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Obviously there exists a list of the finite simple groups, but why should it be a nice list, one that you can write down?

Solomon's AMS article goes some way toward a historical / technical explanation of how work on the proof proceeded. But, though I would like someday to attain some appreciation of the mathematics used in the proof, I'm hoping that there is some plausibility argument out there to convince the non-expert (like me!) that a classification ought to be feasible at all. A few possible lines of thought come to mind:

  • Groups have very simple axioms. So perhaps they should be easy to classify. This seems like not a very convincing argument, but perhaps there is some way to make it more convincing.
  • Lie groups have a nice classification, and many tools are available for their study and that of their finite analogues. And in fact, it turns out that almost all finite simple nonabelian groups fall under this heading. Is it somehow clear a priori that these should be essentially all the examples? What sort of plausibility arguments might lead one to believe this?
  • If there are not currently any good heuristic arguments to convince a non-expert that a classification should be possible, then will this always be the case? Or will we someday understand things better...

There is probably a model-theoretic way to formalize this question. As a total guess, it might be something along the lines of "Do the finite simple groups have a finitely axiomatizable first-order theory?", except probably "finitely axiomatizable first-order theory" doesn't really capture the idea of a classification. If someone could point me towards how to formalize the idea of "classifiable", or "feasibly classifiable", I'd appreciate it.FSGs up to order SEFSGs up to order MO

EDIT: To clarify, what I'd like is an argument that finite simple groups should be classifiable which does not boil down to an outline of the actual classification proof. Joseph O'Rourke asked on StackExchange Why are there only a finite number of sporadic simple groups?. There, Jack Schmidt pointed out the work of Michler towards a uniform construction of the sporadic groups, as reviewed here. Following the citation trail, one finds a 1976 lecture by Brauer in which he says that he's not sure whether there are finitely many or infinitely many sporadic groups, and which he concludes with some historical notes that describe a back-and-forth over the decades: at times it was believed there were infinitely many sporadic groups, and at times that there were only finitely many. So it appears that the answer to my question is no-- at least up to 1976, there was no evidence apart from the classification program as a whole to suggest that there should be only finitely many sporadic groups.

So I'd like to refocus my question: are such lines of argument developing today, or likely to develop in the (near? distant?) future? And has there been further clarification of what exactly is meant by a classification? (Is this too drastic a change? should I start a new thread?)

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An (easier?) question that one should be able to answer first: why should finite dimensional simple Lie algebras (over the complex numbers) be "classifiable"? Whatever heuristic works for groups ought to be applicable there too. –  Sheikraisinrollbank Sep 9 '10 at 10:48
I asked a related question on Math StackExchange: "Why are there only a finite number of sporadic groups?" math.stackexchange.com/questions/2427/… –  Joseph O'Rourke Sep 9 '10 at 11:45
I would say that one of the things that makes groups possible to classify is that they have an enormous amount of structure attached (e.g., representation theory, etc.). On the other hand, there are also classes of objects with very little structure that are also possible to classify; it's the ones with an intermediate amount of structure that seem to be difficult. –  Peter Shor Sep 9 '10 at 12:04
The word "heuristic" refers to a particular method of finding a solution to a problem. Could you clarify exactly what you mean by it as that doesn't seem to fit with the question? My reading of your question is that you mean: "Is there an accessible explanation of why the finite groups are classifiable?". –  Loop Space Sep 9 '10 at 12:27
@Mariano: "Sporadic: Occurring at irregular intervals or only in a few places; scattered or isolated" (Oxford American Dictionary). I don't see why this should imply finiteness. –  Laurent Moret-Bailly Sep 12 '10 at 8:31

5 Answers 5

up vote 35 down vote accepted

It is unlikely that there is any easy reason why a classification is possible, unless someone comes up with a completely new way to classify groups. One problem, as least with the current methods of classification via centralizers of involutions, is that every simple group has to be tested to see if it leads to new simple groups containing it in the centralizer of an involution. For example, when the baby monster was discovered, it had a double cover, which was a potential centralizer of an involution in a larger simple group, which turned out to be the monster. The monster happens to have no double cover so the process stopped there, but without checking every finite simple group there seems no obvious reason why one cannot have an infinite chain of larger and larger sporadic groups, each of which has a double cover that is a centralizer of an involution in the next one. Because of this problem (among others), it was unclear until quite late in the classification whether there would be a finite or infinite number of sporadics.

Any easy way to get around this has been overlooked by about a hundred finite group theorists.

