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

Abstract

Let $G$ be a transitive permutation group on $Ω$ with two points $α, β\inΩ$ such that $G_α\cap G_β=1$. The Saxl graph $Σ(G)$ of the pair $(G,Ω)$ is the graph with vertex set $Ω$, while two vertices $α', β'$ are adjacent if and only if $G_{α'}\cap G_{β'}=1$. It was conjectured by Burness and Giudici that the Saxl graph $Σ(G)$ 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.