Automorphism Groups of a Family of Cayley Graphs of the Alternating Groups

Abstract

Let $A_n$ be the alternating group of degree $n$ with $n\ge 5$. Set $S=\{(1ij),(1ji)\mid 2\le i,j\le n,i\ne j\}$. In this paper, it is shown that the full automorphism group of the Cayley graph $\mathrm{C}\mathrm{a}\mathrm{y}(A_n,S)$ is the semi-product $R(A_n)⋊\mathrm{A}\mathrm{u}\mathrm{t}(A_n,S)$, where $R(A_n)$ is the right regular representation of $A_n$ and $\mathrm{A}\mathrm{u}\mathrm{t}(A_n,S)=\{\varphi \in \mathrm{A}\mathrm{u}\mathrm{t}(A_n)\mid S^{\varphi }=S\}\cong {\mathrm{S}}_{\mathrm{n}-1}$.