but also the Sylow p-subgroup of G together with all fusion among its subgroups, are determined up to isomorphism by the homotopy type of BG ∧ p . Our goal here is to give a direct link between p-local structures and homotopy types which arise from them. This theory tries to make explicit the essence of what it means to be the p-completed classifying space of a finite group, and at the same time yields new spaces which are not of this type, but which still enjoy most of the properties a space of the form BG ∧ p would have. We hope that the ideas presented here will have further applications and generalizations in algebraic topology. But this theory also fits well with certain aspects of modular representation theory. In particular, it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p.A saturated fusion system F over a p-group S consists of a set Hom F (P, Q) of monomorphisms, for each pair of subgroups P, Q ≤ S, which form a category under composition, include all monomorphisms induced by conjugation in S, and satisfy certain other axioms formulated by Puig (Definitions 1.1 and 1.2 below). In particular, these axioms are satisfied by the conjugacy homomorphisms in a finite group. We refer to [Pu] and [Pu2] for more details of Puig's work on saturated

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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]).

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