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

Let $G$ be a transitive permutation group on a set $Ω$, and suppose $G_α\cap G_β=1$ for some distinct $α, β\inΩ$. The Saxl graph $Σ(G)$ of $(G, Ω)$ is defined as the graph with vertex set $Ω$, where two vertices $α’, β’$ are adjacent if and only if $G_{α’}\cap G_{β’}=1$. 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.

💡 Research Summary

The paper addresses the Burness‑Giudici (BG) conjecture for primitive permutation groups whose socle is the simple unitary group PSU(3,q). The conjecture asserts that for any primitive group G with base size b(G)=2, the associated Saxl graph Σ(G) has the property that every pair of vertices shares a common neighbour; equivalently, Σ(G) has diameter two. While the conjecture had already been verified for primitive groups with socle PSL(2,q) and for several other families, the case of PSU(3,q) remained open.

The authors begin by recalling the definition of the Saxl graph: vertices are the points of the permutation domain Ω, and two vertices α,β are adjacent precisely when the point stabilisers G_α and G_β intersect trivially. For groups with b(G)=2, the neighbours of a vertex α are exactly the points lying in the regular suborbits of G_α. The paper’s strategy is to analyse these suborbits for the action of G on the coset space Ω=