EXTENSIONS OF p-LOCAL FINITE GROUPS

C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, AND B. OLIVER

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 manner very similar to the situation for finite groups.

From the point of view of group theory, developing an extension theory for p-local 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 psubgroup 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, §§7.3–7.4]) and central subgroups (described via the Z^{∗}-theorem of Glauberman [Gl]).

From the point of view of homotopy theory, one of the problems which comes up when looking for a general theory of extensions of p-local finite groups is that while an extension of groups 1 → K → Γ → G → 1 always induces a (homotopy) fibration sequence of classifying spaces, it does not in general induce a fibration sequence of p-completed classifying spaces. Two cases where this does happen are those where G is a p-group, and where the extension is central. In both of these cases, BH_{p}^{∧} is the homotopy fiber of the map BΓ_{p}^{∧} ⟶ BG_{p}^{∧}. Thus, also from the point of view of homotopy theory, it is natural to study extensions of p-local finite groups with pgroup quotient, and to study central extensions of p-local finite groups. The third case we study is that of extensions with quotient of order prime to p; and the case of p and p′-group quotients to some extent unify to give a theory of extensions with p-solvable quotient. Recall in this connection that by a previous result of ours [5A1, Proposition C], solvable p-local finite groups all come from p-solvable groups. In all three of these situations, we develop a theory of extensions which parallels the situation for finite groups.

1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D20.

Key words and phrases. Classifying space, p-completion, finite groups, fusion.

C. Broto is partially supported by MCYT grant BFM2001–2035.

N. Castellana is partially supported by MCYT grant BFM2001–2035.

J. Grodal is partially supported by NSF grants DMS-0104318 and DMS-0354633.

R. Levi is partially supported by EPSRC grant GR/M7831.

B. Oliver is partially supported by UMR 7539 of the CNRS.

We now describe the contents of this paper in more detail, stating simplified versions of our main results on extensions. Stronger and more precise versions of some of these theorems will be stated and proven later.

In Section 3, we construct a very general theory of fusion subsystems (Proposition 3.8) and linking subsystems (Theorem 3.9) with quotient a p-group or a group of order prime to p. As a result we get the following theorem (Corollary 3.10), which for a p-local finite group (S, F, L), describes a correspondence between covering spaces of the geometric realization |L| and certain p-local finite subgroups of (S, F, L).

Theorem A. Suppose that (S, F, L) is a p-local finite group. Then there is a normal subgroup H ◁ π_{1}(|L|) which is minimal among all those whose quotient is finite and psolvable. Any covering space of the geometric realization |L| whose fundamental group contains H is homotopy equivalent to |L′| for some p-local finite group (S′, F′, L′), where S′ ≤ S and F′ ⊆ F.

Moreover, the p-local finite group (S′, F′, L′) of Theorem A can be explicitly described in terms of L, as we will explain in Section 3.

In order to use this theorem, it is useful to have ways of finding the finite p-solvable quotients of π_{1}(|L|). This can be done by iteration, using the next two theorems. In them, the maximal p-group quotient of π_{1}(|L|), and the maximal quotient of order prime to p, are described solely in terms of the fusion system F.

When G is an infinite group, we define O^{p}(G) and O^{p′}(G) to be the intersection of all normal subgroups of G of p-power index, or index prime to p, respectively. These clearly generalize the usual definitions for finite G (but are not the only possible generalizations).

Theorem B (Hyperfocal subgroup theorem for p-local finite groups). For a p-local finite group (S, F, L), the natural homomorphism

S ⟶ π_{1}(|L|)/O^{p}(π_{1}(|L|)) ≅ π_{1}(|L|_{p}^{∧})

is surjective, with kernel equal to

O_{F}^{p}(S) =^{def} ⟨g^{−1}α(g) ∈ S | g ∈ P ≤ S, α ∈ O^{p}(Aut_{F}(P))⟩.

For any saturated fusion system F over a p-group S, we let O_{∗}^{p′}(F) ⊆ F be the smallest fusion subsystem of F (in the sense of Definition 1.1) which contains all automorphism groups O^{p′}(Aut_{F}(P)) for P ≤ S. Equivalently, O_{∗}^{p′}(F) is the smallest subcategory of F with the same objects, and which contains all restrictions of all automorphisms in F of p-power order. This subcategory is needed in the statement of the next theorem.

EXTENSIONS OF P -LOCAL FINITE GROUPS 3

Theorem C. For a p-local finite group (S, F, L), the natural map

Out_{F}(S) ⟶ π_{1}(|L|)/O^{p′}(π_{1}(|L|))

is surjective, with kernel equal to

Out_{F}^{0}(S) =^{def} ⟨α ∈ Out_{F}(S) | α|_{P} ∈ Mor_{O_{∗}^{p′}(F)}(P, S), some F-centric P ≤ S⟩.

Theorem B is proved as Theorem 2.5, and Theorem C is proved as Theorem 5.5. In fact, we give a purely algebraic description of these subsystems of “p-power index”

or of “index prime to p” (Definition 3.1), and then show in Sections 4.1 and 5.1 that they in fact all arise as finite covering spaces of |L| (see Theorems 4.4 and 5.5). Similar results were also discovered independently by Puig [Pu4]. Then, in Sections 4.2 and 5.2, we establish converses to these concerning the extensions of a p-local finite group, which include the following theorem.

Theorem D. Let (S, F, L) be a p-local finite group. Suppose we are given a fibration sequence |L|_{p}^{∧} → E → BG, where G is a finite p-group or has order prime to p. Then there exists a p-local finite group (S′, F′, L′) such that |L′|_{p}^{∧} ≃ E_{p}^{∧}.

This is shown as Theorems 4.7 and 5.8. Moreover, when G is a p-group, we give in Theorem 4.7 an explicit algebraic construction of the p-local finite group (S′, F′, L′).

Finally, in Section 6, we develop the theory of central extensions of p-local finite groups. Our main results there (Theorems 6.8 and 6.13) give a more elaborate version of the following theorem. Here, the center of a p-local finite group (S, F, L) is defined to be the subgroup of elements x ∈ Z(S) such that α(x) = x for all α ∈ Mor(F^{c}).

Theorem E. Suppose that A is a central subgroup of a p-local finite group (S, F, L). Then there exists a canonical quotient p-local finite group (S/A, F/A, L/A), and the canonical projection of |L| onto |L/A| is a principal fibration with fiber BA.

Conversely, for any principal fibration E → |L| with fiber BA, where A is a finite abelian p-group, there exists a p-local finite group (S̃, F̃, L̃) such that |L̃| ≃ E. Furthermore, this correspondence sets up a 1 − 1 correspondence between central extensions of L by A and elements in H^{2}(|L|; A).

One motivation for this study was the question of whether extensions of p-local finite groups coming from finite groups can produce exotic p-local finite groups. In the case of central extensions, we are able to show that (S, F, L) comes from a finite group if and only if (S/A, F/A, L/A) comes from a group (Corollary 6.14). For the other types of extensions studied in this paper, this is still an open question. We have so far failed to produce exotic examples in this way, and yet we have also been unable to show that exotic examples cannot occur. This question seems to be related to some rather subtle and interesting group theoretic issues relating local to global structure; see Corollary 4.8 for one partial result in this direction.

This paper builds on the earlier paper [5A1] by the same authors, and many of the results in that paper were originally motivated by this work on extensions.

The authors would like to thank the University of Aberdeen, Universitat Autònoma de Barcelona, Université Paris 13, and Aarhus Universitet for their hospitality. In particular, the origin of this project goes back to a three week period in the spring of 2001, when four of the authors met in Aberdeen.

1. A quick review of p-local finite groups

We first recall the definitions of a fusion system, and a saturated fusion system, in the form given in [BLO2]. For any group G, and any pair of subgroups H, K ≤ G, we set

N_{G}(H, K) = {x ∈ G | xHx^{−1} ≤ K},

let c_{x} denote conjugation by x (c_{x}(g) = xgx^{−1}), and set

Hom_{G}(H, K) = {c_{x} ∈ Hom(H, K) | x ∈ N_{G}(H, K)} ≅ N_{G}(H, K)/C_{G}(H).

By analogy, we also write

Aut_{G}(H) = Hom_{G}(H, H) = {c_{x} ∈ Aut(H) | x ∈ N_{G}(H)} ≅ N_{G}(H)/C_{G}(H).

Definition 1.1 ([Pu2] and [BLO2, Definition 1.1]). A fusion system over a finite pgroup S is a category F, where Ob(F) is the set of all subgroups of S, and which satisfies the following two properties for all P, Q ≤ S:

• Hom_{S}(P, Q) ⊆ Hom_{F}(P, Q) ⊆ Inj(P, Q); and

• each φ ∈ Hom_{F}(P, Q) is the composite of an isomorphism in F followed by an inclusion.

The following additional definitions and conditions are needed in order for these systems to be very useful. If F is a fusion system over a finite p-subgroup S, then two subgroups P, Q ≤ S are said to be F-conjugate if they are isomorphic as objects of the category F.

Definition 1.2 ([Pu2], see [BLO2, Def. 1.2]). Let F be a fusion system over a p-group S.

• A subgroup P ≤ S is fully centralized in F if |C_{S}(P)| ≥ |C_{S}(P′)| for all P′ ≤ S which is F-conjugate to P.

• A subgroup P ≤ S is fully normalized in F if |N_{S}(P)| ≥ |N_{S}(P′)| for all P′ ≤ S which is F-conjugate to P.

• F is a saturated fusion system if the following two conditions hold:

(I) For all P ≤ S which is fully normalized in F, P is fully centralized in F and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

(II) If P ≤ S and φ ∈ Hom_{F}(P, S) are such that φP is fully centralized, and if we

set

N_{φ} = {g ∈ N_{S}(P) | φc_{g}φ^{−1} ∈ Aut_{S}(φP)},

then there is φ ∈ Hom_{F}(N_{φ}, S) such that φ|_{P} = φ.

If G is a finite group and S ∈ Syl_{p}(G), then by [BLO2, Proposition 1.3], the category F_{S}(G) defined by letting Ob(F_{S}(G)) be the set of all subgroups of S and setting Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q) is a saturated fusion system.

An alternative pair of axioms for a fusion system being saturated have been given by Radu Stancu [St]. He showed that axioms (I) and (II) above are equivalent to the two axioms:

(I′) Inn(S) ∈ Syl_{p}(Aut_{F}(S)).

(II′) If P ≤ S and φ ∈ Hom_{F}(P, S) are such that φP is fully normalized, and if we set

N_{φ} = {g ∈ N_{S}(P) | φc_{g}φ^{−1} ∈ Aut_{S}(φP)},

then there is φ ∈ Hom_{F}(N_{φ}, S) such that φ|_{P} = φ.

The following consequence of conditions (I) and (II) above will be needed several times throughout the paper.

Lemma 1.3. Let F be a saturated fusion system over a p-group S. Let P, P′ ≤ S be a pair of F-conjugate subgroups such that P′ is fully normalized in F. Then there is a homomorphism α ∈ Hom_{F}(N_{S}(P), N_{S}(P′)) such that α(P) = P′.

Proof. This is shown in [BLO2, Proposition A.2(b)]. □

In this paper, it will sometimes be necessary to work with fusion systems which are not saturated. This is why we have emphasized the difference between fusion systems, and saturated fusion systems, in the above definitions.

We next specify certain collections of subgroups relative to a given fusion system.

Definition 1.4. Let F be a fusion system over a finite p-subgroup S.

• A subgroup P ≤ S is F-centric if C_{S}(P′) = Z(P′) for all P′ ≤ S which is Fconjugate to P.

• A subgroup P ≤ S is F-radical if Out_{F}(P) is p-reduced; i.e., if O_{p}(Out_{F}(P)) = 1.

• For any P ≤ S which is fully centralized in F, the centralizer fusion system C_{F}(P) is the fusion system over C_{S}(P) defined by setting

Hom_{C_{F} (P)}(Q, Q′) = {α|Q | α ∈ Hom_{F}(QP, Q′P), α|_{P} = Id_{P} , α(Q) ≤ Q′}.

A subgroup P ≤ S is F-quasicentric if for all P′ ≤ S which is F-conjugate to P and fully centralized in F, C_{F}(P′) is the fusion system of the p-group C_{S}(P′).

• F^{c} ⊆ F^{q} ⊆ F denote the full subcategories of F whose objects are the F-centric subgroups, and F-quasicentric subgroups, respectively, of S.

If F = F_{S}(G) for some finite group G, then P ≤ S is F-centric if and only if P is p-centric in G (i.e., Z(P) ∈ Syl_{p}(C_{G}(P))), and P is F-radical if and only if N_{G}(P)/(P ·C_{G}(P)) is p-reduced. Also, P is F-quasicentric if and only if C_{G}(P) contains a normal subgroup of order prime to p and of p-power index.

In fact, when working with p-local finite groups, it suffices to have a fusion system F^{c} defined on the centric subgroups of S, and which satisfies axioms (I) and (II) above for those centric subgroups. In other words, fusion systems defined only on the centric subgroups are equivalent to fusion systems defined on all subgroups, as described in the following theorem.

Theorem 1.5. Fix a p-group S and a fusion system F over S.

(a) Assume F is saturated. Then each morphism in F is a composite of restrictions of morphisms between subgroups of S which are F-centric, F-radical, and fully normalized in F. More precisely, for each P, P′ ≤ S and each φ ∈ Iso_{F}(P, P′), there are subgroups P = P_{0}, P_{1}, …, P_{k} = P′, subgroups Q_{i} ≥ ⟨P_{i−1}, P_{i}⟩ (i = 1, …, k) which are F-centric, F-radical, and fully normalized in F, and automorphisms φ_{i} ∈ Aut_{F}(Q_{i}), such that φ_{i}(P_{i−1}) = P_{i} for all i and φ = φ_{k} ∘ ⋯ ∘ φ_{1}|_{P}.

(b) Assume conditions (I) and (II) in Definition 1.2 are satisfied for all F-centric subgroups P ≤ S. Assume also that each morphism in F is a composite of restrictions of morphisms between F-centric subgroups of S. Then F is saturated.

Proof. Part (a) is Alperin’s fusion theorem for saturated fusion systems, in the form shown in [BLO2, Theorem A.10]. Part (b) is a special case of [5A1, Theorem 2.2]: the case where H is the set of all F-centric subgroups of S. □

Theorem 1.5(a) will be used repeatedly throughout this paper. The following lemma is a first easy application of the theorem, and provides a very useful criterion for a subgroup to be quasicentric or not.

Lemma 1.6. Let F be a saturated fusion system over a p-group S. Then the following hold for any P ≤ S.

(a) Assume that P ≤ Q ≤ P ·C_{S}(P) and Id ≠ α ∈ Aut_{F}(Q) are such that α|_{P} = Id_{P} and α has order prime to p. Then P is not F-quasicentric.

(b) Assume that P is fully centralized in F, and is not F-quasicentric. Then there are P ≤ Q ≤ P ·C_{S}(P) and Id ≠ α ∈ Aut_{F}(Q) such that Q is F-centric, α|_{P} = Id_{P}, and α has order prime to p.

Proof. (a) Fix any P′ which is F-conjugate to P and fully centralized in F. By axiom (II), there is φ ∈ Hom_{F}(Q, S) such that φ(P) = P′; set Q′ = φ(Q). Thus φαφ^{−1}|_{C_{Q′}(P′)} is an automorphism in C_{F}(P′) whose order is not a power of p, so C_{F}(P′) is not the fusion system of C_{S}(P′), and P is not F-quasicentric.

(b) Assume that P is fully centralized in F and not F-quasicentric. Then C_{F}(P) strictly contains the fusion system of C_{S}(P) (since C_{F}(P′) is isomorphic as a category to C_{F}(P) for all P′ which is F-conjugate to P and fully centralized in F). Since C_{F}(P) is saturated by [BLO2, Proposition A.6], Theorem 1.5(a) implies there is a subgroup Q ≤ C_{S}(P) which is C_{F}(P)-centric and fully normalized in C_{F}(P), and such that Aut_{C_{F} (P)}(Q) ≩ Aut_{C_{S}(P)}(Q). Since Q is fully normalized, Aut_{C_{S}(P)}(Q) is a Sylow psubgroup of Aut_{C_{F} (P)}(Q), and hence this last group is not a p-group. Also, by [BLO2, Proposition 2.5(a)], P Q is F-centric since Q is C_{F}(P)-centric. □

Since orbit categories — both of fusion systems and of groups — will play a role in certain proofs in the last three sections, we define them here.

Definition 1.7. (a) If F is a fusion system over a p-group S, then O^{c}(F) (the centric orbit category of F) is the category whose objects are the F-centric subgroups of S, and whose morphism sets are given by

Mor_{O^{c}(F)}(P, Q) = Rep_{F}(P, Q) =^{def} Q\ Hom_{F}(P, Q).

Let Z_{F}: O^{c}(F) → ℤ_{(p)}-mod be the functor which sends P to Z(P) and [φ] (the

class of φ ∈ Hom_{F}(P, Q)) to Z(Q) ⟶^{φ^{−1}} Z(P).

