The Saxl hypergraph of a permutation group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913848061067264 |
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| author | Lee, Melissa Pisani, Anthony |
| author_facet | Lee, Melissa Pisani, Anthony |
| contents | Given a permutation group $G \le \mathrm{Sym}(Ω)$, a subset $B$ of $Ω$ is said to be a base if its pointwise stabiliser in $G$ is trivial, and the base size $b(G)$ is the minimum size of a base. In the notable case $b(G) = 2$, Burness and Giudici define the Saxl graph of $G$ to be the graph on $Ω$ with bases of size 2 as edges. Later work of Freedman et al. extends this notion to any group for which $b(G) \ge 2$, taking the pairs of points contained in bases of size $b(G)$ for edges. We study an alternative generalisation, the Saxl hypergraph, where bases of size $b(G)$ are themselves the edges. In particular, we consider groups with complete Saxl hypergraphs, primitive groups whose Saxl hypergraphs have flag-spanning tours, and appropriate generalisations of Burness and Giudici's Common Neighbour Conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_13849 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Saxl hypergraph of a permutation group Lee, Melissa Pisani, Anthony Group Theory Combinatorics Given a permutation group $G \le \mathrm{Sym}(Ω)$, a subset $B$ of $Ω$ is said to be a base if its pointwise stabiliser in $G$ is trivial, and the base size $b(G)$ is the minimum size of a base. In the notable case $b(G) = 2$, Burness and Giudici define the Saxl graph of $G$ to be the graph on $Ω$ with bases of size 2 as edges. Later work of Freedman et al. extends this notion to any group for which $b(G) \ge 2$, taking the pairs of points contained in bases of size $b(G)$ for edges. We study an alternative generalisation, the Saxl hypergraph, where bases of size $b(G)$ are themselves the edges. In particular, we consider groups with complete Saxl hypergraphs, primitive groups whose Saxl hypergraphs have flag-spanning tours, and appropriate generalisations of Burness and Giudici's Common Neighbour Conjecture. |
| title | The Saxl hypergraph of a permutation group |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2505.13849 |