Arc-transitive pentavalent Cayley graphs with soluble vertex stabilizer on finite nonabelian simple groups

TL;DR

This paper classifies connected arc-transitive pentavalent Cayley graphs with soluble vertex stabilizers on finite nonabelian simple groups, showing most are normal except for specific cases, and constructs a novel example on _{79}.

Contribution

It proves the normality of such Cayley graphs except for two specific alternating groups and constructs the first known non-normal example on _{79}.

Findings

Most such graphs are normal.

Two exceptions are _{39} and _{79}.

Constructed a non-normal 3-arc-transitive Cayley graph on _{79}.

Abstract

A Cayley graph $\Ga=\Cay(G,S)$ is said to be normal if $G$ is normal in $\Aut\Ga$. The concept of normal Cayley graphs was first proposed by M.Y.Xu in [Discrete Math. 182, 309-319, 1998] and it plays an important role in determining the full automorphism groups of Cayley graphs. In this paper, we investigate the normality problem of the connected arc-transitive pentavalent Cayley graphs with soluble vertex stabilizer on finite nonabelian simple groups. We prove that all such graphs $\Ga$ are either normal or $G=\A_{39}$ or $\A_{79}$. Further, a connected arc-transitive pentavalent Cayley graph on $\A_{79}$ is constructed. To our knowledge, this is the first known example of pentavalent 3-arc-transitive Cayley graph on finite nonabelian simple group which is non-normal.