The Burness-Giudici Conjecture on Primitive Groups of Lie-type with Rank One: Part (I)

Abstract: It was conjectured by Burness and Giudici that every primitive permutation group $G$ containing some regular suborbits has the property that $Γ\cap Γ^g\neq \emptyset$ for any $g\in G$, where $Γ$ is the union of all regular suborbits of $G$ relative to $α$. 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. The case of $soc(G)=PSU(3,q)$ is divided into two parts: this paper constitutes Part (I), addressing all primitive groups $G$ with socle $PSU(3,q)$ whose point stabilizers contain $PSO(3,q)$; and the remaining primitive actions will be covered in Part (II). The cases for $soc(G)\in{Ree(q),Sz(q)}$ are treated in Part (III). 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.