# How much of the ATLAS of finite groups is independently checked and/or computer verified?

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).

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.

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.

• and while there were various small errors found in the Atlas over time, you might like to compare them with huge gaps that were found in CFSG over time :-) Aug 19, 2015 at 3:37
• This is an interesting question, and worthy of discussion. But I don't really see why the ATLAS has been picked out here. Notionally it's because "the content has not been completely independently verified". But one could say similar things about results in many other areas of mathematics -- one is really asking how we know whether such-and-such a theorem is true, and this is (at least to some extent) a vexed and deep philosophical question. Actually the ATLAS seems to me to be a particularly bad target because it has stood the test of time so well. The level of accuracy is pretty astonishing. Aug 19, 2015 at 9:50
• Oh, and is there an implication that "old computer calculations" are worse than new ones?!? Aug 19, 2015 at 9:51
• @NickGill it arose in discussion in Serre's talk - that's all I can say as to why it was singled out. As to old code: code that can no longer be run, or even be read from its storage medium cannot be verified or checked! Aug 19, 2015 at 13:05
• @NickGill As I tried to explain in my answer, I think the ATLAS does warrant picking out, because in practice many people rely on its contents, and it can be hard to locate the source of some of this information. So, despite the fact (which I don't dispute) that it has stood the test of time extremely well, I still find it a good target! Aug 19, 2015 at 17:35

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.

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

and

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
• $$N(5B)\simeq 5^{1+6}_+.4.J_2.2$$.
• Jun 17, 2016 at 22:39
• Wow, thanks! This is exactly what I wanted. In particular, I believe they check all the character tables in the ATLAS except for B, 2.B and M. Jun 18, 2016 at 10:39
• Or rather, all but 6 groups have runtimes for the verification shown at math.rwth-aachen.de/~Thomas.Breuer/atlasverify/overview.html, and I suspect that there of those are at least partially checked. Jun 22, 2016 at 1:31
• In arxiv.org/abs/1604.00754 most of the groups with missing runtimes in math.rwth-aachen.de/~Thomas.Breuer/atlasverify/overview.html are checked, with the correctness for the character table for 2.B modulo correctness of the character table for B, the Baby Monster. The outstanding character tables from that long list are then only B and M. Breuer is explicit in each case where he is assuming certain facts about the group structures as given, that presumably were not computer verified at that point in time. May 15, 2017 at 10:36
• Regarding the character table $N(2B)$, I think it might appear in Norton's paper Counting Nets in the Monster, but I don't have access to check. Dec 13, 2021 at 0:52

Unlike Dima, I am inclined to agree with Serre on this point. Although most of the facts recorded in the ATLAS have been proved elsewhere or, in the case of all of the character tables except for those of the very large groups like the Monster, can be easily recomputed in GAP or Magma using standard algorithms for finite groups, it can be very difficult in some cases to track down alternative proofs.

I have recently completed a book (co-authored with John Bray and Colva Roney-Dougal) calculating complete lists of maximal subgroups of almost simple classical groups in dimensions up to $12$, and we were confronted with this problem. Although we cited the ATLAS many times, we tried hard to provide alternative citations or, for facts that could be easily checked by computer, we provided code to do this. In fact nearly all of the facts we required were either about maximal subgroups of groups in the ATLAS or involved entries in character tables. For the sporadic groups there were virtually always alternative papers to cite, which were generally also cited in the ATLAS.

We had more difficulties with things like maximal subgroups of almost simple extensions of some of the more complicated classical groups in the ATLAS, like $U_4(3)$. For these we could not always find alternative sources that gave precise enough information, and we were told informally that some of the information had been originally calculated by unidentified research students or PostDocs. So we tried hard to re-prove these facts.

Having said that, we found remarkably few errors in the ATLAS. I think there might have been one or two very small and minor inaccuracies in some of the structure descriptions, which we reported to the authors, and I think they might have known about them already. I see that there is an "ATLAS 30 Years On" conference coming up in Princeton in November 2015, so perhaps this will lead to more discusssion of these questions.

I should also reinforce the point made by David Roberts that one needs to be very cautious when using Computer Algebra Systems, such as GAP and Magma, to verify facts contained in sources like the ATLAS, because it is possible that the code used is itself relying on these sources. For example $\mathtt{ MaximalSubgroups}$ in Magma will generally look up the maximal subgroups of the group's composition factors in a database, which will have been constructed using the ATLAS. However, a default use of $\mathtt{CharacterTable}$ on a finite group will use a general purpose algorithm (such as Dixon-Schneider), which does not rely on properties of specific (simple) groups.

