Classification of finite simple groups

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four broad classes described below. These groups can be seen as the basic building blocks of all finite groups, in a way 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-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution.Theorem — Every finite simple group is isomorphic to one of the following groups: In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four broad classes described below. These groups can be seen as the basic building blocks of all finite groups, in a way 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-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution. Group theory is central to many areas of pure and applied mathematics and the classification theorem is one of the great achievements of modern mathematics. 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. Gorenstein (d.1992), Lyons, and Solomon are gradually publishing a simplified and revised version of the proof. The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups (and their action on other mathematical objects) can sometimes be reduced to questions about finite simple groups. Thanks to the classification theorem, such questions can sometimes be answered by checking each family of simple groups and each sporadic group. Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups. The completed proof of the classification was announced by Aschbacher (2004) after Aschbacher and Smith published a 1221-page proof for the missing quasithin case. Gorenstein (1982, 1983) wrote two volumes outlining the low rank and odd characteristic part of the proof, and Michael Aschbacher, Richard Lyons, and Stephen D. Smith et al. (2011)wrote a 3rd volume covering the remaining characteristic 2 case. The proof can be broken up into several major pieces as follows: The simple groups of low 2-rank are mostly groups of Lie type of small rank over fields of odd characteristic, together with five alternating and seven characteristic 2 type and nine sporadic groups.