 Classification of finite simple groups

Group theory Group theory Finite groups (classification)Cyclic group Z_{n}
Symmetric group, S_{n}
Dihedral group, D_{n}
Alternating group A_{n}
Mathieu groups M_{11}, M_{12}, M_{22}, M_{23}, M_{24}
Conway groups Co_{1}, Co_{2}, Co_{3}
Janko groups J_{1}, J_{2}, J_{3}, J_{4}
Fischer groups F_{22}, F_{23}, F_{24}
Baby Monster group B
Monster group MSolenoid (mathematics)
Circle group
General linear group GL(n)
Special linear group SL(n)
Orthogonal group O(n)
Special orthogonal group SO(n)
Unitary group U(n)
Special unitary group SU(n)
Symplectic group Sp(n)
Lorentz group
Poincaré group
Conformal group
Diffeomorphism group
Loop group
Infinitedimensional Lie groups O(∞) SU(∞) Sp(∞)v · mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Gorenstein, Lyons, and Solomon are gradually publishing a simplified and revised version of the proof.
Contents
Statement of the classification theorem
Main article: List of finite simple groupsTheorem. Every finite simple group is isomorphic to one of the following groups:
 A cyclic group with prime order;
 An alternating group of degree at least 5;
 A simple group of Lie type, including both
 the classical Lie groups, namely the groups of projective special linear, unitary, symplectic, or orthogonal transformations over a finite field;
 the exceptional and twisted groups of Lie type (including the Tits group which is not strictly a group of Lie type).
 The 26 sporadic simple groups.
The classification theorem has applications in many branches of mathematics, as questions about the structure of finite 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 of quasithin groups. The completed proof of the classification was announced by Aschbacher (2004) after Aschbacher and Smith published a 1221 page proof for the missing quasithin case.
Overview of the proof of the classification theorem
Gorenstein (1982, 1983) wrote two volumes outlining the low rank and odd characteristic part of the proof, and Michael 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:
Groups of small 2rank
The simple groups of low 2rank 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 2rank include:
 Groups of 2rank 0, in other words groups of odd order, which are all solvable by the FeitThompson theorem.
 Groups of 2rank 1. The Sylow 2subgroups are either cyclic, which is easy to handle using the transfer map, or generalized quaternion, which are handled with the BrauerSuzuki theorem: in particular there are no simple groups of 2rank 1.
 Groups of 2rank 2. Alperin showed that the Sylow subgoup must be dihedral, quasidihedral, wreathed, or a Sylow 2subgroup of U_{3}(4). The first case was done by the Gorenstein–Walter theorem which showed that the only simple groups are isomorphic to L_{2}(q) for q odd or A_{7}, the second and third cases were done by the Alperin–Brauer–Gorenstein theorem which implies that the only simple groups are isomorphic to L_{3}(q) or U_{3}(q) for q odd or M_{11}, and the last case was done by Lyons who showed that U_{3}(4) is the only simple possibility.
 Groups of sectional 2rank at most 4, classified by the Gorenstein–Harada theorem.
The classification of groups of small 2rank, 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 2rank at least 5 then MacWilliams showed that its Sylow 2subgroups are connected, and the balance theorem implies that any simple group with connected Sylow 2subgroups is either of component type or characteristic 2 type. (For groups of low 2rank the proof of this breaks down, because theorems such as the signalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.)
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 Btheorem, 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 2local subgroup Y is a 2group. 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 2subgroup, 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 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 the trichotomy 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 the GilmanGriess 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.
Existence and uniqueness of the simple groups
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 the monster totaled about 200 pages, and the identification of the Ree 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 proofs originally used computer calculations, some of which have since been replaced by shorter hand proofs.
History of the proof
Gorenstein's program
In 1972 Gorenstein (1979, Appendix) announced a program for completing the classification of finite simple groups, consisting of the following 16 steps:
 Groups of low 2rank. This was essentially done by Gorenstein and Harada, who classified the groups with sectional 2rank at most 4. Most of the cases of 2rank at most 2 had been done by the time Gorenstein announced his program.
 The semisimplicity of 2layers. The problem is to prove that the 2layer of the centralizer of an involution in a simple group is semisimple.
 Standard form in odd characteristic. If a group has an involution with a 2component that is a group of Lie type of odd characteristic, the goal is to show that it has a centralizer of involution in "standard form" meaning that a centralizer of involution has a component that is of Lie type in odd characteristic and also has a centralizer of 2rank 1.
 Classification of groups of odd type. The problem is to show that if a group has a centralizer of involution in "standard form" then it is a group of Lie type of odd characteristic. This was solved by Aschbacher's classical involution theorem.
