Classification of finite simple groups

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.

Daniel Gorenstein (1923-1992), Richard Lyons, and Ronald Solomon are gradually publishing a simplified and revised version of the proof.

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.

These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2.

This gives a large number of separate problems; for example, the original proofs of existence and uniqueness of the monster group totaled about 200 pages, and the identification of the Ree groups by Thompson and Bombieri was one of the hardest parts of the classification.

In 1972 Gorenstein (1979, Appendix) announced a program for completing the classification of finite simple groups, consisting of the following 16 steps: Many of the items in the table below are taken from Solomon (2001).

As of 2023[update], ten volumes of the second generation proof have been published (Gorenstein, Lyons & Solomon 1994, 1996, 1998, 1999, 2002, 2005, 2018a, 2018b; & Capdeboscq, 2021, 2023).

(This length stems in part from the second generation proof being written in a more relaxed style.)

Aschbacher (2004) has called the work on the classification problem by Ulrich Meierfrankenfeld, Bernd Stellmacher, Gernot Stroth, and a few others, a third generation program.

This section lists some results that have been proved using the classification of finite simple groups.