Saxl graphs of primitive affine groups with sporadic point stabilisers

TL;DR

This paper investigates a conjecture about Saxl graphs of primitive affine groups with sporadic point stabilisers, confirming it in most cases and providing new insights into the structure of these groups.

Contribution

It verifies a conjecture on Saxl graphs for a class of affine groups with sporadic stabilisers, extending understanding of their combinatorial properties.

Findings

Conjecture holds in all but ten cases for the specified groups.

Saxl graphs exhibit the property that every two vertices have a common neighbour in these cases.

Provides classification and verification for groups with sporadic simple point stabilisers.

Abstract

Let $G$ be a permutation group on a set $\Omega$. A base for $G$ is a subset of $\Omega$ whose pointwise stabiliser is trivial, and the base size of $G$ is the minimal cardinality of a base. If $G$ has base size $2$, then the corresponding Saxl graph $\Sigma(G)$ has vertex set $\Omega$ and two vertices are adjacent if they form a base for $G$. A recent conjecture of Burness and Giudici states that if $G$ is a finite primitive permutation group with base size $2$, then $\Sigma(G)$ has the property that every two vertices have a common neighbour. We investigate this conjecture when $G$ is an affine group and a point stabiliser is an almost quasisimple group whose unique quasisimple subnormal subgroup is a covering group of a sporadic simple group. We verify the conjecture under this assumption, in all but ten cases.