are done. Hence, assume that θ is a non-inner automorphism. By Lemma 2.4, we observe that V( ΓGθ ) = 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 E(ΓGθ)c=12|G|(|G|−1)−|E(ΓGθ)|, the result follows. □

4 Independent sets

In the section, we give a description of independent subsets of the graph ΓGθ , which enables us to compute the independence number of ΓGθ . We begin with the easier case of abelian groups. Indeed, utilizing the following lemma, we can determine the structure of ΓGθ 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 x∼ y if and only if xy≠ yx.

  2. If x,y are in different cosets of Fix(θ), then x∼ y 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 ΓGθ ≅ 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 ΓGθ . The following key lemma illustrates this relationship in a much suitable form.

Lemma 4.4

Let I be an independent subset of ΓGθ . 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,w ∈ I. 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 ab ∈ I for some a,b ∈ I. For x ∈ I 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,y ∈ I. Therefore I is abelian, which is a contradiction. □

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

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 ΓGθ 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(Igθ).

Proof

First observe that if I is an independent subset of ΓGθ , then A = 〈I−1I〉 is an abelian subgroup of Fix(Igθ) and I ⊆ gA for every g ∈ I 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 ΓGθ . Clearly, every subset of gA is an independent set in ΓGθ . To complete the proof, we must show that any two independent sets xA ⊆ yB in which A and B are maximal abelian subgroups of Fix(Ixθ) and Fix(Iyθ), respectively, coincide. First observe that A ⊆ x−1yB so that x−1y = b0 ∈ B. Hence A ⊆ B. Now, for every b ∈ B, we have θ(b)=by−1=bb0−1x−1=bx−1, 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 ΓGθ is empty if and only if G is abelian and θ is the identity automorphism, in which case ΓGθ 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( ΓGθ ) ≤ |G|/2 and the equality holds if and only if Fix(Igθ) is an abelian subgroup of G of index 2 for some element g ∈ G.

Proof

If α( ΓGθ ) > |G|/2, then Corollary 4.6 gives an element g ∈ G such that G = Fix(Igθ) is abelian. But then θ = Ig−1 = I, which is a contradiction. Hence α( ΓGθ ) ≤ |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 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 ΓGθ is a left coset to an abelian group, the evaluation of the chromatic number of ΓGθ 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 [G:⋂i=1nHi]≤n!.

Tomkinson’s theorem has the following immediate result connecting the chromatic number of ΓGθ 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 Ii ⊆ giAi and Ai is an abelian subgroup of Fix(Igiθ), for i = 1,…,χ = χ( ΓGθ ). If 1 ∈ Ik, then gk ∈ Ak, which implies that Ak ⊆ Fix(θ). Thus

[G:Fix(θ)]≤[G:Ak]≤[G:⋂i=1nAi]≤χ!.

Now assume the equality holds. Then Fix(θ) = Ak ⊆ Ai, for all i = 1,…,χ. Since Ai are abelian, Fix(θ) commutes with all elements of A1,…,Aχ. On the other hand, as Ak ⊆ Ai, we have a = θ(a) = agi−1 , for all a ∈ Ak 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( ΓGθ ) = 2 if and only if G is abelian and [G:Fix(θ)] = 2.

Proof

Clearly, α( ΓGθ ) ≥ |G|/2. On the other hand, by Corollary 4.6, α( ΓGθ ) ≤ |G|/2, from which it follows that α( ΓGθ ) = |G|/2. Hence G = I1∪ I2, where (I1,I2) is a bipartition of ΓGθ satisfying |I1| = |I2| = |G|/2. Assume 1 ∈ I1. Then, by Theorem 4.5, I1 = g1A1 for some g1 ∈ G in which A1 = Fix(Ig1θ) is an abelian subgroup of G. Since g1 ∈ A1, it follows that A1 = Fix(θ). Clearly, A2 = A1 and g2 ∈ G ∖ A1. Now θ(a)=ag1−1=ag2−1, for all a ∈ A1, 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( ΓGθ ) = 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 x ∈ A ∖ Z(G) and y ∈ G ∖ A 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 = I1∪ I2∪ I3 be a tripartition of G in which |I1| ≥ |I2| ≥ |I3| and Ii ⊆ giAi (i = 1,2,3) for some elements gi ∈ G 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:A1∩ A2∩ A3] ≤ 3! = 6. We distinguish two cases:

Case 1. G = 〈A1,A2,A3〉. Then A1 ∩ A2 ∩ A3 ⊆ Z(G) and we must have G/Z(G)≅ C2× C2 or S3. Hence A1 ∩ A2 ∩ A3 = Z(G). One can verify that 2 = [G:A1] ≥ [G:A2] ≥ 3 and A1 ≠ A2. Since G ≠ A1 ∪ A2 and every element of G/Z(G) has order 1, 2 or 3, one can always find an element g ∈ G ∖ A1 ∪ A2 such that gAi = giAi, for i = 1,2. Thus, for a ∈ Ai we have θ(a)=agi−1=ag−1 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 ΓGθ ≅ΓG has a subgraph isomorphic to ΓS3≅ K5 ∖ K2 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,A3 ⊆ A1. Then A2,A3 ⊂ A1 otherwise A1 = A2 and hence θ(a) = ag1 = ag2 for all a ∈ A1. Since g1A1 ≠ g2A2, it follows that g1 g2−1 ∈ G ∖ A1 commutes with A1 so that G = A1〈g1 g2−1 〉 is abelian, a contradiction. Hence A2,A3 ⊂ A1, from which together with Tomkinson’s result we must have [G:A1] = 2, [A1:A2] = 2 and A2 = A3. Clearly, g2A2 ∪ g3A3 = g1′ A1 where G = g1A1 ∪ g1′ A1. If g1 ∈ A1, then A1 = Fix(θ). As A2 ⊆ A1 we have a = θ(a) = ag2−1 for all a ∈ A2, which implies that A2 = Z(G). Hence we obtain part (2). Next assume that g1 ∉ A1. Then g1′ ∈ A1 and consequently g2,g3 ∈ g1′ A1 = A1. This implies that A2 ⊆ Fix(θ). As A2 ⊆ A1, we have a = θ(a) = ag1−1 for all a ∈ A2 showing that A2 = Z(G). Assuming g2 ∈ A1 ∖ A2, we obtain θ(g2) = g2z for some z ∈ A2. Furthermore, g2z=θ(g2)=g2g1−1=g2g1 as g2 ∈ A1. 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 ΓGθ .