 Quasistandard form
 Central involutions
 Classification of alternating groups. More precisely, show that if a simple group has
 Some sporadic groups
 Thin groups. The simple thin finite groups, those with 2local prank at most 1 for odd primes p, were classified by Aschbacher in 1978
 Groups with a strongly pembedded subgroup for p odd
 The signalizer functor method for odd primes. The main problem is to prove a signalizer functor theorem for nonsolvable signalizer functors. This was solved by McBride in 1982.
 Groups of characteristic p type. This is the problem of groups with a strongly pembedded 2local subgroup with p odd, which was handled by Aschbacher.
 Quasithin groups. A quasithin group is one whose 2local subgroups have prank at most 2 for all odd primes p, and the problem is to classify the simple ones of characteristic 2 type. This was completed by Aschbacher and Smith in 2004.
 Groups of low 2local 3rank. This was essentially solved by Aschbacher's trichotomy theorem for groups with e(G)=3. The main change is that 2local 3rank is replaced by 2local prank for odd primes.
 Centralizers of 3elements in standard form. This was essentially done by the Trichotomy theorem.
 Classification of simple groups of characteristic 2 type. This was handled by the GilmanGriess theorem, with 3elements replaced by pelements for odd primes.
Timeline of the proof
Many of the items in the list below are taken from Solomon (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.
Publication date 1832 Galois introduces normal subgroups and finds the simple groups A_{n} (n≥5) and PSL_{2}(F_{p}) (p≥5) 1854 Cayley defines abstract groups 1861 Mathieu finds the first two Mathieu groups M_{11}, M_{12}, the first sporadic simple groups. 1870 Jordan lists some simple groups: the alternating and projective special linear ones, and emphasizes the importance of the simple groups. 1872 Sylow proves the Sylow theorems 1873 Mathieu finds three more Mathieu groups M_{22}, M_{23}, M_{24}. 1892 Otto Hölder proves that the order of any nonabelian finite simple group must be a product of at least 4 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 nontrivial elementary abelian 2group. 1901 Frobenius proves that a Frobenius group has a Frobenius kernel, so in particular is not simple. 1901 Dickson defines classical groups over arbitrary finite fields, and exceptional groups of type G_{2} over fields of odd characteristic. 1901 Dickson introduces the exceptional finite simple groups of type E_{6}. 1904 Burnside uses character theory to prove Burnside's theorem that the order of any nonabelian finite simple group must be divisible by at least 3 distinct primes. 1905 Dickson introduces simple groups of type G_{2} over fields of even characteristic 1911 Burnside conjectures that every nonabelian finite simple group has even order 1928 Hall proves the existence of Hall subgroups of solvable groups 1933 Hall begins his study of pgroups 1935 Brauer begins the study of modular characters. 1936 Zassenhaus classifies finite sharply 3transitive permutation groups 1938 Fitting introduces the Fitting 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 GL_{2}(F_{q}) as the centralizer of an involution. 1955 The Brauer–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 the Chevalley groups, in particular introducing exceptional simple groups of types F_{4}, E_{7}, and E_{8}. 1956 Hall–Higman theorem 1957 Suzuki shows that all finite simple CA groups of odd order are cyclic. 1958 The Brauer–Suzuki–Wall theorem characterizes the projective special linear groups of rank 1, and classifies the simple CA groups. 1959 Steinberg introduces the Steinberg groups, giving some new finite simple groups, or types ^{3}D_{4} and ^{2}E_{6} (the latter were independently found at about the same time by Jacques Tits). 1959 The Brauer–Suzuki theorem about groups with generalized quaternion Sylow 2subgroups shows in particular that none of them are simple. 1960 Thompson proves that a group with a fixedpointfree automorphism of prime order is nilpotent. 1960 Feit, Hall, and Thompson show that all finite simple CN groups of odd order are cyclic. 1960 Suzuki introduces the Suzuki groups, with types ^{2}B_{2}. 1961 Ree introduces the Ree groups, with types ^{2}F_{4} and ^{2}G_{2}. 1963 Feit and Thompson prove the odd order theorem. 1964 Tits introduces BN pairs for groups of Lie type and finds the Tits group 1965 The Gorenstein–Walter theorem classifies groups with a dihedral Sylow 2subgroup. 1966 Glauberman proves the Z* theorem 1966 Janko introduces the Janko group J1, the first new sporadic group for about a century. 1968 Glauberman proves the ZJ theorem 1968 Higman and Sims introduce the Higman–Sims group 1968 Conway introduces the Conway groups 1969 Walter's theorem classifies groups with abelian Sylow 2subgroups 1969 Introduction of the Suzuki sporadic group, the Janko group J2, the Janko group J3, the McLaughlin group, and the Held group. 1969 Gorenstein introduces signalizer functors based on Thompson's ideas. 1970 Bender introduced the generalized Fitting subgroup 1970 The Alperin–Brauer–Gorenstein theorem classifies groups with quasidihedral or wreathed Sylow 2subgroups, completing the classification of the simple groups of 2rank at most 2 1971 Fischer introduces the three Fischer groups 1971 Thompson classifies quadratic pairs 1971 Bender classifies group with a strongly embedded subgroup 1972 Gorenstein proposes a 16step program for classifying finite simple groups; the final classification follows his outline quite closely. 