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 $S \not =1$ 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 $S$.