Glauberman Z*-theorem

Statement

Let $G$ be a finite group. Denote by $Z^{*}(G)$ the normal subgroup containing $O(G)$ (the Brauer core) such that $Z^{*}(G)/O(G)=Z(G/O(G))$.

Let $S$ be a 2-Sylow subgroup of $G$. Suppose $S$ contains an involution $t$ which is not conjugate (in $G$) to any other element of $S$. Then $t\in Z^{*}(G)$. In particular, the existence of such an involution implies that $G$ is not a simple group.

Relation with other results

Brauer-Suzuki theorem

Further information: Brauer-Suzuki theorem