Normal edge-transitive Cayley graphs on non-abelian simple groups

Abstract

Let Γ be a Cayley graph on a finite group G, and let $N_{\mathrm{Aut}\left(\Gamma \right)}\left(R\right(G\left)\right)$ be the normalizer of $R\left(G\right)$ (the right regular representation of G) in the full automorphism group $\mathrm{Aut}\left(\Gamma \right)$ of Γ. We say that Γ is a normal Cayley graph on G if $N_{\mathrm{Aut}\left(\Gamma \right)}\left(R\right(G\left)\right)=\mathrm{Aut}\left(\Gamma \right)$, and that Γ is a normal edge-transitive Cayley graph on G if $N_{\mathrm{Aut}\left(\Gamma \right)}\left(R\right(G\left)\right)$ acts transitively on the edge set of Γ. In 1999, Praeger proved that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency 3 is normal. As an extension of this, in this paper, we prove that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency p is normal for each prime p. This, however, is not true for composite valency. We give a method to construct connected normal edge-transitive but non-normal Cayley graphs of certain groups, and using this, we prove that if G is either ${\mathrm{PSL}}_2\left(q\right)$ for an odd prime $q\ge 5$, or $A_n$ for $n\ge 5$, then there exists a connected normal edge-transitive but non-normal 8-valent Cayley graph of G.