On the Glauberman-Solomon Theorem

Paul Flavell

group theory

On the Glauberman-Solomon Theorem

arXiv:2607.26740

summary

The paper gives a concise, self‑contained proof that for any nontrivial odd‑prime p‑group S there exists a nontrivial characteristic subgroup W(S) that is normal in every finite Qd(p)-free group of characteristic p having S as a Sylow subgroup.

Abstract

We give a self contained and concise proof of the following result, which was first proved by Glauberman as his $Z J$ -Theorem. Let $p$ be an odd prime and $S \neq = 1$ a $p$ -group. Then there exists a characteristic subgroup $W (S) \neq = 1$ of $S$ with the property: whenever $G$ is a finite $Q d (p)$ -free group of characteristic $p$ and $S$ is a Sylow subgroup of $G$ then $W (S) ⊴ G$ .

5 pages

Topics & keywords

#finite groups #p-groups #characteristic subgroups #Sylow subgroups #Glauberman theoremZJ-TheoremQd(p)-freecharacteristic pW(S)self-contained proof