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

Abstract

Let $G$ be a transitive permutation group on a set $\Omega$, and suppose $G_{\alpha }\cap G_{\beta }=1$ for some distinct $\alpha ,\beta \in \Omega$. 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 every primitive permutation group $G$, its Saxl graph has the property that any two vertices share a common neighbor. We focus on proving the conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$; that is, $soc(G)\in \{PSL(2,q),PSU(3,q),Ree(q),Sz(q)\}$. The case $soc(G)=PSL(2,q)$ has been treated in two earlier papers. The purpose of the present paper is to settle the case $soc(G)=PSU(3,q)$. To finsh this work, we draw on methods from abstract- and permutation- group theory, finite unitary geometry, probabilistic approach, number theory (employing Weil's bound), and, most importantly, algebraic combinatorics, which provides us some key ideas.

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