Classification of finite simple groups

Inmathematics,theclassification of finite simple groupsis a result ofgroup theorystating that everyfinite simple groupis 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 groupis sometimes regarded as a sporadic group because it is not strictly agroup of Lie type,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 numbersare the basic building blocks of thenatural numbers.TheJordan–Hölder theoremis a more precise way of stating this fact about finite groups. However, a significant difference frominteger factorizationis that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphicgroups with the samecomposition seriesor, put in another way, theextension problemdoes not have a unique solution.

Gorenstein(d.1992),Lyons,andSolomonare gradually publishing a simplified and revised version of the proof.

Statement of the classification theorem

Main article:List of finite simple groups

Theorem—Every finitesimple groupisisomorphicto one of the following groups:

• a member of one of four infinite classes of such, namely:
  • thecyclic groupsof prime order,
  • thealternating groupsof degree at least 5,
  • thegroups of Lie type,
  • thederived subgroupof the groups of Lie Type, such as theTits group
• one of 26 groups called the "sporadic groups"

The classification of the finite simple 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 Gorensteinannounced 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.

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, 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:

Groups of small 2-rank

The simple groups of low2-rankare 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:

• Groups of 2-rank 0, in other words groups of odd order, which are allsolvableby theFeit–Thompson theorem.
• Groups of 2-rank 1. The Sylow 2-subgroups are either cyclic, which is easy to handle using the transfer map, or generalizedquaternion,which are handled with theBrauer–Suzuki theorem:in particular there are no simple groups of 2-rank 1 except for the cyclic group of order two.
• Groups of 2-rank 2. Alperin showed that the Sylow subgroup must be dihedral, quasidihedral, wreathed, or a Sylow 2-subgroup ofU₃(4). The first case was done by theGorenstein–Walter theoremwhich showed that the only simple groups are isomorphic toL₂(q) forqodd orA₇,the second and third cases were done by theAlperin–Brauer–Gorenstein theoremwhich implies that the only simple groups are isomorphic toL₃(q) orU₃(q) forqodd orM₁₁,and the last case was done by Lyons who showed thatU₃(4) is the only simple possibility.
• Groups of sectional 2-rank at most 4, classified by theGorenstein–Harada theorem.

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 theoremimplies 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 functortheorem 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 centralizerCof 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.

Groups of characteristic 2 type

A group is of characteristic 2 type if thegeneralized Fitting subgroupF*(Y) of every 2-local subgroupYis 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 theoremand 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 themonster grouptotaled about 200 pages, and the identification of theRee groupsby 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.

History of the proof

Gorenstein's program

In 1972Gorenstein (1979,Appendix) announced a program for completing the classification of finite simple groups, consisting of the following 16 steps:

1. Groups of low 2-rank. This was essentially done by Gorenstein and Harada, who classified the groups with sectional 2-rank at most 4. Most of the cases of 2-rank at most 2 had been done by the time Gorenstein announced his program.
2. The semisimplicity of 2-layers. The problem is to prove that the 2-layer of the centralizer of an involution in a simple group is semisimple.
3. Standard form in odd characteristic. If a group has an involution with a 2-component 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 2-rank 1.
4. 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'sclassical involution theorem.
5. Quasi-standard form
6. Central involutions
7. Classification of alternating groups.
8. Some sporadic groups
9. Thin groups. The simplethin finite groups,those with 2-localp-rank at most 1 for odd primesp,were classified by Aschbacher in 1978
10. Groups with a strongly p-embedded subgroup forpodd
11. The signalizer functor method for odd primes. The main problem is to prove asignalizer functortheorem for nonsolvable signalizer functors. This was solved by McBride in 1982.
12. Groups of characteristicptype. This is the problem of groups with a stronglyp-embedded 2-local subgroup withpodd, which was handled by Aschbacher.
13. Quasithin groups. Aquasithin groupis one whose 2-local subgroups havep-rank at most 2 for all odd primesp,and the problem is to classify the simple ones of characteristic 2 type. This was completed by Aschbacher and Smith in 2004.
14. Groups of low 2-local 3-rank. This was essentially solved by Aschbacher'strichotomy theoremfor groups withe(G)=3. The main change is that 2-local 3-rank is replaced by 2-localp-rank for odd primes.
15. Centralizers of 3-elements in standard form. This was essentially done by theTrichotomy theorem.
16. Classification of simple groups of characteristic 2 type. This was handled by theGilman–Griess theorem,with 3-elements replaced byp-elements for odd primes.