(b) If G is a finite group and S ∈ Syl_{p}(G), then O_{S}^{c}(G) (the centric orbit category of G) is the category whose objects are the subgroups of S which are p-centric in G, and where

Mor_{O_{S}^{c}(G)}(P, Q) = Q\N_{G}(P, Q) ≅ Map_{G}(G/P, G/Q).

Let Z_{G}: O_{S}^{c}(G) → ℤ_{(p)}-mod be the functor which sends P to Z(P) and [g] (the class of g ∈ N_{G}(P, Q)) to conjugation by g^{−1}.

We now turn to linking systems associated to abstract fusion systems.

Definition 1.8 ([BLO2, Def. 1.7]). Let F be a fusion system over the p-group S. A centric linking system associated to F is a category L whose objects are the Fcentric subgroups of S, together with a functor π: L ⟶ F^{c}, and “distinguished” monomorphisms P ⟶^{δ_{P}} Aut_{L}(P) for each F-centric subgroup P ≤ S, which satisfy the following conditions.

(A) π is the identity on objects. For each pair of objects P, Q ∈ L, Z(P) acts freely on Mor_{L}(P, Q) by composition (upon identifying Z(P) with δ_{P} (Z(P)) ≤ Aut_{L}(P)), and π induces a bijection

Mor_{L}(P, Q)/Z(P) ⟶^{≅} Hom_{F}(P, Q).

(B) For each F-centric subgroup P ≤ S and each x ∈ P, π(δ_{P} (x)) = c_{x} ∈ Aut_{F}(P).

(C) For each f ∈ Mor_{L}(P, Q) and each x ∈ P, the following square commutes in L:

\begin{matrix}P & \overline{^{f} →} & Q \\ P & ↓^{δ_{P} (x)} & \overline{^{f} →} & Q. & ↓^{δ_{Q}(π(f)(x))} \end{matrix}

A p-local finite group is defined to be a triple (S, F, L), where S is a finite p-group, F is a saturated fusion system over S, and L is a centric linking system associated to F. The classifying space of the triple (S, F, L) is the p-completed nerve |L|_{p}^{∧}.

For any finite group G with Sylow p-subgroup S, a category L_{S}^{c}(G) was defined in [BLO1], whose objects are the p-centric subgroups of G, and whose morphism sets are defined by

Mor_{L_{S}^{c}(G)}(P, Q) = N_{G}(P, Q)/O^{p}(C_{G}(P)).

Since C_{G}(P) = Z(P) × O^{p}(C_{G}(P)) when P is p-centric in G, L_{S}^{c}(G) is easily seen to satisfy conditions (A), (B), and (C) above, and hence is a centric linking system associated to F_{S}(G). Thus (S, F_{S}(G), L_{S}^{c}(G)) is a p-local finite group, with classifying space |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧} (see [BLO1, Proposition 1.1]).

It will be of crucial importance in this paper that any centric linking system associated to F can be extended to a quasicentric linking system; a linking system with similar properties, whose objects are the F-quasicentric subgroups of S. We first make more precise what this means.

Definition 1.9. Let F be any saturated fusion system over a p-group S. A quasicentric linking system associated to F consists of a category L^{q} whose objects are the Fquasicentric subgroups of S, together with a functor π: L^{q} ⟶ F^{q}, and distinguished monomorphisms

P ·C_{S}(P) ⟶^{δ_{P}} Aut_{L^{q}}(P),

which satisfy the following conditions.

(A)_{q} π is the identity on objects and surjective on morphisms. For each pair of objects P, Q ∈ L^{q} such that P is fully centralized, C_{S}(P) acts freely on Mor_{L^{q}}(P, Q) by composition (upon identifying C_{S}(P) with δ_{P} (C_{S}(P)) ≤ Aut_{L^{q}}(P)), and π induces a bijection

Mor_{L^{q}}(P, Q)/C_{S}(P) ⟶^{≅} Hom_{F}(P, Q).

(B)_{q} For each F-quasicentric subgroup P ≤ S and each g ∈ P, π sends δ_{P} (g) ∈ Aut_{L^{q}}(P) to c_{g} ∈ Aut_{F}(P).

(C)_{q} For each f ∈ Mor_{L^{q}}(P, Q) and each x ∈ P, f ∘ δ_{P} (x) = δ_{Q}(π(f)(x)) ∘ f in Mor_{L^{q}}(P, Q).

(D)_{q} For each F-quasicentric subgroup P ≤ S, there is a morphism ι_{P} ∈ Mor_{L^{q}}(P, S) such that π(ι_{P}) = incl_{P}^{S} ∈ Hom(P, S), and such that for each g ∈ P ·C_{S}(P), δ_{S}(g) ∘ ι_{P} = ι_{P} ∘ δ_{P} (g) in Mor_{L^{q}}(P, S).

If P and P′ are F-conjugate and F-quasicentric, then for any Q ≤ S, Mor_{L^{q}}(P, Q) ≅ Mor_{L^{q}}(P′, Q) and Hom_{F}(P, Q) ≅ Hom_{F}(P′, Q), while the centralizers C_{S}(P) and C_{S}(P′) need not have the same order. This is why condition (A)_{q} makes sense only if we assume that P is fully centralized; i.e., that C_{S}(P) is as large as possible. When P is F-centric, then this condition is irrelevant, since every subgroup P′ which is F-conjugate to P is fully centralized (C_{S}(P′) = Z(P′) ≅ Z(P)).

Note that (D)_{q} is a special case of (C)_{q} when P is F-centric; this is why the axiom is not needed for centric linking systems. We also note the following relation between these axioms:

Lemma 1.10. In the situation of Definition 1.9, axiom (C)_{q} implies axiom (B)_{q}.

Proof. Fix an F-quasicentric subgroup P ≤ S, and an element g ∈ P. We apply (C)_{q} with f = δ_{P} (g). For each x ∈ P, if we set y = π(δ_{P} (g))(x), then δ_{P} (g) ∘ δ_{P} (x) = δ_{P} (y) ∘ δ_{P} (g). Since δ_{P} is an injective homomorphism, this implies that gx = yg, and thus that y = c_{g}(x). So π(δ_{P} (g)) = c_{g}. □

When (S, F, L) is a p-local finite group, and L^{q} is a quasicentric linking system associated to F, then we say that L^{q} extends L if the full subcategory of L^{q} with objects the F-centric subgroups of S is isomorphic to L via a functor which commutes with the projection functors to F and with the distinguished monomorphisms. In [5A1, Propositions 3.4 & 3.12], we constructed an explicit quasicentric linking system L^{q} associated to F and extending L, and showed that it is unique up to an isomorphism of categories which preserves all of these structures. So from now on, we will simply refer to L^{q} as the quasicentric linking system associated to (S, F, L).

Condition (D)_{q} above helps to motivate the following definition of inclusion morphisms in a quasicentric linking system.

Definition 1.11. Fix a p-local finite group (S, F, L), with associated quasicentric linking system L^{q}.

(a) A morphism ι_{P} ∈ Mor_{L^{q}}(P, S) is an inclusion morphim if it satisfies the hypotheses of axiom (D)_{q}: if π(ι_{P}) = incl_{P}^{S}, and if δ_{S}(g) ∘ ι_{P} = ι_{P} ∘ δ_{P} (g) in Mor_{L^{q}}(P, S) for all g ∈ P ·C_{S}(P). If P = S, then we also require that ι_{S} = Id_{S}.

(b) A compatible set of inclusions for L^{q} is a choice of morphisms {ι_{P}^{Q}} for all pairs of F-quasicentric subgroups P ≤ Q, such that ι_{P}^{Q} ∈ Mor_{L^{q}}(P, Q), such that ι_{P}^{R} = ι_{Q}^{R} ∘ι_{P}^{Q} for all P ≤ Q ≤ R, and such that ι_{P}^{S} is an inclusion morphism for each P.

The following properties of quasicentric linking systems were also proven in [5A1].

Proposition 1.12. The following hold for any p-local finite group (S, F, L), with associated quasicentric linking system L^{q}.

(a) The inclusion L ⊆ L^{q} induces a homotopy equivalence |L| ≃ |L^{q}| between geometric realizations. More generally, for any full subcategory L′ ⊆ L^{q} which contains as objects all subgroups of S which are F-centric and F-radical, the inclusion L′ ⊆ L^{q} induces a homotopy equivalence |L′| ≃ |L^{q}|.

(b) Let φ ∈ Mor_{L^{q}}(P, R) and ψ ∈ Mor_{L^{q}}(Q, R) be any pair of morphisms in L^{q} with the same target group such that Im(π(φ)) ≤ Im(π(ψ)). Then there is a unique morphism χ ∈ Mor_{L^{q}}(P, Q) such that φ = ψ ∘ χ.

Proof. The homotopy equivalences |L^{q}| ≃ |L| ≃ |L′| are shown in [5A1, Theorem 3.5]. Point (b) is shown in [5A1, Lemma 3.6]. □

Point (b) above will be frequently used throughout the paper. In particular, it makes it possible to embed the linking system of S (or an appropriate full subcategory) in L^{q}, depending on the choice of an inclusion morphisms ι_{P} as defined above, for each object P. Such inclusion morphisms always exist by axiom (D)_{q}. In the following proposition, L_{S}(S)|_{Ob(L^{q})} denotes the full subcategory of L_{S}(S) whose objects are the F-quasicentric subgroups of S.

In general, for a functor F: C → C′, and objects c, d ∈ Ob(C), we let F_{c,d} denote the map from Mor_{C}(c, d) to Mor_{C′}(F (c), F (d)) induced by F.

Proposition 1.13. Fix a p-local finite group (S, F, L). let L^{q} be its associated quasicentric linking system, and let π: L^{q} ⟶ F^{q} be the projection. Then any choice of inclusion morphisms ι_{P} = ι_{P}^{S} ∈ Mor_{L^{q}}(P, S), for all F-quasicentric subgroups P ≤ S, extends to a unique inclusion of categories

δ: L_{S}(S)|_{Ob(L^{q})} ⟶ L^{q}

such that δ_{P,S}(1) = ι_{P} for all P; and such that

(a) δ_{P,P} (g) = δ_{P} (g) for all g ∈ P · C_{S}(P), and

(b) π(δ_{P,Q}(g)) = c_{g} ∈ Hom(P, Q) for all g ∈ N_{S}(P, Q).

In addition, the following hold.

(c) If we set ι_{P}^{Q} = δ_{P,Q}(1) for all P ≤ Q, then {ι_{P}^{Q}} is a compatible set of inclusions for L^{q}.

(d) For any P ◁ Q ≤ S, where P and Q are both F-quasicentric and P is fully centralized in F, and any morphism ψ ∈ Aut_{L^{q}}(P) which normalizes δ_{P,P} (Q), there is a unique ψ̂ ∈ Aut_{L^{q}}(Q) such that ψ̂ ∘ ι_{P}^{Q} = ι_{P}^{Q} ∘ ψ. Furthermore, for any g ∈ Q, ψδ_{P,P} (g)ψ^{−1} = δ_{P,P} (π(ψ̂)(g)).

(e) Every morphism φ ∈ Mor_{L^{q}}(P, Q) in L^{q} is a composite φ = ι_{P′}^{Q} ∘ φ′ for a unique morphism φ′ ∈ Iso_{L^{q}}(P, P′), where P′ = Im(π(φ)).

Proof. For each P and Q, and each g ∈ N_{S}(P, Q), there is by Proposition 1.12(b) a unique morphism δ_{P,Q}(g) such that

δ_{S}(g) ∘ ι_{P} = ι_{Q} ∘ δ_{P,Q}(g).

This defines δ on morphism sets, and also allows us to define ι_{P}^{Q} = δ_{P,Q}(1). Then by the axioms in Definition 1.9, {ι_{P}^{Q}} is a compatible set of inclusions for L^{q}, and δ is a functor which satisfies (a), (b), and (c). Point (e) is a special case of Proposition 1.12(b) (where P′ = Im(π(φ))).

If δ_{P,Q}(g) = δ_{P,Q}(g′) for g, g′ ∈ N_{G}(P, Q), then δ_{S}(g) ∘ ι_{P} = δ_{S}(g′) ∘ ι_{P}, and hence g = g′ by [5A1, Lemma 3.9]. Thus each δ_{P,Q} is injective.

It remains to prove (d). Set φ = π(ψ) ∈ Aut_{F}(P) for short. Since ψ normalizes δ_{P,P} (Q), for all g ∈ Q there is h ∈ Q such that ψδ_{P,P} (g)ψ^{−1} = δ_{P,P} (h), and this implies the relation φc_{g}φ^{−1} = c_{h} in Aut_{F}(P). Thus Q is contained in N_{φ}. So by axiom (II) (and since P is fully centralized), φ extends to φ ∈ Iso_{F}(Q, Q′) for some P ◁ Q′ ≤ S. Let ψ̂_{0} ∈ Iso_{L^{q}}(Q, Q′) be any lifting of φ to L^{q}.

By axiom (A)_{q} (and since ψ is an isomorphism), there is x ∈ C_{S}(P) such that ι_{P}^{Q′} ∘ δ_{P} (x) ∘ ψ = ψ̂_{0} ∘ ι_{P}^{Q}. By (C)_{q}, ψ̂_{0}δ_{Q}(Q)ψ̂_{0}^{−1} = δ_{Q′}(Q′), and hence after restriction, δ_{P} (x) ∘ ψ conjugates δ_{P,P} (Q) to δ_{P,P} (Q′). Since ψ normalizes δ_{P,P} (Q), this shows that δ_{P} (x) conjugates δ_{P,P} (Q) to δ_{P,P} (Q′), and hence (since δ_{P,P} is injective) that xQx^{−1} = Q′. We thus have the following commutative diagram:

\begin{matrix}P & \overline{^{ψ} →} & P & \overline{^{δ_{P} (x)} →} & P & \overline{^{δ_{P} (x)^{−1}} →} & P \\ Q & ↓^{ι_{P}^{Q}} & \overline{ψ̂_{0} →} & Q′ & ↓^{ι_{P}^{Q′}} & \overline{^{δ_{Q,Q′}(x)^{−1}}→} & Q. & ↓^{ι_{P}^{Q}} \end{matrix}

So if we set ψ̂ = δ_{Q,Q′}(x)^{−1} ∘ ψ̂_{0}, then ψ̂ ∈ Aut_{L^{q}}(Q) and ι_{P}^{Q} ∘ ψ = ψ̂ ∘ ι_{P}^{Q}.

The uniqueness of ψ̂ follows from [5A1, Lemma 3.9]. Finally, for any g ∈ Q, ψ̂δ_{Q}(g)ψ̂^{−1} = δ_{Q}(π(ψ̂)(g)) by (C)_{q}, and hence ψδ_{P,P} (g)ψ^{−1} = δ_{P,P} (π(ψ̂)(g)) since morphisms have unique restrictions (Proposition 1.12(b) again). □

Once we have fixed a compatible set of inclusions {ι_{P}^{Q}} in a linking system L^{q}, then for any φ ∈ Mor_{L^{q}}(P, Q), and any P′ ≤ P and Q′ ≤ Q such that π(φ)(P′) ≤ Q′, there is a unique morphism φ ∈ Mor_{L^{q}}(P′, Q′) such that ι_{Q′}^{Q} ∘ φ = φ ∘ ι_{P′}^{P}. We think of φ as the restriction of φ.

Note, however, that all of this depends on the choice of morphisms ι_{P} ∈ Mor_{L^{q}}(P, S) which satisfy the hypotheses of axiom (D)_{q}, and that not just any lifting of the inclusion incl ∈ Hom_{F}(P, S) can be chosen. To see why, assume for simplicity that P is also fully centralized. From the axioms in Definition 1.9 and Proposition 1.12(b), we see that if ι_{P} , ι′_{P} ∈ Mor_{L^{q}}(P, S) are two liftings of incl ∈ Hom_{F}(P, S), then ι′_{P} = ι_{P} ∘ δ_{P} (g) for some unique g ∈ C_{S}(P). But if ι_{P} satisfies the conditions of (D)_{q}, then ι′_{P} also satisfies those conditions only if g ∈ Z(C_{S}(P)).

One situation where the choice of inclusion morphisms is useful is when describing the fundamental group of |L| or of its p-completion. For any group Γ, we let B(Γ) denote the category with one object, and with morphism monoid the group Γ. Recall that |L| ≃ |L^{q}| (Proposition 1.12(a)), so we can work with either of these categories; we will mostly state the results for |L^{q}|. Let the vertex S be the basepoint of |L^{q}|. For each morphism φ ∈ Mor_{L^{q}}(P, Q), let J(φ) ∈ π_{1}(|L^{q}|) denote the homotopy class of the loop ι_{Q}·φ·ι_{P}^{−1} in |L^{q}| (where paths are composed from right to left). This defines a functor

J: L^{q} ⟶ B(π_{1}(|L^{q}|)),

where all objects are sent to the unique object of B(π_{1}(|L^{q}|)), and where all inclusion morphisms are sent to the identity. Let j: S → π_{1}(|L^{q}|) denote the composite of J with the distinguished monomorphism δ_{S}: S → Aut_{L}(S).

