Z* theorem

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In mathematics, George Glauberman's Z* theorem states that if G is a finite group and T is a Sylow 2-subgroup of G containing an involution not conjugate in G to any other element of T, then the involution lies in Z*(G). The subgroup Z*(G) is the inverse image in G of the center of G/O(G), where O(G) is the maximal normal subgroup of G of odd order.

This generalizes the Brauer–Suzuki theorem (and the proof uses the Brauer-Suzuki theorem to deal with some small cases).

[edit] References

• G. Glauberman. Central elements in core-free groups. J. Algebra 4 1966 403-420.
• E. C. Dade, Character theory of finite groups, in Finite simple groups, ISBN 0-12-563850-7