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

TL;DR

The paper advances the Burness-Giudici conjecture for Saxl graphs by focusing on primitive groups with socle $\mathrm{PSU}(3,q)$ and point-stabilizers not containing $\mathrm{PSO}(3,q)$. It develops a fixed-point-ratio bounding framework, reducing the problem to analyzing maximal subgroups of form $\mathrm{PSU}(3,q').\big(\tfrac{q+1}{q'+1},3\big)$ and its two subcases $c=1$ and $c=3$. Through a careful, case-by-case examination of prime-order subgroups, their normalizers, and fixed-point counts in the coset actions, the authors prove $\check{Q}(G) < \frac{1}{2}$ for these groups, which implies that any two Saxl-graph vertices have a common neighbour. The work completes the PSU$(3,q)$-type portion of the rank-one Lie-type analysis (with PSL$(2,q)$ and the Ree/Sz cases treated separately in other papers), thereby extending the validity of the BG-conjecture in this important family. The techniques combine group-theoretic structure, conjugacy-class information from PGU(3,q), and fixed-point computations to obtain explicit bounds applicable to a broad class of primitive groups.

Abstract

Let $G$ be a transitive permutation group on $Ω$ with two points $α, β\inΩ$ such that $G_α\cap G_β=1$. The Saxl graph $Σ(G)$ of the pair $(G,Ω)$ is the graph with vertex set $Ω$, while two vertices $α', β'$ are adjacent if and only if $G_{α'}\cap G_{β'}=1$. It was conjectured by Burness and Giudici that the Saxl graph $Σ(G)$ of any primitive permutation group $G$ has the property that any two vertices have a common neighbor. 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. This paper will address most cases where $soc(G)=PSU(3,q)$, with the exception of a particularly intricate configuration in which the point stabilizer contains $PSO(3,q)$. That specific configuration has been treated in a separate paper.

Theorems & Definitions (21)

• Conjecture 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Remark 2.4
• Theorem 4.1
• proof
• Lemma 4.2
• Lemma 4.3