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

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 groupss 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 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.

Encyclopedia Home Page

Classification of finite simple groups

The Sporadic Groups

References