mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

finite group theory In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

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

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

''G'' has even

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

''g'' > 2 then it has a proper

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

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 or morphism 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 the ...

there are only a finite number of

finite simple group In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple g ...

s with a given

centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

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 the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups. A simpl ...

s. Later it motivated a part of the

classification of finite simple groups In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...

References

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