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

TL;DR

This paper develops a method to bound the size of vertex stabilizers in finite vertex-transitive graphs, reducing the problem to cases involving nonabelian simple groups and applying to locally primitive/quasiprimitive graphs.

Contribution

It introduces a reduction technique for bounding vertex stabilizer sizes in G-vertex-transitive graphs, focusing on cases where G is quasiprimitive or biquasiprimitive, and applies normal quotient methods.

Findings

Bound on vertex stabilizer size in G-vertex-transitive graphs.

Reduction to nonabelian simple group cases.

Application to G-locally primitive/quasiprimitive graphs.

Abstract

In this paper we discuss a method for bounding the size of the stabiliser of a vertex in a $G$-vertex-transitive graph $\Gamma$. In the main result the group $G$ is quasiprimitive or biquasiprimitive on the vertices of $\Gamma$, 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 $G_\alpha$ of a vertex $\alpha$ acts primitively (respectively quasiprimitively) on the set of vertices adjacent to $\alpha$. 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 $G_\alpha$ is bounded above by some function depending only on the valency of $\Gamma$, when $\Gamma$ is $G$-locally primitive or $G$-locally quasiprimitive, respectively.