| Algebraic structure →Group theory Group theory |
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Infinite dimensional Lie group
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Inmathematics, theclassification of finite simple groups (popularly called theenormous theorem[1][2]) is a result ofgroup theory stating that everyfinite simple group is eithercyclic, oralternating, or belongs to a broad infinite class called thegroups of Lie type, or else it is one of twenty-six exceptions, calledsporadic (theTits group is sometimes regarded as a sporadic group because it is not strictly agroup of Lie type,[3] 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 allfinite groups, reminiscent of the way theprime numbers are the basic building blocks of thenatural numbers. TheJordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference frominteger factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the samecomposition series or, put in another way, theextension problem does not have a unique solution.
Daniel Gorenstein (1923–1992),Richard Lyons, andRonald Solomon are gradually publishing a simplified and revised version of the proof.
Theorem—Every finitesimple group is, up toisomorphism, one of the following groups:
The classification theorem has applications in many branches of mathematics, as questions about the structure offinite groups (and their action on other mathematical objects) can sometimes be reduced to questions about finite simple groups. Thanks to the classification theorem, such questions can sometimes be answered by checking each family of simple groups and each sporadic group.
Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification ofquasithin groups. The completed proof of the classification was announced byAschbacher (2004) after Aschbacher and Smith published a 1221-page proof for the missing quasithin case.
Gorenstein (1982,1983) wrote two volumes outlining the low rank and odd characteristic part of the proof, andMichael Aschbacher, Richard Lyons, and Stephen D. Smith et al. (2011)wrote a 3rd volume covering the remaining characteristic 2 case. The proof can be broken up into several major pieces as follows:
The simple groups of low2-rank are mostly groups of Lie type of small rank over fields of odd characteristic, together with five alternating and seven characteristic 2 type and nine sporadic groups.
The simple groups of small 2-rank include:
The classification of groups of small 2-rank, especially ranks at most 2, makes heavy use of ordinary and modular character theory, which is almost never directly used elsewhere in the classification.
All groups not of small 2 rank can be split into two major classes: groups of component type and groups of characteristic 2 type. This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and thebalance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type. (For groups of low 2-rank the proof of this breaks down, because theorems such as thesignalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.)
A group is said to be of component type if for some centralizerC of an involution,C/O(C) has a component (whereO(C) is the core ofC, 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 theB-theorem, which states that every component ofC/O(C) is the image of a component ofC.
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.
A group is of characteristic 2 type if thegeneralized Fitting subgroupF*(Y) of every 2-local subgroupY 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 notoriousquasithin groups, classified by Aschbacher and Smith. These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2.
Groups of rank at least 3 are further subdivided into 3 classes by thetrichotomy theorem, proved by Aschbacher for rank 3 and by Gorenstein and Lyons for rank at least 4.The three classes are groups of GF(2) type (classified mainly by Timmesfeld), groups of "standard type" for some odd prime (classified by theGilman–Griess theorem and work by several others), and groups of uniqueness type, where a result of Aschbacher implies that there are no simple groups.The general higher rank case consists mostly of the groups of Lie type over fields of characteristic 2 of rank at least 3 or 4.
The main part of the classification produces a characterization of each simple group. It is then necessary to check that there exists a simple group for each characterization and that it is unique. This gives a large number of separate problems; for example, the original proofs of existence and uniqueness of themonster group totaled about 200 pages, and the identification of theRee groups by Thompson and Bombieri was one of the hardest parts of the classification. Many of the existence proofs and some of the uniqueness proofs for the sporadic groups originally used computer calculations, most of which have since been replaced by shorter hand proofs.
In 1972Gorenstein (1979, Appendix) announced a program for completing the classification of finite simple groups, consisting of the following 16 steps:
Many of the items in the table below are taken fromSolomon (2001). The date given is usually the publication date of the complete proof of a result, which is sometimes several years later than the proof or first announcement of the result, so some of the items appear in the "wrong" order.
