What Kind of Thing is a Sporadic Simple Group?

David Corfield discusses some speculation originally from Israel Gelfand:

Sporadic simple groups are not groups, they are objects from a still unknown infinite family, some number of which happened to be groups, just by chance.

(In David’s terminology, that means that sporadic finite simple groups are not a natural kind.)

I used to believe this very same thing, so I find it interesting that others have speculated the same thing. A couple of years ago, though, I came across a remark by Michael Aschbacher that made me rethink my view: the classification of finite simple groups is primarily an asymptotic result. Every sufficiently large finite simple group is either cyclic, alternating, or a group of Lie type.

Results that are true only for large enough parameter values are common enough that the existence of small-value counterexamples does not require special explanation. For example, the classification of simple modular Lie algebras looks completely different over small characteristics than it does over large characteristics. The best known results for number theoretic results such as Waring’s problem and Goldbach’s conjecture are asymptotic. Small numbers are just bad news.

3 thoughts on “What Kind of Thing is a Sporadic Simple Group?”

1. Small numbers are just bad news.

My own speculation involves Kolmogorov complexity, in general agreement of your position. Basically, when things are controlled by small numbers (dimension, size, whatever), there often isn’t enough “room” for generic phenomena to happen. Because of the lack of room, unlikely events are forced. Which turns out to be consistent with Corfield and Gelfand. In other words, I think both explanations can be made consistent with each other.

2. Actually, there is a number of very specific results of asymptotic nature about finite simple groups ; Ashbacher’s thesis is much more than just a philosophy. One of such results is a paper by Larsen and Pink Finite Subgroups of Algebraic Groups. I quote the abstract:

Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of GLn over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic p, a commutative group of order prime to p, and a p-group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.

3. That’s an interesting result. I wonder if people will begin to avoid invoking the classification and prove results directly: there is something unaesthetic about depending on a result whose proof is so long.