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

2025/01/29 by Bailey, R. A., Cameron, P. J., Gavioli, N. +1
#20B05 20D15 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2501.17545

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.

Related

• Groups generated by derangements
• Derangements in non-Frobenius groups
• Orbits of permutation groups with no derangements
• On derangements in simple permutation groups
• Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston-Shalev conjecture