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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.22456 |
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| _version_ | 1866917258719133696 |
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| author | Chen, Huye Du, Shaofei Li, Weicong |
| author_facet | Chen, Huye Du, Shaofei Li, Weicong |
| contents | Let $G$ be a transitive permutation group on $Ω$ with two points $α, β\inΩ$ such that $G_α\cap G_β=1$. 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 $G_{α'}\cap G_{β'}=1$. 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 $soc(G)\in \{PSL(2,q), PSU(3,q), Ree(q), Sz(q)\}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Burness-Giudici Conjecture on Some Primitive Groups with Socle PSU(3,q) Chen, Huye Du, Shaofei Li, Weicong Group Theory Let $G$ be a transitive permutation group on $Ω$ with two points $α, β\inΩ$ such that $G_α\cap G_β=1$. 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 $G_{α'}\cap G_{β'}=1$. 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 $soc(G)\in \{PSL(2,q), PSU(3,q), Ree(q), Sz(q)\}$. 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. |
| title | The Burness-Giudici Conjecture on Some Primitive Groups with Socle PSU(3,q) |
| topic | Group Theory |
| url | https://arxiv.org/abs/2512.22456 |