2025/12/27 by Huye Chen, Chen, Huye, Shaofei Du +3 · 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.2512.22456
openalex publication_date 2025/12/27 · openalex created_date 2025/12/31 · openalex updated_date 2026/07/28
Let G be a transitive permutation group on Ω with two points α, β∈Ω such that Gα∩ Gβ=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 Gα'∩ Gβ'=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)∈ \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.