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

TL;DR

This paper surveys the relationship between derangements subgroups and Frobenius--Wielandt kernels in permutation groups, establishing conditions under which p-groups appear as stabilizers in such groups.

Contribution

It clarifies the conditions for p-groups to serve as stabilizers in transitive non-regular permutation groups with proper derangements subgroups.

Findings

Proper derangements subgroups are Frobenius--Wielandt kernels.

No structural restrictions on p-groups as Frobenius--Wielandt complements.

A p-group appears as a stabilizer iff it meets a specific group-theoretic condition.

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.