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:


  • 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.