| Date | Development |
| 1832 | Galois introduces normal subgroups and finds the simple groups An (n ≥ 5) andPSL2(Fp) (p ≥ 5) |
| 1854 | Cayley defines abstract groups |
| 1861 | Mathieu describes the first twoMathieu groups M11, M12, the first sporadic simple groups, and announces the existence of M24. |
| 1870 | Jordan lists some simple groups: the alternating and projective special linear ones, and emphasizes the importance of the simple groups. |
| 1872 | Sylow proves theSylow theorems |
| 1873 | Mathieu introduces three moreMathieu groups M22, M23, M24. |
| 1892 | Hölder proves that the order of any nonabelian finite simple group must be a product of at least four (not necessarily distinct) primes, and asks for a classification of finite simple groups. |
| 1893 | Cole classifies simple groups of order up to 660 |
| 1896 | Frobenius and Burnside begin the study of character theory of finite groups. |
| 1899 | Burnside classifies the simple groups such that the centralizer of every involution is a non-trivial elementary abelian 2-group. |
| 1901 | Frobenius proves that aFrobenius group has a Frobenius kernel, so in particular is not simple. |
| 1901 | Dickson defines classical groups over arbitrary finite fields, and exceptional groups of typeG2 over fields of odd characteristic. |
| 1901 | Dickson introduces the exceptional finite simple groups of typeE6. |
| 1904 | Burnside uses character theory to proveBurnside's theorem that the order of any non-abelian finite simple group must be divisible by at least 3 distinct primes. |
| 1905 | Dickson introduces simple groups of type G2 over fields of even characteristic |
| 1911 | Burnside conjectures that every non-abelian finite simple group has even order |
| 1928 | Hall proves the existence ofHall subgroups of solvable groups |
| 1933 | Hall begins his study ofp-groups |
| 1935 | Brauer begins the study ofmodular characters. |
| 1936 | Zassenhaus classifies finite sharply 3-transitive permutation groups |
| 1938 | Fitting introduces theFitting subgroup and proves Fitting's theorem that for solvable groups the Fitting subgroup contains its centralizer. |
| 1942 | Brauer describes the modular characters of a group divisible by a prime to the first power. |
| 1954 | Brauer classifies simple groups with GL2(Fq) as the centralizer of an involution. |
| 1955 | TheBrauer–Fowler theorem implies that the number of finite simple groups with given centralizer of involution is finite, suggesting an attack on the classification using centralizers of involutions. |
| 1955 | Chevalley introduces theChevalley groups, in particular introducing exceptional simple groups of typesF4,E7, andE8. |
| 1956 | TheHall–Higman theorem describes the possibilities for theminimal polynomial of an element of prime power order for a representation of ap-solvable group. |
| 1957 | Suzuki shows that all finite simpleCA groups of odd order are cyclic. |
| 1958 | TheBrauer–Suzuki–Wall theorem characterizes the projective special linear groups of rank 1, and classifies the simpleCA groups. |
| 1959 | Steinberg introduces theSteinberg groups, giving some new finite simple groups, of types3D4 and2E6 (the latter were independently found at about the same time by Tits). |
| 1959 | TheBrauer–Suzuki theorem about groups with generalized quaternion Sylow 2-subgroups shows in particular that none of them are simple. |
| 1960 | Thompson proves that a group with a fixed-point-free automorphism of prime order is nilpotent. |
| 1960 | Feit, Marshall Hall, and Thompson show that all finite simpleCN groups of odd order are cyclic. |
| 1960 | Suzuki introduces theSuzuki groups, with types2B2. |
| 1961 | Ree introduces theRee groups, with types2F4 and2G2. |
| 1963 | Feit and Thompson prove theodd order theorem. |
| 1964 | Tits introduces BN pairs for groups of Lie type and finds theTits group |
| 1965 | TheGorenstein–Walter theorem classifies groups with a dihedral Sylow 2-subgroup. |
| 1966 | Glauberman proves theZ* theorem |
| 1966 | Janko introduces theJanko group J1, the first new sporadic group for about a century. |
| 1968 | Glauberman proves theZJ theorem |
| 1968 | Higman and Sims introduce theHigman–Sims group |
| 1968 | Conway introduces theConway groups |
| 1969 | Walter's theorem classifies groups with abelian Sylow 2-subgroups |
| 1969 | Introduction of theSuzuki sporadic group, theJanko group J2, theJanko group J3, theMcLaughlin group, and theHeld group. |
| 1969 | Gorenstein introducessignalizer functors based on Thompson's ideas. |
| 1970 | MacWilliams shows that the 2-groups with no normal abelian subgroup of rank 3 have sectional 2-rank at most 4. (The simple groups with Sylow subgroups satisfying the latter condition were later classified by Gorenstein and Harada.) |
| 1970 | Bender introduced thegeneralized Fitting subgroup |
| 1970 | TheAlperin–Brauer–Gorenstein theorem classifies groups with quasi-dihedral or wreathed Sylow 2-subgroups, completing the classification of the simple groups of 2-rank at most 2 |
| 1971 | Fischer introduces the threeFischer groups |
