Theorem of Brewers Suzuki

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The set of Brewers Suzuki (by Richard Brauer and Michio Suzuki ) is a mathematical theorem from the group theory . It is used in the structure theory of finite simple groups, especially in the characterization of their possible 2-Sylow subsets.

statement

If a finite group has a 2- Sylow group , which is a generalized quaternion group , and if this group also has no nontrivial normal divisor of odd order , its center has order 2. In particular, the group is then not simple.

Inferences

Brauer-Suzuki's theorem is one of several theorems needed to characterize 2-Sylow subsets of simple groups. The final result says that 2-Sylow groups are either dihedral or semi-dihedral or contain a non-cyclic elementary Abelian characteristic subgroup.

literature

  • 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.
  • EC Dade, Character theory of finite groups , in Finite simple groups , ISBN 0-12-563850-7 , contains a detailed proof of the Brauer-Suzuki theorem.