Fixers and derangements of finite permutation groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913908428636160 |
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| author | Huang, Hong Yi Li, Cai Heng Xie, Yi Lin |
| author_facet | Huang, Hong Yi Li, Cai Heng Xie, Yi Lin |
| contents | Let $G\leqslant\mathrm{Sym}(Ω)$ be a finite transitive permutation group with point stabiliser $H$. We say that a subgroup $K$ of $G$ is a fixer if every element of $K$ has fixed points, and we say that $K$ is large if $|K| \geqslant |H|$. There is a special interest in studying large fixers due to connections with Erdős-Ko-Rado type problems. In this paper, we classify up to conjugacy the large fixers of the almost simple primitive groups with socle $\mathrm{PSL}_2(q)$, and we use this result to verify a special case of a conjecture of Spiga on permutation characters. We also present some results on large fixers of almost simple primitive groups with socle an alternating or sporadic group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18753 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fixers and derangements of finite permutation groups Huang, Hong Yi Li, Cai Heng Xie, Yi Lin Group Theory Combinatorics Let $G\leqslant\mathrm{Sym}(Ω)$ be a finite transitive permutation group with point stabiliser $H$. We say that a subgroup $K$ of $G$ is a fixer if every element of $K$ has fixed points, and we say that $K$ is large if $|K| \geqslant |H|$. There is a special interest in studying large fixers due to connections with Erdős-Ko-Rado type problems. In this paper, we classify up to conjugacy the large fixers of the almost simple primitive groups with socle $\mathrm{PSL}_2(q)$, and we use this result to verify a special case of a conjecture of Spiga on permutation characters. We also present some results on large fixers of almost simple primitive groups with socle an alternating or sporadic group. |
| title | Fixers and derangements of finite permutation groups |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2404.18753 |