Classification of finite simple groups: Difference between revisions

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{{shortShort description|MassiveTheorem theorem assigning all but 27classifying finite simple groups to a few infinite families}}

{{Group theory sidebar|Finite}}

In [[mathematics]], the '''classification of finite simple groups''' (popularly called the '''enormous theorem<ref>{{Cite web |author1=Rose Eveleth |date=2011-12-09 |title=The Funniest Theories in Physics |url=https://www.livescience.com/33628-funny-physics-theorems-names.html |access-date=2024-11-16 |website=livescience.com |language=en}}</ref>'''<ref>{{Cite web |date=2015-07-01 |title=Researchers Race to Rescue the Enormous Theorem before Its Giant Proof Vanishes |url=https://www.scientificamerican.com/article/researchers-race-to-rescue-the-enormous-theorem-before-its-giant-proof-vanishes/ |access-date=2024-11-16 |website=Scientific American |language=en}}</ref>) is a result of [[group theory]] stating that every [[List of finite simple groups|finite simple group]] is either [[cyclic group|cyclic]], or [[alternating groups|alternating]], or 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 [[Tits group]] is sometimes regarded as a sporadic group because it is not strictly a [[group of Lie type]],<ref name=Conway>{{harvtxt|Conway|Curtis|Norton|Parker|Wilson|1985|loc=p. viii}}</ref> in which case there would be 27 sporadic groups). 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.

[[Daniel Gorenstein|Gorenstein]] (d.19921923–1992), [[Richard Lyons (mathematician)|Richard Lyons]], and [[Ronald Solomon|Solomon]] are gradually publishing a simplified and revised version of the proof.

==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 threespecific 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>

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

}}

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===Groups of component type===

A group is said to be of component type if for some centralizer ''C'' of an involution, ''C''/''O''(''C'') has a component (where ''O''(''C'') is the core of ''C'', the maximal normal subgroup of odd order). These are more or less the groups of Lie type of odd characteristic of large rank, and alternating groups, together with some sporadic groups. A major step in this case is to eliminate the obstruction of the core of an involution. This is accomplished by the [[B-theorem]], which states that every component of ''C''/''O''(''C'') is the image of a component of ''C''.

These are more or less the groups of Lie type of odd characteristic of large rank, and alternating groups, together with some sporadic groups.

A major step in this case is to eliminate the obstruction of the core of an involution. This is accomplished by the [[B-theorem]], which states that every component of ''C''/''O''(''C'') is the image of a component of ''C''.

The idea is that these groups have a centralizer of an involution with a component that is a smaller quasisimple group, which can be assumed to be already known by induction. So to classify these groups one takes every central extension of every known finite simple group, and finds all simple groups with a centralizer of involution with this as a component. This gives a rather large number of different cases to check: there are not only 26 sporadic groups and 16 families of groups of Lie type and the alternating groups, but also many of the groups of small rank or over small fields behave differently from the general case and have to be treated separately, and the groups of Lie type of even and odd characteristic are also quite different.

===Groups of characteristic 2 type===

A group is of characteristic 2 type if the [[generalized Fitting subgroup]] ''F''*(''Y'') of every 2-local subgroup ''Y'' is a 2-group. As the name suggests these are roughly the groups of Lie type over fields of characteristic 2, plus a handful of others that are alternating or sporadic or of odd characteristic. Their classification is divided into the small and large rank cases, where the rank is the largest rank of an odd abelian subgroup normalizing a nontrivial 2-subgroup, which is often (but not always) the same as the rank of a Cartan subalgebra when the group is a group of Lie type in characteristic 2.

A group is of characteristic 2 type if the [[generalized Fitting subgroup]] ''F''*(''Y'') of every 2-local subgroup ''Y'' is a 2-group.

As the name suggests these are roughly the groups of Lie type over fields of characteristic 2, plus a handful of others that are alternating or sporadic or of odd characteristic. Their classification is divided into the small and large rank cases, where the rank is the largest rank of an odd abelian subgroup normalizing a nontrivial 2-subgroup, which is often (but not always) the same as the rank of a Cartan subalgebra when the group is a group of Lie type in characteristic 2.

