SUBGROUP FAMILIES CONTROLLING p-LOCAL FINITE GROUPS

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. A p-local finite group consists of a p-group S, together with a pair of categories (F ,L) of which F is modeled on the conjugacy in a Sylow subgroup of a finite group. The second category L is an extension of the first, 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. The main purpose of this paper is to study how the sets of objects for these categories can be restricted or enlarged without changing the homotopy type of the p-completed nerve of L. The tools introduced simplify the construction and manipulation of p-local finite groups in many cases. We first recall the fusion and linking categories associated to a finite group. Fix a prime p, a finite group G, and a Sylow p-subgroup S of G. A p-fusion category for G is a category F = FH S (G), whose object set is a set H of subgroups of S, and whose morphisms are the homomorphisms between subgroups in H induced by conjugation in G. The associated linking category L = LS (G) has the same objects, and morphisms from P to Q are given by the formula MorL(P,Q) = {x ∈ G | xPx−1 ≤ Q}/O(CG(P )). Here, Op(−) is the subgroup generated by elements of order prime to p. There is a canonical quotient functor LS (G) −−−→ FH S (G) which sends the class of x to conjugation by x. It was shown in [BLO1] that the homotopy theory of the nerve |LS (G)| (for the right choice of H) is closely related to the p-local homotopy theory of BG. Fusion and linking categories were designed to a large extent to capture the “p-local structure” of finite groups, blocks, and p-completed classifying spaces in a way which 1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D20.