Classification of finite simple groups

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G2 F4 E6 E7 E8
Lorentz group
Poincaré group
Conformal group
Diffeomorphism group
Loop group
Infinite-dimensional Lie groups O(∞) SU(∞) Sp(∞)

In mathematics, the classification of the finite simple groups classifies all finite simple groups. These groups can be seen as the basic building blocks of all finite groups, in much the same way as 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.

The proof of theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Gorenstein, Lyons, and Solomon are gradually publishing a simplified and revised version of the proof.


Statement of the classification theorem

Theorem. Every finite simple group is isomorphic to one of the following groups:

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