THE AUTOMORPHISM GROUP OF COMMUTING GRAPH OF A FINITE GROUP

چکیده مقاله

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 6= 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.