On the Glauberman-Solomon Theorem

Abstract

We give a self contained and concise proof of the following result, which was first proved by Glauberman as his $ZJ$-Theorem. Let $p$ be an odd prime and $S \not= 1$ a $p$-group. Then there exists a characteristic subgroup $W(S) \not= 1$ of $S$ with the property: whenever $G$ is a finite $Qd(p)$-free group of characteristic $p$ and $S$ is a Sylow subgroup of $G$ then $W(S) \unlhd G$.