Brauer–Fowler theorem

In mathematical finite group theory, the Brauer–Fowler theorem, proved by Brauer & Fowler (1955), 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 there are only a finite number of finite simple groups 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 groups. Later it motivated a part of the classification of finite simple groups.

References

Brauer, R.; Fowler, K. A. (1955), "On groups of even order", Annals of Mathematics. Second Series 62: 565–583, ISSN 0003-486X, JSTOR 1970080, MR 0074414

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