In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. Group theory is central to many areas of pure and applied mathematics and the classification theorem has been called one of the great intellectual achievements of humanity.[1] The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.
Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphicgroups with the same composition series or, put in another way, the extension problemdoes not have a unique solution.
Gorenstein (d.1992), Lyons, and Solomon are gradually publishing a simplified and revised version of the proof.
https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups
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