Z* theorem

Remarks

The theorem is due to George Glauberman, and its proof uses modular representation theory, in particular Brauer's theory of blocks. The notation comes from $Z^*(G)$, the subgroup of $G$ containing $O(G)$ with $Z^*(G)/O(G) = Z(G/O(G))$; the conclusion says $t \in Z^*(G)$.

Example: $G = S_3$ and $t = (1\,2)$ in $T = \set{e, (1\,2)}$. No other element of $T$ is conjugate to $t$, and $O(S_3) = A_3$. Indeed $S_3 / A_3$ has order $2$, so it is abelian and $t A_3$ is central.

Corollary: if $G$ is a nonabelian simple group, then every involution $t$ in a Sylow $2$-subgroup $T$ is conjugate in $G$ to another element of $T$. Indeed, $G$ has even order because it contains $t$, so $O(G) \ne G$; hence $O(G) = \set{e}$ by simplicity, and the centre of $G$ is trivial, so $t$ cannot be central.

Going deeper

For a student

The Z* theorem is a result about fusion of $2$-elements, like the focal subgroup theorem but for single involutions. It was an important tool in the classification of finite simple groups, where it rules out isolated involutions. The proof by modular character theory is a striking case where representation theory is needed for a statement about conjugacy in a group.