Brauer–Suzuki theorem

The Brauer–Suzuki theorem states that a finite group with generalized quaternion Sylow 2-subgroups has a nontrivial center. Consequently, no finite nonabelian simple group has such Sylow subgroups.

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Sylow theorems: Theorems describing the number, conjugacy, and containment of maximal prime-power subgroups in finite groups. They define the Sylow 2-subgroups whose structure triggers the theorem.
Involutions in group theory: Elements of order two in a group. The theorem's proof analyzes the behavior and centralizers of involutions.
Richard Brauer: A German-American mathematician who helped establish modular representation theory and modern finite group theory. Brauer contributed the methods and structural results behind the theorem.
Nonabelian simple group: A simple group that is not abelian. No such group can have generalized quaternion Sylow 2-subgroups.
Generalized quaternion group: A finite 2-group with a cyclic subgroup of index two and a specific quaternion-like presentation. These are the precise Sylow 2-subgroups covered by the theorem.
Centralizer: The subgroup of elements that commute with a specified element or subset. Centralizers expose how involutions interact with the rest of the group.
Michio Suzuki: A Japanese mathematician known for work on finite groups and the classification of finite simple groups. Suzuki co-established the theorem and studied groups with restricted Sylow subgroups.
Quaternion group: The nonabelian group of order eight whose elements include six elements of order four. It is the smallest generalized quaternion group, so the theorem covers this case.
Center of a group: The subgroup of elements that commute with every element of a group. The theorem's conclusion is that this subgroup is nontrivial.
Normal subgroup: A subgroup invariant under conjugation by every element of its group. The structural argument forces central behavior incompatible with nonabelian simplicity.