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If there was an infinite series of "sporadic" groups starting with the baby monster like described by you, why should this be a obstacle in classifying all finite simple groups? Wouldn't it be just another (besides being less elementary) infinite series like the alternating or the Lie-type groups reducing the number of sporadics to 24? –  j.p. Sep 9 '10 at 14:05
If you knew that there was only one finite simple group with that property and you had a recipe for constructing them, then yes you could just call that a new infinite family and say you had a classification. But you could imagine the situation being much worse, maybe when you look for simple groups whose centralizer of an involution is a double cover of the previous group then you find 50 of them? –  Noah Snyder Sep 9 '10 at 14:38
The approach of looking at centralizers of an element of order 2 in groups of Lie type with odd characteristic p is for sure not optimal. Parts of the classification are being redone by focusing on finite groups of local characteristic p (see mth.msu.edu/~meier/Preprints/CGP/cgp_abstract.html). The classification of only these groups is still very messy (in the sense that one has to look at many special cases), but additionally there seems to be no idea at all yet how to "reduce" to the case of local characteristic p. –  j.p. Sep 9 '10 at 15:59
In the last few years there's also been a programme (led by Aschbacher as far as I can tell) to recast most of the local analysis involved in the classification in the context of fusion systems, in the hope that this will make the proof clearer. I don't know how much headway has been made, but people might find the following of interest: www.math.ku.dk/english/research/.../fusion2007/aschbacher.cph2007.pdf/ –  Colin Reid Sep 9 '10 at 16:36
Can someone explain the mysterious last sentence in this answer? –  daveh Jun 12 '12 at 18:26

There is a paper of Larsen and Pink (Update: It has appeared: Larsen, Michael; Pink, Richard: Subgroups of algebraic groups. J. Amer. Math. Soc. 24 (2011), 1105–1158.)dating back to 1998 (but still in the process of getting published - long story there) that gives a conceptual proof (based on algebraic geometry methods, primarily) that any sufficiently large finite simple group that has a bounded rank linear model (i.e. it is isomorphic to a subgroup of $GL_d(k)$ for some field k and some bounded d) is basically of Lie type. So this, combined with the classification of simple groups of Lie type, gives an answer to the question in the bounded rank case. Unfortunately this isn't the whole story because one can certainly let the rank go to infinity, and then there are also the pesky alternating groups which are not of Lie type at all (except, perhaps, over the field of one element, whatever that means...).

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This seems to be a mod p version of the classical theorem of Jordan that there is a bound on the orders of the simple groups with a complex representation of given degree. As you say, the catch is that one needs to bound the degree of the representation. –  Richard Borcherds Sep 9 '10 at 23:40

I can't tell you much about finite groups, but I can tell you that unfortunately there is no general model-theoretic result along the lines of "If a collection of objects has a simple axiomatization, then it must be easy to classify." In fact, considering some examples, I believe that no such general result could exist even in principle.

Finitely axiomatizable, but hard to classify: The class of all linearly-ordered sets. There are many ways to make precise the idea that these are "hard to classify:" for any uncountable cardinal $\kappa$, there are many (i.e. $2^\kappa$) pairwise nonisomorphic orderings of size $\kappa$; there is no way to characterize arbitrary linear orderings up to isomorphism by a fixed set of cardinal-number invariants; and there are many large families of linear orderings which "look similar" but are nonisomorphic (where "looks similar" could be mean various things: bi-embeddable by maps preserving the truth of all first-order formulas, or "there is a forcing extension of the universe of set theory that preserves all cardinal numbers, adds no new subsets of $\mathbb{R}$, and in which the two structures are isomorphic," etc...)

Easy to classify, but not finitely axiomatizable: For example, the set of all algebraically-closed fields. These are axiomatizable by an infinite list of axioms: take all the field axioms, plus, for each natural number $n > 0$, an axiom saying "every degree-$n$ polynomial has at least one root." However, a simple argument using the compactness theorem shows that this class cannot be finitely axiomatizable. Also, these structures are "very easy to classify" in the sense that they are characterized, up to isomorphism, by just two cardinal numbers: the characteristic and the transcendence degree (over the prime subfield). (And hence any two such fields that ``look similar'' in the senses I mentioned above must actually be isomorphic.)

In fact, it's worse than these examples suggest. There is a theorem in model theory due to Cherlin, Harrington, and Lachlan saying that any axiomatizable class that is ``easy to classify'' in the sense that there is just one member (up to isomorphism) of size $\kappa$ for any infinite cardinal $\kappa$ cannot be finitely axiomatizable!

There is a well-studied notion of "classifiability" in model theory which concerns how hard it is to characterize all the models of a given theory by a "reasonable set of invariants." The main reference is Shelah's monograph Classification Theory. In general, classifiability of a theory has no logical relation with how hard it is to axiomatize the theory (e.g. whether it is finitely axiomatizable, computably axiomatizable, etc.). But Shelah's classification theory only treats theories with only infinite models, so I'm not sure that it can answer your question about finite simple groups.

