A new characteristic subgroup for finite 𝑝-groups

Abstract

An important result with many applications in the theory of finite groups is the following: Let S ≠ 1 S \not =1 be a finite p-group for some prime p. Then S S contains a characteristic subgroup W ( S ) ≠ 1 W(S) \not = 1 with the property that W ( S ) W(S) is normal in every finite group G G of characteristic p p with S ∈ S y l p ( G ) S \in Syl_p(G) that does not possess a section isomorphic to the semi-direct product of S L 2 ( p ) SL_{2}(p) with its natural module. For odd primes, this was first established by Glauberman with his celebrated Z J ZJ -Theorem. The case p = 2 p=2 proved more elusive and was only established much later by the second author. Unlike the Z J ZJ -Theorem, it was not possible to give an explicit description of the subgroup W ( S ) W(S) in terms of the internal structure of S S . In this paper we introduce the notion of “almost quadratic action” and show that in each finite p-group S S there exists a unique maximal elementary abelian characteristic subgroup W ( S ) W(S) which contains Ω 1 ( Z ( S ) ) \Omega _1(Z(S)) and does not allow non-trivial almost quadratic action from S S . We show that W ( S ) W(S) is normal in all finite groups G G of characteristic p with S ∈ S y l p ( G ) S \in Syl_p(G) provided that G G does not possess a section isomorphic to the semi-direct product of S L 2 ( p ) SL_{2}(p) with its natural module. This provides a unified approach for all primes p p and gives a concrete description of the subgroup W ( S ) W(S) in terms of the internal structure of S S .