The next proposition describes how J is universal among functors of this type, and also includes some other technical results for later use about the structure of π_{1}(|L|).

Proposition 1.14. Let (S, F, L) be a p-local finite group, and let L^{q} be the associated quasicentric linking system. Assume a compatible set of inclusions {ι_{P}^{Q}} has been chosen for L^{q}. Then the following hold.

(a) For any group Γ, and any functor λ: L^{q} ⟶ B(Γ) which sends inclusions to the identity, there is a unique homomorphism λ̄: π_{1}(|L^{q}|) ⟶ Γ such that λ = B(λ̄) ∘ J.

(b) For g ∈ P ≤ S with P F-quasicentric, J(δ_{P} (g)) = J(δ_{S}(g)). In particular, J(δ_{P} (g)) = 1 in π_{1}(|L^{q}|) if and only if δ_{P} (g) is nulhomotopic as a loop based at the vertex P of |L^{q}|.

(c) If α ∈ Mor_{L^{q}}(P, Q), and π(α)(x) = y, then j(y) = J(α)j(x)J(α)^{−1} in π_{1}(|L^{q}|).

(d) If x and y are F-conjugate elements of S, then j(x) and j(y) are conjugate in π_{1}(|L^{q}|).

Proof. Clearly, any functor λ: L^{q} → B(Γ) induces a homomorphism λ̄ = π_{1}(|λ|) between the fundamental groups of their geometric realizations. If λ sends inclusion morphisms to the identity, then λ = B(λ̄) ∘ J by definition of J.

The other points follow easily, using condition (C)_{q} for quasicentric linking systems. Point (d) is shown by first reducing to a map between centric subgroups of S which sends x to y. □

We finish this introductory section with two unrelated results which will be needed later in the paper. The first is a standard, group theoretic lemma.

Lemma 1.15. Let Q ◁ P be p-groups. If α is a p′-automorphism of P which acts as the identity on Q and on P/Q, then α = Id_{P}. Equivalently, the group of all automorphisms of P which restrict to the identity on Q and on P/Q is a p-group.

Proof. See [Go, Corollary 5.3.3]. □

The following proposition will only be used in Section 4, but we include it here because it seems to be of wider interest. Note, for any fusion system F over S, any subgroup P ≤ S fully normalized in F, and any P′ which is S-conjugate to P, that P′ is also fully normalized in F since N_{S}(P′) is S-conjugate to N_{S}(P).

Proposition 1.16. Let F be a saturated fusion system over a p-group S. Then for any subgroup P ≤ S, the set of S-conjugacy classes of subgroups F-conjugate to P and fully normalized in F has order prime to p.

Proof. By [BLO2, Proposition 5.5], there is an (S, S)-biset Ω which, when regarded as a set with (S × S)-action, satisfies the following three conditions.

(a) The isotropy subgroup of each point in Ω is of the form

P_{φ} =^{def} {(x, φ(x)) | x ∈ P }

for some P ≤ S and some φ ∈ Hom_{F}(P, S).

(b) For each P ≤ S and each φ ∈ Hom_{F}(P, S), the two structures of (S × P)-set on Ω obtained by restriction and by Id × φ are isomorphic.

(c) |Ω|/|S| ≡ 1 (mod p).

Note that by (a), the actions of S × 1 and 1 × S on Ω are both free.

Now fix a subgroup P ≤ S. Set S_{2} = 1 × S for short, and let Ω_{0} ⊆ Ω be the subset such that Ω_{0}/S_{2} = (Ω/S_{2})^{P}. In other words, Ω_{0} is the set of all x ∈ Ω such that for each g ∈ P, there is some h ∈ S satisfying (g, h)·x = x. Since the action of S_{2} on Ω is free, this element h ∈ S is uniquely determined for each x ∈ Ω_{0} and g ∈ P. Let θ(x): P → S denote the function such that for each g ∈ P, (g, θ(x)(g))·x = x. The isotropy subgroup at x of the (P × S)-action is thus the subgroup P_{θ(x)} = {(g, θ(x)(g)) | g ∈ P }; and by (a), θ(x) ∈ Hom_{F}(P, S). This defines a map

θ: Ω_{0} → Hom_{F}(P, S).

By definition, for each φ ∈ Hom_{F}(P, S), θ^{−1}(φ) is the set of elements of Ω fixed by P_{φ}. By condition (b) above, the action of P × P on Ω induced by the homomorphism 1 × φ ∈ Hom(P × P, S × S) is isomorphic to the action defined by restriction, and thus |θ^{−1}(φ)| = |θ^{−1}(incl)|. This shows that the point inverses of θ all have the same fixed order k.

Now let Rep_{F}(P, S) = Hom_{F}(P, S)/Inn(S): the set of S-conjugacy classes of morphisms from P to S. Let P be the set of S-conjugacy classes of subgroups F-conjugate to P, and let P_{fn} ⊆ P be the subset of classes of subgroups fully normalized in F. If x ∈ Ω_{0} and θ(x) = φ, then for all s ∈ S and g ∈ P,

(1, s)·x = (g, sφ(g))·x = (g, c_{s} ∘ φ(g))·(1, s)·x,

and this shows that θ((1, s)·x) = c_{s} ∘ θ(x). Thus θ induces a map

θ̄: (Ω/S_{2})^{P} = Ω_{0}/S_{2} ⟶^{θ/S_{2}} Rep_{F}(P, S) ⟶^{Im(−)} P,

where Rep_{F}(P, S) = Hom_{F}(P, S)/Inn(S). Furthermore, |Ω/S_{2}| ≡ 1 (mod p) by (c), and thus |(Ω/S_{2})^{P} | ≡ 1 (mod p).

For each P′ which is F-conjugate to P, there are |S|/|N_{S}(P′)| distinct subgroups in the S-conjugacy class [P′]. Hence there are

| Aut_{F}(P)|·|S|/|N_{S}(P′)|

elements of Hom_{F}(P, S) whose image lies in [P′]. Since each of these is the image of k elements in Ω_{0}, this shows that

|θ̄^{−1}([P′])| = k·| Aut_{F}(P)|/|N_{S}(P′)|.

Thus |N_{S}(P′)||k·| Aut_{F}(P)| for all P′ F-conjugate to P, and so θ̄^{−1}([P′]) has order a multiple of p if P′ is not fully normalized in F (if [P′] ∈ P∖P_{fn}). Hence

|θ̄^{−1}(P_{fn})| ≡ |(Ω/S_{2})^{P} | ≡ 1 (mod p) .

So if we set m = |N_{S}(P′)| for [P′] ∈ P_{fn} (i.e., the maximal value of |N_{S}(P′)| for P′ F-conjugate to P), then

|θ̄^{−1}(P_{fn})| = |P_{fn}|·(\frac{k·| Aut_{F}(P)|}{m}),

and thus |P_{fn}| is prime to p. □

Using a similar argument, one can also show that the set Rep_{F}^{fc}(P, S) of elements of Rep_{F}(P, S) whose image is fully centralized also has order prime to p.

2. The fundamental group of |L|_{p}^{∧}

The purpose of this section is to give a simple description of the fundamental group of |L|_{p}^{∧}, for any p-local finite group (S, F, L), purely in terms of the fusion system F. The result is analogous to the (hyper-) focal subgroup theorem for finite groups, as we explain below.

In Section 1, we defined a functor J: L^{q} → B(π_{1}(|L|)), for any p-local finite group (S, F, L), and a homomorphism j = J ∘ δ_{S} from S to π_{1}(|L|). Let τ: S → π_{1}(|L|_{p}^{∧}) be the composite of j with the natural homomorphism from π_{1}(|L|) to π_{1}(|L|_{p}^{∧}).

In [BLO2, Proposition 1.12], we proved that τ: S → π_{1}(|L|_{p}^{∧}) is a surjection. In this section, we will show that Ker(τ) is the hyperfocal subgroup of F, defined by Puig [Pu4] (see also [Pu3]).

Definition 2.1. For any saturated fusion system F over a p-group S, the hyperfocal subgroup of F is the normal subgroup of S defined by

O_{F}^{p}(S) = ⟨g^{−1}α(g) | g ∈ P ≤ S, α ∈ O^{p}(Aut_{F}(P))⟩ .

We will prove, for any p-local finite group (S, F, L), that π_{1}(|L|_{p}^{∧}) ≅ S/O_{F}^{p}(S). This is motivated by Puig’s hyperfocal theorem, and we will also need that theorem in order to prove it. Before stating Puig’s theorem, we first recall the standard focal subgroup theorem. If G is a finite group and S ∈ Syl_{p}(G), then this theorem says that S ∩ [G, G] (the focal subgroup) is the subgroup generated by all elements of the form x^{−1}y for x, y ∈ S which are G-conjugate (cf. [Go, Theorem 7.3.4] or [Suz2, 5.2.8]).

The quotient group S/(S ∩ [G, G]) is isomorphic to the p-power torsion subgroup of G/[G, G], and can thus be identified as a quotient group of the maximal p-group quotient G/O^{p}(G). Since G/O^{p}(G) is a p-group, G = S·O^{p}(G), and hence G/O^{p}(G) ≅ S/(S ∩ O^{p}(G)). Hence S ∩ O^{p}(G) ≤ S ∩ [G, G]. This subgroup S ∩ O^{p}(G) is what Puig calls the hyperfocal subgroup, and is described by the hyperfocal subgroup theorem in terms of S and fusion.

For S ∈ Syl_{p}(G) as above, let O_{G}^{p}(S) be the normal subgroup of S defined by

O_{G}^{p}(S) = O_{F_{S}(G)}^{p}(S) = ⟨g^{−1}α(g) | g ∈ P ≤ S, α ∈ O^{p}(Aut_{G}(P))⟩
= ⟨[g, x] | g ∈ P ≤ S, x ∈ N_{G}(P) of order prime to p⟩ .

Lemma 2.2 ([Pu3]). Fix a prime p, a finite group G, and a Sylow subgroup S ∈ Syl_{p}(G). Then O_{G}^{p}(S) = S ∩ O^{p}(G).

Proof. This is stated in [Pu3, §1.1], but the proof is only sketched there, and so we elaborate on it here. Following standard notation, for any P ≤ S and any A ≤ Aut(P), we write [P, A] = ⟨x^{−1}α(x) | x ∈ P, α ∈ A⟩. Thus O_{G}^{p}(S) is generated by the subgroups [P, O^{p}(Aut_{G}(P))] for all P ≤ S. It is clear that O_{G}^{p}(S) ≤ S ∩ O^{p}(G); the problem is to prove the opposite inclusion.

Set G_{∗} = O^{p}(G) and S_{∗} = S ∩ G_{∗} for short. Then [G_{∗}, G_{∗}] has index prime to p in G_{∗}, so it contains S_{∗}. By the focal subgroup theorem (cf. [Go, Theorem 7.3.4]), applied to S_{∗} ∈ Syl_{p}(G_{∗}), S_{∗} is generated by all elements of the form x^{−1}y for x, y ∈ S_{∗} which are G_{∗}-conjugate. Combined with Alperin’s fusion theorem (in Alperin’s original version [Al] or in the version of Theorem 1.5(a)), this implies that S_{∗} is generated by all subgroups [P, N_{G_{∗}}(P)] for P ≤ S_{∗} such that N_{S_{∗}}(P) ∈ Syl_{p}(N_{G_{∗}}(P)). (This last condition is equivalent to P being fully normalized in F_{S_{∗}}(G_{∗}).) Also, N_{G_{∗}}(P) is generated by O^{p}(N_{G_{∗}}(P)) and the Sylow subgroup N_{S_{∗}}(P), so [P, N_{G_{∗}}(P)] is generated by [P, O^{p}(N_{G_{∗}}(P))] ≤ O_{G}^{p}(S) and [P, N_{S_{∗}}(P)] ≤ [S_{∗}, S_{∗}]. Thus S_{∗} = ⟨O_{G}^{p}(S), [S_{∗}, S_{∗}]⟩. Since O_{G}^{p}(S) is normal in S (hence also normal in S_{∗}), this shows that S_{∗}/O_{G}^{p}(S) is equal to its commutator subgroup, which for a p-group is possible only if S_{∗}/O_{G}^{p}(S) is trivial, and hence S_{∗} = O_{G}^{p}(S). □

By Proposition 1.14, the key to getting information about π_{1}(|L|) is to construct functors from L or L^{q} to B(Γ), for a group Γ, which send inclusions to the identity. The next lemma is our main inductive tool for doing this. Whenever α ∈ Mor_{L^{q}}(P, P′) and β ∈ Mor_{L^{q}}(Q, Q′) are such that P ≤ Q, P′ ≤ Q′, we write α = β|_{P} to mean that α is the restriction of β in the sense defined in Section 1; i.e., ι_{P′}^{Q′} ∘ α = β ∘ ι_{P}^{Q}.

Lemma 2.3. Fix a p-local finite group (S, F, L), and let L^{q} be its associated quasicentric linking system. Assume a compatible set of inclusions {ι_{P}^{Q}} has been chosen for L^{q}. Let H_{0} be a set of F-quasicentric subgroups of S which is closed under Fconjugacy and overgroups. Let P be an F-conjugacy class of F-quasicentric subgroups maximal among those not in H_{0}, set H = H_{0} ∪ P, and let L^{H_{0}} ⊆ L^{H} ⊆ L^{q} be the full subcategories with these objects. Assume, for some group Γ, that

λ_{0} : L^{H_{0}} ⟶ B(Γ)

is a functor which sends inclusions to the identity. Fix some P ∈ P which is fully normalized in F, and fix a homomorphism λ_{P} : Aut_{L^{q}}(P) → Γ. Assume that

(∗) for all P ≨ Q ≤ N_{S}(P) such that Q is fully normalized in N_{F}(P), and for all α ∈ Aut_{L^{q}}(P) and β ∈ Aut_{L^{q}}(Q) such that α = β|_{P}, λ_{P} (α) = λ_{0}(β).

Then there is a unique extension of λ_{0} to a functor λ: L^{H} → B(Γ) which sends inclusions to the identity, and such that λ(α) = λ_{P} (α) for all α ∈ Aut_{F}(P).

Proof. The uniqueness of the extension is an immediate consequence of Theorem 1.5(a) (Alperin’s fusion theorem).

To prove the existence of the extension λ, we first show that (∗) implies the following (a prori stronger) statement:

(∗∗) for all Q, Q′ ≤ S which strictly contain P, and for all β ∈ Mor_{L^{q}}(Q, Q′) and α ∈ Aut_{L^{q}}(P) such that α = β|_{P}, λ_{P} (α) = λ_{0}(β).

To see this, note first that it suffices to consider the case where P is normal in Q and Q′. By assumption, π(β)(P) = π(α)(P) = P, hence π(β)(N_{Q}(P)) ≤ N_{Q′}(P), and therefore β restricts to a morphism β ∈ Mor_{L^{q}}(N_{Q}(P), N_{Q′}(P)) by Proposition 1.12(b) (applied with φ = β ∘ ι_{N_{Q}(P)}^{Q} and ψ = ι_{N_{Q′}(P)}^{Q′}). Since N_{Q}(P), N_{Q′}(P) ≩ P, by the induction

hypothesis, λ_{0}(β) = λ_{0}(β), so we are reduced to proving that λ_{P} (α) = λ_{0}(β).

Thus β ∈ Mor_{N_{L^{q}} (P)}(Q, Q′). We now apply Alperin’s fusion theorem (Theorem 1.5(a)) to the morphism π(β) in the fusion system N_{F}(P) (which is saturated by [BLO2, Proposition A.6]). Thus π(β) = φ_{k} ∘⋯ ∘φ_{1}, where each φ_{i} ∈ Hom_{N_{F} (P)}(Q_{i−1}, Q_{i}) is the restriction to Q_{i−1} of an automorphism φ̂_{i} ∈ Aut_{N_{F} (P)}(R_{i}), where R_{i} ≥ Q_{i−1}, Q_{i} is an N_{F}(P)-centric subgroup of N_{S}(P) which is fully normalized in N_{F}(P), and where Q = Q_{0} and Q′ = Q_{k}. Each R_{i} contains P, and hence is F-centric by [BLO2, Lemma 6.2]. For each φ̂_{i}, also regarded as an automorphism in F, we choose a lifting β̂_{i} ∈ Aut_{L}(R_{i}), and let β_{i} ∈ Mor_{L^{q}}(Q_{i−1}, Q_{i}) be its restriction. By (A)_{q}, β = β_{k} ∘ ⋯ ∘ β_{1} ∘ δ_{Q}(g) for some g ∈ C_{S}(Q), and hence

λ_{0}(β) = λ_{0}(β_{k}) ⋯ λ_{0}(β_{1}) · λ_{0}(δ_{Q}(g)) = λ_{0}(β̂_{k}) ⋯ λ_{0}(β̂_{1}) · λ_{0}(δ_{N_{S}(P)}(g))
= λ_{P} (β̂_{k}|_{P}) ⋯ λ_{P} (β̂_{1}|_{P}) · λ_{P} (Id_{P}) = λ_{P} (β|_{P}) = λ_{P} (α) ,

where the third equality follows from (∗). This finishes the proof of (∗∗).

