On the generalised Saxl graphs of permutation groups
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914287831744512 |
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| author | Freedman, Saul D. Huang, Hong Yi Lee, Melissa Rekvényi, Kamilla |
| author_facet | Freedman, Saul D. Huang, Hong Yi Lee, Melissa Rekvényi, Kamilla |
| contents | A base for a finite permutation group $G \le \mathrm{Sym}(Ω)$ is a subset of $Ω$ with trivial pointwise stabiliser in $G$, and the base size of $G$ is the smallest size of a base for $G$. Motivated by the interest in groups of base size two, Burness and Giudici introduced the notion of the Saxl graph. This graph has vertex set $Ω$, with edges between elements if they form a base for $G$. We define a generalisation of this graph that encodes useful information about $G$ whenever $b(G) \ge 2$: here, the edges are the pairs of elements of $Ω$ that can be extended to bases of size $b(G)$. In particular, for primitive groups, we investigate the completeness and arc-transitivity of the generalised graph, and the generalisation of Burness and Giudici's Common Neighbour Conjecture on the original Saxl graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22613 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the generalised Saxl graphs of permutation groups Freedman, Saul D. Huang, Hong Yi Lee, Melissa Rekvényi, Kamilla Group Theory A base for a finite permutation group $G \le \mathrm{Sym}(Ω)$ is a subset of $Ω$ with trivial pointwise stabiliser in $G$, and the base size of $G$ is the smallest size of a base for $G$. Motivated by the interest in groups of base size two, Burness and Giudici introduced the notion of the Saxl graph. This graph has vertex set $Ω$, with edges between elements if they form a base for $G$. We define a generalisation of this graph that encodes useful information about $G$ whenever $b(G) \ge 2$: here, the edges are the pairs of elements of $Ω$ that can be extended to bases of size $b(G)$. In particular, for primitive groups, we investigate the completeness and arc-transitivity of the generalised graph, and the generalisation of Burness and Giudici's Common Neighbour Conjecture on the original Saxl graph. |
| title | On the generalised Saxl graphs of permutation groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2410.22613 |