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  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  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 .
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.
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
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〉≅ C3 and θ is the nontrivial automorphism of G, then Zθ(G) = ∅ and
Let G be a group and θ be an automorphism of G. Then
(1) = Fix(θ);
(x) is a subgroup of G for all x ∈ G; (x) is a subgroup of G if and only if x2 ∈ Fix(θ); and
(x)| divides |G|.
- It is obvious.
- Let y,z ∈
(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 ∈ (x), that is, (x−1y)(x−1z) ∈ x−1 (x). On the other hand, xy−1x ∈ (x), from which it follows that (x−1y)−1 = x−1xy−1x ∈ x−1 (x). Therefore, x−1 (x) is a subgroup of G.
- From (2) it follows that
(x) is a subgroup of G if and only if x−1 ∈ (x) and this holds if and only if x2 ∈ Fix(θ).
- It follows from (2). □
Let G be a group and θ be an automorphism of G. Then
- Zθ(G)≠∅ if and only if θ ∈ Inn(G); and
- 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.
- 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).
- 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
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.
Let G be a group and θ be an automorphism of G. The θ-non-commuting graph of G, denoted by
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
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
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. □
For every x ∈ G we have deg(x) ≥ |G|/2.
We have diam(
We have girth(
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.
An automorphism θ of G is called class preserving if θ(gG) = gG for every conjugacy class gG of 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
from which, in conjunction with the fact that
4 Independent sets
In the section, we give a description of independent subsets of the graph
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−1x ∈ Fix(θ). Thus
- We have y−1x∉Fix(θ) and consequently
as required. □
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
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
- I−1I is an abelian set; and
- if I is non-abelian, then I is a product-free set.
- Let x,y,z,w ∈ I. Then
Similarly, we have θ(z−1w) = (z−1w)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 have
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
Let G be a group and I be a subset of G. Then I is an independent (resp. a maximal independent) subset of
First observe that if I is an independent subset of
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(
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 ). Let G be covered by some cosets giHi for i = 1,…,n. If the cover is irredundant, then
Tomkinson’s theorem has the following immediate result connecting the chromatic number of
For any group G, we have
and the equality holds only if Fix(θ) ⊆ Z(G).
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(
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) =
In the sequel, we shall characterize those graphs having small chromatic numbers.
Let G be a finite group. Then χ(
Let G be a finite group. Then χ(
- G/Z(G)≅ C2× C2 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) = xy.
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
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
The converse is straightforward. □
We conclude this section with a characterization of complete multipartite-ness of the graphs
First assume that
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
Which other graph theoretical properties of
The rest of paper is devoted to the study of independent sets in the graph
In contrast to our investigations on independent sets one may ask:
How can cliques of
Finally, a fundamental question to ask is:
Suppose θ1 and θ2 are automorphisms of groups G1 and G2, respectively.
- Under which conditions on (G1,θ1) and (G2,θ2) are two graphs
isomorphic (in particular when G1 = G2)?
- How are the pairs (G1,θ1) and (G2,θ2) related, provided that
The authors would like to thank the referees for their kind comments and suggestions.
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Gorenstein D., Finite Groups, Harper & Row, Publishers, New York–London, 1968.
Robinson D.J.S., A Course in the Theory of Groups, Second Edition, Springer-Verlag, New York, 1996.