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.

The classification

If correct, the classification shows every finite simple group to be one of the following types:

• 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.

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:

• 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.

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.

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due to the sheer length and complexity of the published work and the fact that parts of the supposed proof remain unpublished.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.

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 nineties, also unpublished. Aschbacher and Steve Smith have published a rewritten version.

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.

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.)

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.

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.