mathematical 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 ...

finite group theory 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 ...

, the Brauer–Fowler theorem, proved by , states that if a group ''G'' has even order ''g'' > 2 then it has a proper subgroup of order greater than ''g''^1/3. The technique of the proof is to count involutions (elements of order 2) in G. Perhaps more important is another result that the authors derive from the same count of involutions, namely that up to

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

there are only a finite number of finite

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

s with a given centralizer of an involution. This suggested that finite simple groups could be classified by studying their centralizers of involutions, and it led to the discovery of several

sporadic group In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...

s. Later it motivated a part of the

classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...

References

* Theorems about finite groups {{abstract-algebra-stub