Glauberman Z*-theorem

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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