1972 Lyons introduces the Lyons group 1973 Rudvalis introduces the Rudvalis group 1973 Fisher discovers the baby monster group (unpublished), which Fischer and Griess use to discover the monster group, which in turn leads Thompson to the Thompson sporadic group and Norton to the Harada–Norton group (also found in a different way by Harada). 1974 Thompson classifies Ngroups, groups all of whose local subgroups are solvable. 1974 The Gorenstein–Harada theorem classifies the simple groups of sectional 2rank 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 with BN pairs of rank at least 3 are groups of Lie type 1974 Aschbacher classifies the groups with a proper 2generated core 1975 Gorenstein and Walter prove the Lbalance theorem 1976 Glauberman proves the solvable signalizer functor theorem 1976 Aschbacher proves the component 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 the O'Nan group 1976 Janko introduces the Janko group J4, the last sporadic group to be discovered 1977 Aschbacher characterizes the groups of Lie type of odd characteristic in his classical 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 O_{2} extraspecial theorem, breaking the classification of groups of GF(2)type into several smaller problems. 1978 Aschbacher classifies the thin 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 of Ree groups, one of the hardest steps of the classification. 1982 McBride proves the signalizer functor theorem for all finite groups. 1982 Griess constructs the monster group by hand 1983 The Gilman–Griess theorem classifies groups 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 the uniqueness case, one of the three cases given by the trichotomy theorem for groups of characteristic 2 type. 1983 Gorenstein and Lyons prove the trichotomy 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. 1994 Gorenstein, Lyons, and Solomon begin publication of the revised classification 2004 Aschbacher and Smith publish their work on quasithin groups (which are mostly groups of Lie type of rank at most 2 over fields of even characteristic), filling the last (known) gap in the classification. Secondgeneration 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 secondgeneration classification proof. This effort, called "revisionism", was originally led by Daniel Gorenstein.
As of 2005, six volumes of the second generation proof have been published (Gorenstein, Lyons & Solomon 1994, 1996, 1998, 1999, 2002, 2005), with most of the balance of the proof in manuscript. It is estimated that the new proof will eventually fill approximately 5,000 pages. (This length stems in part from second generation proof being written in a more relaxed style.) 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.
 The most important is that the correct, final statement of the theorem is now known. Simpler techniques can be applied that are known to be adequate for the types of groups we know to be finite simple. In contrast, those who worked on the first generation proof did not know how many sporadic groups there were, and in fact some of the sporadic groups (e.g., the Janko groups) were discovered while proving other cases of the classification theorem. As a result, many of the pieces of the theorem were proved using techniques that were overly general.
 Because the conclusion was unknown, the first generation proof consists of many standalone theorems, dealing with important special cases. Much of the work of proving these theorems was devoted to the analysis of numerous special cases. Given a larger, orchestrated proof, dealing with many of these special cases can be postponed until the most powerful assumptions can be applied. The price paid under this revised strategy is that these first generation theorems no longer have comparatively short proofs, but instead rely on the complete classification.
 Many first generation theorems overlap, and so divide the possible cases in inefficient ways. As a result, families and subfamiles of finite simple groups were identified multiple times. The revised proof eliminates these redundancies by relying on a different subdivision of cases.
 Finite group theorists have more experience at this sort of exercise, and have new techniques at their disposal.
Aschbacher (2004) has called the work on the classification problem by Ulrich Meierfrankenfeld, Bernd Stellmacher, Gernot Stroth, and a few others, a third generation program. One goal of this is to treat all groups in characteristic 2 uniformly using the amalgam method.
References
 Aschbacher, Michael (2004). "The Status of the Classification of the Finite Simple Groups". Notices of the American Mathematical Society. http://www.ams.org/notices/200407/feaaschbacher.pdf.
 Aschbacher, Michael; Lyons, Richard; Smith, Stephen D.; Solomon, Ronald (2011), The Classification of Finite Simple Groups: Groups of Characteristic 2 Type, Mathematical Surveys and Monographs, 172, ISBN 9780821853368, http://www.ams.org/bookstore?fn=20&ikey=SURV172
 Conway, John Horton; Curtis, Robert Turner; Norton, Simon Phillips; Parker, Richard A; Wilson, Robert Arnott (1985), Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Oxford University Press, ISBN 0198531990
 Gorenstein, D. (1979), "The classification of finite simple groups. I. Simple groups and local analysis", American Mathematical Society. Bulletin. New Series 1 (1): 43–199, doi:10.1090/S027309791979145518, ISSN 00029904, MR513750
 Gorenstein, D. (1982), Finite simple groups, University Series in Mathematics, New York: Plenum Publishing Corp., ISBN 9780306407796, MR698782
 Gorenstein, D. (1983), The classification of finite simple groups. Vol. 1. Groups of noncharacteristic 2 type, The University Series in Mathematics, Plenum Press, ISBN 9780306413056, MR746470
 Daniel Gorenstein (1985), "The Enormous Theorem", Scientific American, vol. 253, no. 6, pp. 104–115.
