It has been estimated that the original proof of the CFSG spans around 15,000 journal pages written by hundreds of authors over most of the 20th century. The GLS project attempted to simplify this original proof, with a target length around 5000 pages, but this involved a variety of changes to the original proof style, and from what I understand this project has stalled (the last volume was published in 2004 and five more are planned), possibly due to the mounting difficulties of changing the structure of such a large proof.

Has anyone attempted to present an argument for the full CFSG, encoding the high level structure of the proof using journal references as necessary to establish the various subcases? That is, the part of the proof that was happening in the minds of specialists who felt comfortable declaring the problem "solved" after the last journal article had been published (Aschbacher-Smith Quasithin theorem, 1996).

R. Solomon's article above is near to such a project, with a very comprehensive reference list, but it lacks the rigorous presentation of the proof itself, and the division into subcases except in an illustrative sense.

  • $\begingroup$ According to this article in this year's February Issue of Spektrum der Wissenschaft (the German version of Scientific American), the major danger for completing the work on a simplified proof is that the experts working on it are coming to age. $\endgroup$ Oct 3, 2016 at 15:27
  • $\begingroup$ @ManfredWeis This is totally true. Many of the great minds involved in the original proof are already dead. Additionally, there is a notable lack of younger generation workers in the field, because no one wants to spend a PhD on an area that is already "solved" for the most part. $\endgroup$ Oct 3, 2016 at 16:11
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    $\begingroup$ The article mentioned by @ManfredWeis has appeared in english in the July 2015 issue of Scientific American (Vol. 313, no. 1). $\endgroup$ Oct 4, 2016 at 14:23
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    $\begingroup$ FWIW, the second generation proof that you mention as having 'stalled' is in fact steaming ahead. See my answer at mathoverflow.net/a/217397/4177 $\endgroup$
    – David Roberts
    Sep 22, 2021 at 4:57

1 Answer 1


There are two books which together have the purpose of answering this question.

  1. D. Gorenstein, The Classification of Finite Simple Groups. Volume 1: The Noncharacteristic 2 Type Case. Plenum Press, 1983. (Gorenstein died without writing Volume 2.)

  2. M. Aschbacher, R. Lyons, S.D. Smith, R. Solomon, The Classification of Finite Simple Groups: Groups of Charateristic 2 Type. A.M.S. Mathematical Surveys and Monographs 172, 2011.

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    $\begingroup$ Welcome to MO, Professor Lyons! $\endgroup$
    – Todd Trimble
    Oct 2, 2016 at 23:28

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