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
*M*_{11},*M*_{12},*M*_{22},*M*_{23},*M*_{24} - Janko groups
*J*_{1},*J*_{2},*J*_{3},*J*_{4} - Conway groups
*Co*_{1},*Co*_{2},*Co*_{3} - Fischer groups
*F*_{22},*F*_{23},*F*_{24} - 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.