We can now extend λ to be defined on all morphisms in L^{H} not in L^{H_{0}}. Fix such a morphism φ ∈ Mor_{L^{q}}(P_{1}, Q). Set P_{2} = π(φ)(P_{1}) ≤ Q; then P_{1}, P_{2} ∈ P, and φ = ι_{P_{2}}^{Q} ∘ φ′ for some unique φ′ ∈ Iso_{L^{q}}(P_{1}, P_{2}). By Lemma 1.3 (and then lifting to the linking category), there are isomorphisms φ_{i} ∈ Iso_{L^{q}}(N_{S}(P_{i}), N_{i}), for some N_{i} ≤ N_{S}(P) containing P, which restrict to isomorphisms φ_{i} ∈ Iso_{L^{q}}(P_{i}, P). Set ψ = φ_{2} ∘φ′ ∘φ_{1}^{−1} ∈ Aut_{L^{q}}(P). We have thus decomposed φ′ as the composite φ_{2}^{−1} ∘ψ ∘φ_{1}, and can now define

λ(φ) = λ(φ′) = λ_{0}(φ_{2})^{−1}·λ_{P} (ψ)·λ_{0}(φ_{1}).

Now let φ′ = (φ′_{2})^{−1}∘ψ′∘φ′_{1} be another such decomposition, where φ′_{i} is the restriction of φ′_{i} ∈ Iso_{L^{q}}(N_{S}(P_{i}), N′_{i}). We thus have a commutative diagram

\begin{aligned}P & ← ^{φ_{1}} & P_{1} & \overline{^{φ′_{1}} →} & P \\ P & ↓^{ψ} & ← ^{φ_{2}} & P_{2} & ↓^{φ′} & \overline{^{φ′_{2}} →} & P , & ↓^{ψ′} \end{aligned}

where for each i, φ_{i} and φ′_{i} are restrictions of isomorphisms φ_{i} and φ′_{i} defined on N_{S}(P_{i}). To see that the two decompositions give the same value of λ(φ), it remains to show that

λ_{P} (ψ′)·λ_{0}(φ′_{1} ∘ (φ_{1})^{−1}) = λ_{0}(φ′_{2} ∘ (φ_{2})^{−1})·λ_{P} (ψ).

And this holds since λ_{0}(φ′_{i} ∘ (φ_{i})^{−1}) = λ_{P} (φ′_{i} ∘ (φ_{i})^{−1}) by (∗∗).

We have now defined λ on all morphisms in L^{H}, and it sends inclusion morphisms to the identity by construction. By construction, λ sends composites to products, and thus the proof is complete. □

Lemma 2.3 provides the induction step when proving the following proposition, which is the main result needed to compute π_{1}(|L|_{p}^{∧}).

Proposition 2.4. Fix a p-local finite group (S, F, L), and let L^{q} be its associated quasicentric linking system. Assume a compatible set of inclusions {ι_{P}^{Q}} has been chosen for L^{q}. Then there is a unique functor

λ: L^{q} ⟶ B(S/O_{F}^{p}(S))

which sends inclusions to the identity, and such that λ(δ_{S}(g)) = g for all g ∈ S.

Proof. The functor λ will be constructed inductively, using Lemma 2.3. Let H_{0} ⊆ Ob(L^{q}) be a subset (possibly empty) which is closed under F-conjugacy and overgroups. Let P be an F-conjugacy class of F-quasicentric subgroups maximal among those not in H_{0}, set H = H_{0} ∪ P, and let L^{H_{0}} ⊆ L^{H} ⊆ L^{q} be the full subcategories with these objects. Assume that

λ_{0} : L^{H_{0}} ⟶ B(S/O_{F}^{p}(S))

has already been constructed, such that λ_{0}(δ_{S}(g)) = g for all g ∈ S (if S ∈ H_{0}), and such that λ_{0} sends inclusions to the identity.

Fix P ∈ P which is fully normalized in F, and let δ_{P,P} : N_{S}(P) ⟶ Aut_{L^{q}}(P) be the homomorphism of Proposition 1.13. Then Im(δ_{P,P}) is a Sylow p-subgroup of Aut_{L^{q}}(P), since Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) by axiom (I). We identify N_{S}(P) as a subgroup of Aut_{L^{q}}(P) to simplify notation. Then

Aut_{L^{q}}(P)/O^{p}(Aut_{L^{q}}(P)) ≅ N_{S}(P)/(N_{S}(P) ∩ O^{p}(Aut_{L^{q}}(P))) = N_{S}(P)/N_{0},

where by Lemma 2.2, N_{0} is the subgroup generated by all commutators [g, x] for g ∈ Q ≤ N_{S}(P), and x ∈ N_{Aut_{L^{q}} (P)}(Q) of order prime to p. In this situation, conjugation by x lies in Aut_{F}(Q) by Proposition 1.13(d), and thus [g, x] = g·c_{x}(g)^{−1} ∈ O_{F}^{p}(S). We conclude that N_{0} ≤ O_{F}^{p}(S), and hence that the inclusion of N_{S}(P) into S extends to a homomorphism

λ_{P} : Aut_{L^{q}}(P) ⟶ S/O_{F}^{p}(S).

We claim that condition (∗) in Lemma 2.3 holds for λ_{0} and λ_{P}. To see this, fix P ≨ Q ≤ S such that P ◁ Q and Q is fully normalized in N_{F}(P), and fix α ∈ Aut_{L^{q}}(P) and β ∈ Aut_{L^{q}}(Q) such that α = β|_{P}. We must show that λ_{P} (α) = λ_{0}(β). Upon replacing α by α^{k} and β by β^{k} for some appropriate k ≡ 1 (mod p), we can assume that both automorphisms have order a power of p. Since Q is fully normalized, Aut_{N_{S}(P)}(Q) is a Sylow subgroup of Aut_{N_{F} (P)}(Q). Hence (since any two Sylow p-subgroups of a finite group G are conjugate by an element of O^{p}(G)), there is an automorphism γ ∈ O^{p}(Aut_{N_{L^{q}} (P)}(Q)) such that γβγ^{−1} = δ_{Q}(g) for some g ∈ N_{S}(Q) ∩ N_{S}(P). In particular, λ_{0}(γ) = 1 since it is a composite of automorphisms of order prime to p. Set γ = γ|_{P}; then γ ∈ O^{p}(Aut_{L^{q}}(P)) and hence λ_{P} (γ) = 1. Using axiom (C)_{q} and Lemma 1.12, we see that γαγ^{−1} = δ_{P} (g); and thus (since λ_{P} (γ) = 1 and λ_{0}(γ) = 1) that

λ_{0}(β) = λ_{0}(δ_{Q}(g)) = g = λ_{P} (δ_{P} (g)) = λ_{P} (α).

Thus, by Lemma 2.3, we can extend λ_{0} to a functor defined on L^{H}. Upon continuing this procedure, we obtain a functor λ defined on all of L^{q}. □

For any finite group G, π_{1}(BG_{p}^{∧}) ≅ G/O^{p}(G). (This is implicit in [BK, §VII.3], and shown explicitly in [BLO1, Proposition A.2].) Hence our main result in this section can be thought of as the hyperfocal theorem for p-local finite groups.

Theorem 2.5 (Hyperfocal subgroup theorem for p-local finite groups). Let (S, F, L) be a p-local finite group. Then

π_{1}(|L|_{p}^{∧}) ≅ S/O_{F}^{p}(S).

More precisely, the natural map τ: S → π_{1}(|L|_{p}^{∧}) is surjective, and Ker(τ) = O_{F}^{p}(S).

Proof. Let λ: L ⟶ B(S/O_{F}^{p}(S)) be the functor of Proposition 2.4, and let |λ| be the induced map between geometric realizations. Since |B(S/O_{F}^{p}(S))| is the classifying space of a finite p-group and hence p-complete, |λ| factors through the p-completion |L|_{p}^{∧}. Consider the following commutative diagram:

S  \overline{^{j} →} π_{1}(|L|)
[figure: τ; π_{1}(|λ|)]
π_{1}(|L|_{p}^{∧}) ↓^{(−)_{p}^{∧}}  \overline{→^{\begin{matrix}λ̄= \\ π_{1}(|λ|_{p}^{∧}) \end{matrix}}} S/O_{F}^{p}(S) .

Here, τ is surjective by [BLO2, Proposition 1.12]. Also, by construction, the composite λ̄ ∘ τ = π_{1}(|λ|) ∘ j is the natural projection. Thus Ker(τ) ≤ O_{F}^{p}(S), and it remains to show the opposite inclusion.

Fix g ∈ P ≤ S and α ∈ Aut_{F}(P) such that α has order prime to p; we want to show that g^{−1}α(g) ∈ Ker(τ). Since Ker(τ) is closed under F-conjugacy (Proposition 1.14(d)), for any φ ∈ Iso_{F}(P, P′), g^{−1}α(g) is in Ker(τ) if φ(g^{−1}α(g)) is in Ker(τ). In particular, since φ(g^{−1}α(g)) = φ(g)^{−1}·(φαφ^{−1})(φ(g)), we can assume that P is fully centralized in F. Then, upon extending α to an automorphism of P ·C_{S}(P), which can be assumed also to have order prime to p (replace it by an appropriate power if necessary), we can assume that P is F-centric. In this case, by Proposition 1.14(c), j(g) and j(α(g)) are conjugate in π_{1}(|L|) by an element of order prime to p, and hence are equal in π_{1}(|L|_{p}^{∧}) since this is a p-group. This shows that g^{−1}α(g) ∈ Ker(τ), and finishes the proof of the theorem. □

The following result, which will be useful in Section 5, is of a similar nature, but much more elementary.

Proposition 2.6. Fix a p-local finite group (S, F, L), and let L^{q} be its associated quasicentric linking system. Then the induced maps

π_{1}(|L|) ⟶ π_{1}(|F^{c}|)  and  π_{1}(|L^{q}|) ⟶ π_{1}(|F^{q}|)

are surjective, and their kernels are generated by elements of p-power order.

Proof. We prove this for |L^{q}| and |F^{q}|; a similar argument applies to |L| and |F^{c}|. Recall from Section 1 that we can regard π_{1}(|L^{q}|) as the group generated by the loops J(α), for α ∈ Mor(L^{q}), with relations given by composition of morphisms and making inclusion morphisms equal to 1. In a similar way, we regard π_{1}(|F^{q}|) as being generated by loops J(φ) for φ ∈ Mor(F^{q}). Since every morphism α ∈ Hom_{F^{q}}(P, Q) has a lifting to a morphism of L^{q}, the map π_{#} : π_{1}(|L^{q}|) → π_{1}(|F^{q}|) induced by the projection functor π: L^{q} → F^{q} is an epimorphism.

Write JC for the normal subgroup of π_{1}(|L^{q}|) generated by the loops j(g) = J(δ_{P} (g)), for all F-quasicentric subgroups P ≤ S and all g ∈ C_{S}(P). In particular, JC is generated by elements of p-power order. Since π_{#}(j(g)) = 1, JC is contained in the kernel of π_{#}, and we have a factorization π̂_{#} : π_{1}(|L^{q}|)/JC → π_{1}(|F^{q}|).

Define an inverse s: π_{1}(|F^{q}|) → π_{1}(|L^{q}|)/JC as follows. Given a morphism α ∈ Hom_{F^{q}}(P, Q), choose a lifting ᾱ in L^{q}, and set s(α) = [J(ᾱ)]. If ᾱ, ᾱ′ ∈ Hom_{L^{q}}(P, Q) are two liftings of α, then there is an isomorphism χ: P′ → P in L^{q} where P′ is fully centralized, and an element g ∈ C_{S}(P′), such that ᾱ′ = ᾱ ∘ (χ ∘ δ_{P′}(g) ∘ χ^{−1}). Since χ∘δ_{P′}(g)∘χ^{−1} represents a loop that belongs to JC, [J(ᾱ)] = [J(ᾱ′)]. Thus the definition of s does not depend on the choice of ᾱ. It remains to show that s preserves the relations among the generators. But we clearly have that s([J(incl_{P}^{Q})]) = [J(ι_{P}^{Q})] = 1. Also, if α and β are composable morphisms of F^{q}, and ᾱ and β̄ are liftings to L^{q}, then ᾱ ∘ β̄ is a lifting of α ∘ β, and hence s(α)s(β) = [J(ᾱ)][J(β̄)] = [J(ᾱ ∘ β̄)] = s(α ∘ β).

This shows that π_{#} induces an isomorphim π_{1}(|L^{q}|)/JC ≅ π_{1}(|F^{q}|). □

3. Subsystems with p-solvable quotient

In this section, we prove some general results about subsystems of saturated fusion systems and p-local finite subgroups: subsystems with p-group quotient or quotient of order prime to p. These will then be used in the next two sections to prove some more specific theorems.

It will be convenient in this section to write “p′-group” for a finite group of order prime to p. Recall that a p-solvable group is a group G with normal series 1 = H_{0} ◁ H_{1} ◁ ⋯ ◁ H_{k} = G such that each H_{i}/H_{i−1} is a p-group or a p′-group. As one consequence of the results of this section, we show that for any p-local finite group (S, F, L), and any homomorphism θ from π_{1}(|L|) to a finite p-solvable group, there is another p-local finite group (S_{0}, F_{0}, L_{0}) such that |L_{0}| is homotopy equivalent to the covering space of |L| with fundamental group Ker(θ).

We start with some definitions. Recall that for any finite group G, O^{p′}(G) and O^{p}(G) are the smallest normal subgroups of G of index prime to p and of p-power index, respectively. Equivalently, O^{p′}(G) is the subgroup generated by elements of ppower order in G, and O^{p}(G) is the subgroup generated by elements of order prime to p in G.

We want to identify the fusion subsystems of a given fusion system which are analogous to subgroups of G which contain O^{p′}(G) or O^{p}(G). This motivates the following definitions.

Definition 3.1. Let F be a saturated fusion system over a p-group S. Let (S′, F′) ⊆ (S, F) be a saturated fusion subsystem; i.e., S′ ≤ S is a subgroup, and F′ ⊆ F is a subcategory which is a saturated fusion system over S′.

(a) (S′, F′) is of p-power index in (S, F) if S′ ≥ O_{F}^{p}(S), and Aut_{F′}(P) ≥ O^{p}(Aut_{F}(P)) for all P ≤ S′. Equivalently, a saturated fusion subsystem F′ ⊆ F over S′ ≥ O_{F}^{p}(S) has p-power index if it contains all F-automorphisms of order prime to p of subgroups of S′.

(b) (S′, F′) is of index prime to p in (S, F) if S′ = S, and Aut_{F′}(P) ≥ O^{p′}(Aut_{F}(P)) for all P ≤ S. Equivalently, a saturated fusion subsystem F′ ⊆ F over S has index prime to p if it contains all F-automorphisms of p-power order.

This terminology has been chosen for simplicity. Subsystems “of p-power index” or “of index prime to p” are really analogous to subgroups H ≤ G which contain normal subgroups of G of p-power index or index prime to p, respectively. For example, if F is a saturated fusion system over S, and F′ ⊆ F is the fusion system of S itself (i.e., the minimal fusion system over S), then F′ is not in general a subsystem of index prime to p under the above definition, despite the inclusion F′ ⊆ F being analogous to the inclusion of a Sylow p-subgroup in a group.

Over the next three sections, we will classify all saturated fusion subsystems of ppower index, or of index prime to p, in a given saturated fusion system. In both cases, there will be a minimal such subsystem, denoted O^{p}(F) or O^{p′}(F); and the saturated fusion subsystems of the given type will be in bijective correspondence with the subgroups of a given p-group or p′-group.

The following terminology will be useful for describing some of the categories we have to work with.

Definition 3.2. Fix a finite p-group S.

(a) A restrictive category over S is a category F such that Ob(F) is the set of subgroups of S, such that all morphisms in F are group monomorphisms between the subgroups, and with the following additional property: for each P′ ≤ P ≤ S and Q′ ≤ Q ≤ S, and each φ ∈ Hom_{F}(P, Q) such that φ(P′) ≤ Q′, φ|_{P′} ∈ Hom_{F}(P′, Q′).

(b) A restrictive category F over S is normalized by an automorphism α ∈ Aut_{F}(S) if for each P, Q ≤ S, and each φ ∈ Hom_{F}(P, Q), αφα^{−1} ∈ Hom_{F}(α(P), α(Q)).

(c) For any restrictive category F over S and any subgroup A ≤ Aut(S), ⟨F, A⟩ is the smallest restrictive category over S which contains F together with all automorphisms in A and their restrictions.

By definition, any restrictive category is required to contain all inclusion homomorphisms (restrictions of Id_{S}). The main difference between a restrictive category over S and a fusion system over S is that the restrictive subcategory need not contain F_{S}(S).

When F is a fusion system over S, then an automorphism α ∈ Aut(S) normalizes F if and only if it is fusion preserving in the sense used in [BLO1]. For the purposes of this paper, it will be convenient to use both terms, in different situations.

We next define the following subcategories of a given fusion system F.

