The derangements subgroup in a finite permutation group and the Frobenius--Wielandt Theorem
R. A. Bailey, P. J. Cameron, N. Gavioli, C. M. Scoppola
TL;DR
The paper surveys the interplay between derangements subgroups and Frobenius–Wielandt (FW) theory in finite transitive groups, aiming to characterize when a finite $p$-group can occur as a one-point stabiliser with a proper derangements subgroup. It establishes the FW framework, introduces key invariants $(\Delta_G(H), D_G(H), U_G(H), W_G(H), K_G(H))$, and reviews Wielandt’s generalization that yields FW kernels and complements, including their permutation-action implications. A central result shows that, for a transitive non-regular group with a $p$-group stabiliser and $D_G(H)<G$, there exists a cyclic quotient structure $C/E$ within $H$ governing which elements have nontrivial powers in $C\setminus E$, tying the existence of a derangements subgroup to concrete group-theoretic conditions. The paper further notes that while any finite $p$-group can occur as an FW complement in some FW group, realising a $p$-group as a one-point stabiliser with a proper derangements subgroup imposes specific structural constraints, with affine-type constructions and modular-dimension-subgroup realizations providing explicit realizations and a clear characterization of the possible $p$-group stabilisers.
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.