Brauer–Suzuki theorem

In mathematics, the Brauer–Suzuki theorem, proved by Brauer & Suzuki (1959), Suzuki (1962), Brauer (1964), 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.

References

• Brauer, R. (1964), "Some applications of the theory of blocks of characters of finite groups. II", Journal of Algebra, 1: 307–334, doi:10.1016/0021-8693(64)90011-0, ISSN 0021-8693, MR 0174636 
• Brauer, R.; Suzuki, Michio (1959), "On finite groups of even order whose 2-Sylow group is a quaternion group", Proceedings of the National Academy of Sciences of the United States of America, 45: 1757–1759, doi:10.1073/pnas.45.12.1757, ISSN 0027-8424, JSTOR 90063, MR 0109846, PMC 222795, PMID 16590569 
• Dade, Everett C. (1971), "Character theory pertaining to finite simple groups", in Powell, M. B.; Higman, Graham, Finite simple groups. Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: Academic Press, pp. 249–327, ISBN 978-0-12-563850-0, MR 0360785  gives a detailed proof of the Brauer–Suzuki theorem.
• Suzuki, Michio (1962), "Applications of group characters", in Hall, M., 1960 Institute on finite groups: held at California Institute of Technology, Proc. Sympos. Pure Math., VI, American Mathematical Society, pp. 101–105, ISBN 978-0-8218-1406-2