# An extension of the Glauberman ZJ-Theorem

## TL;DR

This paper extends Glauberman's ZJ-Theorem by proving new theorems on the normality and control of fusion related to Thompson subgroups in p-groups, with applications to p-nilpotency criteria.

## Contribution

It introduces an extension of Glauberman's replacement theorem and establishes new normality and fusion control results for Thompson subgroups in p-stable and Qd(p)-free groups.

## Key findings

- Proved an extension of Glauberman's replacement theorem.
- Established normality of certain subgroups in p-stable groups.
- Showed control of fusion by normalizers of Thompson subgroup centers.

## Abstract

Let $p$ be an odd prime and let $J_o(X)$, $J_r(X)$ and $J_e(X)$ denote the three different versions of Thompson subgroups for a $p$-group $X$. In this article, we first prove an extension of Glauberman's replacement theorem. Secondly, we prove the following: Let $G$ be a $p$-stable group and $P\in Syl_p(G)$. Suppose that $C_G(O_{p}(G))\leq O_{p}(G)$. If $D$ is a strongly closed subgroup in $P$, then $Z(J_o(D))$, $\Omega(Z(J_r(D)))$ and $\Omega(Z(J_e(D)))$ are normal subgroups of $G$. Thirdly, we show the following: Let $G$ be a $\text{Qd}(p)$-free group and $P\in Syl_p(G)$. If $D$ is a strongly closed subgroup in $P$, then the normalizers of the subgroups $Z(J_o(D))$, $\Omega(Z(J_r(D)))$ and $\Omega(Z(J_e(D)))$ control strong $G$-fusion in $P$. We also prove a similar result for a $p$-stable and $p$-constrained group. Lastly, we give a $p$-nilpotency criteria, which is an extension of Glauberman-Thompson $p$-nilpotency theorem.