• Do you claim that CFSG is more reliable than Atlas? Or just that Atlas isn't really 100% correct? Aug 19, 2015 at 23:21
• It's not so much a question of reliability or correctness, but rather that it can be difficult in some cases to find the source of or citations for information in the ATLAS> Aug 20, 2015 at 7:41
• Do you know any specific examples where there are no other sources to corroborate facts stated in the ATLAS? Aug 26, 2015 at 8:19
• This hidden reliance on ATLAS is a very interesting point of "under specification" : The software does not say in some clear and well associated documentation what is presupposed. It should say so, otherwise it is useless as is any program without specification of what it does (not how it does it).. Aug 26, 2015 at 22:47
• Structure of maximal subgroups was the specific point that concerned us when writing the book. We could generally find other sources for the sporadic groups, but not always for the more complicated classical groups like $U_4(3)$ or $U_6(2)$ (where there is a lot of information about the almost simple extensions), and in these cases we generally decided to reprove things. Aug 27, 2015 at 9:10

The Atlas was a work of scholarship, not research. Our aim in those days was to collect information together for convenience, and the large character tables, and all the other data, was all proved. The fact that there were so many errors is down to human error - both in our sources and our own work.

Nevertheless it would be possible to re-compute much of it now fairly easily, and in particular computing the character tables of explicitly given matrices or permutations would be less heroic now than 30 years ago. The result would be that there exists a group with the proven properties. If anyone wants to have a go at that, I'd love to help.

The "uniqueness" part is more difficult. There is much published work on properties of simple groups, and the definition of the group studied would need to be somehow be connected to the (re)computed data. This would require further work. For example one would need to show for the first Conway group that the 24x24 matrices used to define the group fix an even unimodular lattice in 24 dimensions with minimum norm 4 and that there is an involution centralizer of the form 2.1+8.O8+(2) and so on.

There is no doubt that the project is feasible, and I had a short discussion with Serre on the subject. Let's hope someone cares enough to actually do it!

• Thanks for commenting, Richard! And welcome to MO. I did not mean to impugn what was obviously a landmark piece of mathematics, I was enthralled when reading for the first time about the work you and your colleagues put in. But my hope is that people take the legacy seriously and push on with the computational side of things, redoing and improving on the ATLAS, not just coding it into packages as received wisdom. Jul 2, 2016 at 12:54

At the recent 'Conway and the Atlases' conference in Princeton it was remarked that computers can easily check everything in the Atlas up to about J4. With regards to references, it is worth noting that whilst the bibliography in the original Atlas is now thirty years old, a very comprehensive update was given in the Atlas of Brauer Characters by Jansen, Lux, Parker and Wilson, published ten years later. It's also worth noting that whilst oldest edition of the Atlas does indeed contain numerous errors, more recent editions contain an 'addenda and corrigenda' section that corrects many of them, the most serious being the omission of some columns from one character table.

• So any sporadic group up to the size of J4 or up to the entry of J4 in the ATLAS? Also, I presume the Lie-type groups, including Chevalley groups are all good? Not to mention the easy infinite families. Dec 21, 2015 at 10:18
• For reference, the discussion panel/group math.arizona.edu/~grouptheory/princeton/group1.txt may have discussed Serre's comments; did the above remark come during that session? Dec 21, 2015 at 10:34

I must disagree with Serre (or your way to interpret his words) here. There are no papers that rely on "the" Atlas, they inevitably only use a tiny part, e.g. the character table of a particular group, information about the structure of a particular subgroup in a particular group, etc. As such, it's totally OK for such a paper; indeed there are probably few places in Atlas which were not checked independently, but these are probably almost never used.

Atlas is an atlas - like all the maps, there are few tiny errors here and there, but still it is more reliable than a theorem (CFSG) with a proof so long that one can probably count people who understand it in full detail on fingers of one hand...

And saying that one must not rely on Atlas for information that is well-known, e.g. maximal subgroups of "smallish" sporadic simple groups, something that was independently checked in many sources, is just silly. As well, many computations of character tables were re-done independently, by people working on GAP and Cayley (Magma), and many tables in the Atlas for non-sporadic groups follow from general theory (e.g. character tables of alternating groups aren't some kind of thing one cannot find elsewhere).

• "probably almost never used." well, there's the sticking point. What if I suddenly need to use one of these parts in a paper. How would I know I need to go through and check things in detail for myself? The point is this: where can I find a list of what is known to be independently verified and what is not? Aug 19, 2015 at 5:08
• there are references, Atlas isn't exactly a book of fairy tales; many things in Atlas can be computed with GAP (or Magma, if you are lucky to have a license), etc. Anyway I think this kind of issue equally applies to every sufficiently long book; e.g. can we trust the authors/referees etc etc. E.g. can we trust every lemma in a proof of CFSG? Aug 19, 2015 at 5:27
• I agree CFSG needs its updated proof finished. Serre does say (roughly) 'proofs stated "modulo the Classification" are ok' Aug 19, 2015 at 5:38
• In the spirit of fairness, here are the errata for the in-progress CFSG II : math.rutgers.edu/~lyons/cfsg Sep 4, 2015 at 5:41