Math 2.0

Group theory and generalizations Finite groups Theorem

Brauer–Fowler theorem

Builds on Simple groupCentraliserOrder of an element

Statement

An involution of a group is an element of order 22. Let nn be a positive integer. Then, up to isomorphism, there are only finitely many finite simple groups GG that contain an involution tt whose centraliser CG(t)C_G(t) has order nn.

Remarks

The proof bounds the order of GG by a function of nn alone, using counting arguments with involutions and their products. In particular, for a given finite group HH, only finitely many finite simple groups have an involution with centraliser isomorphic to HH. The bound obtained is far too large to be used directly. But the finiteness result showed that a simple group can in principle be recognised from the centraliser of an involution.

History

The theorem was proved by Richard Brauer and K. A. Fowler in 1955.

Going deeper

For a student

By the Feit–Thompson theorem, every non-abelian finite simple group has involutions, so the theorem applies to all of them. It started the programme of characterising simple groups by the centralisers of their involutions. This programme became a central part of the classification of finite simple groups; for example, several sporadic groups were first predicted from a possible centraliser of an involution.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Centraliser, concept, no evidence levelCentraliserconceptOrder of an element, concept, no evidence levelOrder of an elementconceptSimple group, concept, no evidence levelSimple groupconceptBrauer–Fowler theorem, statement, no evidence levelBrauer–Fowler theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Centraliser concept
  2. Order of an element concept
  3. Simple group concept
  4. Brauer–Fowler theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Brauer%E2%80%93Fowler_theorem
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Kindstatement, theorem
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Changed2 November 2026

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