I read the following paragraph from Serre's book (Topics in Galois Theory).

Although the proof of the classification theorem has been announced, described, and advertised since 1980, it is not yet clear whether it is complete or not: the part on "quasi-thin" groups has never been published

I am confused as I have read that classification of finite simple groups is complete.

See the first four lines on Wikipedia.

Theorem — Every finite simple group is isomorphic to one of the following groups:

    a member of one of three infinite classes of such, namely:
        the cyclic groups of prime order,
        the alternating groups of degree at least 5,
        the groups of Lie type
    one of 26 groups called the "sporadic groups"
    the Tits group (which is sometimes considered a 27th sporadic group).

Even if I assume that Wikipedia is wrong on this matter, what does he mean by saying that we don't know whether it is complete or not.

  • 4
    $\begingroup$ There is now a published proof of the quasi thin case by Aschbacher and Smith $\endgroup$ – Geoff Robinson Apr 12 '17 at 18:44
  • 10
    $\begingroup$ When an authors says that a problem is open, it means that it's open at the time when the author wrote. So the first thing you should have done is to read the publication year of Serre's book. $\endgroup$ – YCor Apr 12 '17 at 19:21

I think this question is answered by the Wikipedia page quoted in the question:

Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups. The completed proof of the classification was announced by Aschbacher (2004) after Aschbacher and Smith published a 1221-page proof for the missing quasithin case.


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