This post over at the n-category cafe has got me thinking about the classification of finite simple groups. As I said in the comments over there, the thing that fascinates me most about the story of classification (aside from the fact that I heart finite groups) is sociologically what went wrong with finite group theory that made the energy from classification dissipate so that very few young researchers keep its ideas and spirit alive.
When I was thinking about this I did some poking around trying to understand a little more about the odd order theorem (which is something I would genuinely like to understand in detail at some point in my life). The wikipedia article on Feit-Thompson is very high quality, and Stephen Harris sent me a link to an expository article by Glauberman. The impression that these articles give is that the outline of the Feit-Thompson argument isn’t too difficult to understand, but that the details get quite annoying and the argument splits into a number of cases. This got me wondering: “maybe the classification theorem could aspire to become more like the four-color theorem?”
Bear with me for a second here, I know that many people hate the computer proof of the four color theorem, and certainly it’d be nice to have a short conceptual proof of it (see Week 22). Nonetheless I think there are a lot of advantages to the four color theorem over parts of the classification theorem. First off, it’s not that hard to understand how the argument goes, there’s some simple nice ideas and then a huge number of cases any one of which is understandable. My impression is that a mathematician who put her mind to it could pretty much understand this proof in at most a couple weeks (the same is not true of the odd order theorem, let alone the full classification). Secondly, the computer proof of the 4-color theorem is quite reliable and much much more likely to be correct and true than the classification theorem. (Some people seem to consider computer proofs as bad because they are unreliable, these people are either lying or crazy. There are lots of issues with computer proofs, but reliability relative to a long computation in a journal article is not one of them.)
So the idea I’m proposing here is that a different way to clarify the classification project (as opposed to shortening the traditional argument, or making a computer verifiable proof) would be to try to reduce it to a collection of algorithmic tests which can eliminate groups satisfying certain properties. You explain each of these algorithms, you then have a computer package which can apply all applicable tests until it kills off a case. In the end you have a computer program that applies these algorithms to a huge set of cases, and in the end spits out a classification of simple groups. Because the arguments that the computer is doing implicitly wouldn’t end up in the final write-up you could hope to get a much shorter human understandable book that together with the computer program would give a proof of classification. I think in this form it would be much easier for people to understand and use the ideas in this project.
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It is certainly difficult to believe that the proof of the classification is absolutely correct, but I note that the fact that the statement is true seems to be much less controversial (when Serre was one of the people, a few years ago, saying that he didn’t believe the proof was complete, I don’t think he ever said he believed there were other simple finite groups). Moreover, for what it is used, the statement is extremely robust, and would have to be extraordinarily false for the conclusions that are derived from it to be invalidated: any finite number of exceptional groups would be fine, and a new infinite series, if it behaves at all similarly to the ones known, would presumably not be a problem either. (For instance, in the book of Lubotzky and Segal on “Subgroup growth”, one of the uses of classification is to show that there are at most 2 non-abelian finite simple groups of a given order, and the 2 could be worsened considerably before the consequences they derive break down, e.g., any finite bound would do; it would be a considerably different picture than the one we have if this number was unbounded).
Nice idea! I suspect I’ve talked to you about my interest in the notion of what makes these various borderline cases into accepted proofs. The only type you didn’t mention is probabilistic proofs (which of course are just on the other side of the borderline).
The problem with the classification was political, in the dirtiest sense of the word: at some point it became obvious to the people who ran the show that getting more grants for classification is not possible, and it would damage their political and academic standing to keep going. So the classification was declared complete, even though it was quite far from this. Needless to say, for a number of young people in the project it was a huge blow, which rendered them poorly employable, and forced some all together out of academia.
A very limited version of the program you propose was actually carried out, to and extent, long ago, for groups of order, say, less than
But the general case seems to be extremely tough…
Matthew Emerton, on E. Kowalski’s blog, makes a nice point:
I think part of what is so frustrating about the classification of finite groups is that no one has told me such a story.
For the Weil conjectures, one can sketch the proof of all but the Riemmann hypothesis by saying “this looks like the Lefschitz fixed point theorem” and one can motivate the Riemann hypothesis by (i) the analogy to number theory and (ii) an analogy to the Hard Lefschitz principle. (Although my understanding is that neither of these are very close to the actual proof of the RH.)
For the Poincare conjecture, one can say “toplogy gets simpler under Ricci flow”. For Fermat, one can say “Look at the elliptic curve
. The Galois representation on the 
-torsion has extremely simple ramification, by computing with a Tate uniformization, so it should come from a modular form of level 2, and there are none.” For the four-color theorem, one can say “It’s like the proof of the five color theorem — use Euler’s formula to show that a certain family of subgraphs is unavoidable and use chain arguments to color each of those subgraphs — but the list of unavoidable graphs has thousands of elements instead of just three.”
Glauberman’s article (linked by Noah above) does a good job summarizing the proof of the odd order theorem on this level. But I have never seen anyone even try to summarize classification.
Update Edit to make clear when I stop quoting Matt and start speaking for myself.
A sort of follow up to David’s question is: the classification of finite simple groups has a similar flavor to that of simple Lie algebras, but a lot more complicated.
The objects that classify Lie algebras have some independent life in mathematics (McKay correspondence, etc.), so their weird quirks seem less random; E_6, E_7 and E_8 correspond to the duality classes of the Platonic solids, for example. What about those that classify finite simple groups?
perhaps not entirely what you want but the platonic triple corresponding to the E6-8 triple has a corresponding triple for sporadic simples with E8 corresponding to the monster, see the comment of McKay next to the list of the Dynkins.
If you look at the introduction of some of Gorenstein’s books he does a good job of trying to build a narrative around the full classification. He breaks it down into two main steps, techniques for showing that a group must resemble one of the known simples, and then recognition theorems that allow you to identify the group as automatically one of the known simples. He explains this analogously to the classification of lie algebras, where (as long as you’re in characteristic 0) the existence of the Killing form makes the first step really easy.
I’m probably butchering his explanation, and partly this is because the narrative is not as nice as the ones in Matt’s comment. But it’s not as bad as one might have guessed.
Thanks! By the way, only my first paragraph is quoting Matt. I’ll edit my comment to make this more clear.
Dear Secret Bloggers,
When I wrote my comment on Emmanuel’s blog, I was thinking
that the classification of simple Lie algebras could be used as
an analogy for the classification of finite simple groups. And if
I remember correctly from my reading of the introductions of
Gorenstein’s books (a long time ago now), he does explain
the analogy in some detail. (For example, normalizers of Sylow subgroups are analogous to Borel subgroups.)
But Ben has a valid point: the combinatorics of the classification
of simple Lie algebras somehow looms larger in the life of
mathematics than does that of the finite simple groups. But I wonder
if it has to be this way? (In other words, is this a mathematical
phenomenon, i.e. are finite simple groups really more removed
from the rest of maths than finite dimensional simple Lie algebras, or is it just a cultural phenomenon.)
Cheers,
Matt