In mathematical finite group theory, the Brauer–Fowler theorem, proved by , states that if a group has even order g > 2 then it has a subgroup of order greater than g^1/3. This implies that up to isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

  there are only a finite number of finite groups with a given centralizer of an involution

Involution

In mathematics, an involution, or an involutary function, is a function f that is its own inverse:-General properties:Any involution is a bijection.The identity map is a trivial example of an involution...

, which suggested that finite simple groups could be classified by studying their centralizers of involutions, a program that was later realized in the classification of finite simple groups

Classification of finite simple groups

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...

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