The derangements subgroup in a finite permutation group and the Frobenius--Wielandt Theorem
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| Format: | Preprint |
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2025
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| _version_ | 1866929690756775936 |
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| author | Bailey, R. A. Cameron, P. J. Gavioli, N. Scoppola, C. M. |
| author_facet | Bailey, R. A. Cameron, P. J. Gavioli, N. Scoppola, C. M. |
| contents | It is known that if the derangements subgroup of a transitive non-regular permutation group is a proper subgroup, then it is a Frobenius--Wielandt kernel, and, conversely, minimal Frobenius--Wielandt kernels are proper derangements subgroups. We present here a short survey of the literature on this topic, and we show that, although there are no restrictions on the structure of the $p$-groups appearing as Frobenius--Wielandt complements, a $p$-group appears as a one-point stabiliser in a transitive non-regular permutation group with a proper derangements subgroup if and only if it satisfies a certain group-theoretic condition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_17545 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The derangements subgroup in a finite permutation group and the Frobenius--Wielandt Theorem Bailey, R. A. Cameron, P. J. Gavioli, N. Scoppola, C. M. Group Theory 20B05 20D15 It is known that if the derangements subgroup of a transitive non-regular permutation group is a proper subgroup, then it is a Frobenius--Wielandt kernel, and, conversely, minimal Frobenius--Wielandt kernels are proper derangements subgroups. We present here a short survey of the literature on this topic, and we show that, although there are no restrictions on the structure of the $p$-groups appearing as Frobenius--Wielandt complements, a $p$-group appears as a one-point stabiliser in a transitive non-regular permutation group with a proper derangements subgroup if and only if it satisfies a certain group-theoretic condition. |
| title | The derangements subgroup in a finite permutation group and the Frobenius--Wielandt Theorem |
| topic | Group Theory 20B05 20D15 |
| url | https://arxiv.org/abs/2501.17545 |