Classification of finite simple groups - Definition and Overview

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.

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due to the sheer length and complexity of the published work and the fact that parts of the supposed proof remain unpublished.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 classification of quasithin groups. Aschbacher and Steve Smith now claim to have filled this gap, but it is possible that there may be other gaps not yet detected.

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 (also known as the Hall-Janko group HJ), J₃, J₄
• Conway groups Co₁, Co₂, Co₃
• Fischer groups F₂₂, F₂₃, F₂₄
• Higman-Sims group HS
• McLaughlin group McL
• Held group He
• Rudvalis group Ru
• Suzuki sporadic group Suz
• O'Nan group O'N
• Harada-Norton group HN
• Lyons group Ly
• Thompson group Th
• Baby Monster group B
• Fischer-Griess Monster group M

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

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 (led by Daniel Gorenstein) in trying to find a simpler proof. This is the so-called second-generation classification proof. One reason that some mathematicians believe that a simpler proof is possible is that the result to be proved is known, which was not the case for the earlier proof. In particular, during the original proof, nobody knew how many sporadic groups there would be, and in fact some of the sporadic groups (for example, the Janko groups) were discovered in the process of trying to prove the classification theorem.

It is thought (as of 2004) that it might be possible to write a complete proof in about 5,000 pages.

References

• Michael Aschbacher, The Status of the Classificatin of the Finite Simple Groups (http://www.ams.org/notices/200407/fea-aschbacher.pdf), Notices of the American Mathematical Society, August 2004
• Ron Solomon: On Finite Simple Groups and their Classification (http://www.ams.org/notices/199502/solomon.pdf), 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 (http://www.eleves.ens.fr:8080/home/madore/math/simplegroups.html)