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

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:Part (I) addressed all primitive groups $G$ with socle $PSU(3,q)$ whose point stabilizers contain $PSO(3,q)$ and based on it, this Part (II) will finish all cases of $soc(G)=PSU(3,q)$. The cases for $soc(G)\in{Ree(q),Sz(q)}$ will be treated in Part (III).