The derangements subgroup in a finite permutation group and the Frobenius--Wielandt Theorem

R. A. Bailey; P. J. Cameron; N. Gavioli; C. M. Scoppola

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arXiv · 2501.17545

The derangements subgroup in a finite permutation group and the Frobenius--Wielandt Theorem

R. A. Bailey·P. J. Cameron·N. Gavioli·C. M. Scoppola

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Abstract

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.

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R. A. Bailey, P. J. Cameron, N. Gavioli, C. M. Scoppola. 2025-01-29. The derangements subgroup in a finite permutation group and the Frobenius--Wielandt Theorem. https://arxiv.org/abs/2501.17545

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