The Burness-Giudici Conjecture on Primitive Groups with Socle $Ree(q)$ and $Sz(q)$

Abstract

Let $G$ be a transitive permutation group on $Ω$ containing two points $α, β$ such that $G_α\cap G_β=1$. The Saxl graph $Σ(G)$ of $(G, Ω)$ is defined as the graph with vertex set $Ω$, where two vertices $α', β'$ are adjacent if and only if $G_{α'}\cap G_{β'}=1$. Burness and Giudici conjectured that for any primitive permutation group $G$, its Saxl graph $Σ(G)$ satisfies the property that any two vertices share a common neighbor. We study this common-neighbor property for primitive groups whose socle is a simple group of Lie type of rank one, namely $PSL(2,q)$, $PSU(3,q)$, $Ree(q)$ or $Sz(q)$. In this paper, we establish the property for groups with socle $Ree(q)$ or $Sz(q)$, except possibly when $soc(G)=Ree(q)\lneq G $ and a point stabilizer $M$ satisfies $M\cap soc(G)\cong Z_2\times PSL(2,q)$.