[FOM] finite simple groups: no rumor known to me (fwd)

John Baldwin jbaldwin at uic.edu
Sun Nov 2 11:31:53 EST 2008

Barany asked about possible gaps in the proof of the classification of 
finite simple groups.

I consulted with Steve Smith who with Aschenbacher completed the analysis
of the `quasithin' case which filled the last gap in the `first
generation' proof.

It seems easiest to just forward his response.

To: jbaldwin at math.uic.edu
Subject: no rumor known to me

Hi John,

(I guess you could quote me, emphasizing my remarks are informal)

I know of no rumor about any new class of groups.  I expect someone
would have told me.  I'm in pretty regular contact with Aschbacher,
Solomon, Lyons (and a pretty vast group theory community).

There are some remarks on the status of the CFSG in the introduction(s)
to the Aschbacher-Smith quasithin volumes.  Better, Aschbacher lectured
on the status (finished) at an AMS meeting a few years ago;
and this was published (with "status" in the title) in the Notices.

Concerning "writing up in single volume": that is the SECOND round
proof. (the Gorenstein-Lyons-Solomon project in vol 40 of SurvAMS,
now well more than half published).

   In contrast, the first round proof is considered complete with the
publication of quasithin.  Conceivably one might, on close
examination of the first, find bits that were not properly published.
This kind of examination is part of a current project
(fairly well advanced now) of giving an outline of
the CFSG (the "even" part, complementing Gorenstein's
1983 outline of the odd part)---by Aschbacher, Lyons,
Solomon, and Smith).

   Rumors about incompleteness seem to circulate about
   every couple of years; not based on any foundation known
   to me.  I've pretty much assigned it by now to some
   kind of would-be schadenfreude in the math community.

Of course, I (and all the other experts) could be wrong!
But it seems like even a new family would only arise
from some particular "local" error in the proof---and could
thereafter be incorporated into the inductive structure
of the whole proof.
   This kind of speculation (eg by Aschbacher)
was mentioned in Bryan Davies' article in the Notices
(concerning various long proofs in mathematics),
which I think appeared in later 2005.
And that gave rise to the most recent spate
of rumors that I can remember (and may be what your
correspondent is referring to).  The rumors did gives some
impetus to the outline project I mentioned above!  ...and
the four of us wrote a letter to Notices trying to clarify
some points that were perhaps unclear in Davies' text.
(For example, confusion between the original proof and
the GLS second-effort---this confusion seems to still be
present in the rumor reported to you...)

Of course, do let me know if you get any actual
followable leads on rumors, new classes...


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