Introduction

I'm going to make a number of observations about the art of mathematics, the people who practice that art, and the ways such people differ from other people. Perhaps enough such observations, here and in other articles in this volume, will add up to some semi-coherent picture of mathematics and mathematicians.

Despite some existing stereotypes, I don't think mathematicians as a class are all that different from the general run of humanity. But it is true that any small group, selected on the basis of the ability to perform some special task, is not unlikely to be a bit different, at least statistically. Here are three differences which I'll discuss later in more detail: Mathematicians learn faster than most people and are better able to cope with novelty. Mathematicians are better able to concentrate deeply for extended periods of time. Some mathematicians have the ability to carry out thought processes at an “intuitive” level, in order to reach conclusions that would be unattainable, or attainable only very slowly, using linear thought processes.

Some of these tendencies, when carried to extremes, can result in behavior associated with the stereotypical absent-minded professor or mad scientist. To the extent that the more random person on the street has any picture of a typical mathematician (and most are totally unaware of mathematics and mathematicians), the picture is probably of a schizophrenic à la A Beautiful Mind or Proof, or an absent-minded eccentric. Of course it must be admitted that mathematics does have its share of eccentrics, but to my mind that speaks well of the tolerance of the mathematical community and its priorities; the eccentric who produces good mathematics is welcome in the community.

Here are some other things I'll do. I'll describe how students move through the U.S. educational system to become mathematicians. I'll discuss the international community of mathematicians. I'll give my own take on how a mathematician produces mathematics. I hope that this loosely connected collection of small essays will give at least a partial picture of mathematicians and their community.

Mathematics and Mathematicians

To understand mathematicians, it is perhaps useful to begin with a definition or description of mathematics; here is one description with which you may, or may not, agree.

Part II of the book is intended to be an introduction to the local theory of saturated fusion systems. By the “local theory of fusion systems” we mean an extension of some part of the local theory of finite groups to the setting of saturated fusion systems on finite p-groups.

Recall the notions of a p-local finite group and its associated linking system defined in Definitions III.4.1 and III.4.4. One can ask: Why deal with saturated fusion systems rather than p-local finite groups? There are two reasons for this choice. First, it is not known whether to each saturated fusion system there is associated a unique p-local finite group. Thus it remains possible that the class of saturated fusion systems is larger than the class of p-local finite groups. But more important, to date there is no accepted notion of a morphism of p-local finite groups, and hence no category of p-local groups. (The problem arises already for fusion systems and linking systems of groups, since a group homomorphism α from G to H need not send p-centric subgroups of G to p-centric subgroups of H, so it is not clear how to associate to α a map which would serve as a morphism of the p-local group of G with the p-local group of H.) The local theory of finite groups is inextricably tied to the notion of group homomorphism and factor group, so to extend the local theory of finite groups to a different category, we must at the least be dealing with an actual category.

We collect here, for reference, some of the standard results in group theory which have been used throughout the book. We begin with some of the standard elementary properties of p-groups and Sylow subgroups.

Lemma A.1For any pair of p-groups P ≤ Q, if NQ(P) = P, then Q = P.

Proof. This is a general property of nilpotent groups (finite or not). If G is nilpotent and H < G, then by definition, there is K > H such that [G, K] ≤ H. Hence H < K ≤ NG(H). That all p-groups are nilpotent is shown, e.g., in [A4, 9.8] or [G1, Theorem 2.3.3].

Lemma A.2Fix a p-group P, and an automorphism α ∈ Aut(P) of order prime to p. Assume 1 = P0 ⊴ P1 ⊴ … ⊴ Pm = P is a sequence of subgroups all normal in P, such that for each 1 ≤ i ≤ m,α|Pi ≡ IdPi (mod Pi-1). Thenα = IdP.

Proof. See, for example, [G1, Theorem 5.3.2]. It suffices by induction to prove this when m = 2, and when the order of α is a prime q ≠ p. In this case, for each g ∈ P, α acts on the coset gP1 with fixed subset of order ≡ |gP1| (mod q). Since |gP1| = |P1| is a power of p, this shows that α fixes at least one element in gP1. Thus α is the identity on P1 and on at least one element in each coset of P1, and so α = IdP.

Let G be a finite group, p a prime, and S a Sylow p-subgroup of G. Subsets of S are said to be fused in G if they are conjugate under some element of G. The term “fusion” seems to have been introduced by Brauer in the fifties, but the general notion has been of interest for over a century. For example, in his text The Theory of Groups of Finite Order [Bu] (first published in 1897), Burnside proved that if S is abelian then the normalizer in G of S controls fusion in S. (A subgroup H of G is said to control fusion in S if any pair of tuples of elements of S which are conjugate in G are also conjugate under H.)

Initially, information about fusion was usually used in conjunction with transfer, as in the proof of the normal p-complement theorems of Burnside and Frobenius, which showed that, under suitable hypotheses on fusion, G possesses a normal p-complement: a normal subgroup of index |S| in G. But in the sixties and seventies more sophisticated results on fusion began to appear, such as Alperin's Fusion Theorem [Al1], which showed that the family of normalizers of suitable subgroups of S control fusion in S, and Goldschmidt's Fusion Theorem [Gd3], which determined the groups G possessing a nontrivial abelian subgroup A of S such that no element of A is fused into S\A.

In the early nineties, Lluis Puig abstracted the properties of G-fusion in a Sylow subgroup S, in his notion of a Frobenius category on a finite p-group S, by discarding the group G and focusing instead on isomorphisms between subgroups of S.