Timeline of the proof

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 groupsM11,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 groupsM22,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 grouphas a Frobenius kernel, so in particular is not simple.
1901	Dickson defines classical groups over arbitrary finite fields, and exceptional groups of typeG2over fields of odd characteristic.
1901	Dickson introduces the exceptional finite simple groups of typeE6.
1904	Burnside uses character theory to proveBurnside's theoremthat 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 G2over fields of even characteristic
1911	Burnside conjectures that every non-abelian finite simple group has even order
1928	Hall proves the existence ofHall subgroupsof 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 subgroupand 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 theoremimplies 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 theoremdescribes the possibilities for theminimal polynomialof an element of prime power order for a representation of ap-solvable group.
1957	Suzuki shows that all finite simpleCA groupsof odd order are cyclic.
1958	TheBrauer–Suzuki–Wall theoremcharacterizes 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 types3D4and2E6(the latter were independently found at about the same time by Tits).
1959	TheBrauer–Suzuki theoremabout 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 groupsof odd order are cyclic.
1960	Suzuki introduces theSuzuki groups,with types2B2.
1961	Ree introduces theRee groups,with types2F4and2G2.
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 theoremclassifies 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	Conwayintroduces theConway groups
1969	Walter's theoremclassifies 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 functorsbased 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 theoremclassifies 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 groupand 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 theoremclassifies 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 pairsof 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 functortheorem
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 O2extraspecial theorem, breaking the classification ofgroups of GF(2)-typeinto 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 theoremfor all finite groups.
1982	Griess constructs themonster groupby hand
1983	TheGilman–Griess theoremclassifies 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 theoremfor 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 Groupswith 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 theoremusing theCoqproof assistant.

Second-generation classification

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.

As of 2023,ten volumes of the second generation proof have been published (Gorenstein, Lyons & Solomon 1994, 1996, 1998, 1999, 2002, 2005, 2018a, 2018b; & 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.

• The most important thing 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., theJanko 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 stand-alone 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 subfamilies 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, athird generation program.One goal of this is to treat all groups in characteristic 2 uniformly using the amalgam method.

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 ofcompact Lie groups.

• The most obvious reason is that the list of simple groups is quite complicated: with 26 sporadic groups there are likely to be many special cases that have to be considered in any proof. So far no one has yet found a clean uniform description of the finite simple groups similar to the parameterization of the compact Lie groups byDynkin diagrams.
• Atiyah and others have suggested that the classification ought to be simplified by constructing some geometric object that the groups act on and then classifying these geometric structures. The problem is that no one has been able to suggest an easy way to find such a geometric structure associated with a simple group. In some sense, the classification does work by finding geometric structures such asBN-pairs,but this only comes at the end of a very long and difficult analysis of the structure of a finite simple group.
• Another suggestion for simplifying the proof is to make greater use ofrepresentation theory.The problem here is that representation theory seems to require very tight control over the subgroups of a group in order to work well. For groups of small rank, one has such control and representation theory works very well, but for groups of larger rank no-one has succeeded in using it to simplify the classification. In the early days of the classification, there was a considerable effort made to use representation theory, but this never achieved much success in the higher rank case.

Consequences of the classification

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

• TheSchreier conjecture
• TheSignalizer functor theorem
• TheB conjecture
• TheSchur–Zassenhaus theoremfor all groups (though this only uses theFeit–Thompson theorem).
• A transitive permutation group on a finite set with more than 1 element has a fixed-point-free element of prime power order.
• The classification of2-transitive permutation groups.
• The classification ofrank 3 permutation groups.
• TheSims conjecture
• Frobenius's conjectureon the number of solutions ofxⁿ= 1.

