# 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:

## 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.