The Burness-Giudici Conjecture on Some Primitive Groups with Socle PSU(3,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 ${\alpha }^{\prime },{\beta }^{\prime }$ are adjacent if and only if $G_{{\alpha }^{\prime }}\cap G_{{\beta }^{\prime }}=1$. 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.