Extensions of linking systems and fusion systems

Abstract

Abstract. 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 ∈ 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. 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 L 0 via isotypical automorphisms (Definition 5). Without this assumption, it need not induce an action on the fusion system F 0 . The error in the proof of the theorem occurs in Step 4. In that step, a certain property of

Reduced, tame and exotic fusion systems

“…If H is a set of subgroups of S, then T H (G) ⊆ T S (G) denotes the full subcategory with object set H. Definition 1.9 [17,Definition 3]. Let F be a fusion system over a finite p-group S. A linking system associated to F is a finite category L, together with a pair of functors…”

“…Proof. Most of this is contained in [17,Proposition 4]. Point (a ) follows from (a), which implies that if ψ ∈ Mor L (P, P ) and π P (ψ) = Id P , then ψ is an automorphism.…”

“…In practice, it seems that any reduced indecomposable fusion system which is not simple (which has no proper normal fusion subsystems in the sense of Definition 1.18 or of Aschbacher [1, § 6]) has to be over a p-group of very large order. The smallest example of this type we know of is the fusion system of A 6 A 5 , over a group of order 2 17 .…”

Fusion systems

“…In fact, the above statement is a shortened version of the main theorem in [O3], where conditions are also given to be able to extend L 0 to a linking system L associated to F and containing L 0 as a normal linking subsystem in the sense of [O3,Definition 8] or [AOV1, Definition 1.27]. More generally, the proof of the theorem as stated above involves constructing (in all cases) a transporter system T associated to F such that L 0 T ; this is stated and proven explicitly in [BLO5,Theorem 5.4].…”

The rank of fusion systems

“…Here, we introduce the definition of "weakly normal" by Linckelmann [8] and Oliver [10] since "normal" refers to the definition of normality given by Aschbacher in [1].…”