Variations on Glauberman's ZJ Theorem

TL;DR

This paper presents a new proof of Glauberman's ZJ Theorem that clarifies the proof process and introduces two new ZJ-type subgroups of a p-group, expanding the classical framework.

Contribution

It provides a clearer proof of Glauberman's ZJ Theorem and introduces two new ZJ-type subgroups that often surpass the classical ones in size.

Findings

New proof clarifies the choices involved in Glauberman's ZJ Theorem.

Introduction of two new ZJ-type subgroups of a p-group.

These subgroups often contain the classical ZJ_r(S) and ZJ_o(S).

Abstract

We give a new proof of Glauberman's ZJ Theorem, in a form that clarifies the choices involved and offers more choices than classical treatments. In particular, we introduce two new ZJ-type subgroups of a $p$-group $S$, that contain $ZJ_r(S)$ and $ZJ_o(S)$ respectively and are often strictly larger.