Automorphism group of the complete alternating group graph

Abstract

Let $S_n$ and $A_n$ denote the symmetric group and alternating group of degree $n$ with $n\geq 3$, respectively. Let $S$ be the set of all $3$-cycles in $S_n$. The \emph{complete alternating group graph}, denoted by $CAG_n$, is defined as the Cayley graph $\mathrm{Cay}(A_n,S)$ on $A_n$ with respect to $S$. In this paper, we show that $CAG_n$ ($n\geq 4$) is not a normal Cayley graph. Furthermore, the automorphism group of $CAG_n$ for $n\geq 5$ is obtained, which equals to $\mathrm{Aut}(CAG_n)=(R(A_n)\rtimes \mathrm{Inn}(S_n))\rtimes \mathbb{Z}_2\cong (A_n\rtimes S_n)\rtimes \mathbb{Z}_2$, where $R(A_n)$ is the right regular representation of $A_n$, $\mathrm{Inn}(S_n)$ is the inner automorphism group of $S_n$, and $\mathbb{Z}_2=\langle h\rangle$, where $h$ is the map $\alpha\mapsto\alpha^{-1}$ ($\forall \alpha\in A_n$).