Mahsa Mirzargar, Peter P. Pach, and A. R. Ashrafi University of Kashan, E\"{o}tv\"{o}s Lor\'{a}nd University, University of Kashan

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 $x, y \in X$ ($x \ne y$) joined by an edge whenever they commute. In the case that $X = G$, this graph is named commuting graph of $G$, denoted by $\Delta(G)$. The aim of this paper is to study the automorphism group of the commuting graph. It is proved that ${\rm Aut}(\Delta(G))$ is abelian if and only if $|G| \leq 2$; $|{\rm Aut}(\Delta(G))|$ is of prime power if and only if $|G| \leq 2$, and $|{\rm Aut}(\Delta(G))|$ is square-free if and only if $|G| \leq 3$. Some new graphs that are useful in studying the automorphism group of $\Delta(G)$ are presented and their main properties are investigated.

Keywords : commuting graph, automorphism group, extra special group