Derangements in non-Frobenius groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916416824803328 |
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| author | Garzoni, Daniele |
| author_facet | Garzoni, Daniele |
| contents | We prove that if $G$ is a transitive permutation group of sufficiently large degree $n$, then either $G$ is primitive and Frobenius, or the proportion of derangements in $G$ is larger than $1/(2n^{1/2})$. This is sharp, generalizes substantially bounds of Cameron--Cohen and Guralnick--Wan, and settles conjectures of Guralnick--Tiep and Bailey--Cameron--Giudici--Royle in large degree. We also give an application to coverings of varieties over finite fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03305 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Derangements in non-Frobenius groups Garzoni, Daniele Group Theory 20B05, 20G40, 14G05 We prove that if $G$ is a transitive permutation group of sufficiently large degree $n$, then either $G$ is primitive and Frobenius, or the proportion of derangements in $G$ is larger than $1/(2n^{1/2})$. This is sharp, generalizes substantially bounds of Cameron--Cohen and Guralnick--Wan, and settles conjectures of Guralnick--Tiep and Bailey--Cameron--Giudici--Royle in large degree. We also give an application to coverings of varieties over finite fields. |
| title | Derangements in non-Frobenius groups |
| topic | Group Theory 20B05, 20G40, 14G05 |
| url | https://arxiv.org/abs/2409.03305 |