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| Autori principali: | , |
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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2512.22461 |
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| _version_ | 1866918328459591680 |
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| author | Chen, Huye Du, Shaofei |
| author_facet | Chen, Huye Du, Shaofei |
| contents | 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 $soc(G)\in\{Ree(q), Sz(q)\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22461 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Burness-Giudici Conjecture on Primitive Groups with Socle $Ree(q)$ and $Sz(q)$ Chen, Huye Du, Shaofei Group Theory 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 $soc(G)\in\{Ree(q), Sz(q)\}$. |
| title | The Burness-Giudici Conjecture on Primitive Groups with Socle $Ree(q)$ and $Sz(q)$ |
| topic | Group Theory |
| url | https://arxiv.org/abs/2512.22461 |