# 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. This controversy stems from the sheer length and complexity of the proof and the fact that parts of it remain unpublished.

If correct, 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)
- One of 26 left-over groups known as the
**sporadic groups**

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2 A Second-Generation Classification 3 References |

## 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. Several of these groups were predicted to exist before they were constructed. Each group is named after the mathematician(s) who first predicted the its existence. The full list is:

- Mathieu groups
*M*_{11},*M*_{12},*M*_{22},*M*_{23},*M*_{24} - Janko groups
*J*_{1},*J*_{2}(also known as the Hall-Janko group*HJ*),*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
*O'N* - Harada-Norton group
*HN* - Lyons group
*Ly* - Thompson group
*Th* - Baby Monster group
*B* - Fischer-Griess Monster group
*M*

## A Second-Generation Classification

## 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.