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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2512.22459 |
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| _version_ | 1866918331732197376 |
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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 a set $Ω$, and suppose $G_α\cap G_β=1$ for some distinct $α, β\inΩ$. 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 every primitive permutation group $G$, its Saxl graph has the property that any two vertices share a common neighbor.
We focus on proving the conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$; that is, $soc(G)\in \{PSL(2,q),PSU(3,q), Ree(q),Sz(q)\}$. The case $soc(G)=PSL(2,q)$ has been treated in two earlier papers. The purpose of the present paper is to settle the case $soc(G)=PSU(3,q)$. To finsh this work, we draw on methods from abstract- and permutation- group theory, finite unitary geometry, probabilistic approach, number theory (employing Weil's bound), and, most importantly, algebraic combinatorics, which provides us some key ideas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22459 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Burness-Giudici Conjecture on Primitive Groups with Socle PSU(3,q) Chen, Huye Du, Shaofei Li, Weicong Group Theory Let $G$ be a transitive permutation group on a set $Ω$, and suppose $G_α\cap G_β=1$ for some distinct $α, β\inΩ$. 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 every primitive permutation group $G$, its Saxl graph has the property that any two vertices share a common neighbor. We focus on proving the conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$; that is, $soc(G)\in \{PSL(2,q),PSU(3,q), Ree(q),Sz(q)\}$. The case $soc(G)=PSL(2,q)$ has been treated in two earlier papers. The purpose of the present paper is to settle the case $soc(G)=PSU(3,q)$. To finsh this work, we draw on methods from abstract- and permutation- group theory, finite unitary geometry, probabilistic approach, number theory (employing Weil's bound), and, most importantly, algebraic combinatorics, which provides us some key ideas. |
| title | The Burness-Giudici Conjecture on Primitive Groups with Socle PSU(3,q) |
| topic | Group Theory |
| url | https://arxiv.org/abs/2512.22459 |