Heuristic argument that finite simple groups _ought_ to be "classifiable"?

Question

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?)

Answer 1

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.

Answer 2

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...).

Answer 3

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.

Answer 4

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.

Answer 5

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.