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](https://mathoverflow.net/questions/114943/where-are-the-second-and-third-generation-proofs-of-the-classification-of-fin)) 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 G. Mason claimed a proof in an unpublished manuscript in 1981, but this was found to contain serious gaps.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](http://homepages.math.uic.edu/%7Esmiths/papers/quasithin/quasithin.pdf), [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](https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups)).


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.


${}$


References:


[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](https://zbmath.org/?q=an:1065.20023).]


[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](https://zbmath.org/?q=an:1065.20024).


[3] *Harada, Koichiro; Solomon, Ronald*, [**Finite groups having a standard component (L) of type $\widehat M\_{12}$ or $\widehat M\_{22}$.**](http://dx.doi.org/10.1016/j.jalgebra.2006.09.034), J. Algebra 319, No. 2, 621-628 (2008). [ZBL1135.20009](https://zbmath.org/?q=an:1135.20009).



Known and fixed gaps in the proof of the CFSG
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.


