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 $\alpha', \beta'$ are adjacent if and only if $G_{\alpha'}\cap G_{\beta'}=1$. 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 focused on proving this conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$; that is, groups with $soc(G)\in \{PSL(2,q), PSU(3,q), Ree(q), Sz(q)\}$. The case $soc(G)=PSL(2,q)$ has been published in two papers. In this paper, we treat the cases where $soc(G)\in\{Ree(q), Sz(q)\}$.