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Classification of finite simple groups

The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which classifies all of the finite simple groups. In all, the work comprises about 10,000 - 15,000 pages in 500 journal articles by some 100 authors. 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)
or one of 26 left-over groups known as the sporadic groups

The Sporadic Groups

5 of the sporadic groups were discovered by Mathieu in the 1860's and the other 21 were found between 1965 and 1975. The full list is:

Mathieu groups[?] M₁₁, M₁₂, M₂₂, M₂₃, M₂₄
Janko groups[?] J₁, J₂, J₃, J₄
Conway groups[?] Co₁,Co₂,Co₂
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[?] ON
Harada-Norton group[?] HN
Lyons group[?] Ly
Thompson group[?] Th
Baby Monster group[?]
Monster group M

External links and references:

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.

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