On θ-commutators and the corresponding non-commuting graphs

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.


Introduction
The commutator of two elements x and y of a group G is de ned usually as [x, y] := x − y − xy. The in uence of commutators in the theory of groups is inevitable and the analogy of computations encouraged some authors to de ne and study modi cations of the ordinary commutators to include automorphisms or more generally endormorphisms of the underlying groups. The rst 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 − 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 − Iy(x), Iy being the inner automorphism associate to y, one may generalize them in a natural way to [x, θ] = x − 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 rst in Gorenstein's book [3, p. 33] while it rst appears in practice in the pioneering papers [4,5] of Hegarty. According to Ree's de nition 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 − y − xθ(y). One observes that [x, y] θ = if and only if θ(y) = y x . Hence [x, y] θ = does not guarantee in general that [y, x] θ = . The aim of this paper is to introduce a new generalization of commutators, as a minor modi cation to that of Ree, in order to obtain a new commutator behaving more like the ordinary commutators. Indeed, we de ne the conjugation of x by y via θ as θ(y) − θ(x)y, which is simply the image of y − xθ − (y), the conjugate of x by y via θ − in the sense of Ree's, under θ. Hence the corresponding commutators, we call them the θ-commutators, will be [x, θ y] := x − θ(y) − θ(x)y and we observe that [x, θ y] = if and only if [y, θ x] = . This property of θcommutators, 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 de ne the θ-noncommuting graph associated to θ-commutators of a group and describe some of its basic properties and its correlations with other notions, namely xed-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].

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 de ned as [x, θ y] := x − θ(y) − θ(x)y. Also, the autocommutator of x and θ is known to be [θ, x] − = [x, θ] := x − θ(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
The θ-centralizer of elements as well as the θ-center of a group can be de ned analogously as follows:

De nition 2.2. Let G be a group and θ be an automorphism of G. The θ-centralizer of an element x
Utilizing θ-centralizers, the θ-center of G is de ned 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 and θ is the nontrivial automorphism of G, then Z θ (G) = ∅ and C θ 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
and this holds if and only if x ∈ Fix(θ).

As a result, Z θ (G) is a subgroup of G if and only if θ is the identity automorphism.
Proof Hence gx ∈ Z(G) and consequently Z θ (G) ⊆ Z(G)g − . Conversely, if x ∈ Z(G)g − , then gx ∈ Z(G) and the above argument shows that [x, θ y] = for all y ∈ G. Therefore, Z(G)g − ⊆ 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.

The θ-non-commuting graphs
Having de ned the θ-commutators, we can now de ne 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.
De nition 3.1. Let G be a group and θ be an automorphism of G. The θ-non-commuting graph of G, denoted by Γ θ G , 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] ̸ = .
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 : , presents an isomorphism between Γ θ G and Γ Ig θ G . 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. Proof. Assume that [x, θ y] = for all y ∈ G \ X. We claim that x − (G \ X) is a proper subgroup of G. Suppose on the contrary that x − (G \ X) = G. One can easily see that θ(x − y) = (x − y) x − for all y ∈ G \ X. Hence θ = I x − , 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 − (G \ X) | ≤ |G|/ and consequently |X| = |G|/ , as required.
In what follows, we obtain some criterion for an automorphism to be xed-point-free (or regular) or classpreserving. Remind that an automorphism θ of G is xed-point-free if the only xed point of θ is the trivial element, that is, Fix(θ) = is the trivial subgroup of G.

Theorem 3.6. The graph Γ θ G is complete if and only if θ is a xed-point-free automorphism of G.
Proof. Assume Γ θ G is a complete graph. Then θ is non-inner and [x, θ y] ̸ = for all vertices x and y in Γ θ G . If θ is not xed-point-free, then there exists an element x ∈ G such that θ(x) = x. But then [x, θ ] = , which is impossible. Thus θ is xed-point-free. Conversely, suppose that θ is a xed-point-free automorphism. By [6, 10.5 for all distinct vertices x and y, which implies that [x, θ y] ̸ = , 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 G for every conjugacy class g G of G. Proof. If θ is an inner automorphism, then |E(Γ θ G )| = |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(Γ θ G ) = G. Then from which, in conjunction with the fact that E (Γ θ G ) c = |G|(|G| − ) − |E(Γ θ G )|, the result follows.

Independent sets
In the section, we give a description of independent subsets of the graph Γ θ G , which enables us to compute the independence number of Γ θ G . We begin with the easier case of abelian groups. Indeed, utilizing the following lemma, we can determine the structure of Γ θ G precisely when G is an abelian group. Proof.
(1) By assumption y − x ∈ Fix(θ). Thus (2) We have y − x / ∈ Fix(θ) and consequently as required. As we have seen in Lemma 4.1, there is a close relationship between independence and commutativity of vertices in the graph Γ θ G . The following key lemma illustrates this relationship in a much suitable form.

Lemma 4.4. Let I be an independent subset of Γ θ G . Then (1) I − I 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 Similarly, we have θ(z − w) = (z − w) y − , 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 from which we get θ(x) = x. Hence [x, y] = [x, θ y] = 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 Γ θ G .

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 Γ θ G 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(Ig θ).
Proof. First observe that if I is an independent subset of Γ θ G , then A = I − I is an abelian subgroup of Fix(Ig 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(Ig θ) whenever I is a maximal independent subset of Γ θ G . Clearly, every subset of gA is an independent set in Γ θ G . 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 − yB so that , which implies that B ⊆ Fix(Ixθ). The maximality of A yields A = B and consequently xA = yB, as required.
Corollary 4.7. The graph Γ θ G is empty if and only if G is abelian and θ is the identity automorphism, in which case Γ θ G is the null graph.

Corollary 4.8. Let G be a nite group and θ be an automorphism of G. If either G is non-abelian or θ is nonidentity, then α(Γ θ G ) ≤ |G|/ and the equality holds if and only if Fix(Ig θ) is an abelian subgroup of G of index for some element g ∈ G.
Proof. If α(Γ θ G ) > |G|/ , then Corollary 4.6 gives an element g ∈ G such that G = Fix(Ig θ) is abelian. But then θ = I g − = I, which is a contradiction. Hence α(Γ θ G ) ≤ |G|/ . Now if the equality holds, by using Corollary 4.6 once more, we observe that Fix(Ig θ) is an abelian subgroup of G of index for some g ∈ G. The converse is straightforward.

Chromatic number
The results of section 4 on the independence number can be applied to study the chromatic number of θnon-commuting graphs. Since every maximal independent set in Γ θ G is a left coset to an abelian group, the evaluation of the chromatic number of Γ θ G 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 g i H i for i = , . . . , n. If the cover is irredundant, then [G : n i= H i ] ≤ n!.
Tomkinson's theorem has the following immediate result connecting the chromatic number of Γ θ G to the number of xed points of θ.