---
tag: K5CQM8MYYG
kind: statement
role: theorem
name: Brauer–Fowler theorem
msc: [20D05, 20D06]
uses: [0YMYMNQ0QZ, 02YYV36BFM, 0WE2NHZJ7B]
about: [0YMYMNQ0QZ, 02YYV36BFM]
sources:
  - url: https://en.wikipedia.org/wiki/Brauer%E2%80%93Fowler_theorem
same_as:
  wikidata: Q4958229
  wikipedia: Brauer–Fowler theorem
---

## Statement

An involution of a group is an element of [[order|0WE2NHZJ7B]] $2$.
Let $n$ be a positive integer.
Then, up to isomorphism, there are only finitely many finite [[simple groups|0YMYMNQ0QZ]] $G$ that contain an involution $t$ whose [[centraliser|02YYV36BFM]] $C_G(t)$ has order $n$.

## Remarks

The proof bounds the order of $G$ by a function of $n$ alone, using counting arguments with involutions and their products.
In particular, for a given finite group $H$, only finitely many finite simple groups have an involution with centraliser isomorphic to $H$.
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.

## Explanations

### L1 human

An involution is an element that undoes itself.
The theorem says that knowing the elements that commute with one involution limits a simple group to finitely many possibilities.

### L2 human

By the [[Feit–Thompson theorem|6VK9RSG13Q]], 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|06J6P03MG7]]; for example, several [[sporadic groups|PZ49Z4CBFZ]] were first predicted from a possible centraliser of an involution.