# 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.

• Do you know of any examples where "sporadic" is actually used to describe an infinite family? I'm kind of thinking that if there were infinitely many sporadic groups, we would find a cofinite number of them which share some given property and define a new "family" out of those... – Jason DeVito Aug 14 '10 at 1:43
• Take a look at my answer to this previous question math.stackexchange.com/questions/1423/… and also Richard Borcherds' answer to the same question asked on MO: mathoverflow.net/questions/34424/… Is there anything in your question which is not addressed by these answers? – Pete L. Clark Aug 14 '10 at 1:55
• In other words, the only reason we know that there are finitely many sporadic simple groups is the full CFSG. There is no other "reason" for it, and indeed even with the proof in hand (so to speak!) it looks rather miraculous. – Pete L. Clark Aug 14 '10 at 1:58
• @Pete: Thanks for the related links! I accept your judgment that there is essentially no reason aside from the full classification theorem. Or at least no reason discernible to mathematicians today! – Joseph O'Rourke Aug 14 '10 at 12:16

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).

• Thank you, Jack, this is exactly what I was seeking! – Joseph O'Rourke Aug 16 '10 at 13:51

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.

• Joseph is almost asking whether there is a short proof (much less than 10^4 pages) of the statement: "If $G$ is a finite simple group and $|G|$ is large enough, and not prime, then G is an alternating or of Lie type." This is still an interesting question (even though one wants the list of sporadics for so many applications). – Glasby Feb 6 at 2:25

What do we mean by 'sporadic' ? I suggest that we may find some extension of the Chevalley programme (Tohoku J 1950+) that brings in the sporadics. This is about construction - not the classification as we know it.

• @John: Although I appreciate your reply, I am afraid I do not understand it; my fault not yours. I don't know, for example, what the "Chevalley programme" is, nor did a quick Google search enlighten me... – Joseph O'Rourke Oct 14 '10 at 14:59
• @Joseph : John is referring to Claude Chevalley's article in the Tohoku Journal : Sur certains groupes simples, Tohoku Math. J. (2) Volume 7, Number 1-2 (1955), pp14-66. You can find it on a library or Project Euclid. – ogerard May 19 '11 at 8:54

I can't give a mathematical explanation. But I can give a philosophical explanation...

It has been found by "Monstrous Moonshine" that some of the sporadic groups are related to symmetries in string theory which is thought to be a fairly good model of our Universe (in that it predicts gravity and some other forces).

If we imagine that a description of the Universe involves some sort of algebra with a complicated symmetry then in order for it to be unique it is likely to be of the sporadic kind rather than one in an infinite family.

Since groups are related to symmetries, an infinite number of sporadic symmetries might correspond to an infinite number of types of Universe possible. Conceptually, the probability that our Universe would correspond to one of the smaller symmetry groups would be zero.

Hence, if there were an infinite number of sporadic groups,... the Universe shouldn't exist. Or at least it shouldn't be understandable to anyone in it!