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

Abstract

A graph $Γ$ is called $(G, s)$-arc-transitive if $G \le \mathrm{Aut}(Γ)$ is transitive on the set of vertices of $Γ$ and the set of $s$-arcs of $Γ$, where for an integer $s \ge 1$ an $s$-arc of $Γ$ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots,v_s)$ of $Γ$ 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 $1 \le i \le s-1$. $Γ$ is called 2-transitive if it is $(\mathrm{Aut}(Γ), 2)$-arc-transitive but not $(\mathrm{Aut}(Γ), 3)$-arc-transitive. A Cayley graph $Γ$ of a group $G$ is called normal if $G$ is normal in $\mathrm{Aut}(Γ)$ and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if $Γ$ is a tetravalent 2-transitive Cayley graph of a finite simple group $G$, then either $Γ$ 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 $Γ$ 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.