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

2025/12/27 by Huye Chen, Chen, Huye, Shaofei Du +1 · 2 citations
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.48550/arxiv.2512.22461

openalex publication_date 2025/12/27 · openalex created_date 2025/12/31 · openalex updated_date 2026/07/28

Abstract

Let G be a transitive permutation group on Ω containing two points α, β such that G_α∩ 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'∩ 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)∈ \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)∈\Ree(q), Sz(q)\.

Citations

• The Burness-Giudici Conjecture on Some Primitive Groups with Socle PSU(3,q)
• The Saxl hypergraph of a permutation group
• On the generalised Saxl graphs of permutation groups
• On base sizes for primitive groups of product type
• Saxl graphs of primitive affine groups with sporadic point stabilisers
• On the Saxl graphs of primitive groups with soluble stabilisers
• On the Burness-Giudici Conjecture

Cited by

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

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• The Primitive Permutation Groups of Degree Less Than 4096
• Primitive pseudo-finite permutation groups of finite SU-rank