Subgroup families controlling p-local finite 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 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.

Extensions of linking systems and fusion systems

“…The existence of restriction morphisms in L 0 (Proposition 4(b)) carries over easily to the existence of restriction morphisms in L 1 , and they are unique by (4).…”

“…The main differences between this definition and those in [3] and [4] are that it is more flexible on the set of objects in L, and that we define here δ as a functor on the transporter category of S. That δ can be defined on T Ob(L) (S) follows as a consequence of the earlier definitions (see [3,Proposition 1.11] and [4,Lemma 3.7]), and including it in the definition allows us to drop axiom (D) q in [4,Definition 3.3]. We will see shortly (in Proposition 4(g)) that all objects in a linking system L must be quasicentric.…”

“…By Step 4, F is H-saturated; i.e., it satisfies the saturation axioms for subgroups in H. It is also H-generated by definition: each morphism in F is a composite of restrictions of morphisms between subgroups in H. So by [4,Theorem A], to prove F is saturated, it suffices to show the following holds for all P ≤ S:…”

“…The reason for assuming L contains all subgroups which are F-centric and Fradical originates with [4,Theorem 3.5], which says that if L ⊆ L are two linking systems associated to the same fusion system, such that Ob(L ) contains all Fcentric F-radical subgroups and Ob(L) is contained in the set of all F-quasicentric subgroups, then the geometric realizations of these two categories are homotopy equivalent. In other words, this seems to be the minimal set of objects needed to get the information which one needs from a linking system.…”

Unstable Adams operations acting onp–local compact groups and fixed points

“…The notation comes from Broto et al [3], where the authors explore certain subsets of objects of a saturated fusion system which control saturation. The first property (generation) means that morphisms in F i are compositions of restrictions of morphisms among the objects in Ob.L i /, and the second property (saturation) means that the objects in Ob.L i / satisfy the saturation axioms.…”