Extensions of Linking Systems and Fusion Systems

We correct two errors in the statement and proof of a theorem in an earlier paper (2007), and at the same time extend that result to a more general theorem about extensions of p-local finite groups. 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]. When G is a finite group and S ∈ Sylp(G), the fusion category of G is the category FS(G) whose objects consist of all subgroups of S, and where MorFS(G)(P,Q) = HomG(P,Q) def = {cg ∈ Hom(P,Q) | g ∈ G, gPg−1 ≤ Q} . This 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. The main theorem in this paper (Theorem 9) describes how to construct certain types of extensions of abstract fusion and linking systems. The special case shown in [5, Theorem 4.6] shows how to extend a linking system by a p-group of outer automorphisms. 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 L0 via isotypical automorphisms (Definition 5). Without this assumption, it need not induce an action on the fusion system F0. The error in the proof of the theorem occurs in Step 4. In that step, a certain property of Received by the editors November 21, 2008 and, in revised form, February 13, 2009. 2000 Mathematics Subject Classification. Primary 55R35; Secondary 20D20, 20E22.