The rank 1 groups are the thin groups, classified by Aschbacher, and the rank 2 ones are the notorious [[quasithin group]]s, classified by Aschbacher and Smith. These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2.

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|-

|style="vertical-align:top"|2008

|Harada and Solomon fill a minor gap in the classification by describing groups with a standard component that is a cover of the [[Mathieu group M22|Mathieu group M₂₂]], a case that was accidentally omitted from the proof of the classification due to an error in the calculation of the Schur multiplier of M22M₂₂.

|-

|style="vertical-align:top"|2012

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==Second-generation classification==

The proof of the theorem, as it stood around 1985 or so, can be called ''first generation''. Because of the extreme length of the first generation proof, much effort has been devoted to finding a simpler proof, called a '''second-generation classification proof'''. This effort, called "revisionism", was originally led by [[Daniel Gorenstein]], and coauthored with [[Richard Lyons (mathematician)|Richard Lyons]] and [[Ronald Solomon]].

{{As of|2023}}, ten volumes of the second generation proof have been published (Gorenstein, Lyons & Solomon 1994, 1996, 1998, 1999, 2002, 2005, 2018a, 2018b; & [[Inna Capdeboscq]], 2021, 2023). In 2012 Solomon estimated that the project would need another 5 volumes, but said that progress on them was slow. It is estimated that the new proof will eventually fill approximately 5,000 pages. (This length stems in part from the second generation proof being written in a more relaxed style.) However, with the publication of volume 9 of the GLS series, and including the Aschbacher–Smith contribution, this estimate was already reached, with several more volumes still in preparation (the rest of what was originally intended for volume 9, plus projected volumes 10 and 11). Aschbacher and Smith wrote their two volumes devoted to the quasithin case in such a way that those volumes can be part of the second generation proof.

Gorenstein and his collaborators have given several reasons why a simpler proof is possible.

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===Length of proof===

Gorenstein has discussed some of the reasons why there might not be a short proof of the classification similar to the classification of [[compact Lie group]]s.

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==Consequences of the classification==

This section lists some results that have been proved using the classification of finite simple groups.

*A breakthrough in the best known theoretical algorithm for the [[graph isomorphism problem]] in 1982<ref>{{Cite journal |last=Luks |first=Eugene M. |date=1982-08-01 |title=Isomorphism of graphs of bounded valence can be tested in polynomial time |url=https://www.sciencedirect.com/science/article/pii/0022000082900095 |journal=Journal of Computer and System Sciences |volume=25 |issue=1 |pages=42–65 |doi=10.1016/0022-0000(82)90009-5 |issn=0022-0000|url-access=subscription }}</ref>

*The [[Schreier conjecture]]

*The [[Signalizer functor theorem]]

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==See also==

*[[O'Nan–Scott theorem]]

==Notes==

<references group="note" />

==Citations==

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* Madore, David (2003) ''[http://www.eleves.ens.fr:8080/home/madore/math/simplegroups.html Orders of nonabelian simple groups.] {{Webarchive|url=https://web.archive.org/web/20050404210024/http://www.eleves.ens.fr:8080/home/madore/math/simplegroups.html |date=2005-04-04 }}'' Includes a list of all nonabelian simple groups up to order 10¹⁰.

* [https://mathoverflow.net/q/180355 In what sense is the classification of all finite groups “impossible”?]

* {{Cite journal|last=Ornes|first=Stephen|title=Researchers Race to Rescue the Enormous Theorem before Its Giant Proof Vanishes|url=https://www.scientificamerican.com/article/researchers-race-to-rescue-the-enormous-theorem-before-its-giant-proof-vanishes/|journal=[[Scientific American]]|date=2015|language=en|doi=10.1038/scientificamerican0715-68|title-link=doi|volume=313|issue=1|pages=68–75|pmid=26204718|url-access=subscription}}

* {{Cite web |title=Where are the second- (and third-)generation proofs of the classification of finite simple groups up to? |url=https://mathoverflow.net/questions/114943/where-are-the-second-and-third-generation-proofs-of-the-classification-of-fin|website=[[MathOverflow]] |language=en}} (Last updated in February 2024)