The Burness-Giudici Conjecture on Primitive Groups with Socle $Ree(q)$ and $Sz(q)$
Huye Chen, Shaofei Du
TL;DR
The paper advances the Burness-Giudici conjecture by validating it for primitive groups whose socle is a rank-1 Lie-type simple group, specifically ${\rm Sz}(q)$ and ${\rm Ree}(q)$. The authors employ a fixed-point–ratio framework, bounding the quantity ${\check{Q}(G)}$ via detailed analyses of regular suborbits and normalizers of prime-order subgroups in explicit subgroup structures. They develop and apply extensive combinatorial and group-theoretic computations to show ${\check{Q}(G)<\tfrac{1}{2}}$, which yields the Saxl graph ${\Sigma(G)}$ property that any two vertices share a common neighbor. In the Sz$(q)$ case, they use a Suzuki-model representation and suborbit counts; in the Ree$(q)$ case, they treat two major intersection scenarios for maximal subgroups, providing uniform bounds and a Magma check for a small exceptional instance. Overall, the results complete the rank-1 Lie-type cases for base-2 primitive groups, extending the scope of BG-Conjecture verification beyond ${\rm PSL}(2,q)$ and enriching methods for fixed-point computations on complex finite groups.”
Abstract
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 $soc(G)\in\{Ree(q), Sz(q)\}$.