mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Brauer–Suzuki theorem, proved by , , , states that if a

finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

has a generalized quaternion

Sylow 2-subgroup In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed ...

and no non-trivial

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...

s of

odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...

order, then the group has a

center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...

of order 2. In particular, such a group cannot be

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

. A generalization of the Brauer–Suzuki theorem is given by Glauberman's

Z* theorem In mathematics, George Glauberman's Z* theorem is stated as follows: Z* theorem: Let ''G'' be a finite group, with ''O''(''G'') being its maximal normal subgroup of odd order. If ''T'' is a Sylow 2-subgroup of ''G'' containing an involution no ...

References

* * * gives a detailed proof of the Brauer–Suzuki theorem. * Theorems about finite groups {{Abstract-algebra-stub