As the "second-generation" proof of the Classification of Finite Simple Groups is being written up in the volumes by Gorenstein, Lyons, Aschbacher, Smith, Solomon, and others (see e.g. this question) certainly much of the work will go into fixing minor (and possibly major) issues and gaps in the proof, since the first announcement in 1983.

Here are two such gaps:

  1. The classification of quasithin groups. G. Mason claimed a proof in an unpublished manuscript in 1981, but this was found to contain serious gaps. It would not be until 2004 that this gap would be fixed (see this behemoth, [1,2]).

  2. In 2008, Harada and Solomon [3] filled a minor gap in the classification by describing groups with a standard component that is a cover of the Mathieu group $M_{22}$, a case that was accidentally omitted from the proof of the classification due to an error in the calculation of the Schur multiplier of $M_{22}$ (from the Wikipedia page for CFSG).

I would like to see a longer such list! Thus:

What other (major or minor) gaps have been discovered, and subsequently fixed, in the proof of the CFSG, since the announcement in 1983?

Of course, if there are any gaps that are "known, but with a known fix" (but which have not yet made it into the aforementioned second-generation proof), then these would also be interesting to know.



[1] Aschbacher, Michael; Smith, Stephen D., The classification of quasithin groups. I: Structure of strongly quasithin $\mathcal K$-groups., Mathematical Surveys and Monographs 111. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3410-X/hbk). xiv, 477 p. (2004). ZBL1065.20023.]

[2] Aschbacher, Michael; Smith, Stephen D., The classification of quasithin groups. II: Main theorems: the classification of simple QTKE-groups., Mathematical Surveys and Monographs 112. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3411-8/hbk). xii, pp. 479-1221. (2004). ZBL1065.20024.

[3] Harada, Koichiro; Solomon, Ronald, Finite groups having a standard component (L) of type $\widehat M_{12}$ or $\widehat M_{22}$., J. Algebra 319, No. 2, 621-628 (2008). ZBL1135.20009.

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    $\begingroup$ Richard Lyons occasionally contributes here. He is immersed in the 2G version of CFSG, and I am sure he will be aware of (at least) the majority of whatever issues have been resolved, and which still exist. $\endgroup$ Sep 22, 2021 at 11:28
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    $\begingroup$ @GeoffRobinson Let us hope he sees this question, then! :-) $\endgroup$ Sep 22, 2021 at 17:13
  • $\begingroup$ Richard Lyons maintains a list of errata for the second-generation proof. $\endgroup$ Apr 29, 2022 at 22:30

1 Answer 1


Here is an answer from my point of view, immersed as I am -- Geoff is right -- in the second generation project. First, a few general comments. Our overriding purpose has been to expound a coherent proof of CFSG that is supported completely by what we call ``Background Results,'' an explicit and restricted list of published books and papers, plus the assertion that every one of the $26$ sporadic groups is determined up to isomorphism, as a finite simple group, by its so-called centralizer-of-involution pattern. This list has changed over the years. In our first volume it is explicitly listed as we conceived it at the time (1990's). Further additions, mostly of post-first-generation publications, are noted as they have been adopted in subsequent volumes. (Some of these additions are characterizations of some sporadic groups--for example, the Monster and Baby Monster--by weaker data than centralizer-of-involution pattern, so that they supplant the earlier Background Results characterizing those groups.) The biggest additions, by far, are Aschbacher and Smith's monumental books on the quasi-thin problem, since we were hardly going to do it as well ourselves, let alone better. Whatever errors may be in the second-generation proof, therefore, are either in the Background Results or in our series.

Naturally, we have taken ideas and arguments from many papers and books outside the Background Results in formulating our proof. Occasionally in the course of understanding these results, or adapting them for our purposes, we have uncovered gaps. None of these is at all comparable in scope (by orders of magnitude) to the well-known quasi-thin gap that Aschbacher and Smith bridged; in that sense, they could be called ``minor.'' To deal with these gaps, when they threatened our proof, we have either found alternative arguments ourselves, or asked the authors for help. In every case, so far, the gap has been closed in one of these two ways. However, and unfortunately for the purposes of answering your question, we have not kept a log of these incidents. Nor have we by any means intended to examine every paper needed in the first-generation proof this way. We are guided just by what we need in the second generation.

Here is an example of a minor gap that came to our attention in the preparation of volume $9$. We needed a certain characterization of the $7$- and $8$-dimensional orthogonal groups over the field of $3$ elements. We were guided by an important paper by Aschbacher that had appeared relatively late and without much fanfare in the first generation. There was an apparent gap -- very technical -- in the paper, and Professor Aschbacher promptly supplied us with a correction.

Another example that I know well, from before the CFSG, came in $1972$ in my paper pointing to the possible existence of the sporadic group $Ly$. I asserted that if such a group existed, then every nonidentity element of order a power of $5$ actually would have order $5$. Koichiro Harada wrote me shortly thereafter that on the contrary, there would be elements of order $25$. He was right; I had miscalculated. Luckily, the miscalculation did not affect the rest of the paper.

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    $\begingroup$ Though it's unfortunate (for this question) that no log was kept, this is still an excellent answer. Thank you! (When you mentioned it, I now recall seeing the order $5$ vs $25$ issue mentioned in an erratum paper). $\endgroup$ Sep 23, 2021 at 19:05
  • $\begingroup$ How much of this proof has been verified so far using proof assistant programs? Is anyone working on that actively? (I ask as an outsider, but I was a grad student in Cambridge when the Atlas was published.) $\endgroup$ Sep 25, 2021 at 8:56
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    $\begingroup$ The Odd order theorem has been verified. Beyond that I am the wrong person to ask. $\endgroup$ Sep 25, 2021 at 13:38
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    $\begingroup$ @PaulTaylor Here are two relevant questions. See Ben Steinberg's answer to the first. I think the answer to your question boils down to: the statement of the CFSG is not even yet formalised, let alone the proof. (But I say this as an outsider both to CFSG and proof assistants). $\endgroup$ Sep 25, 2021 at 20:15

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