The automorphism group of commuting graph of a finite group
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Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and x, y ∈ X (x ≠ y) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by △(G). The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(△(G)) is abelian if and only if |G| ≤ 2 |Aut(△(G))| is of prime power if and only if |G| ≤ 2, and |Aut(△(G))| is square-free if and only if |G| ≤ 3. Some new graphs that are useful in studying the automorphism group of △(G) are presented and their main properties are investigated.
Abstract
Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and x, y ∈ X (x ≠ y) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by △(G). The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(△(G)) is abelian if and only if |G| ≤ 2 |Aut(△(G))| is of prime power if and only if |G| ≤ 2, and |Aut(△(G))| is square-free if and only if |G| ≤ 3. Some new graphs that are useful in studying the automorphism group of △(G) are presented and their main properties are investigated.
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