Brauer–Suzuki theorem

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In mathematics, the Brauer-Suzuki theorem states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups of odd order, then the group has a centre of order 2. In particular, such a group cannot be simple.

A generalization of the Brauer-Suzuki theorem is given by Glauberman's Z* theorem.

The theorem is named after the mathematicians Michio Suzuki and Richard Brauer.

[edit] References

• R. Brauer, M. Suzuki, On finite groups of even order whose 2-Sylow subgroup is a quaternion group, Proc. Nat. Acad. Sci. 45 (1959) 1757-1759.
• E. C. Dade, Character theory of finite groups, in Finite simple groups, ISBN 0-12-563850-7, gives a detailed proof of the Brauer–Suzuki theorem.