A p-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a one-to-one correspondence between connected p-compact groups and finite reflection groups over the p-adic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as p-compact groups by their Weyl groups seen as finite reflection groups over the p-adic integers. Our approach in fact gives a largely self-contained proof of the entire classification theorem for p odd.

A p-local finite group consists of a finite p-group S, together with a pair of categories which encode "conjugacy" relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system F over a finite p-group S is saturated can be determined by just looking at smaller classes of subgroups of S. We also prove that the homotopy type of the classifying space of a given p-local finite group is independent of the family of subgroups used to define it, in the sense that it remains unchanged when that family ranges from the set of F -centric F -radical subgroups (at a minimum) to the set of F -quasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to p-constrained finite groups, and prove that they in fact all arise from groups.

Abstract. A p-local finite group consists of a finite p-group S, together with a pair of categories which encode "conjugacy" relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we study and classify extensions of p-local finite groups, and also compute the fundamental group of the classifying space of a p-local finite group.A p-local finite group consists of a finite p-group S, together with a pair of categories (F, L), of which F is modeled on the conjugacy (or fusion) in a Sylow subgroup of a finite group. The category L is essentially an extension of F and contains just enough extra information so that its p-completed nerve has many of the same properties as p-completed classifying spaces of finite groups. We recall the precise definitions of these objects in Section 1, and refer to [BLO2] and [5A1] for motivation for their study.In this paper, we study extensions of saturated fusion systems and of p-local finite groups. This is in continuation of our more general program of trying to understand to what extent properties of finite groups can be extended to properties of p-local finite groups, and to shed light on the question of how many (exotic) p-local finite groups there are. While we do not get a completely general theory of extensions of one p-local finite group by another, we do show how certain types of extensions can be described in a manner very similar to the situation for finite groups.From the point of view of group theory, developing an extension theory for plocal finite groups is related to the question of to what extent the extension problem for groups is a local problem, i.e., a problem purely described in terms of a Sylow p-subgroup and conjugacy relations inside it. In complete generality this is not the case. For example, strongly closed subgroups of a Sylow p-subgroup S of G need not correspond to normal subgroups of G. However, special cases where this does happen include the case of existence of p-group quotients (the focal subgroup theorems; see [Go,) and central subgroups (described via the Z * -theorem of Glauberman [Gl]).

We prove that any connected 2 2 –compact group is classified by its 2 2 –adic root datum, and in particular the exotic 2 2 –compact group DI ⁡ ( 4 ) \operatorname {DI}(4) , constructed by Dwyer–Wilkerson, is the only simple 2 2 –compact group not arising as the 2 2 –completion of a compact connected Lie group. Combined with our earlier work with Møller and Viruel for p p odd, this establishes the full classification of p p –compact groups, stating that, up to isomorphism, there is a one-to-one correspondence between connected p p –compact groups and root data over the p p –adic integers. As a consequence we prove the maximal torus conjecture, giving a one-to-one correspondence between compact Lie groups and finite loop spaces admitting a maximal torus. Our proof is a general induction on the dimension of the group, which works for all primes. It refines the Andersen–Grodal–Møller–Viruel methods by incorporating the theory of root data over the p p –adic integers, as developed by Dwyer–Wilkerson and the authors. Furthermore we devise a different way of dealing with the rigidification problem by utilizing obstruction groups calculated by Jackowski–McClure–Oliver in the early 1990s.

A collection C of subgroups of a finite group G can give rise to three different standard formulas for the cohomology of G in terms of either: the subgroups in C; or their centralizers; or their normalizers. We give a short but systematic study of the relationship among such formulas for nine standard collections C of p-subgroups, obtaining some new formulas in the process. To do this, we exhibit some sufficient conditions on the poset C which imply comparison results.Comment: 12 pages. Extra counterexamples added. To appear in Advances in Mat