# Glauberman Z*-theorem

From Groupprops

## Statement

Let be a finite group. Denote by the normal subgroup containing (the Brauer core) such that .

Let be a 2-Sylow subgroup of . Suppose contains an involution which is not conjugate (in ) to any other element of . Then . In particular, the existence of such an involution implies that is not a simple group.

## Relation with other results

### Brauer-Suzuki theorem

`Further information: Brauer-Suzuki theorem`