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

Abstract

Let $G$ be a transitive permutation group on $\Omega$ containing two points $\alpha ,\beta$ such that $G_{\alpha }\cap G_{\beta }=1$. The Saxl graph $\Sigma (G)$ of $(G,\Omega )$ is defined as the graph with vertex set $\Omega$, where two vertices ${\alpha }^{\prime },{\beta }^{\prime }$ are adjacent if and only if $G_{{\alpha }^{\prime }}\cap G_{{\beta }^{\prime }}=1$. Burness and Giudici conjectured that for any primitive permutation group $G$, its Saxl graph $\Sigma (G)$ 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 $soc(G)=Ree(q)⪇G$ and a point stabilizer $M$ satisfies $M\cap soc(G)\cong Z_2\times PSL(2,q)$.