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

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.