Classification of finite simple groups: Difference between revisions

Revision as of 22:03, 13 April 2025

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==Statement of the classification theorem==

{{Main|List of finite simple groups}}

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

* a member of one18 of fourspecific infinite classesfamilies of suchsimple groups, namely:,

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

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

** 16 other infinite families known as the [[List_of_finite_simple_groups#Groups_of_Lie_type | simple groups of Lie type]],

* or one of 26 specific groups called the "[[sporadic groups]]".

** the [[commutator subgroup|derived subgroup]] of the groups of Lie Type, such as the [[Tits group]]<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>

* or one of 26 groups called the "[[sporadic groups]]"

}}