We will take as a starting point the important paper of Solomon [Sol74], which proves that, for a 2-Sylow subgroup P of Spin7(3), there is a particular pattern of the fusion of involutions in P that, while not internally inconsistent, is not consistent with living inside a finite group.
Benson [Ben98a] constructed a topological space that should be the 2-completed classifying space of a finite group whose fusion pattern matched that which Solomon considered. Since such a group does not exist, this space can be thought of as the shadow cast by an invisible group. Benson predicted that this topological space is but one facet of a general theory, a prediction that was confirmed with the development of p-local finite groups.
… we have Oliver’s proof [Oli04][Oli06] of the Martino-Priddy conjecture [MP96], which states that two finite groups have homotopy equivalent p-completed classifying spaces if and only if the fusion systems are isomorphic.
These three quotes, from Craven’s Preface to The Theory of Fusion Systems, suggest something very exciting, namely a new interaction between topology and finite group theory capable of shedding light on new mysteries pertaining to the latter. Let’s ask the obvious first questions: what are these mysteries like, and what are these fusion systems that elucidate them? Well, a first run at an answer to the former question is this (also from the Preface):
Various results that could be considered part of local finite group theory (the study of p-subgroups, normalizers, conjugacy, and so on) were extended to p-blocks in the 1990s and early part of the twenty-first century, but at the time were not viewed as taking place in the more general setting of fusion systems.
We know what p-groups are, of course, but what is a p-block? We find on p. 27 that in decomposing an algebra over a Noetherian ring into indecomposable two-sided ideals the latter are, by definition, the blocks of the algebra; and then it’s quickly on to Brauer theory. (Consider, e.g., the beautiful theorem on p.31 due to Brauer: For G a finite group and k an algebraically closed field of characteristic p, “[t]he number of simple kG-modules is equal to the number of conjugacy classes of elements whose order is prime to p.”) And we subsequently encounter fusion systems on p.47: “Having spent a long time discussing Brauer pairs and defect groups, we are at last at a point where we can introduce the fusion system of a block. If b is a block idempotent, then the fusion system of b is a fusion system on a defect group D of b, consisting of the b-Brauer pairs (P, eP) with P≤D.” But this calls for definitions of Brauer pairs, defect groups, block idempotents, and so forth, and we’re quickly getting awfully technical.
However, most of us have met some Brauer theory along the way at some point (algebraists being insiders, of course), so there should be some resonance, even without dotting i’s and crossing t’s: what we’re dealing with here is based on some very classical stuff. Says Craven, “It is difficult to pinpoint the origins of the theory of fusion systems: it could be argued that they stretch back to Burnside and Frobenius, with arguments about the fusion of p-elements of finite groups.” It is generally agreed however that the scholars who gave this subject its kick-start include Lluís Puig and the aforementioned Bob Oliver, together with Charles Broto and Ran Levi, and the papers in question are all relatively recent, with some of them appearing in the last decade. Once again, here is Craven: “As this is a young subject, still in development, the foundations of the theory have not yet been solidified; indeed, there is some debate as to the correct definition of a fusion system! (It should be noted that the definitions are all equivalent, and so the choice is only apparent).”
This having been said, Craven presents the reader with a well structured introduction to this exciting new material, presented in two parts. The first part of the book, titled “Motivation,” is concerned with examples of methods in finite group theory that relate in some way to fusion systems, viz. fusion in finite groups, fusion in representation theory, and fusion in topology, and there is a good deal of Brauer theory present in this discussion. The book’s second part, “The Theory,” is explicitly concerned with fusion systems as such, with the two closing chapters being, respectively, “local theory of fusion systems” and “exotic fusion systems.”
The book evinces fine scholarship as well as a serious concern with the audience’s genuine participation in this pedagogical project: Craven presents a lot of mostly very young mathematics quite coherently and very carefully, and he includes a great number of exercises. It strikes me that thorough coverage of The Theory of Fusion Systems will soon be a non-negotiable requisite for any neophyte wishing to enter into this subject which appears to be a very exciting new chapter in the theory of groups.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.