On 2-Arc-Transitive Cayley Graphs of Nonabelian Characteristically Simple Groups

Abstract

The classification of symmetric Cayley graphs has attracted considerable attention. Results have been obtained for finite abelian groups and nonabelian simple groups. However, the problem remains open for finite nonabelian characteristically simple groups, although some results on normality for small valencies are known. In this paper, we determine the possible structures of automorphism groups $X$ of connected 2-arc-transitive Cayley graphs $Γ$ of $G\cong T^k$, where $T$ is a finite nonabelian simple group and $k\geqslant2$. In the quasiprimitive case, we prove that $G\unlhd X$, $Γ$ is complete, or $X$ is of product action type with restricted parameters. For the non-quasiprimitive case, we determine the possible structures of the quasiprimitive normal quotients of $Γ$.