Definition 3.3. Let F be any fusion system over a p-group S.

(a) O_{∗}^{p}(F) ⊆ F denotes the smallest restrictive subcategory of F whose morphism set contains O^{p}(Aut_{F}(P)) for all subgroups P ≤ S.

(b) O_{∗}^{p′}(F) ⊆ F denotes the smallest restrictive subcategory of F whose morphism set contains O^{p′}(Aut_{F}(P)) for all subgroups P ≤ S.

By definition, for subgroups P, Q, a morphism φ ∈ Hom_{F}(P, Q) lies in O_{∗}^{p′}(F) if and only if it is a composite of morphisms which are restrictions of elements of O^{p′}(Aut_{F}(R)) for subgroups R ≤ S. Morphisms in O_{∗}^{p}(F) are described in a similar way.

The subcategory O_{∗}^{p}(F) is not, in general, a fusion system — and this is why we had to define restrictive categories. The subcategory O_{∗}^{p′}(F) is always a fusion system (since Aut_{S}(P) ≤ O^{p′}(Aut_{F}(P)) for all P ≤ S), but it is not, in general, saturated. The subscripts “∗” have been put in as a reminder of these facts, and as a contrast with the notation O^{p}(F) and O^{p′}(F) which will be used to denote certain minimal saturated fusion systems.

We now check some of the basic properties of the subcategories O_{∗}^{p}(F) and O_{∗}^{p′}(F), stated in terms of these definitions.

Lemma 3.4. The following hold for any fusion system F over a p-group S.

(a) O_{∗}^{p}(F) and O_{∗}^{p′}(F) are normalized by Aut_{F}(S).

(b) If F is saturated, then F = ⟨O_{∗}^{p}(F), Aut_{F}(S)⟩ = ⟨O_{∗}^{p′}(F), Aut_{F}(S)⟩.

(c) If F′ ⊆ F is any restrictive subcategory normalized by Aut_{F}(S) and such that F = ⟨F′, Aut_{F}(S)⟩, then for each P, Q ≤ S and φ ∈ Hom_{F}(P, Q), there are morphisms α ∈ Aut_{F}(S), φ′ ∈ Hom_{F′}(α(P), Q), and φ′′ ∈ Hom_{F′}(P, α^{−1}Q), such that φ = φ′ ∘ α|_{P} = α ∘ φ′′.

Proof. To simplify notation in the following proofs, for α ∈ Aut_{F}(S), we write α in composites of morphisms between subgroups of S, rather than specifying the appropriate restriction each time.

(a) Set F′ = O_{∗}^{p}(F) or O_{∗}^{p′}(F). Let α ∈ Aut_{F}(S) and let φ ∈ Hom_{F′}(P, Q). We must show that αφα^{−1} ∈ Hom_{F′}(α(P), α(Q)). By definition of F′, there are subgroups

P = P_{0}, P_{1}, …, P_{k} = φ(P) ≤ Q,

subgroups P_{1}, …, P_{k}, and automorphisms χ_{i} ∈ O^{p}(Aut_{F}(P_{i})) (if F′ = O_{∗}^{p}(F)) or χ_{i} ∈ O^{p′}(Aut_{F}(P_{i})) (if F′ = O_{∗}^{p′}(F)), such that P_{i−1}, P_{i} ≤ P_{i}, χ_{i}(P_{i−1}) = P_{i}, and

φ = incl_{P_{k}}^{Q} ∘(χ_{k}|_{P_{k−1}}) ∘ ⋯ ∘ (χ_{2}|_{P_{1}}) ∘ (χ_{1}|_{P_{0}}).

Let P′_{i} = α(P_{i}), P ′_{i} = α(P_{i}), and χ′_{i} = αχ_{i}α^{−1} ∈ O^{p}(Aut_{F}(P ′_{i})) or O^{p′}(Aut_{F}(P ′_{i})). Then

αφα^{−1} = incl_{P′_{k}}^{α(Q)} ∘(χ′_{k}|_{P′_{k−1}}) ∘ ⋯ ∘ (χ′_{2}|_{P′_{1}}) ∘ (χ′_{1}|_{P′_{0}})

is in Hom_{F′}(α(P), α(Q)), as required.

(b1) Now assume that F is saturated. We show here that F = ⟨O_{∗}^{p′}(F), Aut_{F}(S)⟩; i.e., that each φ ∈ Mor(F) is a composite of morphisms in O_{∗}^{p′}(F) and in Aut_{F}(S). By Alperin’s fusion theorem for saturated fusion systems (Theorem 1.5(a)), it suffices to prove this when φ ∈ Aut_{F}(P) for some P ≤ S which is F-centric and fully normalized. The result clearly holds if P = S. So we can assume that P ≨ S, and also assume inductively that the lemma holds for every automorphism of any subgroup P′ ≤ S such that |P′| > |P |.

Consider the subgroup

K = φ Aut_{S}(P)φ^{−1} = {φc_{g}φ^{−1} ∈ Aut_{F}(P) | g ∈ N_{S}(P)}.

Then K is a p-subgroup of Aut_{F}(P), and hence K ≤ Aut_{O_{∗}^{p′}(F)}(P). Since P is fully normalized in F, Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)). Thus K and Aut_{S}(P) are both Sylow p-subgroups of Aut_{O_{∗}^{p′}(F)}(P), and there is some χ ∈ Aut_{O_{∗}^{p′}(F)}(P) such that χKχ^{−1} ≤ Aut_{S}(P). In other words,

N_{χφ} =^{def} {g ∈ N_{S}(P) | (χφ)c_{g}(χφ)^{−1} ∈ Aut_{S}(P)} = N_{S}(P).

So by condition (II) in Definition 1.2, χ ∘ φ can be extended to a homomorphism ψ ∈ Hom_{F}(N_{S}(P), S). By the induction hypothesis (and since N_{S}(P) ≩ P), ψ, and hence χ ∘ φ, are composites of morphisms in N_{F}(S) and in O_{∗}^{p′}(F). Also, χ ∈ Aut_{O_{∗}^{p′}(F)}(P) by assumption, and hence φ is a composite of morphisms in these two subcategories.

(b2) To see that F = ⟨O_{∗}^{p}(F), Aut_{F}(S)⟩ when F is saturated, it again suffices to restrict to the case where φ ∈ Aut_{F}(P) for some P ≤ S which is F-centric and fully normalized. But in this case, Aut_{F}(P) is generated by its Sylow p-subgroup Aut_{S}(P) ≤ Aut_{N_{F} (S)}(P), together with O^{p}(Aut_{F}(P)) ≤ Aut_{O_{∗}^{p}(F)}(P).

(c) Since F = ⟨F′, Aut_{F}(S)⟩, every morphism in F is a composite of morphisms in F′ and restrictions of automorphisms in Aut_{F}(S). Assume φ ∈ Hom_{F}(P, Q) is the composite

P = P_{0} ⟶^{α_{0}} Q_{0} ⟶^{φ_{1}} P_{1} ⟶^{α_{1}} Q_{1} ⟶ … ⟶ P_{n} ⟶^{α_{n}} Q_{n} = Q,

where α_{i} ∈ Aut_{F}(S), α_{i}(P_{i}) = Q_{i}, and φ_{i} ∈ Hom_{F′}(Q_{i−1}, P_{i}). Write α_{j,i} = α_{j} ⋯ α_{i} for any i ≤ j, and set α = α_{n,0} = α_{n} ⋯ α_{0}. Then

φ = α ∘ (α_{n−1,0}^{−1}φ_{n}α_{n−1,0}) ⋯ (α_{0,0}^{−1}φ_{1}α_{0,0}) = (α_{n,n}φ_{n}α_{n,n}^{−1}) ⋯ (α_{n,1}φ_{1}α_{n,1}^{−1}) ∘ α;

where each α_{n,i}φ_{i}α_{n,i}^{−1} and each α_{i,0}^{−1}φ_{i+1}α_{i,0} is a morphism in F′ since F′ is normalized by Aut_{F}(S). □

The following lemma will also be needed later.

Lemma 3.5. Let F be a saturated fusion system over a p-group S. Fix a normal subgroup S_{0} ◁ S which is strongly F-closed; i.e., no element of S_{0} is F-conjugate to any element of S∖S_{0}. Let (S_{0}, F_{0}) be a saturated fusion subsystem of (S, F). Then for any P ≤ S which is F-centric and F-radical, P ∩ S_{0} is F_{0}-centric.

Proof. Assume P ≤ S is F-centric and F-radical, and set P_{0} = P ∩S_{0} for short. Choose a subgroup P′_{0} ≤ S_{0} which is F-conjugate to P_{0} and fully normalized in F. In particular, by (I), P′_{0} is fully centralized in F. By Lemma 1.3, there is φ ∈ Hom_{F}(N_{S}(P_{0}), N_{S}(P′_{0})) such that φ(P_{0}) = P′_{0}. Set P′ = φ(P); thus P′ is F-conjugate to P (so P′ is also F-centric and F-radical), and P′_{0} = P′ ∩ S_{0} since S_{0} is strongly closed. For any P′′_{0} ≤ S_{0} which is F_{0}-conjugate to P_{0} (hence F-conjugate to P′_{0}), there is a morphism ψ ∈ Hom_{F}(P′′_{0}·C_{S}(P′′_{0}), P′_{0}·C_{S}(P′_{0})) such that ψ(P′′_{0}) = P′_{0} (by axiom (II)), and then ψ(C_{S_{0}}(P′′_{0})) ≤ C_{S_{0}}(P′_{0}). So if C_{S_{0}}(P′_{0}) = Z(P′_{0}), then C_{S_{0}}(P′′_{0}) = Z(P′′_{0}) for all P′′_{0} which is F_{0}-conjugate to P_{0}, and P_{0} is F_{0}-centric.

We are thus reduced to showing that C_{S_{0}}(P′_{0}) = Z(P′_{0}); and without loss of generality, we can assume that P′ = P and P′_{0} = P_{0}. Since S_{0} is strongly closed, every α ∈ Aut_{F}(P) leaves P_{0} invariant. Let Aut_{F}^{0}(P) ≤ Aut_{F}(P) be the subgroup of elements which induce the identity on P_{0} and on P/P_{0}. This is a normal subgroup of Aut_{F}(P) since all elements of Aut_{F}(P) leave P_{0} invariant, and is also a p-subgroup by Lemma 1.15. Thus Aut_{F}^{0}(P) ≤ O_{p}(Aut_{F}(P)), and hence Aut_{F}^{0}(P) ≤ Inn(P) since P is Fradical.

We want to show that C_{S_{0}}(P_{0}) = Z(P_{0}). Fix any x ∈ C_{S_{0}}(P_{0}), and assume first that the coset x·Z(P_{0}) ∈ C_{S_{0}}(P_{0})/Z(P_{0}) is fixed by the conjugation action of P. Thus x ∈ S_{0}, [x, P_{0}] = 1, and [x, P] ≤ Z(P_{0}), so c_{x} ∈ Aut_{F}^{0}(P) ≤ Inn(P), and xg ∈ C_{S}(P) for some g ∈ P. Since P is F-centric, this implies that xg ∈ P, so x ∈ C_{S_{0}}(P_{0})∩P = Z(P_{0}). In other words, [C_{S_{0}}(P_{0})/Z(P_{0})]^{P} = 1, so C_{S_{0}}(P_{0})/Z(P_{0}) = 1, and thus C_{S_{0}}(P_{0}) = Z(P_{0}). □

The motivation for the next definition comes from considering the situation which arises when one is given a saturated fusion system F with an associated quasicentric linking system L^{q}, and a functor L^{q} → B(Γ) which sends inclusions to the identity (or equivalently a homomorphism from π_{1}(|L^{q}|) to Γ) for some group Γ. Such a functor is equivalent to a function Θ: Mor(L^{q}) → Γ which sends composites to products and sends inclusions to the identity; and for any H ≤ Γ, Θ^{−1}(H) = Mor(L_{H}^{q}) for some subcategory L_{H}^{q} ⊆ L^{q} with the same objects. Let F_{H}^{q} ⊆ F^{q} be the image of L_{H}^{q} under the canonical projection; then in some sense (to be made precise later), F_{H}^{q} is a subsystem of F^{q} of index [Γ:H].

What we now need is to make sense of such “inverse image subcategories” of the fusion system F, when we are not assuming that we have an associated linking system. Let 𝔖𝔲𝔟(Γ) denote the set of nonempty subsets of Γ. Given a function Θ as above, there is an obvious associated function Θ: Mor(F^{q}) → 𝔖𝔲𝔟(Γ), which sends a morphism α ∈ Mor(F^{q}) to Θ(π^{−1}(α)). Here, π denotes the natural projection from L^{q} to F^{q}. Moreover, Θ also induces a homomorphism θ = Θ ∘ δ_{S} from S to Γ. The maps Θ and θ are closely related to each other, and satisfy certain properties, none of which depend on the existence (or choice) of a quasicentric linking system associated to F. In fact, we will see that the data encoded in such a pair of functions, if they satisfy the appropriate conditions, suffices to describe precisely what is meant by “inverse image subcategories” of fusion systems; and to show that under certain restrictions, these categories are (or generate) saturated fusion subsystems.

Definition 3.6. Let F be a saturated fusion system over a p-group S, and let F_{0} ⊆ F^{q} be any full subcategory such that Ob(F_{0}) is closed under F-conjugacy. A fusion mapping triple for F_{0} consists of a triple (Γ, θ, Θ), where Γ is a group, θ: S ⟶ Γ is a homomorphism, and

Θ: Mor(F_{0}) ⟶ 𝔖𝔲𝔟(Γ),

is a map which satisfies the following conditions for all subgroups P, Q, R ≤ S which lie in F_{0}:

(i) For all P ⟶^{φ} Q ⟶^{ψ} R in F_{0}, and all x ∈ Θ(ψ), Θ(ψφ) = x·Θ(φ).

(ii) If P is fully centralized in F, then Θ(Id_{P}) = θ(C_{S}(P)).

(iii) If φ = c_{g} ∈ Hom_{F}(P, Q), where g ∈ N_{S}(P, Q), then θ(g) ∈ Θ(φ).

(iv) For all φ ∈ Hom_{F}(P, Q), all x ∈ Θ(φ), and all g ∈ P, xθ(g)x^{−1} = θ(φ(g)).

For any fusion mapping triple (Γ, θ, Θ) and any H ≤ Γ, we let F_{H}^{•} ⊆ F be the smallest restrictive subcategory which contains all φ ∈ Mor(F^{q}) such that Θ(φ) ∩ H ≠ ∅. Let F_{H} ⊆ F_{H}^{•} be the full subcategory whose objects are the subgroups of θ^{−1}(H).

When θ is the trivial homomorphism (which is always the case when Γ is a p′-group), then a fusion mapping triple (Γ, θ, Θ) on a subcategory F_{0} ⊆ F^{q} is equivalent to a functor from F_{0} to B(Γ); i.e., Θ(φ) contains just one element for all φ ∈ Mor(F_{0}). By (i), it suffices to show this for identity morphisms; and by (ii), |Θ(Id_{P})| = 1 if P is fully centralized. For arbitrary P, it then follows from (i), together with the assumption (included in the definition of 𝔖𝔲𝔟(Γ)) that Θ(φ) ≠ ∅ for all φ.

The following additional properties of fusion mapping triples will be needed.

Lemma 3.7. Fix a saturated fusion system F over a p-group S, let F_{0} be a full subcategory such that Ob(F_{0}) is closed under F-conjugacy, and let (Γ, Θ, θ) be a fusion mapping triple for F_{0}. Then the following hold for all P, Q, R ∈ Ob(F_{0}).

(v) Θ(Id_{P}) is a subgroup of Γ, and Θ restricts to a homomorphism

Θ_{P} : Aut_{F}(P) → N_{Γ}(Θ(Id_{P}))/Θ(Id_{P}).

Thus Θ_{P} (α) = Θ(α) (as a coset of Θ(Id_{P})) for all α ∈ Aut_{F}(P).

(vi) For all P ⟶^{φ} Q ⟶^{ψ} R in F_{0}, and all x ∈ Θ(φ), Θ(ψφ) ⊇ Θ(ψ)·x, with equality if φ(P) = Q. In particular, if P ≤ Q, then Θ(ψ|_{P}) ⊇ Θ(ψ).

(vii) Assume S ∈ Ob(F_{0}). Then for any φ ∈ Hom_{F}(P, Q), any α ∈ Aut_{F}(S), and any x ∈ Θ(α), Θ(αφα^{−1}) = xΘ(φ)x^{−1}, where αφα^{−1} ∈ Hom_{F}(α(P), α(Q)).

Proof. (v) By (i), for any α, β ∈ Aut_{F}(P) and any x ∈ Θ(α), Θ(αβ) = x·Θ(β). When applied with α = β = Id_{P}, this shows that Θ(Id_{P}) is a subgroup of Γ. (Note here that Θ(Id_{P}) ≠ ∅ by definition of 𝔖𝔲𝔟(Γ).) When applied with β = α^{−1}, this shows that x^{−1} ∈ Θ(α^{−1}) if x ∈ Θ(α). Hence Θ(α) = x·Θ(Id_{P}) implies that Θ(α^{−1}) = Θ(Id_{P})·x^{−1} and Θ(α^{−1}) = x^{−1}·Θ(Id_{P}), and thus that each Θ(α) is a right coset as well as a left coset. Thus Θ(α) ⊆ N_{Γ}(Θ(Id_{P})) for all α ∈ Aut_{F}(P), and the induced map Θ_{P} is clearly a homomorphism.

