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  "body": "\n## Statement\n\nAn involution of a group is an element of [[order|0WE2NHZJ7B]] $2$.\nLet $n$ be a positive integer.\nThen, 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 $n$.\n\n## Remarks\n\nThe proof bounds the order of $G$ by a function of $n$ alone, using counting arguments with involutions and their products.\nIn particular, for a given finite group $H$, only finitely many finite simple groups have an involution with centraliser isomorphic to $H$.\nThe bound obtained is far too large to be used directly.\nBut the finiteness result showed that a simple group can in principle be recognised from the centraliser of an involution.\n\n## History\n\nThe theorem was proved by Richard Brauer and K. A. Fowler in 1955.\n\n## Explanations\n\n### L1 human\n\nAn involution is an element that undoes itself.\nThe theorem says that knowing the elements that commute with one involution limits a simple group to finitely many possibilities.\n\n### L2 human\n\nBy the [[Feit–Thompson theorem|6VK9RSG13Q]], every non-abelian finite simple group has involutions, so the theorem applies to all of them.\nIt started the programme of characterising simple groups by the centralisers of their involutions.\nThis 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.\n",
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