Symmetric Cayley graphs on non-abelian simple groups of valency 7

Abstract

Let $\Gamma$ be a connected $7$-valent symmetric Cayley graph on a finite non-abelian simple group $G$. If $\Gamma$ is not normal, Li {\em et al.} [On 7-valent symmetric Cayley graphs of finite simple groups, J. Algebraic Combin. 56 (2022) 1097-1118] characterised the group pairs $(\mathrm{soc}(\mathrm{Aut}(\Gamma )/K),GK/K)$, where $K$ is a maximal intransitive normal subgroup of $\mathrm{Aut}(\Gamma )$. In this paper, we improve this result by proving that if $\Gamma$ is not normal, then $\mathrm{Aut}(\Gamma )$ contains an arc-transitive non-abelian simple normal subgroup $T$ such that $G<T$ and $(T,G)=({\mathrm{A}}_n,{\mathrm{A}}_{n-1})$ with $n=7$, $3\cdot 7$, $3^2\cdot 7$, $2^2\cdot 3\cdot 7$, $2^3\cdot 3\cdot 7$, $2^3\cdot 3^2\cdot 5\cdot 7$, $2^4\cdot 3^2\cdot 5\cdot 7$, $2^6\cdot 3\cdot 7$, $2^7\cdot 3\cdot 7$, $2^6\cdot 3^2\cdot 7$, $2^6\cdot 3^4\cdot 5^2\cdot 7$, $2^8\cdot 3^4\cdot 5^2\cdot 7$, $2^7\cdot 3^4\cdot 5^2\cdot 7$, $2^{10}\cdot 3^2\cdot 7$, $2^{24}\cdot 3^2\cdot 7$. Furthermore, $\mathrm{soc}(\mathrm{Aut}(\Gamma )/R)=(T\times R)/R$, where $R$ is the largest solvable normal subgroup of $\mathrm{Aut}(\Gamma )$.