Theorem 5.5

The graph ΓGθ is a complete multipartite graph if and only if Fix(Igθ) is abelian for all g ∈ G.

Proof

First assume that ΓGθ 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 g ∈ G. Now assume that Fix(Igθ) is abelian for all g ∈ G. Let x ∈ G. By assumption, xFix(Ixθ) is an independent subset of ΓGθ . On the other hand, for y∉ xFix(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 ΓGθ , it follows the sets gFix(Igθ) partition G and hence ΓGθ 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 ΓGθ associated to θ-commutators of a group G and established some connections between graph theoretical properties of ΓGθ and group theoretical properties of the automorphism θ. For instance, it is proved, among other results, that ΓGθ is complete as a graph if and only if θ is fixed-point-free and that ΓGθ receives minimum number of edges if and only if θ is class preserving.

Question

Which other graph theoretical properties of ΓGθ 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 ΓGθ . 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 g ∈ G. This result provided us with another partial answer to the above question: the graph ΓGθ 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( ΓGθ ) of ΓGθ , i.e. the minimum number of independent sets to cover all vertices of ΓGθ , is revealed and a lower bound for χ( ΓGθ ) in terms of Fix(θ) is deduced. Also, the structure of G or properties of θ when ΓGθ admits special colorings is obtained.

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

Question

How can cliques of ΓGθ 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 ΓG1θ1andΓG2θ2 isomorphic (in particular when G1 = G2)?

  2. How are the pairs (G1,θ1) and (G2,θ2) related, provided that ΓG1θ1andΓG2θ2 are isomorphic?

Acknowledgement

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

References

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[3] Gorenstein D., Finite Groups, Harper & Row, Publishers, New York–London, 1968.Search in Google Scholar

[4] Hegarty P., The absolute center of a group, J. Algebra, 1994, 169, 929–935.10.1006/jabr.1994.1318Search in Google Scholar

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Received: 2017-5-5
Accepted: 2017-9-27
Published Online: 2017-12-29

© 2017 Shalchi et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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Keywords for this article
Commutator; Automorphism; Independence number; Chromatic number; Multipartite graph
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  241. Topical Issue on Topological and Algebraic Genericity in Infinite Dimensional Spaces
  242. Linear dynamics of semigroups generated by differential operators
  243. Special Issue on Recent Developments in Differential Equations
  244. Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces
  245. Special Issue on Recent Developments in Differential Equations
  246. Determination of a diffusion coefficient in a quasilinear parabolic equation
  247. Special Issue on Recent Developments in Differential Equations
  248. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables
  249. Special Issue on Recent Developments in Differential Equations
  250. A nonlinear plate control without linearization
  251. Special Issue on Recent Developments in Differential Equations
  252. Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations
  253. Special Issue on Recent Developments in Differential Equations
  254. Inverse problem for a physiologically structured population model with variable-effort harvesting
  255. Special Issue on Recent Developments in Differential Equations
  256. Existence of solutions for delay evolution equations with nonlocal conditions
  257. Special Issue on Recent Developments in Differential Equations
  258. Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities
  259. Special Issue on Recent Developments in Differential Equations
  260. Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application
  261. Special Issue on Recent Developments in Differential Equations
  262. Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations
  263. Special Issue on Recent Developments in Differential Equations
  264. Integro-differential systems with variable exponents of nonlinearity
  265. Special Issue on Recent Developments in Differential Equations
  266. Elliptic operators on refined Sobolev scales on vector bundles
  267. Special Issue on Recent Developments in Differential Equations
  268. Multiplicity solutions of a class fractional Schrödinger equations
  269. Special Issue on Recent Developments in Differential Equations
  270. Determining of right-hand side of higher order ultraparabolic equation
  271. Special Issue on Recent Developments in Differential Equations
  272. Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain
  273. Topical Issue on Metaheuristics - Methods and Applications
  274. Learnheuristics: hybridizing metaheuristics with machine learning for optimization with dynamic inputs
  275. Topical Issue on Metaheuristics - Methods and Applications
  276. Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison
  277. Topical Issue on Cyber-security Mathematics
  278. Monomial codes seen as invariant subspaces
  279. Topical Issue on Cyber-security Mathematics
  280. Expert knowledge and data analysis for detecting advanced persistent threats
  281. Topical Issue on Cyber-security Mathematics
  282. Feedback equivalence of convolutional codes over finite rings
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