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?)
Best Answer
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.