Classification of finite simple groups: Difference between revisions

Revision as of 20:35, 2 August 2023

Revision as of 17:56, 27 September 2023

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{{Group theory sidebar|Finite}}

In [[mathematics]], the '''classification of finite simple groups''' is a result of [[group theory]] stating that every [[List of finite simple groups|finite simple group]] is either [[cyclic groupsgroup|cyclic]], or [[alternating groups|alternating]], or it belongs to a broad infinite class called the [[groups of Lie type]], or else it is one of twenty-six or twenty-seven exceptions, called [[sporadic groups|sporadic]]. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

Simple groups can be seen as the basic building blocks of all [[finite group]]s, reminiscent of the way the [[prime number]]s are the basic building blocks of the [[natural number]]s. The [[Jordan–Hölder theorem]] is a more precise way of stating this fact about finite groups. However, a significant difference from [[integer factorization]] is that such "building blocks" do not necessarily determine a unique group, since there might be many non-[[isomorphic]] groups with the same [[composition series]] or, put in another way, the [[group extension#Extension problem|extension problem]] does not have a unique solution.

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{{math_theorem|Every finite [[simple group]] is [[isomorphic]] to one of the following groups:

* a member of one of three infinite classes of such, namely:

** the [[cyclic groupsgroup]]s of prime order,

** the [[alternating groups]] of degree at least 5,

** the [[groups of Lie type]]<ref group="note" name="tits">The infinite family of [[Ree group#Ree groups of type 2F4|Ree groups of type {{math|²F₄(2^2''n''+1)}}]] contains only finite groups of Lie type. They are simple for {{math|''n''≥1}}; for {{math|''n''{{=}}0}}, the group {{math|²F₄(2)}} is not simple, but it contains the simple [[commutator subgroup]] {{math|²F₄(2)′}}. So, if the infinite family of commutator groups of type {{math|²F₄(2^2''n''+1)′}} is considered a systematic infinite family (all of Lie type except for {{math|''n''{{=}}0}}), the Tits group {{math|T :{{=}} ²F₄(2)′}} (as a member of this infinite family) is not sporadic.</ref>