Classification of tetravalent $2$-transitive non-normal Cayley graphs of finite simple groups

TL;DR

This paper classifies tetravalent 2-transitive non-normal Cayley graphs of finite simple groups, identifying the specific groups that produce such graphs and determining their automorphism groups.

Contribution

It proves that only the group M11 yields connected tetravalent 2-transitive non-normal Cayley graphs among four candidate groups, and fully characterizes these graphs.

Findings

Only M11 produces such graphs.

Exactly two non-isomorphic graphs are identified.

Automorphism groups of these graphs are determined.

Abstract

A graph $\Gamma$ is called $(G, s)$-arc-transitive if $G \le \mathrm{Aut}(\Gamma)$ is transitive on the set of vertices of $\Gamma$ and the set of $s$-arcs of $\Gamma$, where for an integer $s \ge 1$ an $s$-arc of $\Gamma$ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots,v_s)$ of $\Gamma$ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$. $\Gamma$ is called 2-transitive if it is $(\mathrm{Aut}(\Gamma), 2)$-arc-transitive but not $(\mathrm{Aut}(\Gamma), 3)$-arc-transitive. A Cayley graph $\Gamma$ of a group $G$ is called normal if $G$ is normal in $\mathrm{Aut}(\Gamma)$ and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if $\Gamma$ is a tetravalent 2-transitive Cayley graph of a finite simple group $G$, then either $\Gamma$ is normal or $G$ is one of the groups $\mathrm{PSL}_2(11)$, $M_{11}$, $M_{23}$ and $A_{11}$. However, it was unknown whether $\Gamma$ is normal when $G$ is one of these four groups. In the present paper we answer this question by proving that among these four groups only $M_{11}$ produces connected tetravalent 2-transitive non-normal Cayley graphs. We prove further that there are exactly two such graphs which are non-isomorphic and both determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.