(vi) By (i), Θ(ψφ) ⊇ Θ(ψ)·Θ(φ) for any pair of composable morphisms φ, ψ in F_{0}. In particular, Θ(ψφ) ⊇ Θ(ψ)·x if x ∈ Θ(φ). If φ is an isomorphism, then 1 ∈ Θ(Id_{P}) = x·Θ(φ^{−1}) by (v) and (i), so x^{−1} ∈ Θ(φ^{−1}). This gives the inclusions

Θ(ψ) = Θ(ψφφ^{−1}) ⊇ Θ(ψφ)·x^{−1} ⊇ Θ(ψ)xx^{−1},

and hence these are both equalities. The last statement is the special case where P ≤ Q and φ = incl_{P}^{Q}; 1 ∈ Θ(incl_{P}^{Q}) by (iii). (vii) For x ∈ Θ(α), Θ(αφ) = x·Θ(φ) = Θ(αφα^{−1})·x by (i) and (vi). □

We are now ready to prove the main result about fusion mapping triples.

Proposition 3.8. Let F be a saturated fusion system over a finite p-group S. Let (Γ, θ, Θ) be any fusion mapping triple on F^{q}, where Γ is a p-group or a p′-group, and

θ: S ⟶ Γ  and  Θ: Mor(F^{q}) ⟶ 𝔖𝔲𝔟(Γ),

Then the following hold for any subgroup H ≤ Γ, where we set S_{H} = θ^{−1}(H).

(a) F_{H} is a saturated fusion system over S_{H}.

(b) If Γ is a p-group, then a subgroup P ≤ S_{H} is F_{H}-quasicentric if and only if it is F-quasicentric. Also, F_{1}^{•} ⊇ O_{∗}^{p}(F).

(c) If Γ is a p′-group, then S_{H} = S. A subgroup P ≤ S is F_{H}-centric (fully centralized in F_{H}, fully normalized in F_{H}) if and only if it is F-centric (fully centralized in F, fully normalized in F). Also, F_{1}^{•} ⊇ O_{∗}^{p′}(F).

Proof. Throughout the proof, (i)–(vii) refer to the conditions in Definition 3.6 and Lemma 3.7. If X is any set of morphisms of F, then we let Θ(X) be the union of the sets Θ(α) for α ∈ X. Condition (v) implies that for any P ≤ S and any subgroup A ≤ Aut_{F}(P), Θ(A) is a subgroup of Γ.

We first prove the following two additional properties of these subcategories:

(1) For each pair of subgroups P, Q ≤ S, and each φ ∈ Hom_{F}(P, Q), there are Q′ ≤ S, φ′ ∈ Hom_{F_{1}^{•}}(P, Q′), and α ∈ Aut_{F}(S) such that α(Q′) = Q and φ = (α|_{Q′}) ∘ φ′.

(2) For all P ≤ S there exists P′ ≤ S which is fully normalized in F, and φ ∈ Hom_{F_{1}^{•}}(N_{S}(P), N_{S}(P′)) such that φ(P) = P′.

By (vii), for any α ∈ Aut_{F}(S), any x ∈ Θ(α), and any φ ∈ Hom_{F}(P, Q), Θ(αφα^{−1}) = xΘ(φ)x^{−1}. In particular, Θ(αφα^{−1}) = 1 if and only if Θ(φ) = 1, and thus F_{1}^{•} is normalized by Aut_{F}(S).

Let O_{∗}^{p′}(F)^{c} ⊆ O_{∗}^{p′}(F) and O_{∗}^{p}(F)^{c} ⊆ O_{∗}^{p}(F) be the full subcategories whose objects are the F-centric subgroups of S. By (v), for each F-quasicentric subgroup P ≤ S, Aut_{F_{1}^{•}}(P) = Ker(Θ_{P}) where Θ_{P} is a homomorphism to a subquotient of Γ. Hence Aut_{F_{1}^{•}}(P) contains O^{p′}(Aut_{F}(P)) if Γ is a p′-group or O^{p}(Aut_{F}(P)) if Γ is a p-group. This shows that F_{1}^{•} contains either O_{∗}^{p′}(F)^{c} (if Γ is a p′-group) or O_{∗}^{p}(F)^{c} (if Γ is a pgroup). Thus F^{c} ⊆ ⟨F_{1}^{•}, Aut_{F}(S)⟩ by Lemma 3.4(b), and so F = ⟨F_{1}^{•}, Aut_{F}(S)⟩ since F is the smallest restrictive category over S which contains F^{c} by Theorem 1.5(a) (Alperin’s fusion theorem). Point (1) now follows from Lemma 3.4(c).

To see (2), recall that by Lemma 1.3, if P′′ is any subgroup F-conjugate to P and fully normalized in F, then there is a morphism φ′ ∈ Hom_{F}(N_{S}(P), N_{S}(P′′)) such that φ′(P) = P′′. By (1), φ′ = α ∘ φ for some α ∈ Aut_{F}(S) and some φ ∈ Hom_{F_{1}^{•}}(N_{S}(P), N_{S}(α^{−1}(P′′))). Also, the subgroup P′ = α^{−1}(P′′) is fully normalized in F since P′′ is.

(b) Assume Γ is a p-group, and fix H ≤ Γ. Consider the following set of subgroups of S_{H}:

Q_{H} = {P ≤ S_{H} | ∀ P′ F_{H}-conjugate to P, ∀ P′ ≤ Q ≤ P′·C_{S_{H}}(P′),

∀ α ∈ Aut_{F_{H}}(Q) such that α|_{P′} = Id, |α| is a power of p}.

This set clearly contains all F_{H}-centric subgroups of S_{H}. By Lemma 1.6, if F_{H} is saturated, then Q_{H} is precisely the set of F_{H}-quasicentric subgroups. We prove (b) here with “F_{H}-quasicentric” replaced by “element of Q_{H}”. This is all that will be needed for the proof that F_{H} is saturated; and once that is shown then (b) (in its original form) will follow.

Assume P ≤ S_{H} and P ∉ Q_{H}. Fix P′ which is F_{H}-conjugate to P, Q ≤ P′·C_{S_{H}}(P′) which contains P′, and Id_{Q} ≠ α ∈ Aut_{F_{H}}(Q) of order prime to p, such that α|_{P′} = Id_{P′}. Then P′, and hence P, is not F-quasicentric by Lemma 1.6(a).

We have shown that if P is F-quasicentric, then P ∈ Q_{H}, and it remains to check the converse. Assume P is not F-quasicentric: fix P′ ≤ Q ≤ P′·C_{S}(P′) and α ∈ Aut_{F}(Q) as in Lemma 1.6(b). In particular, Q is F-centric, and hence F-quasicentric. Set Q_{1} = Q ∩ S_{1} (where S_{1} = θ^{−1}(1)). Since 1 ∈ Θ(α) by (v) (and since |α| is prime to p), θ(g) = θ(α(g)) for all g ∈ Q by (iv), and thus α(Q_{1}) = Q_{1} and (since g^{−1}α(g) ∈ Ker(θ) = S_{1} for g ∈ Q) α induces the identity on Q/Q_{1}. Since |α| is not a power of p, it cannot be the identity on both Q_{1} and Q/Q_{1} (Lemma 1.15), and hence α|_{Q_{1}} ≠ Id_{Q_{1}}. Thus P′ ≤ P′Q_{1} ≤ P′·C_{S_{H}}(P′), α|_{P′Q_{1}} is a nontrivial automorphism of P′Q_{1} of order prime to p whose restriction to P′ is the identity.

Finally, by (1), P is F_{H}-conjugate to a subgroup P′′ for which there is some φ ∈ Iso_{F}(P′, P′′) which is the restriction of some φ ∈ Aut_{F}(S). Since P ≤ S_{H} and is F_{H}- conjugate to P′′, P′′ ≤ S_{H} by (iv) (applied with α ∈ Iso_{F_{H}}(P, P′′) and x ∈ Θ(α) ∩ H). Set Q′′ = φ(P′Q_{1}) = P′′·φ(Q_{1}) ≤ S_{H} and α′′ = φαφ^{−1} ∈ Aut_{F}(Q′′). Since Γ is a p-group and |α′′| is prime to p, (v) implies that Θ(α′′) = Θ(Id_{Q}), and hence that 1 ∈ Θ(α′′) and hence α′′ ∈ Aut_{F_{H}}(Q′′). But then P′′ ∉ Q_{H}, and hence P ∉ Q_{H}.

It remains to show that F_{1}^{•} ⊇ O_{∗}^{p}(F); i.e., to show that

Aut_{F_{1}^{•}}(P) ≥ O^{p}(Aut_{F}(P))  (3)

for each P ≤ S. If P ≤ S is F-quasicentric, then (3) holds by (v): Aut_{F_{1}^{•}}(P) is the kernel of a homomorphism from Aut_{F}(P) to a p-group. If P is not F-quasicentric but is fully centralized in F, then every automorphism α ∈ Aut_{F}(P) of order prime to p extends to an automorphism α ∈ Aut_{F}(P ·C_{S}(P)), which (after replacing it by an appropriate power) can also be assumed to have order prime to p, and hence in O^{p}(Aut_{F}(P ·C_{S}(P))). Thus (3) holds for P, since it holds for the F-centric subgroup P ·C_{S}(P). Finally, by (1), every subgroup of S is F_{1}^{•}-conjugate to a subgroup which is fully centralized in F, and thus (3) holds for all P ≤ S.

(c) This point holds, in fact, without assuming that the fusion system F be saturated. Assume Γ is a p′-group; then θ is the trivial homomorphism, and so S_{H} = θ^{−1}(H) = S for all H. Fix H ≤ Γ, and let P be any subgroup of S. Since each F_{H}-conjugacy class is contained in some F-conjugacy class, any subgroup which is fully centralized (fully normalized) in F is also fully centralized (fully normalized) in F_{H}. By the same reasoning, any F-centric subgroup P ≤ S is also F_{H}-centric.

Conversely, assume P is not fully centralized in F, and let P′ be a subgroup in the Fconjugacy class of P such that |C_{S}(P′)| > |C_{S}(P)|. Fix some φ ∈ Iso_{F}(P, P′). By (1), there are a subgroup P′′ ≤ S and isomorphisms α ∈ Aut_{F}(S) and φ′ ∈ Iso_{F_{1}}(P, P′′), such that α^{−1}(P′′) = P′ and φ = (α|_{P′′}) ∘ φ′. Thus P′′ is F_{H}-conjugate to P, and is F-conjugate to P′ via a restriction of an F-automorphism of S. Hence |C_{S}(P′′)| = |C_{S}(P′)| > |C_{S}(P)|, and this shows that P is not fully centralized in F_{H}. A similar argument shows that if P is not fully normalized in F (or not F-centric), then it is not fully normalized in F_{H} (or not F_{H}-centric).

The proof that F_{1}^{•} ⊇ O_{∗}^{p′}(F) is identical to the proof of the corresponding result in (b).

(a) Fix H ≤ Γ. Clearly, F_{H} is a fusion system over S_{H}; we must prove it is saturated. By (b) or (c), each F_{H}-centric subgroup of S_{H} is F-quasicentric. Hence by Theorem 1.5(b), it suffices to prove conditions (I) and (II) in Definition 1.2 for F-quasicentric subgroups P ≤ S_{H}. Thus we will be working only with subgroups P ≤ S_{H} for which Θ is defined on Hom_{F}(P, S).

Proof of (I): Assume Γ is a p′-group; thus S_{H} = S. If P is fully normalized in F_{H}, then by (c), P is fully normalized in F, hence it is fully centralized in F and in F_{H}. Also, Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)), and hence Aut_{S}(P) is also a Sylow p-subgroup of Aut_{F_{H}}(P). This proves (I) in this case.

Now assume Γ is a p-group. Fix P ≤ S_{H} which is F-quasicentric and fully normalized in F_{H}, and let α ∈ Hom_{F_{1}^{•}}(N_{S}(P), N_{S}(P′)) be as in (2). In particular, 1 ∈ Θ(α) by definition of F_{1}^{•}; so by (iv), θ(α(N_{S_{H}}(P))) = θ(N_{S_{H}}(P)) ≤ H. Hence α(N_{S_{H}}(P)) ≤ S_{H} ∩ N_{S}(P′) = N_{S_{H}}(P′), and this is an equality since P is fully normalized in F_{H}. In particular, this shows that P′ ≤ S_{H}. The conclusion of (I) holds for P (i.e., P is fully centralized in F_{H} and Aut_{S_{H}}(P) ∈ Syl_{p}(Aut_{F_{H}}(P))) if the conclusion of (I) holds for P′. So we can assume that P = P′ is fully normalized in F and in F_{H}.

Fix Q ≤ S_{H} which is F_{H}-conjugate to P and fully centralized in F_{H}, and choose an isomorphism ψ ∈ Iso_{F_{H}}(P, Q). After applying (2) again, we can assume that Q is also fully normalized in F, and hence also fully centralized. Hence ψ extends to some ψ_{1} ∈ Hom_{F}(P ·C_{S}(P), Q·C_{S}(Q)), which is an isomorphism since P and Q are both fully centralized in F. Fix h ∈ Θ(ψ^{−1}) ∩ H and g ∈ Θ(ψ_{1}). Then gh ∈ Θ(Id_{Q}) = θ(C_{S}(Q)) by (i) and (ii). Let a ∈ C_{S}(Q) be such that θ(a) = gh, and set

ψ_{2} = c_{a}^{−1} ∘ ψ_{1} ∈ Iso_{F}(P ·C_{S}(P), Q·C_{S}(Q)).

Thus ψ_{2}|_{P} = ψ_{1}|_{P} = ψ since a ∈ C_{S}(Q), and h^{−1} = θ(a)^{−1}g ∈ Θ(ψ_{2}) by (iii) and (i). By (iv), for all g ∈ P ·C_{S}(P), θ(ψ_{2}(g)) = h^{−1}θ(g)h, and hence ψ_{2}(g) ∈ H if and only if g ∈ H. Thus ψ_{2} sends C_{S_{H}}(P) onto C_{S_{H}}(Q). Since Q is fully centralized in F_{H}, so is P.

It remains to show that Aut_{S_{H}}(P) ∈ Syl_{p}(Aut_{F_{H}}(P)). Set P_{Θ} = Θ(Id_{P}) = θ(C_{S}(P)) for short (see (ii)). By (v), Θ restricts to a homomorphism

Θ_{P} : Aut_{F}(P) ⟶ N_{Γ}(P_{Θ})/P_{Θ}.

Set

Ĥ = {hP_{Θ} | h ∈ H ∩ N_{Γ}(P_{Θ})} ≤ N_{Γ}(P_{Θ})/P_{Θ};

then Aut_{F_{H}}(P) = Θ_{P}^{−1}(Ĥ) by definition of F_{H}. For any φ = c_{a} ∈ Aut_{S}(P)∩Aut_{F_{H}}(P) (where a ∈ N_{S}(P)), Θ(φ) = θ(a)·θ(C_{S}(P)) by (ii) and (iii); and thus Θ(φ) contains some h ∈ H where h = θ(b) for some b ∈ a·C_{S}(P). Hence b ∈ S_{H}, and φ = c_{b} ∈ Aut_{S_{H}}(P). This shows that

Aut_{S_{H}}(P) = Aut_{S}(P) ∩ Aut_{F_{H}}(P) = Aut_{S}(P) ∩ Θ_{P}^{−1}(Ĥ).

Also, Θ_{P} (Aut_{S}(P)) = Θ_{P} (Aut_{F}(P)), since Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) and Γ is a pgroup, and hence

[Aut_{S}(P) : Aut_{S_{H}}(P)] = [Im(Θ_{P}) : Ĥ] = [Aut_{F}(P) : Aut_{F_{H}}(P)].

Since Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) (i.e., [Aut_{F}(P) : Aut_{S}(P)] is prime to p), this implies that Aut_{S_{H}}(P) ∈ Syl_{p}(Aut_{F_{H}}(P)).

Proof of (II): Fix a morphism φ ∈ Iso_{F_{H}}(P, Q), for some F-quasicentric P, Q ≤ S_{H} such that Q = φ(P) is fully centralized in F_{H}, and set N_{φ} = {g ∈ N_{S_{H}}(P) | φc_{g}φ^{−1} ∈ Aut_{S_{H}}(Q)}. By (2), there is a subgroup Q′ fully normalized in F, and a morphism ψ ∈ Hom_{F_{1}^{•}}(N_{S}(Q), N_{S}(Q′)), such that ψ(Q) = Q′. By condition (II) for the saturated fusion system F, there is φ_{1} ∈ Hom_{F}(N_{φ}, N_{S}(Q′)) such that φ_{1}|_{P} = ψ ∘ φ. Fix some

x ∈ Θ(φ_{1}) ⊆ Θ(ψφ) = Θ(φ)

(where the last equality holds since 1 ∈ Θ(ψ)). Since φ ∈ Mor(F_{H}), there is h ∈ H such that h ∈ Θ(φ), and thus

hx^{−1} ∈ Θ(Id_{Q′}) = θ(C_{S}(Q′)).

