Classification of finite simple groups

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The classification of the finite simple groups, also called the enormous theorem, 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.

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[edit] The classification

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

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.

Sometimes the Tits group is regarded as a sporadic group (in which case there are 27 sporadic groups) because it is not strictly a group of Lie type.

[edit] 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:

Matrix representations over finite fields for all the sporadic groups have been computed, except for the Monster.

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.

[edit] Table of the sporadic group orders

Group Order Factorized order 1SF
F1 or M 808017424794512875886459904961710757005754368000000000 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 ≈ 8 × 1053
F2 or B 4154781481226426191177580544000000 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47 ≈ 4 × 1033
F3 or Th 90745943887872000 ≈ 9 × 1016
Ly 51765179004000000 28 · 37 · 56 · 7 · 11 · 31 · 37 · 67 ≈ 5 × 1016
F5 or HN 273030912000000 214 · 36 · 56 · 7 · 11 · 19 ≈ 3 × 1014
Fi24 or Fi24' 1255205709190661721292800 221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 29 ≈ 1 × 1024
Fi23 4089470473293004800 218 · 313 · 52 · 7 · 11 · 13 · 17 · 23 ≈ 4 × 1018
Fi22 64561751654400 217 · 39 · 52 · 7 · 11 · 13 ≈ 6 × 1013
Co1 4157776806543360000 ≈ 4 × 1018
Co2 42305421312000 ≈ 4 × 1013
Co3 495766656000 ≈ 5 × 1011
O'N 460815505920 29 · 34 · 5 · 73 · 11 · 19 · 31 ≈ 5 × 1011
Suz 448345497600 ≈ 4 × 1011
Ru 145926144000 214 · 33 · 53 · 7 · 13 · 29 ≈ 1 × 1011
He 4030387200 210 · 33 · 52 · 73 · 17 ≈ 4 × 109
McL 898128000 ≈ 9 × 108
HS 44352000 ≈ 4 × 107
J4 86775571046077562880 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 ≈ 9 × 1019
J3 or HJ 50232960 27 · 35 · 5 · 17 · 19 ≈ 5 × 107
J2 or HJM 604800 27 · 33 · 52 · 7 ≈ 6 × 105
J1 175560 23 · 3 · 5 · 7 · 11 · 19 ≈ 2 × 105
M24 244823040 210 · 33 · 5 · 7 · 11 · 23 ≈ 2 × 108
M23 10200960 27 · 32 · 5 · 7 · 11 · 23 ≈ 1 × 107
M22 443520 27 · 32 · 5 · 7 · 11 ≈ 4 × 105
M12 95040 26 · 33 · 5 · 11 ≈ 1 × 105
M11 7920 24 · 32 · 5 · 11 ≈ 8 × 103

[edit] 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 due to Geoff Mason. Gorenstein announced the classification of finite simple groups in 1983, based partly on the impression that the quasithin case was finished. Aschbacher filled this gap in the early 90s, also unpublished. Aschbacher and Steve Smith have published a different proof comprising two volumes of about 1300 pages.

[edit] 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.

[edit] A third-generation classification

There is also a third-generation program involving a number of people, particularly Ulrich Meierfrankenfeld, Bernd Stellmacher, and Gernot Stroth.

[edit] References

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