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

DOI10.1017/S0004972720001446arXiv1611.06308MaRDI QIDQ6279898

Xin Gui Fang, Jie Wang, Sanming Zhou

Publication date: 19 November 2016

Abstract: A graph $Gamma$ is called $(G,s)$-arc-transitive if $GlemathrmAut(Gamma)$ is transitive on the set of vertices of $Gamma$ and the set of $s$-arcs of $Gamma$, where for an integer $sge1$ 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 $1leiles$ and $v_{i-1}e{v}_{i+1}$ for $1leiles-1$. $Gamma$ is called 2-transitive if it is $(mathrmAut(Gamma),2)$-arc-transitive but not $(mathrmAut(Gamma),3)$-arc-transitive. A Cayley graph $Gamma$ of a group $G$ is called normal if $G$ is normal in $mathrmAut(Gamma)$ 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.

Mathematics Subject Classification ID

Finite automorphism groups of algebraic, geometric, or combinatorial structures (20B25) Graphs and abstract algebra (groups, rings, fields, etc.) (05C25)

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