| 1971 | Thompson classifiesquadratic pairs |
| 1971 | Bender classifies group with astrongly embedded subgroup |
| 1972 | Gorenstein proposes a 16-step program for classifying finite simple groups; the final classification follows his outline quite closely. |
| 1972 | Lyons introduces theLyons group |
| 1973 | Rudvalis introduces theRudvalis group |
| 1973 | Fischer discovers thebaby monster group (unpublished), which Fischer and Griess use to discover themonster group, which in turn leads Thompson to theThompson sporadic group and Norton to theHarada–Norton group (also found in a different way by Harada). |
| 1974 | Thompson classifiesN-groups, groups all of whose local subgroups are solvable. |
| 1974 | TheGorenstein–Harada theorem classifies the simple groups of sectional 2-rank at most 4, dividing the remaining finite simple groups into those of component type and those of characteristic 2 type. |
| 1974 | Tits shows that groups withBN pairs of rank at least 3 are groups of Lie type |
| 1974 | Aschbacher classifies the groups with a proper2-generated core |
| 1975 | Gorenstein and Walter prove theL-balance theorem |
| 1976 | Glauberman proves the solvablesignalizer functor theorem |
| 1976 | Aschbacher proves thecomponent theorem, showing roughly that groups of odd type satisfying some conditions have a component in standard form. The groups with a component of standard form were classified in a large collection of papers by many authors. |
| 1976 | O'Nan introduces theO'Nan group |
| 1976 | Janko introduces theJanko group J4, the last sporadic group to be discovered |
| 1977 | Aschbacher characterizes the groups of Lie type of odd characteristic in hisclassical involution theorem. After this theorem, which in some sense deals with "most" of the simple groups, it was generally felt that the end of the classification was in sight. |
| 1978 | Timmesfeld proves the O2 extraspecial theorem, breaking the classification ofgroups of GF(2)-type into several smaller problems. |
| 1978 | Aschbacher classifies thethin finite groups, which are mostly rank 1 groups of Lie type over fields of even characteristic. |
| 1981 | Bombieri uses elimination theory to complete Thompson's work on the characterization ofRee groups, one of the hardest steps of the classification. |
| 1982 | McBride proves thesignalizer functor theorem for all finite groups. |
| 1982 | Griess constructs themonster group by hand |
| 1983 | TheGilman–Griess theorem classifies groups of characteristic 2 type and rank at least 4 with standard components, one of the three cases of the trichotomy theorem. |
| 1983 | Aschbacher proves that no finite group satisfies the hypothesis of theuniqueness case, one of the three cases given by the trichotomy theorem for groups of characteristic 2 type. |
| 1983 | Gorenstein and Lyons prove thetrichotomy theorem for groups of characteristic 2 type and rank at least 4, while Aschbacher does the case of rank 3. This divides these groups into 3 subcases: the uniqueness case, groups of GF(2) type, and groups with a standard component. |
| 1983 | Gorenstein announces the proof of the classification is complete, somewhat prematurely as the proof of the quasithin case was incomplete. |
| 1985 | Conway, Curtis, Norton, Parker, Wilson and Thackray publish theAtlas of Finite Groups with basic information about 93 finite simple groups. |
| 1994 | Gorenstein, Lyons, and Solomon begin publication of the revised classification |
| 2004 | Aschbacher and Smith publish their work onquasithin groups (which are mostly groups of Lie type of rank at most 2 over fields of even characteristic), filling the last gap in the classification known at that time. |
| 2008 | Harada and Solomon fill a minor gap in the classification by describing groups with a standard component that is a cover of theMathieu 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 M22. |
| 2012 | Gonthier and collaborators announce a computer-checked version of theFeit–Thompson theorem using theRocq (then:Coq)proof assistant.[4] |
The proof of the theorem, as it stood around 1985 or so, can be calledfirst generation. Because of the extreme length of the first generation proof, much effort has been devoted to finding a simpler proof, called asecond-generation classification proof. This effort, called "revisionism", was originally led byDaniel Gorenstein, and coauthored withRichard Lyons andRonald Solomon.
As of 2023[update], 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.
Aschbacher (2004) has called the work on the classification problem by Ulrich Meierfrankenfeld, Bernd Stellmacher, Gernot Stroth, and a few others, athird generation program. One goal of this is to treat all groups in characteristic 2 uniformly using the amalgam method.
Gorenstein has discussed some of the reasons why there might not be a short proof of the classification similar to the classification ofcompact Lie groups.
This section lists some results that have been proved using the classification of finite simple groups.