The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. In all, the work comprises tens of thousands of pages in 500 journal articles by some 100 authors. Mathematics is the study of quantity, structure, space and change. ... 1955 is a common year starting on Saturday of the Gregorian calendar. ... 1983 is a common year starting on Saturday of the Gregorian calendar. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a simple group is a group G such that G is not the trivial group and the only normal subgroups of G are the trivial group and G itself. ...

The classification

If correct, the classification shows every finite simple group to be one of the following types: In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type (including the Tits group, which strictly speaking is not of Lie type), or one of 26 sporadic groups. ...

• A cyclic group with prime order
• An alternating group of degree at least 5
• A "classical group" (projective special linear, symplectic, orthogonal or unitary group over a finite field)
• An exceptional or twisted group of Lie type (including the Tits group)
• One of 26 left-over groups known as the sporadic groups (listed below)

The theorem has widespread applications in many branches of mathematics, as questions about finite groups can often be reduced to questions about finite simple groups, which by the classification can be reduced to an enumeration of cases. In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics an alternating group is the group of even permutations of a finite set. ... The projective linear group of a vector space V over a field F is the quotient group PGL(V) = GL(V)/Z(V) where GL(V) is the general linear group on V and Z(V) is the group of all nonzero scalar transformations of V. The projective special linear... In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, the unitary group of degree n over the field F (which is either the field R of real numbers or the field C of complex numbers) is the group of n by n unitary matrices with entries from F, with the group operation that of matrix multiplication. ... In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ... In mathematics, a group of Lie type is a finite group related to the points of a simple algebraic group with values in a finite field. ... The Tits group 2F4(2) is a finite simple group of order 17971200 named for the Belgian mathematician Jacques Tits. ... Mathematics is the study of quantity, structure, space and change. ...

The sporadic groups

Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list is: Events and trends Italian unification under King Victor Emmanuel II. Wars for expansion and national unity continue until the incorporation of the Papal States (March 17, 1861 - September 20, 1870). ... 1965 was a common year starting on Friday (link goes to calendar). ... 1975 was a common year starting on Wednesday (the link is to a full 1975 calendar). ...

• Mathieu groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄
• Janko groups J₁, J₂ or HJ, J₃ or HJM, J₄
• Conway groups Co₁, Co₂, Co₃
• Fischer groups Fi₂₂, Fi₂₃, Fi₂₄ or Fi₂₄′
• Higman-Sims group HS
• McLaughlin group McL
• Held group He or F₇
• Rudvalis group Ru
• Suzuki sporadic group Suz
• O'Nan group O'N
• Harada-Norton group HN or F₅
• Lyons group Ly
• Thompson group Th or F₃
• Baby Monster group B or F₂
• Fischer-Griess Monster group M or F₁

Matrix representations over finite fields for all the sporadic groups have been computed. In mathematics, the Mathieu groups are five finite simple groups discovered by the French mathematician Emile Léonard Mathieu. ... In mathematics, the Janko groups J1, J2, J3 and J4 are four of the twenty-six sporadic groups; their respective orders are: J1 The smallest Janko group, J1 of order 175560, has a presentation in terms of two generators a and b and c = abab-1 as It can also... In mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway. ... In mathematics, the term Fischer groups usually refers to the three finite groups denoted Fi22, Fi23, and Fi24, all of which are simple groups, and constitute three of the 26 sporadic groups. ... In mathematics, the Higman-Sims group is a finite sporadic simple group of order 44352000. ... In mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway. ... In mathematics, the Held group, He, is the unique finite simple sporadic group of order . ... The Rudvalis group, Ru, is the sporadic simple group of order . It is named for Arunas Rudvalis. ... In mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway. ... In mathematics, the ONan group, ON is the sporadic simple group of order . It is named for Michael ONan who discovered it. ... In mathematics, the Harada-Norton group, HN is the sporadic simple group of order . It is named for Koichiro Harada and Simon Norton. ... In mathematics, the Lyons group, Ly, is a finite sporadic simple group of order It can be characterized as the unique simple group where the centralizer of an involution, and hence of all the involutions, is isomorphic to the nontrivial central extension of the cyclic group C2 by the alternating... This page is about the sporadic finite simple group Th F, T and V see Thompson groups. ... In mathematics, the Baby Monster group B (or just Baby Monster) is a group of order    241 313 56 72 11 13 17 19 23 31 47 = 4154781481226426191177580544000000 ≈ 4 1033. ... In mathematics, the Monster group M is a group of order    246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808017424794512875886459904961710757005754368000000000 ≈ 8 · 1053. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...