Fix a ∈ C_{S}(Q′) such that hx^{−1} = θ(a), and set

φ_{2} = c_{a} ∘ φ_{1} ∈ Hom_{F}(N_{φ}, N_{S}(Q′)).

Then by (i), h ∈ Θ(φ_{2}), so φ_{2} ∈ Hom_{F_{H}}(N_{φ}, N_{S_{H}}(Q′)); and φ_{2}|_{P} = φ_{1}|_{P} since a ∈ C_{S}(Q′). Since Q is fully centralized in F_{H}, ψ sends C_{S_{H}}(Q) isomorphically onto C_{S_{H}}(Q′); and hence (by definition of N_{φ}) φ_{2}(N_{φ}) ≤ Im(ψ). So φ =^{def} ψ^{−1} ∘ φ_{2} sends N_{φ} into N_{S_{H}}(Q) and extends φ. □

We next extend Proposition 3.8 to a result about linking systems and p-local finite groups. The main point of the following theorem is that for any p-local finite group

(S, F, L) and any epimorphism π_{1}(|L|) θ̂ ; −↠ Γ, where Γ is a finite p-group or p′-group, there is another p-local finite group (S_{H}, F_{H}, L_{H}) for each subgroup H ≤ Γ, such that |L_{H}| is homotopy equivalent to the covering space of |L| whose fundamental group is θ̂^{−1}(H).

Recall that for any p-local finite group (S, F, L) with associated quasicentric linking system L^{q}, j: S → π_{1}(|L|) ≅ π_{1}(|L^{q}|) denotes the homomorphism induced by the distinguished monomorphism δ_{S}: S → Aut_{L}(S), and J: L^{q} → B(π_{1}(|L^{q}|)) is the functor which sends morphisms to loops as defined in Section 1. Let θ̂ be a homomorphism from π_{1}(|L^{q}|) to a group Γ, and set

θ = θ̂ ∘ j ∈ Hom(S, Γ) and Θ̂ = B(θ̂) ∘ J: L^{q} ⟶ B(Γ).

Note that these depend on a choice of a compatible set of inclusions {ι_{P}^{Q}} for L^{q} (since J depends on such a choice). For any subgroup H ≤ Γ, let L_{H}^{•} ⊆ L^{q} be the subcategory with the same objects and with morphism set Θ̂^{−1}(H); and let L_{H}^{q} ⊆ L_{H}^{•} be the full subcategory obtained by restricting to subgroups of S_{H} =^{def} θ^{−1}(H). Finally, let F_{H} be the fusion system over S_{H} generated by π(L_{H}^{q}) ⊆ F^{q} and restrictions of morphisms, and let L_{H} ⊆ L_{H}^{q} be the full subcategory on those objects which are F_{H}-centric.

Theorem 3.9. Let (S, F, L) be a p-local finite group, let L^{q} be its associated quasicentric linking system, and let π: L^{q} → F be the projection. Assume a compatible set of inclusions {ι_{P}^{Q}} has been chosen for L^{q}. Fix a finite group Γ which is a p-group or a p′-group, and a surjective homomorphism

θ̂: π_{1}(|L^{q}|) −↠ Γ.

Set θ = θ̂∘j: S ⟶ Γ. Fix H ≤ Γ, and set S_{H} = θ^{−1}(H). Then (S_{H}, F_{H}, L_{H}) is also a p-local finite group, and (via the inclusion of L_{H} into L^{q}) |L_{H}| is homotopy equivalent to the covering space of |L^{q}| ≃ |L| with fundamental group θ̂^{−1}(H).

Proof. Define Θ: Mor(F^{q}) → 𝔖𝔲𝔟(Γ) by setting Θ(α) = Θ̂(π^{−1}(α)), where Θ̂ = B(θ̂) ∘ J as above. Then Θ and θ satisfy hypotheses (i)–(iv) of Definition 3.6: points (i) and (ii) follow from (A)_{q}, while (iii) follows from Proposition 1.13 and (iv) from (C)_{q}. Thus (Γ, θ, Θ) is a fusion mapping triple on F^{q}, and F_{H} is a saturated fusion system over S_{H} by Proposition 3.8.

Let O_{∗}^{p}(F) and O_{∗}^{p′}(F) be the categories of Definition 3.3, and let O_{∗}^{p′}(F)^{q} ⊆ O_{∗}^{p′}(F) and O_{∗}^{p}(F)^{q} ⊆ O_{∗}^{p}(F) be the full subcategories whose objects are the F-quasicentric subgroups of S. By Proposition 3.8(b,c), π(L_{1}^{•}) contains O_{∗}^{p′}(F)^{q} (if Γ is a p′-group) or O_{∗}^{p}(F)^{q} (if Γ is a p-group). Hence in either case, by Lemma 3.4(b), all morphisms in L^{q} are composites of morphisms in L_{1}^{•} and restrictions of morphisms in Aut_{L}(S). Since L_{1}^{•} = Θ̂^{−1}(1), by definition, and since Θ̂(α) = Θ̂(β) whenever α is a restriction of β, this shows that Θ̂ restricts to a surjection of Aut_{L}(S) onto Γ. In particular, this implies that

∀ P ∈ Ob(L^{q}) and ∀ g ∈ Γ, ∃ P′ ≤ S and α ∈ Iso_{L^{q}}(P, P′) such that Θ̂(α) = g; (1)

where in fact, α can always be chosen to be the restriction of an automorphism of S.

We start by proving that L_{H}^{q} is a linking system associated to F_{H} (for its set of objects), and hence that L_{H} is a centric linking system. Let P ≤ S_{H} be a F-quasicentric subgroup, and choose g ∈ P ·C_{S}(P). By construction, Θ̂(δ_{S}(g)) = θ(g), and Θ̂(ι_{P}) = 1. In particular, the inclusion morphisms are in L_{H}. Also, ι_{P} ∘ δ_{P} (g) = δ_{S}(g) ∘ ι_{P} by definition of an inclusion morphism (Definition 1.11). Hence Θ̂(δ_{P} (g)) = θ(g) in this situation; and in particular δ_{P} (g) ∈ Aut_{L_{H}^{q}}(P) if and only if g ∈ S_{H}. Thus δ_{P} restricts to a distinguished monomorphism P ·C_{S_{H}}(P) ⟶ Aut_{L_{H}^{q}}(P) for L_{H}^{q} and axiom (D)_{q} is satisfied. Moreover, if f, f′ ∈ Mor_{L_{H}^{q}}(P, Q) are such that π(f) = π(f′) in Hom_{F}(P, Q), and g ∈ C_{S}(P) is the unique element such that f′ = f ∘ δ_{P} (g) (using axiom (A)_{q} for L^{q}), then Θ̂(δ_{P} (g)) = 1, and hence g ∈ S_{H}. This shows that axiom (A)_{q} for L_{H}^{q} holds. Axioms (B)_{q} and (C)_{q} for L_{H}^{q} also follow immediately from the same properties for L^{q}.

We have now shown that (S_{H}, F_{H}, L_{H}) is a p-local finite group. It remains to show that |L_{H}| is homotopy equivalent to a certain covering space of |L^{q}| ≃ |L|. We show this by first choosing certain full subcategories L^{x} ⊆ L^{q} and L_{H}^{x} ⊆ L_{H}^{q} such that |L^{x}| ≃ |L^{q}| ≃ |L| and |L_{H}^{x}| ≃ |L_{H}^{q}| ≃ |L_{H}|, and then proving directly that |L_{H}^{x}| is a covering space of |L^{x}|.

If Γ is a p′-group, then for any P ≤ S_{H} = S, P is F_{H}-centric if and only if it is F-centric (Proposition 3.8(c)). By the above remarks, L_{H} is a centric linking system associated to F_{H}. Set L^{x} = L and L_{H}^{x} = L_{H} in this case.

If Γ is a p-group, then for any P ≤ S_{H}, P is F_{H}-quasicentric if and only if it is F-quasicentric (Proposition 3.8(b)). So by what was just shown, L_{H}^{q} is a quasicentric linking system associated to F_{H} which extends L_{H}. Let L^{x} ⊆ L^{q} be the full subcategory whose objects are those subgroups P ≤ S such that P ∩ S_{H} is F-quasicentric. Set S_{1} = Ker(θ), and let F_{1} ⊆ F be the saturated fusion system over S_{1} defined in Proposition 3.8. The definition of θ as a restriction of Θ̂ ensures that θ(g) and θ(g′) are Γ-conjugate whenever g, g′ are F-conjugate; in particular, no element of S_{1} is Fconjugate to any element of S∖S_{1}. Hence by Lemma 3.5, for each P ≤ S which is F-centric and F-radical, P ∩ S_{1} is F_{1}-centric, hence F-quasicentric, so P ∩ S_{H} is Fquasicentric, and thus P ∈ Ob(L^{x}). So the inclusion of |L^{x}| in |L^{q}| is a homotopy equivalence by Proposition 1.12.

Still assuming Γ is a p-group, let L_{H}^{x} ⊆ L^{x} be the subcategory with the same objects, where Mor(L_{H}^{x}) = Θ̂^{−1}(H). For each P ∈ Ob(L_{H}^{x}) = Ob(L^{x}), P ∩S_{H} is F-quasicentric by assumption, hence F_{H}-quasicentric; and by Proposition 1.13, each φ ∈ Mor_{L_{H}^{x}}(P, Q) restricts to a unique morphism φ_{H} ∈ Mor_{L_{H}^{x}}(P ∩S_{H}, Q∩S_{H}). These restrictions define a deformation retraction from |L_{H}^{x}| to |L_{H}^{q}|, and thus the inclusion of categories induces a homotopy equivalence |L_{H}^{x}| ≃ |L_{H}^{q}| ≃ |L_{H}|.

Thus in both cases, we have chosen categories L_{H}^{x} ⊆ L^{x} with the same objects, where L^{x} is a full subcategory of L^{q} and Mor(L_{H}^{x}) = Mor(L^{x}) ∩ Θ̂^{−1}(H), and where |L^{x}| ≃ |L| and |L_{H}^{x}| ≃ |L_{H}|. Let E_{Γ}(Γ/H) be the category with object set Γ/H, and with a morphism ǧ from aH to gaH for each g ∈ Γ and aH ∈ Γ/H. Thus Aut_{E_{Γ}(Γ/H)}(1·H) ≅ H, and |E_{Γ}(Γ/H)| = EG/H ≃ BH.

Let L̃ be the pullback category in the following square:

\begin{aligned}L̃ & \overline{→} & E_{Γ}(Γ/H) \\ L^{x} & ↓ & \overline{→} & B(Γ) , & ↓ \end{aligned}

Thus Ob(L̃) = Ob(L^{x}) × Γ/H, and Mor(L̃) is the set of pairs of morphisms in L^{x} and E_{Γ}(Γ/H) which get sent to the same morphism in B(Γ). Then L_{H}^{x} can be identified with the full subcategory of L̃ with objects the pairs (P, 1·H) for P ∈ Ob(L^{x}). By (1), each object in L̃ is isomorphic to an object in L_{H}^{x}, and so |L_{H}^{x}| ≃ |L̃|. By construction, |L̃| is the covering space over |L^{x}| with fundamental group θ^{−1}(H). Since |L̃| ≃ |L_{H}| and |L^{x}| ≃ |L|, this finishes the proof of the last statement. □

The following is an immediate corollary to Theorem 3.9.

Corollary 3.10. For any p-local finite group (S, F, L), any finite p-solvable group Γ, and any homomorphism

θ̂: π_{1}(|L|) ⟶ Γ,

there is a p-local finite group (S_{0}, F_{0}, L_{0}) such that |L_{0}| is homotopy equivalent to the covering space of |L| with fundamental group Ker(θ̂). Furthermore, this can be chosen such that S_{0} is a subgroup of S and F_{0} is a subcategory of F.

For any p-local finite group (S, F, L), since F is a finite category, Corollary 3.10 implies that there is a unique maximal p-solvable quotient group of π_{1}(|L|), which is finite. In contrast, if we look at arbitrary finite quotient groups of the fundamental group, they can be arbitrarily large.

As one example, consider the case where p = 2, S ∈ Syl_{2}(A_{6}) (so S ≅ D_{8}), F = F_{S}(A_{6}), and L = L_{S}^{c}(A_{6}). It is not hard to show directly, using Van Kampen’s theorem, that π_{1}(|L|) ≅ Σ_{4}∗_{S}Σ_{4}: the amalgamated free product of two copies of Σ_{4} intersecting

in S, where each of the two subgroups C_{2}^{2} in D_{8} is normalized by one of the Σ_{4}. Thus π_{1}(|L|) surjects onto any finite group Γ which is generated by two copies of Σ_{4} intersecting in the same way. This is the case when Γ = A_{6}, and also when Γ = P SL_{2}(q) for any q ≡ ±9 (mod 16). However, the kernel of any such homomorphism defined on π_{1}(|L|) is torsion free (and infinite), and hence cannot be the fundamental group of the geometric realization of any centric linking system. In fact, in this case, there is no nontrivial homomorphism from π_{1}(|L|) to a finite 2-solvable group.

4. Fusion subsystems and extensions of p-power index

Recall that for any saturated fusion system F over a p-group S, we defined O_{F}^{p}(S) ◁ S to be the subgroup generated by all elements of the form g^{−1}α(g), for g ∈ P ≤ S, and α ∈ Aut_{F}(P) of order prime to p. In this section, we classify all saturated fusion subsystems of p-power index in a given saturated fusion system F over S, and show that there is a one-to-one correspondence between such subsystems and the subgroups of S which contain O_{F}^{p}(S). In particular, there is a unique minimal subsystem O^{p}(F) of this type, which is a fusion system over O_{F}^{p}(S). We then look at extensions, and describe the procedure for finding all larger saturated fusion systems of which F is a fusion subsystem of p-power index.

\underline{4.1 Subsystems of p-power index} In this subsection, we classify all saturated fusion subsystems of p-power index in a given saturated fusion system F, and show that there is a unique minimal subsystem O^{p}(F) of this type. We also show that there is a bijective correspondence between subgroups T of the finite p-group

Γ_{p}(F) =^{def} S/O_{F}^{p}(S),

and fusion subsystems F_{T} of p-power index in F. We have already seen that Γ_{p}(F) = S/O_{F}^{p}(S) is isomorphic to π_{1}(|L|_{p}^{∧}) for any centric linking system L associated to F, and our result also shows that all (connected) covering spaces of |L|_{p}^{∧} are realized as classifying spaces of subsystems of p-power index. The index of F_{T} in F can then be defined to be the index of T in Γ_{p}(F), or equivalently the covering degree of the covering space.

The main step in doing this is to construct a fusion mapping triple (Γ_{p}(F), θ, Θ) for F^{q}, where θ is the canonical surjection of S onto Γ_{p}(F). This construction parallels very closely the construction in Proposition 2.4 of a functor from L^{q} to B(Γ_{p}(F)), when L is a linking system associated to F. In fact, we could in principal state and prove the two results simultaneously, but the extra terminology which that would require seemed to add more complications than would be saved by combining the two.

The following lemma provides a very general, inductive tool for constructing explicit fusion mapping triples.

Lemma 4.1. Fix a saturated fusion system F over a p-group S. Let H_{0} be a set of F-quasicentric subgroups of S which is closed under F-conjugacy and overgroups. Let P be an F-conjugacy class of F-quasicentric subgroups maximal among those not in H_{0}, set H = H_{0} ∪ P, and let F^{H_{0}} ⊆ F^{H} ⊆ F^{q} be the full subcategories with these objects. Fix a group Γ and a homomorphism θ: S → Γ, and let

Θ_{0} : Mor(F^{H_{0}}) ⟶ 𝔖𝔲𝔟(Γ)

be such that (Γ, θ, Θ_{0}) is a fusion mapping triple for F^{H_{0}}. Let P ∈ P be fully normalized in F, and fix a homomorphism

Θ_{P} : Aut_{F}(P) ⟶ N_{Γ}(θ(C_{S}(P)))/θ(C_{S}(P))

such that the following two conditions hold:

(+) xθ(g)x^{−1} = θ(α(g)) for all g ∈ P, α ∈ Aut_{F}(P), and x ∈ Θ_{P} (α).

(∗) For all P ≨ Q ≤ S such that P ◁ Q and Q is fully normalized in N_{F}(P), and for all α ∈ Aut_{F}(P) and β ∈ Aut_{F}(Q) such that α = β|_{P}, Θ_{P} (α) ⊇ Θ_{0}(β).

Then there is a unique extension of Θ_{0} to a fusion mapping triple (Γ, θ, Θ) for F^{H} such that Θ(α) = Θ_{P} (α) for all α ∈ Aut_{F}(P).

