I want to learn the classification of finite simple groups. But it is often commented that it is a theorem spanning tens of thousands of pages of research papers. So it is quite intimidating to an outsider like me. Can someone please point me where to start and trace out atleast the first few basic must reads in order to get started in this business?

My current background: Representation Theory (of finite groups, semisimple Lie algebras; only very basic stuff)

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    $\begingroup$ You can find some overviews of the proof in Aschbacher's "Finite Group Theory" and Gorenstein's "Finite Simple Groups". $\endgroup$
    – S. Carnahan
    Sep 14, 2020 at 16:14
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    $\begingroup$ Do you want to understand the final answer, understand how it was arrived at (or both)? $\endgroup$ Sep 15, 2020 at 9:20
  • $\begingroup$ Maybe you are beyond this stage already, but the introductory material in the ATLAS of Finite Simple Groups is a great place to start. $\endgroup$
    – IJL
    Sep 15, 2020 at 10:18
  • $\begingroup$ @GeoffRobinson both. I want to understand the groups involved as well as why those are all. $\endgroup$
    – ArB
    Sep 15, 2020 at 14:52
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    $\begingroup$ @user165332 : Wilson's book, mentioned in Mare's answer, is a good reference for the groups involved, and the GLS project, mentioned in my answer, is a good reference for the proof that these are all. See also Richard Lyons's answer for an overview of the "first generation proof" of the classification. $\endgroup$ Sep 15, 2020 at 21:48

2 Answers 2


The book "The Finite Simple Groups" by Robert Wilson gives a great overview and every chapter ends with a section on further reading. I think this might be the best start to this big project of understanding the classification.

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    $\begingroup$ Wilson's book describes and constructs (most of) the finite simple groups, but doesn't say anything about what most people think of as the classification theorem, namely the proof that there aren't any other finite simple groups. $\endgroup$ Sep 14, 2020 at 22:53

A natural place to start would be Volume 1 of the so-called GLS project.

Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The Classification of the Finite Simple Groups, Mathematical Surveys and Monographs, Volume 40, Number 1, American Mathematical Society, 1994.

The GLS project is a series of books that is intended to provide a complete proof of the Classification Theorem, modulo a relatively short list of references (such as Aschbacher and Smith's work on quasithin groups) containing crucial pieces of the proof that the GLS authors felt were already satisfactorily written up in a self-contained manner. The GLS project is expected to comprise 12 volumes, of which 8 have already been published. Volume 1 above contains an outline of the entire proof, as it was envisaged in 1994. There have been some changes to the plan since then, but nothing too radical, so Volume 1 still serves as a good starting point. Also you will probably want to read Ron Solomon's progress report in the June/July 2018 issue of the Notices of the AMS, written shortly before Volume 8 was published.


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