See also

• O'Nan–Scott theorem

References

• Aschbacher, Michael(2004)."The Status of the Classification of the Finite Simple Groups"(PDF).Notices of the American Mathematical Society.Vol. 51, no. 7. pp. 736–740.
• 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, vol. 172,ISBN978-0-8218-5336-8
• 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,ISBN978-0-19-853199-9
• Gorenstein, D.(1979), "The classification of finite simple groups. I. Simple groups and local analysis",Bulletin of the American Mathematical Society,New Series,1(1): 43–199,doi:10.1090/S0273-0979-1979-14551-8,ISSN0002-9904,MR0513750
• Gorenstein, D.(1982),Finite simple groups,University Series in Mathematics, New York: Plenum Publishing Corp.,ISBN978-0-306-40779-6,MR0698782
• Gorenstein, D.(1983),The classification of finite simple groups. Vol. 1. Groups of noncharacteristic 2 type,The University Series in Mathematics, Plenum Press,ISBN978-0-306-41305-6,MR0746470
• Daniel Gorenstein(1985), "The Enormous Theorem",Scientific American,December 1, 1985, vol. 253, no. 6, pp. 104–115.
• Gorenstein, D.(1986), "Classifying the finite simple groups",Bulletin of the American Mathematical Society,New Series,14(1): 1–98,doi:10.1090/S0273-0979-1986-15392-9,ISSN0002-9904,MR0818060
• Gorenstein, D.;Lyons, Richard;Solomon, Ronald(1994),The classification of the finite simple groups,Mathematical Surveys and Monographs, vol. 40, Providence, R.I.:American Mathematical Society,ISBN978-0-8218-0334-9,MR1303592
  • Gorenstein, D.;Lyons, Richard;Solomon, Ronald(1996),The classification of the finite simple groups, Number 2,Mathematical Surveys and Monographs, vol. 40, Providence, R.I.:American Mathematical Society,ISBN978-0-8218-0390-5,MR1358135
  • Gorenstein, D.;Lyons, Richard;Solomon, Ronald(1998),The classification of the finite simple groups, Number 3,Mathematical Surveys and Monographs, vol. 40, Providence, R.I.:American Mathematical Society,ISBN978-0-8218-0391-2,MR1490581
  • Gorenstein, D.;Lyons, Richard;Solomon, Ronald(1999),The classification of the finite simple groups, Number 4. Part II, Chapters 1-4: Uniqueness Theorems,Mathematical Surveys and Monographs, vol. 40, Providence, R.I.:American Mathematical Society,ISBN978-0-8218-1379-9,MR1675976
  • Gorenstein, D.;Lyons, Richard;Solomon, Ronald(2002),The classification of the finite simple groups, Number 5,Mathematical Surveys and Monographs, vol. 40, Providence, R.I.:American Mathematical Society,ISBN978-0-8218-2776-5,MR1923000
  • Gorenstein, D.;Lyons, Richard;Solomon, Ronald(2005),The classification of the finite simple groups, Number 6: Part IV: The Special Odd Case,Mathematical Surveys and Monographs, vol. 40, Providence, R.I.:American Mathematical Society,ISBN978-0-8218-2777-2,MR2104668
  • Gorenstein, D.;Lyons, Richard;Solomon, Ronald(2018),The classification of the finite simple groups, Number 7: Part III, Chapters 7–11: The Generic Case, Stages 3b and 4a,Mathematical Surveys and Monographs, vol. 40, Providence, R.I.:American Mathematical Society,ISBN978-0-8218-4069-6,MR3752626
  • Gorenstein, D.;Lyons, Richard;Solomon, Ronald(2018),The Classification of the Finite Simple Groups, Number 8: Part III, Chapters 12–17: The Generic Case, Completed,Mathematical Surveys and Monographs, vol. 40, Providence, R.I.:American Mathematical Society,ISBN978-1-4704-4189-0,MR3887657
  • Capdeboscq, Inna;Gorenstein, D.;Lyons, Richard;Solomon, Ronald(2021),The Classification of the Finite Simple Groups, Number 9: Part V, Chapters 1-8: Theorem$C_{5}$and Theorem$C_{6}$,Stage 1,Mathematical Surveys and Monographs, vol. 40, Providence, R.I.:American Mathematical Society,ISBN978-1-4704-6437-0,MR4244365
  • Capdeboscq, Inna;Gorenstein, D.;Lyons, Richard;Solomon, Ronald(2023),The Classification of the Finite Simple Groups, Number 10: Part V, Chapters 9-17: Theorem$C_{6}$and Theorem$C_{4}^{*}$,Case A,Mathematical Surveys and Monographs, vol. 40, Providence, R.I.:American Mathematical Society,ISBN978-1-4704-7553-6,MR4656413
• Mark Ronan,Symmetry and the Monster,ISBN978-0-19-280723-6,Oxford University Press, 2006. (Concise introduction for lay reader)
• Marcus du Sautoy,Finding Moonshine,Fourth Estate, 2008,ISBN978-0-00-721461-7(another introduction for the lay reader. American edition published in 2009 as Symmetry: A Journey into the Patterns of Nature)
• Ron Solomon(1995) "On Finite Simple Groups and their Classification,"Notices of the American Mathematical Society.(Not too technical and good on history. American version published in 2009 as Symmetry: A Journey into the Patterns of Nature)
• Solomon, Ronald (2001),"A brief history of the classification of the finite simple groups"(PDF),Bulletin of the American Mathematical Society,New Series,38(3): 315–352,doi:10.1090/S0273-0979-01-00909-0,ISSN0002-9904,MR1824893,archived(PDF)from the original on 2001-06-15– article wonLevi L. Conant prizefor exposition
• Thompson, John G.(1984), "Finite nonsolvable groups", in Gruenberg, K. W.; Roseblade, J. E. (eds.),Group theory. Essays for Philip Hall,Boston, MA:Academic Press,pp. 1–12,ISBN978-0-12-304880-6,MR0780566
• Wilson, Robert A.(2009),The finite simple groups,Graduate Texts in Mathematics251, vol. 251, Berlin, New York:Springer-Verlag,doi:10.1007/978-1-84800-988-2,ISBN978-1-84800-987-5,Zbl1203.20012

External links

• ATLAS of Finite Group Representations.Searchable database ofrepresentationsand 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.Archived2005-04-04 at theWayback MachineIncludes a list of all nonabelian simple groups up to order 10¹⁰.
• In what sense is the classification of all finite groups “impossible”?
• Ornes, Stephen (2015)."Researchers Race to Rescue the Enormous Theorem before Its Giant Proof Vanishes".Scientific American.313(1): 68–75.doi:10.1038/scientificamerican0715-68.PMID26204718.
• "Where are the second- (and third-)generation proofs of the classification of finite simple groups up to?".MathOverflow.(Last updated in February 2024)