Bounding the size of a vertex-stabiliser in a finite vertex-transitive graph

Abstract

In this paper we discuss a method for bounding the size of the stabiliser of a vertex in a $G$-vertex-transitive graph $Γ$. In the main result the group $G$ is quasiprimitive or biquasiprimitive on the vertices of $Γ$, and we obtain a genuine reduction to the case where $G$ is a nonabelian simple group. Using normal quotient techniques developed by the first author, the main theorem applies to general $G$-vertex-transitive graphs which are $G$-locally primitive (respectively, $G$-locally quasiprimitive), that is, the stabiliser $G_α$ of a vertex $α$ acts primitively (respectively quasiprimitively) on the set of vertices adjacent to $α$. We discuss how our results may be used to investigate conjectures by Richard Weiss (in 1978) and the first author (in 1998) that the order of $G_α$ is bounded above by some function depending only on the valency of $Γ$, when $Γ$ is $G$-locally primitive or $G$-locally quasiprimitive, respectively.