Z* theorem

Z* theorem

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).

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


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