The n-Category Café

In 2001, Ronald Solomon wrote in his A Brief History of the Classification of Finite Simple Groups, Bulletin of the American Mathematical Society 38(3) 315-382:

Is there a completely new and revolutionary approach to the Classification waiting to be discovered? In Thompson’s eloquent article “Finite Non-Solvable Groups” written around 1982 [T4], he says:

“… the classification of finite simple groups is an exercise in taxonomy. This is obvious to the expert and to the uninitiated alike. To be sure, the exercise is of colossal length, but length is a concomitant of taxonomy. Those of us who have been engaged in this work are the intellectual confreres of Linnaeus. Not surprisingly, I wonder if a future Darwin will conceptualize and unify our hard won theorems. The great sticking point, though there are several, concerns the sporadic groups. I find it aesthetically repugnant to accept that these groups are mere anomalies…Possibly…The Origin of Groups remains to be written, along lines foreign to those of Linnean outlook.”

I doubt that any developments of the past two decades would change Thompson’s summary of the state of the field. We are still waiting and wondering. Are the finite simple groups, like the prime numbers, jewels strung on an as-yet invisible thread? And will this thread lead us out of the current labyrinthine proof to a radically new proof of the Classification Theorem? (p. 345)

[T4] J. G. Thompson, Finite non-solvable groups, pp. 1-12 in K. W. Gruenberg and J. E. Roseblade, Group Theory: Essays for Philip Hall, Academic Press, London, 1984.

I’ve had some very interesting comments about this from Alexandre Borovik, a colleague of Israel Gelfand. In an initial e-mail he told me about the classification that:

1. There is a school of thought that sporadic groups are not groups, they are representatives of some wider class of objects, only finitely many of which have happened, by chance, to be groups.

2. Some “non-sporadic groups” are actually sporadic; they just happened to be isomorphic to groups in “classical series”. $PSL(3,4)$ is the most notorious example; it is actually $M_{21}$; its properties are truly pathological.

3. The general theory of “taxonomy” of finite objects described by relational languages is developed in model theory (Cherlin-Lachlan Theorem). It is based on CFSG. In that sense, CFSG is the mother of all taxonomies. This is one of the factors which contribute to the huge metamathematical importance of CFSG.”

In a follow-up message, he wrote:

I believe I mentioned to you Israel Gelfand’s prophecy:

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.

A number of “classical” simple groups are in fact “sporadic” in a sense that they behave in an absolutely bizarre and pathological way. For example, $Alt(6)=PSL(2,9)$ and $PSL(3,4)$; acting on 9+1 = 10 and 16+4+1 = 21 points of their projective geometries, they appear in the stabilisers of points, $M_{10}$ and $M_{21}$ in the Mathieu groups $M_{11}$ and $M_{22}$. $PSL(3,4)$ probably holds the world record for the most bizarre Schur multiplier.

$M_{13}$ is an exciting object; maybe it provides a tantalising glance into some new possibilities, maybe not - much more research is needed.

For more on $M_{13}$, see The Mathieu group $M_{12}$ and its pseudogroup extension $M_{13}$.

See what the claim is? Finite simple sporadic groups do not form a natural kind. First, as they are currently defined they exclude close relations which are non-sporadic, i.e., which appear in the infinite families of the classification. Second, even with these additional groups they are part of a much larger kind, the ones appearing in the classification just ‘happening’ to be groups.