The Burness-Giudici Conjecture on Primitive Groups with Socle $Ree(q)$ and $Sz(q)$

Let $G$ be a transitive permutation group on $Ω$ containing two points $α, β$ such that $G_α\cap G_β=1$. 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 any primitive permutation group $G$, its Saxl graph $Σ(G)$ satisfies the property that any two vertices share a common neighbor. We focused on proving this conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$; that is, groups with $soc(G)\in {PSL(2,q), PSU(3,q), Ree(q), Sz(q)}$. The case $soc(G)=PSL(2,q)$ has been published in two papers. In this paper, we treat the cases where $soc(G)\in{Ree(q), Sz(q)}$.

💡 Research Summary

The paper addresses the Burness‑Giudici conjecture concerning Saxl graphs of primitive permutation groups with base size two. For a transitive group G on a finite set Ω, the Saxl graph Σ(G) connects two points α, β whenever their point‑stabilisers intersect trivially (G_α∩G_β = 1). The conjecture asserts that for any primitive group G with base size b(G)=2, every pair of vertices in Σ(G) shares a common neighbour; equivalently, Σ(G) has diameter at most two.

Previous work settled the conjecture for many families: symmetric and alternating groups, PSL(2,q), PSU(3,q), and several product‑type groups. The remaining rank‑1 groups of Lie type are the Suzuki groups Sz(q) (q=2^m, m odd) and the Ree groups Ree(q) (q=3^{2m+1}). The present article completes the picture by proving the conjecture for these two families.

The authors work with G≤Aut(T) where T is either Sz(q) or Ree(q). They consider the action of G on the coset space Ω=