Why are there only a finite number of sporadic simple groups?

Is there any overarching reason why, after excluding the infinite classes of finite simple groups (cyclic, alternating, Lie-type), what remains---the sporadic, exceptional finite simple groups, is in fact a finite list (just 26)? In some sense, the prime numbers can be viewed as "sporadic," but there is an infinite supply. Is there some principle that indicates that there must be only a finite number of these exceptional groups, and the "only issue" (to minimize a huge, multi-year community effort) was to identify them?

I ask in relative ignorance of modern group theory, and apologize in advance for the naiveness of my question.

Solution 1:

Gerhard Michler has worked on a research program to show fairly convincingly that the possibility of infinitely many sporadic groups (with a uniform construction, but highly non-uniform properties) was quite real. Roughly speaking the second round of sporadic groups was discovered looking for special configurations of centralizers of involutions, and he shows how this search can be continued, how it constructs almost all of the sporadic simple groups in a uniform fashion, and how it does not obviously stop there.

This is discussed in some detail in his books MR2266036 and MR2583258, the Theory of Finite Simple Groups, volumes I and II.

So, at least according to him, it should not be taken for granted that there are only finitely many sporadic groups, as there is a fairly reasonable procedure for possibly producing an infinite collection of basically unrelated finite simple groups (at least, nowhere near as related as groups of a fixed Lie type and rank).

Solution 2:

Chevalley (in the paper cited above), described what we call the Lie-Chevalley finite simple groups to which we add their fixed-point subgroups. There are 26 other finite simple groups constructed outside this program and they are called the sporadics. The classification CFSG tells us that the list is complete.

To understand the sporadics, we need to find other constructions with an aim of capturing all FSGs.

There is a host of established mathematics that may be needed. CFSG does not stray far from finite groups and their geometry but exploring areas as diverse as integrable systems, symplectic geometry, characteristic classes, and KMS BC systems may all shed light on the problem of why we have sporadics. If we can uniformly construct all FSGs then we may find as yet undiscovered common properties and a common 'reason' for them.