 Gorenstein, D. (1986), "Classifying the finite simple groups", American Mathematical Society. Bulletin. New Series 14 (1): 1–98, doi:10.1090/S027309791986153929, ISSN 00029904, MR818060
 Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1994), The classification of the finite simple groups, Mathematical Surveys and Monographs, 40, Providence, R.I.: American Mathematical Society, ISBN 9780821803349, MR1303592, http://www.ams.org/online_bks/surv401
 Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1996), The classification of the finite simple groups. Number 2. Part I. Chapter G, Mathematical Surveys and Monographs, 40, Providence, R.I.: American Mathematical Society, ISBN 9780821803905, MR1358135, http://www.ams.org/online_bks/surv402
 Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1998), The classification of the finite simple groups. Number 3. Part I. Chapter A, Mathematical Surveys and Monographs, 40, Providence, R.I.: American Mathematical Society, ISBN 9780821803912, MR1490581
 Gorenstein, D.; Lyons, Richard; Solomon, Ronald (1999), The classification of the finite simple groups. Number 4. Part II. Chapters 1–4, Mathematical Surveys and Monographs, 40, Providence, R.I.: American Mathematical Society, ISBN 9780821813799, MR1675976
 Gorenstein, D.; Lyons, Richard; Solomon, Ronald (2002), The classification of the finite simple groups. Number 5. Part III. Chapters 1–6, Mathematical Surveys and Monographs, 40, Providence, R.I.: American Mathematical Society, ISBN 9780821827765, MR1923000
 Gorenstein, D.; Lyons, Richard; Solomon, Ronald (2005), The classification of the finite simple groups. Number 6. Part IV, Mathematical Surveys and Monographs, 40, Providence, R.I.: American Mathematical Society, ISBN 9780821827772, MR2104668
 Mark Ronan, Symmetry and the Monster, ISBN 9780192807236, Oxford University Press, 2006. (Concise introduction for lay reader)
 Marcus du Sautoy, Finding Moonshine, Fourth Estate, 2008, ISBN 9780007214617 (another introduction for the lay reader)
 Ron Solomon (1995) "On Finite Simple Groups and their Classification," Notices of the American Mathematical Society. (Not too technical and good on history)
 Solomon, Ronald (2001), "A brief history of the classification of the finite simple groups", American Mathematical Society. Bulletin. New Series 38 (3): 315–352, doi:10.1090/S0273097901009090, ISSN 00029904, MR1824893, http://www.ams.org/bull/20013803/S0273097901009090/S0273097901009090.pdf – article won Levi L. Conant prize for exposition
 Thompson, John G. (1984), "Finite nonsolvable groups", in Gruenberg, K. W.; Roseblade, J. E., Group theory. Essays for Philip Hall, Boston, MA: Academic Press, pp. 1–12, ISBN 9780123048806, MR780566
 Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251, 251, Berlin, New York: SpringerVerlag, doi:10.1007/9781848009882, ISBN 9781848009875, Zbl 05622792
External links
 ATLAS of Finite Group Representations. Searchable database of representations and other data for many finite simple groups.
 Elwes, Richard, "An enormous theorem: the classification of finite simple groups," Plus Magazine, Issue 41, December 2006. For laypeople.
 Madore, David (2003) Orders of nonabelian simple groups. Includes a list of all nonabelian simple groups up to order 10^{10}.
Categories: Group theory
 Sporadic groups
 Finite groups
 Theorems in algebra
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 Classification of finite simple groups

Group theory Group theory Finite groups (classification)Cyclic group Z_{n}
Symmetric group, S_{n}
Dihedral group, D_{n}
Alternating group A_{n}
Mathieu groups M_{11}, M_{12}, M_{22}, M_{23}, M_{24}
Conway groups Co_{1}, Co_{2}, Co_{3}
Janko groups J_{1}, J_{2}, J_{3}, J_{4}
Fischer groups F_{22}, F_{23}, F_{24}
Baby Monster group B
Monster group MSolenoid (mathematics)
Circle group
General linear group GL(n)
Special linear group SL(n)
Orthogonal group O(n)
Special orthogonal group SO(n)
Unitary group U(n)
Special unitary group SU(n)
Symplectic group Sp(n)
Lorentz group
Poincaré group
Conformal group
Diffeomorphism group
Loop group
Infinitedimensional Lie groups O(∞) SU(∞) Sp(∞)