The Burness-Giudici Conjecture on Some Primitive Groups with Socle PSU(3,q)

TL;DR

This paper investigates a conjecture about the connectivity properties of Saxl graphs for primitive groups with socle PSU(3,q), providing partial proofs for the conjecture in these cases.

Contribution

It proves the Burness-Giudici conjecture for most primitive groups with socle PSU(3,q), advancing understanding of their Saxl graph structure.

Findings

Confirmed the conjecture for most cases with socle PSU(3,q)

Identified a specific configuration with $PSO(3,q)$ where the conjecture remains unproven

Extended previous results on groups with socle PSL(2,q)

Abstract

Let $G$ be a transitive permutation group on $\Omega$ with two points $\alpha, \beta\in\Omega$ such that $G_{\alpha}\cap G_{\beta}=1$. The Saxl graph $\Sigma(G)$ of the pair $(G,\Omega)$ is the graph with vertex set $\Omega$, while two vertices $\alpha', \beta'$ are adjacent if and only if $G_{\alpha'}\cap G_{\beta'}=1$. It was conjectured by Burness and Giudici that the Saxl graph $\Sigma(G)$ of any primitive permutation group $G$ has the property that any two vertices have a common neighbor. We focused on proving the conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$, that is, those with $soc(G)\in \{PSL(2,q), PSU(3,q), Ree(q), Sz(q)\}$. The case of $soc(G)=PSL(2,q)$ has been published in two papers. This paper will address most cases where $soc(G)=PSU(3,q)$, with the exception of a particularly intricate configuration in which the point stabilizer contains $PSO(3,q)$. That specific configuration has been treated in a separate paper.