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 $n\ge 3$, respectively. Let $S$ be the set of all $3$-cycles in $S_n$. The \emph{complete alternating group graph}, denoted by $CA{G}_n$, is defined as the Cayley graph $\mathrm{Cay}(A_n,S)$ on $A_n$ with respect to $S$. In this paper, we show that $CA{G}_n$ ($n\ge 4$) is not a normal Cayley graph. Furthermore, the automorphism group of $CA{G}_n$ for $n\ge 5$ is obtained, which equals to $\mathrm{Aut}(CA{G}_n)=(R(A_n)⋊\mathrm{Inn}(S_n))⋊{\mathbb{Z}}_2\cong (A_n⋊S_n)⋊{\mathbb{Z}}_2$, 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 ${\mathbb{Z}}_2=⟨h⟩$, where $h$ is the map $\alpha \mapsto {\alpha }^{-1}$ ($\forall \alpha \in A_n$).

9 pages, 1 figure

Cited by in corpus (2)

• The second largest eigenvalues of some Cayley graphs on alternating groups
• Automorphism groups of a class of cubic Cayley graphs on symmetric groups