On Valency Problems of Saxl Graphs

TL;DR

This paper develops a method to compute the valency of Saxl graphs for finite transitive groups and applies it to classify valencies in various primitive group cases, extending previous results.

Contribution

It introduces a general technique for calculating Saxl graph valency and applies it to primitive groups with specific stabilisers, expanding understanding of their valency properties.

Findings

Exact valency calculations for primitive groups with Frobenius stabilisers

Valency determination for almost simple primitive groups with alternating socle

Extension of previous results on prime-power and odd valency cases

Abstract

Let $G$ be a permutation group on a set $\Omega$ and recall that a base for $G$ is a subset of $\Omega$ such that its pointwise stabiliser is trivial. In a recent paper, Burness and Giudici introduced the Saxl graph of $G$, denoted $\Sigma(G)$, with vertex set $\Omega$ and two vertices adjacent if they form a base. If $G$ is transitive, then $\Sigma(G)$ is vertex-transitive and it is natural to consider its valency (which we refer to as the valency of $G$). In this paper we present a general method for computing the valency of any finite transitive group and we use it to calculate the exact valency of every primitive group with stabiliser a Frobenius group with cyclic kernel. As an application, we calculate the valency of every almost simple primitive group with an alternating socle and soluble stabiliser and we use this to extend results of Burness and Giudici on almost simple primitive groups with prime-power or odd valency.