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How hard is it to classify all big finite simple groups, i.e., all finite simple groups larger than some sufficiently large constant? Alternatively - how hard is it to classify all finite simple groups up to finitely many exceptions?

Is any known proof of such a classification significantly easier or shorter than the classification of (all) finite simple groups?

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The answers to these related questions lead me to think that the answer is "no" : mathoverflow.net/questions/38161/… and mathoverflow.net/questions/34424/… –  Andy Putman Feb 13 '11 at 21:20
If it is "no", is it felt to be a very difficult project? A rather worthwhile one? –  H A Helfgott Feb 14 '11 at 0:10
The impression I have (and I'm not an expert) is that this is thought to be no easier than the classification itself (ie quite hard). Of course, that means that a breakthrough here would be great! –  Andy Putman Feb 14 '11 at 1:10
I agree with Andy that the conventional wisdom is that it would be essentially as hard as the full classification. –  Noah Snyder Feb 14 '11 at 3:15
See also Terence Tao's answer to this question: mathoverflow.net/questions/38161/… –  Matthew Kahle Feb 14 '11 at 16:19

3 Answers 3

As Andy says, people didn't know until fairly late in the classification whether there were finitely or infinitely many sporadic finite simple groups. Actually "sporadic" has a fairly specific operational meaning; it means finite simple groups that are not prime cyclic, alternating, or Chevalley type. (The finite simple groups of Chevalley type are basically Lie groups over finite fields, with the twist that there are some extra ones in characterestic 2 and 3. The Tits group is usually counted with these even though it's not part of an infinite sequence. Arguably a prime cyclic group is also a Lie group over a finite field.)

Another example is the theorem on highly transitive permutation groups. The classification implies that there are no 6-transitive permutation groups on $n$ points other than $A_n$ and $S_n$. I have heard that without the classification, there is no bound which is uniform in $n$.

People certainly think that it is a good project to improve the classification of finite simple groups in general. In fact Gorenstein's announcement that the classification was complete was controversial, because there was clearly interesting mathematics left to be discovered even though there was sort-of enough at that time to believe the classification. But you have to study the classification to know what needs to be improved. If people do not have an a priori argument that there are only finitely many sporadic groups, then it would be great to have one, but I don't see how you can know in advance that you should look for that.

Note that in the modern classifications of complex simple Lie algebras or compact simple Lie groups, you do not prove that there are only finitely many exceptional ones before proving the full classification using root systems and Dynkin diagrams. On the contrary, both in the classification and in the representation theory that comes after, people are the happiest when they can treat all of the root systems uniformly, i.e., when the exceptional Lie algebras are not treated as exceptions. The fact that sporadic groups are more exceptional than exceptional Lie algebras could be one reason that the current classification of finite simple groups seems unsatisfying.

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Arguably an alternating group is also a Lie group over a "finite field"...! –  Qiaochu Yuan Feb 14 '11 at 10:10
@Qiaochu: According to Tits' dictum, symmetric or alternating groups behave like Chevalley groups over a "field of one element", the point being that these are doubly transitive permutation groups and thus have a BN-pair (of rank 1). –  Jim Humphreys Feb 14 '11 at 13:06
As Greg explained, for a large number of group-theoretic problems, there is a huge gap between things that can be proved based on the classification and things that can be proved without it, (For many of the applications the identity of the sporadic groups does not matter.) Therefore, any shortcut of the kind asked in the question will have vast applications. An attempt towards such or even much weaker shortcut may be based on trying to prove substantial consequences of the classification theorem without using the classification theorem. Indeed... –  Gil Kalai Feb 14 '11 at 18:14
(cont) Indeed, applications to permutation groups are where people have looked I am (vaguely) aware of certain results on permutation groups based on the classification for which model theoretic classification-free proofs were found. And vaguely remember that this gave 1-2 decades ago some hopes for a cheaper, less precise, (perhaps much much less precise), version of the classification theorem, based perhaps on model theory. But I dont remember details or what have happend in this direction since. –  Gil Kalai Feb 14 '11 at 18:14
Can people give their opinion of how hard an open problem they would think proving some of these consequences without the classification theorem would be? For example - do you think it extremely hard to prove that a 1000-transitive permutation group must be Alt(n) or Sym(n)? –  H A Helfgott Feb 14 '11 at 22:13

To expand a little more on the original question and Greg's solid answer, "larger than some sufficiently large constant" already strikes me as ambiguous. To start with a specified constant seems impossible, while to leave it indeterminate makes the question fuzzy. How to identify or prove the existence of a suitable constant without already having the substantial part of the classification under control?

"Is any known proof of such a classification significantly easier or shorter than the classification of (all) finite simple groups?" This is easy to answer negatively.

All ongoing efforts to understand the classification better start with the understanding that the finiteness of the number of simple groups which were not known decades ago is a fundamental mystery. Much work up to 1980 went into narrowing the possibilities for unknown groups (via possible centralizers of involutions, etc.). But the eventual number 26 of "sporadic" groups was arrived at only after the fact. There are admirable ongoing efforts to write down all details while streamlining and improving the proofs (and while there are still people around who can do it). For me and other outsiders there remains the problem of making the end result look more conceptually natural and inevitable.

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There's nothing fuzzy about proving a theorem of the form "there exists an n such that all groups of size larger than n is one of the following kinds", possibly by non-constructive methods that leave n unknown. –  Henry Towsner Jul 23 at 15:12

Hrushovski asks roughly this question in his paper On Pseudo-Finite Dimensions. In section 6 he discusses the "classification of large finite simple groups" (in analogy to some of the related results which can be proven by model theoretic means, and end up only giving information about sufficiently large finite objects), and notes that the only known proof is via the full classification of finite simple groups.

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