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. However, there is a controversy in the mathematical community on whether these articles provide a complete and correct proof.

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

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

Mathieu groupss M₁₁, M₁₂, M₂₂, M₂₃, M₂₄

Janko groups J₁, J₂, 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 ON

Harada-Norton group HN

Lyons group Ly

Thompson group Th

Baby Monster group B
Monster group M

References

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.