Extensions of $p$-local finite groups

Abstract

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]).

Extensions of linking systems and fusion systems

“…Other special cases of this theorem have already been shown in two later papers, so we feel it will be useful to have this more general result in the literature. This paper has two purposes: to correct some errors in the statement and proof of a theorem in the earlier paper [5], and also to prove a more general version of this theorem, describing (very roughly) how to construct extensions of fusion and linking systems by groups of outer automorphisms. Special cases of this construction have been used in at least two papers written since [5].…”

“…This paper has two purposes: to correct some errors in the statement and proof of a theorem in the earlier paper [5], and also to prove a more general version of this theorem, describing (very roughly) how to construct extensions of fusion and linking systems by groups of outer automorphisms. Special cases of this construction have been used in at least two papers written since [5].When G is a finite group and S ∈ Syl p (G), the fusion category of G is the category F S (G) whose objects consist of all subgroups of S, and whereThis gives a means of encoding the p-local structure of G: the conjugacy relations among the p-subgroups of G. The centric linking category of G is a closely related category which (among other things) provides a link between the fusion in G and the homotopy type of its p-completed classifying space. These categories motivated the definition by Puig [10] of abstract fusion systems, and by Broto, Levi, and Oliver [3] of abstract linking systems.…”

“…Other special cases were used by Castellana and Libman [6] to construct wreath products of linking systems, and by Andersen, Oliver, and Ventura [1] to construct exotic fusion and linking systems under certain hypotheses. Since all three of these constructions have very similar proofs, it should be useful to have one reference which covers all of these cases, and hopefully any others which might be needed in the future.There was an omission in the statement of [5, Theorem 4.6], in that the group S must be assumed to act on L 0 via isotypical automorphisms (Definition 5). Without this assumption, it need not induce an action on the fusion system F 0 .…”

A cohomological characterization of nilpotent fusion systems

“…For the reverse implication, set π = S/hyp(F ) and consider the universal covering space [5,Theorem 4.4],…”

Fusion systems and amalgams

“…By [5a2,Theorems 4.3 & 5.4], each saturated fusion system F over a finite p-group S contains a unique minimal saturated fusion subsystem O p (F) of p-power index (over hyp(F)), and a unique minimal saturated fusion subsystem O p (F) of index prime to p (over S). Furthermore: Proposition 1.8.…”