ATLAS of Finite Groups – How Much Is Independently Checked?

Question

In a recent talk Finite groups, yesterday and today Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the CFSG and said so was ok, but a proof that relied on the ATLAS was not so ok, because the content has not been completely independently verified, and has as its basis old computer and other calculations that have only been done once.

Given that (numerous?) little errors have been found over time, it would be good to have a definitive record as to which bits have either been calculated or proved elsewhere, or formally verified in the case of the computer calculations, and if so where and by whom (with code for the latter case). Note that merely being able to do some calculations in GAP is not quite enough, since, as the documentation says

Part of the constructions have been documented in the literature on almost simple groups, or the results have been used in such publications, see for example the references in [CCNPW85] and [BN95].

where CCNPW85 is the ATLAS and BN95 is Breuer and Norton's Improvements to the Atlas (in an appendix of Atlas of Brauer Characters).

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EDIT Since it may not have been clear, I was after statements modeled on the following:

• "All results about classical groups of Lie type are well-known and documented elsewhere"
• "All results about conjugacy classes of the sporadic groups except Janko 4 [say] are calculated afresh and contained in X computer package"
• "The results on [blah] about [some group] are only contained in the ATLAS, and no papers or independently written software have reproved/recalculated them"

If it's easier to specify what is only in the ATLAS then that would be good, since clearly a lot of the classical material would be known and calculated long before.

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EDIT May 2017 In a recent talk (50 years ago: a great time for number theory — the first few minutes only before the main talk) Serre mentions his comments discussed here, the fact people got worked up about it, and the paper Breuer, Malle, and O'Brien – Reliability and reproducibility of Atlas information in Farrokh Shirjian's answer, which he feels addresses his complaints.

Answer 1

It may also worth to look at the paper

• T. Breuer, G. Malle, and E. A. O'Brien, Reliability and reproducibility of Atlas information, Contemporary Mathematics 694 (2017) pp 21–31, doi:10.1090/conm/694/13960, arXiv:1603.08650

in which is discussed the reliability and reproducibility of much of the information contained in the Atlas of Finite Groups.

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(Added by David Roberts) There are the follow-up papers

• Thomas Breuer, Constructing the ordinary character tables of some Atlas groups using character theoretic methods, https://arxiv.org/abs/1604.00754,

and

• Thomas Breuer, Kay Magaard, Robert Wilson, Verification of the ordinary character table of the Baby Monster, https://arxiv.org/abs/1902.07758,

which bring the verification project to the point of double checking all the ATLAS character tables except that of the Monster, $\mathbb{M}$.

Finally, I found the slides Verifying the Character Table of the Monster, from a presentation on 12 December 2020, giving the work of Breuer, Magaard and Wilson on independently constructing the Monster's character table, modulo checking the tables for the following normaliser subgroups of $\mathbb{M}$:

• $N(2B)\simeq 2^{1+24}_+.\mathrm{Co}_1$, due to Norton;
• $N(3B) \simeq 3^{1+12}_+.2.\mathrm{Suz}.2$, due to Barraclough and Wilson in The Character Table of a Maximal Subgroup of the Monster; and
• $N(5B)\simeq 5^{1+6}_+.4.J_2.2$.