A new characteristic subgroup for finite p-groups.

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