Of the 26 sporadic groups, 20 of them can be seen inside the Monster group as subgroups or quotients of subgroups. The 6 exceptions are J₁, J₃, J₄, O'N, Ru and Ly. These 6 groups are sometimes known as the pariahs. In mathematics, the Monster group M is a group of order    246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808017424794512875886459904961710757005754368000000000 ≈ 8 · 1053. ...

So far, there has been little progress in providing a convincing unification for the sporadic groups.

Remaining skepticism on the proof

Some doubts remain on whether these articles provide a complete and correct proof, due to the sheer length and complexity of the published work and the fact that parts of the supposed proof remain unpublished. Jean-Pierre Serre is a notable skeptic of the claim of a proof. Such doubts were justified to an extent as gaps were later found and eventually fixed. Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ...

For over a decade, experts have known of a "serious gap" (according to Michael Aschbacher) in the (unpublished) classification of quasithin groups. Aschbacher filled this gap in the early 90s, also unpublished. Aschbacher and Steve Smith have published a rewritten version. Michael Aschbacher (born April 8, 1944) is Shaler Arthur Hanisch Professor of Mathematics at the California Institute of Technology. ... // Events and trends The 1990s are generally classified as having moved slightly away from the more conservative 1980s, but keeping the same mind-set. ...

A second-generation classification

Because of the extreme length of the proof of the classification of finite simple groups, there has been a lot of work, called "revisionism", originally led by Daniel Gorenstein, in finding a simpler proof. This is the so-called second-generation classification proof. Daniel Gorenstein (January 1, 1923–August 6, 1992) was an American mathematician. ...

Six volumes have been published as of 2005, and manuscripts exist for most of the rest. The two Aschbacher and Smith volumes were written to provide a proof for the quasithin case that would work with both the first- and second-generation proof. It is estimated that the new proof will be approximately 5,000 pages when complete. (It should be noted that the newer proofs are being written in a more generous style.) 2005 is a common year starting on Saturday of the Gregorian calendar. ...

Gorenstein and his collaborators have given several reasons why a simpler proof is possible. The most important is that the correct, final statement is now known. Techniques can be applied that will suffice for the actual groups. In contrast, during the original proof, nobody knew how many sporadic groups there were, and in fact some of the sporadic groups (for example, the Janko groups) were discovered in the process of trying to prove cases of the classification theorem. As a result, overly general techniques were applied. In mathematics, the Janko groups J1, J2, J3 and J4 are four of the twenty-six sporadic groups; their respective orders are: J1 The smallest Janko group, J1 of order 175560, has a presentation in terms of two generators a and b and c = abab-1 as It can also...

Again, because the conclusion was unknown, and for a long time not even conceivable, the original proof consisted of many separate complete theorems, classifying important special cases. These proofs, in order to reach their own final statements, had to analyze numerous special cases. Often, most of the work was in these exceptions. As part of a larger, orchestrated proof, many of these special cases can be bypassed, to be handled when the most powerful assumptions can be applied. The price paid is that these original theorems, in the revised strategy, no longer have comparatively short proofs, but depend on the complete classification.

Nor were these separate theorems efficient regarding the subdivision of cases. Numerous target groups were identified multiple times as a result. The revised proof relies on a different subdivision of cases, eliminating these redundancies.

Finally, finite group theorists have more experience and new techniques.

References

• Michael Aschbacher, The Status of the Classification of the Finite Simple Groups, Notices of the American Mathematical Society, August 2004
• Daniel Gorenstein, Richard Lyons, Ronald Solomon The Classification of the Finite Simple Groups (volume 1), AMS, 1994 (volume 2), AMS,
• Ron Solomon: On Finite Simple Groups and their Classification, Notices of the American Mathematical Society, February 1995
• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England 1985.
• Orders of non abelian simple groups: includes a list of all non-abelian simple groups up to order 10,000,000,000.
• Atlas of Finite Group Representations: contains representations and other data for many finite simple groups, including all the sporadic groups except the Monster group.