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Hi John. But the question is in the setting of finite model theory, so your heuristic does not apply (I am not saying it is false). For example, in the finite setting, it is trivial to classify linear orderings of a given size. The boundary of easy/hard needs to be redefined in the finite context. Hrushovski and Cherlin have done some work on model theory of specific finite structures, using non-standard analysis to be able to carry out the standard arguments and obtain meaningful results. –  Andres Caicedo Sep 17 '10 at 22:28
@Andrés: I wonder if anyone has defined an easy-to-classify/hard-to-classify boundary for axiomatizable classes of finite structures? I can't recall hearing about such a thing, but it could be interesting if it exists. –  John Goodrick Sep 18 '10 at 6:58

There is an interesting result of U. Felgner (MR1107758, see also MR1477188): simplicity is an elementary statement in the class of finite non-abelian groups. E.i. he showed that there is a first order sentence $\sigma$ such that $G\models \sigma$, where $G$ is finite if and only if G is non-abelian and simple. However the proof uses the classification of finite simple groups.

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This ought to have been a comment, but it's too long.

Let me try to address the model theory part of your question. A direct consequence of the compactness theorem in FOL is the following: no set of FO axioms can capture a class of finite structures with arbitrarily large members, without also including an infinite structure.

Since the simple alternating groups get arbitrarily large, you can't hope to exactly capture the f.s.g. in FOL. (Much less with a finite set of sentences.) Worse still, even if you settle for capturing the f.s.g. plus some infinite simple groups, simplicity is not captured by FOL.

One can prove this with an ultraproduct construction. Ultraproducts are widely used in model theory because of Łoś's theorem. It basically says that you can take any collection of FO structures S_i and make a sort of limit structure S* which models a given FO sentence iff "most" of the S_i model it. (So S* is a kind of "generic S_i".) In particular, if there were a set A of sentences s.t. all models of A were simple, the abelian groups $\mathbb{Z}_p$ would be models of A. Then any ultraproduct of all $\mathbb{Z}_p$ would also be a model of A. But this is an infinite abelian group, hence not simple. (We've tacitly used the fact that commutativity is FO-expressible.)

A standard move around the finiteness issue is to "cheat" and restrict one's attention to finite structures. I.e. one has a set of senteces A and looks at the class of finite models of A. This avoids both the compactness and ultraproduct barriers, since if an ultraproduct of finite structures is finite, it is necessarily isomorphic (not only elementarily equivalent) to one of the factors. (Exercise!)

I don't know if simplicity becomes expressible in the finite-models-only context.

In any case, having a finite axiomatization is not synonimous with "classification" as it is normally used. For example, groups themselves have a finite axiomatization which is pretty low in logical complexity, but it seems that "classifying all groups" in anything like the detail of CSFG is essentially hopeless.

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Well, this is true within the language of groups, but surely in the richer language of set theory (e.g. ZFC), one can (and does) capture the notion of a finite simple group? (OK, so one would now have nonstandard models of ZFC, coming for instance from the ultraproduct construction, in which the notion of finite is itself nonstandard, but this is largely irrelevant for most mathematical purposes as one can just work with the standard model throughout.) –  Terry Tao Sep 11 '10 at 15:43
Indeed I imagine that one could even phrase the classification of finite simple groups in Peano Arithmetic, given that one can already model finite sets in that language. –  Terry Tao Sep 11 '10 at 15:44
I would like to learn more about the broader model-theoretic aspects of this question: how to define what is meant by a "classification". I think I may ask about it here as a separate question later. Thanks for these initial observations. –  Tim Campion Sep 13 '10 at 16:55
Terry has a point: I should have stated at the outset that, when talking about expressibility here, especially in the ultraproduct construction, I assume the language of groups. I wasn't sure how familiar the OP was with the basic results of logic, so I thought I'd tell him about the better-known weaknesses of FOL in capturing classes of structures, and I find them easier to understand in the "natural language" of a structure. –  Pietro KC Sep 15 '10 at 2:01
Of course, one can express the notion of f.s.g. in ZF. But besides the compactness issue, which cannot be avoided syntactically (so we have to "consider only finite models", or "work in the standard model"), one may note that expressibility in ZF is a very weak property. In particular, the finite axiomatizability of f.s.g. in ZF would not immediately lead us to expect the CFSG. On the other hand, Tim's question about finite axiomatizability in a simple language is a pretty good initial guess for an explanation of CFSG. –  Pietro KC Sep 15 '10 at 3:18

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