Proof. Note that (+) is just point (v) of Lemma 3.7 applied to the subgroup P, while (∗) is just point (vi) applied to restrictions to P. So both of these conditions are necessary if we want to be able to extend Θ_{0} and Θ_{P} to a fusion mapping triple for F^{H}.

The uniqueness of the extension is an immediate consequence of Alperin’s fusion theorem, in the form of Theorem 1.5(a). The proof of existence is almost identical to the proof of Lemma 2.3, so we just sketch it here briefly.

We first show that we can replace (∗) by the following (a priori stronger) statement:

(∗∗) for all Q, Q′ ≤ S which strictly contain P, and for all β ∈ Hom_{F}(Q, Q′) and α ∈ Aut_{F}(P) such that α = β|_{P}, Θ_{P} (α) ⊇ Θ_{0}(β).

It suffices to show this when P is normal in Q and Q′, since otherwise we can replace Q and Q′ by N_{Q}(P) and N_{Q′}(P). In this case, β ∈ Hom_{N_{F} (P)}(Q, Q′), and by Theorem 1.5(a) (Alperin’s fusion theorem), it is a composite of restrictions of automorphisms of subgroups fully normalized in N_{F}(P). So it suffices to prove (∗∗) when β is such an automorphism, and this is what is assumed in (∗).

Now fix any morphism φ ∈ Hom_{F}(P_{1}, Q) which lies in F^{H} but not in F^{H_{0}}; thus P_{1} ∈ P. Set P_{2} = φ(P_{1}) ≤ Q, and let φ′ ∈ Iso_{F}(P_{1}, P_{2}) be the “restriction” of φ. By Lemma 1.3, there are isomorphisms φ_{i} ∈ Iso_{F^{q}}(N_{S}(P_{i}), N_{i}), for some N_{i} ≤ N_{S}(P) containing P, which restrict to isomorphisms φ_{i} ∈ Iso_{F^{q}}(P_{i}, P). Fix elements x_{i} ∈ Θ_{0}(φ_{i}). Set ψ = φ_{2} ∘ φ′ ∘ φ_{1}^{−1} ∈ Aut_{F^{q}}(P). Thus φ′ = φ_{2}^{−1} ∘ ψ ∘ φ_{1}, and we define

Θ(φ) = Θ(φ′) = x_{2}^{−1}·Θ_{P} (ψ)·x_{1}.

This is independent of the choice of x_{i}, since Θ_{0}(φ_{i}) ⊆ θ(C_{S}(P))·x_{i} by axioms (i) and (ii) in the definition of a fusion mapping triple. It is independent of the choice of φ_{1} and φ_{2} by the same argument as was used in the proof of Lemma 2.3 (and this is where we need point (∗∗)). Conditions (i)–(iv) are easily checked. For example, (iv) — the condition that xθ(g)x^{−1} = θ(α(g)) whenever g ∈ P_{1}, φ ∈ Hom_{F}(P_{1}, P_{2}), and x ∈ Θ(φ)

— holds when φ can be extended to a larger subgroup since (Γ, θ, Θ_{0}) is already a fusion mapping triple, holds for φ ∈ Aut_{F}(P) by (+), and thus holds in the general case since Θ(φ) was defined via a composition of such morphisms. Thus (Γ, θ, Θ) is a fusion mapping triple on F^{H}. □

The construction of a fusion mapping triple to Γ_{p}(F) in the following lemma is a first application of Lemma 4.1. Another application will be given in the next section.

Lemma 4.2. Let F be a saturated fusion system over a p-group S, and let

θ: S ⟶ Γ_{p}(F) = S/O_{F}^{p}(S)

be the projection. Then there is a fusion mapping triple (Γ_{p}(F), θ, Θ) on F^{q}.

Proof. The function Θ will be constructed inductively, using Lemma 4.1. Let H_{0} ⊆ Ob(F^{q}) be a subset (possibly empty) which is closed under F-conjugacy and overgroups. Let P be an F-conjugacy class of F-quasicentric subgroups maximal among those not in H_{0}, set H = H_{0} ∪ P, and let F^{H_{0}} ⊆ F^{H} ⊆ F^{q} be the full subcategories with these objects. Assume we have already constructed a fusion mapping triple (Γ_{p}(F), θ, Θ_{0}) for F^{H_{0}}.

We recall the notation of Lemma 2.2. If G is any finite group, and S ∈ Syl_{p}(G), then

O_{G}^{p}(S) =^{def} ⟨[g, x] | g ∈ P ≤ S, x ∈ N_{G}(P) of order prime to p⟩.

By Lemma 2.2, O_{G}^{p}(S) = S ∩ O^{p}(G), and hence G/O^{p}(G) ≅ S/O_{G}^{p}(S).

Fix P ∈ P which is fully normalized in F. Let N_{0} be the subgroup generated by commutators [g, x] for g ∈ N_{S}(P) and x ∈ N_{Aut_{F} (P)}(N_{S}(P)) of order prime to p. Then Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) and Aut_{N_{0}}(P) = O_{Aut_{F} (P)}^{p}(Aut_{S}(P)), and by Lemma 2.2,

Aut_{F}(P)/O^{p}(Aut_{F}(P)) ≅ Aut_{S}(P)/Aut_{N_{0}}(P) ≅ N_{S}(P)/⟨N_{0}, C_{S}(P)⟩.

Also, N_{0} ≤ O_{F}^{p}(S), and so the inclusion of N_{S}(P) into S induces a homomorphism

Θ_{P} : Aut_{F}(P) −↠ Aut_{F}(P)/O^{p}(Aut_{F}(P))_{≅N_{S}(P)/⟨N_{0},C_{S}(P)⟩} ⟶ N_{Γ_{p}(F)}(θ(C_{S}(P)))/θ(C_{S}(P))_{≅N_{S}(C_{S}(P)·S_{0})/(C_{S}(P)·S_{0})}.

Here, we write S_{0} = O_{F}^{p}(S) for short; thus Γ_{p}(F) = S/S_{0} and θ(C_{S}(P)) = C_{S}(P)·S_{0}/S_{0}.

Point (+) in Lemma 4.1 holds by the construction of Θ_{P}. So it remains only to prove that condition (∗) in Lemma 4.1 holds.

To see this, fix P ≨ Q ≤ S such that P ◁ Q and Q is fully normalized in N_{F}(P), and fix α ∈ Aut_{F^{q}}(P) and β ∈ Aut_{F^{q}}(Q) such that α = β|_{P}. We must show that Θ_{P} (α) ⊇ Θ_{0}(β). Upon replacing α by α^{k} and β by β^{k} for some appropriate k ≡ 1 (mod p), we can assume that both automorphisms have order a power of p. Since Q is fully normalized, Aut_{N_{S}(P)}(Q) is a Sylow subgroup of Aut_{N_{F} (P)}(Q); and hence there are automorphisms γ ∈ Aut_{F}(Q) and γ ∈ Aut_{F}(P) of order prime to p such that γ = γ|_{P} and γβγ^{−1} = c_{g}|_{Q} for some g ∈ N_{S}(Q) ∩ N_{S}(P). Then γαγ^{−1} = c_{g}|_{P}. Also, 1 ∈ Θ_{P} (γ) and 1 ∈ Θ_{0}(γ), since both automorphisms have order prime to p. So

Θ_{0}(β) = Θ(c_{g}|_{Q}) = g·θ(C_{S}(Q)) ⊆ g·θ(C_{S}(P)) = Θ_{P} (c_{g}|_{P}) = Θ_{P} (α).

Thus, by Lemma 2.3, we can extend Θ_{0} to a fusion mapping triple on F^{H}. Upon continuing this procedure, we obtain a fusion mapping triple defined on all of F^{q}. □

We now apply Lemma 4.2 to classify fusion subsystems of p-power index. Recall that by definition, if F is a saturated fusion system over S and F_{0} ⊆ F is a fusion subsystem over S_{0} ≤ S, then F has p-power index if and only if S_{0} ≥ O_{F}^{p}(S), and Aut_{F_{0}}(P) ≥ O^{p}(Aut_{F}(P)) for all P ≤ S_{0}.

Theorem 4.3. Fix a saturated fusion system F over a p-group S. Then for each subgroup T ≤ S containing O_{F}^{p}(S), there is a unique saturated fusion system F_{T} ⊆ F over T with p-power index. For each such T, F_{T} has the properties:

(a) a subgroup P ≤ T is F_{T} -quasicentric if and only if it is F-quasicentric; and

(b) for each pair P, Q ≤ T of F-quasicentric subgroups,

Hom_{F_{T}} (P, Q) = {φ ∈ Hom_{F}(P, Q) | Θ(φ) ∩ (T/O_{F}^{p}(S)) ≠ ∅}.

Here,

Θ: Mor(F^{q}) ⟶ 𝔖𝔲𝔟(Γ_{p}(F)) = 𝔖𝔲𝔟(S/O_{F}^{p}(S))

is the map of Lemma 4.2.

Proof. Let F_{T} ⊆ F be the fusion system over T defined on F-quasicentric subgroups by the formula in (b), and then extended to arbitrary subgroups by taking restrictions and composites. (This is the fusion system denoted F_{T/O_{F}^{p} (S)} in Proposition 3.8, but we simplify the notation here.) By Proposition 3.8(a,b) (applied with Γ = Γ_{p}(F) and H = T/O_{F}^{p}(S)), F_{T} is saturated, a subgroup P ≤ T is F_{T} -quasicentric if and only if it is F-quasicentric, and Aut_{F_{T}} (P) ≥ O^{p}(Aut_{F}(P)) for all P ≤ T.

Now let F′_{T} ⊆ F be another saturated subsystem over the same subgroup T which also has p-power index. We claim that F′_{T} = F_{T}, and thus that F_{T} is the unique subsystem with these properties. By assumption, for each P ≤ T, Aut_{F_{T}} (P) and Aut_{F′_{T}} (P) both contain O^{p}(Aut_{F}(P)), and hence each is generated by O^{p}(Aut_{F}(P)) and any one of its Sylow p-subgroups. So if P is fully normalized in F_{T} and F′_{T} both, then

Aut_{F_{T}} (P) = ⟨O^{p}(Aut_{F}(P)), Aut_{T} (P)⟩ = Aut_{F′_{T}} (P).  (1)

In particular, Aut_{F_{T}} (T) = Aut_{F′_{T}} (T). Set p^{k} = |T |, fix 0 ≤ m < k, and assume inductively that Hom_{F_{T}} (P, Q) = Hom_{F′_{T}} (P, Q) for all P, Q ≤ T of order > p^{m}. By Alperin’s fusion theorem for saturated fusion systems (Theorem 1.5(a)), if |P | = p^{m}, |Q| ≥ p^{m}, and P ≠ Q, then all morphisms in Hom_{F_{T}} (P, Q) and Hom_{F′_{T}} (P, Q) are composites of restrictions of morphisms between subgroups of order > p^{m}, and hence Hom_{F_{T}} (P, Q) = Hom_{F′_{T}} (P, Q) by the induction hypothesis. In particular, two subgroups of order p^{m} are F_{T} -conjugate if and only if they are F′_{T} -conjugate. So for any P ≤ T of order p^{m}, P is fully normalized in F_{T} if and only if it is fully normalized in F′_{T}. In either case, Aut_{F_{T}} (P) = Aut_{F′_{T}} (P): by (1) if P is fully normalized, and by Alperin’s fusion theorem again (and the induction hypothesis) if it is not. □

We can now define O^{p}(F) as the minimal fusion subsystem of F of p-power index: the unique fusion subsystem over O_{F}^{p}(S) of p-power index. The next theorem will show that when F has an associated linking system L, then O^{p}(F) has an associated linking system O^{p}(L), and that |O^{p}(L)|_{p}^{∧} is the universal cover of |L|_{p}^{∧}.

Theorem 4.4. Fix a p-local finite group (S, F, L). Then for each subgroup T ≤ S containing O_{F}^{p}(S), there is a unique p-local finite subgroup (T, F_{T} , L_{T}) such that F_{T} has p-power index in F, and such that L_{T}^{q} = π^{−1}(F_{T}^{q}) where π is the usual projection of L^{q} onto F^{q}. Furthermore, |L_{T} | is homotopy equivalent, via the inclusion of |L_{T}^{q} | ≃ |L_{T} | into |L^{q}| ≃ |L|, to a covering space of |L| of degree [S : T]. Hence the classifying space |L_{T} |_{p}^{∧} of (T, F_{T} , L_{T}) is homotopy equivalent to the covering space of |L|_{p}^{∧} with fundamental group T/O_{F}^{p}(S).

Proof. Assume a compatible set of inclusions {ι_{P}^{Q}} has been chosen for L^{q}. By Proposition 2.4, there is a functor λ: L^{q} → B(Γ_{p}(F)) which sends inclusions to the identity, and such that λ(δ_{S}(g)) = g for all g ∈ S. Hence by Theorem 3.9, (T, F_{T} , L_{T}) is a p-local finite group, and |L_{T} | is a covering space of |L|. Also, if we write L_{1} for the linking system over O_{F}^{p}(S), then the fibration sequences

|L_{1}| → |L| → BΓ_{p}(F) and |L_{1}| → |L_{T} | → B(T/O_{F}^{p}(S))

are still fibration sequences after p-completion [BK, II.5.2(iv)], and hence |L_{T} |_{p}^{∧} is the covering space of |L|_{p}^{∧} with fundamental group T/O_{F}^{p}(S). The uniqueness follows from Theorem 4.3. □

Thus there is a bijective correspondence between fusion subsystems of (S, F), or p-local finite subgroups of (S, F, L), of p-power index, and subgroups of S/O_{F}^{p}(S) ≅ π_{1}(|L|_{p}^{∧}). The classifying spaces of the p-local finite subgroups of (S, F, L) of p-power index are (up to homotopy) just the covering spaces of the classifying space of (S, F, L).

\underline{4.2 Extensions of p-power index} We next consider the opposite problem: how to construct extensions of p-power index of a given p-local finite group. In the course of this construction, we will see that the linking system really is needed to construct an extension of the fusion system. The following definition will be useful.

Definition 4.5. Fix a saturated fusion system F over a p-group S. An automorphism α ∈ Aut(S) is fusion preserving if it normalizes F; i.e., if it induces an automorphism of the category F by sending P to α(P) and φ ∈ Mor(F) to αφα^{−1} ∈ Mor(F). Let Aut_{fus}(S, F) ≤ Aut(S) denote the group of all fusion preserving automorphisms, and set

Out_{fus}(S, F) = Aut_{fus}(S, F)/ Aut_{F}(S).

We first describe the algebraic data needed to determine extensions of p-power index. Fix a p-local finite group (S, F, L), let L^{q} be the associated quasicentric linking system, and let {ι_{P}^{Q}} be a compatible set of inclusions. Then for any g ∈ S, g acts on the set Mor(L^{q}) by composing on the left or right with δ_{S}(g) and its restrictions. Thus for any φ ∈ Mor_{L^{q}}(P, Q), we set

gφ = δ_{Q,gQg^{−1}}(g) ∘ φ ∈ Mor_{L^{q}}(P, gQg^{−1})

and

φg = φ ∘ δ_{g^{−1}P g,P} (g) ∈ Mor_{L^{q}}(g^{−1}P g, Q).

This defines natural left and right actions of S on the set Mor(L^{q}). The resulting conjugation action φ ↦ gφg^{−1} extends to an action on the category, where g sends an object P to gP g^{−1}. The functor π: L^{q} → F^{q} is equivariant with respect to the conjugation action of S on L^{q} and the action of Inn(S) ≤ Aut_{fus}(S, F) on F.

If (S_{0}, F_{0}, L_{0}) is contained in (S, F, L) with p-power index, and S_{0} ◁ S, then the S action on L clearly restricts to an S-action on L_{0}. The following theorem provides a converse to this. Given a p-local finite group (S_{0}, F_{0}, L_{0}), an extension S of S_{0}, and an S-action on L which satisfies certain obvious compatibility conditions, this data always determines a p-local finite group which contains (S_{0}, F_{0}, L_{0}) with p-power index.

Theorem 4.6. Fix a p-local finite group (S_{0}, F_{0}, L_{0}), and assume that a compatible set of inclusions {ι_{P}^{Q}} has been chosen for L_{0}. Fix a p-group S such that S_{0} ◁ S and Aut_{S}(S_{0}) ≤ Aut_{fus}(S_{0}, F_{0}), and an action of S on L_{0} which:

(a) extends the conjugation action of S_{0} on L_{0};

(b) makes the canonical monomorphism δ_{S_{0}} : S_{0} ⟶ Aut_{L_{0}}(S_{0}) S-equivariant; (c) makes the projection π: L_{0} ⟶ F_{0} S-equivariant with respect to the Aut_{S}(S_{0})-

action on F_{0}; and

(d) sends inclusion morphisms in L_{0} to inclusion morphisms.

Then there is a p-local finite group (S, F, L) such that F ⊇ F_{0}, L^{q} ⊇ L_{0}, the conjugation action of S on L^{q} restricts to the given S-action on L_{0}, and (S_{0}, F_{0}, L_{0}) is a subgroup of p-power index in (S, F, L).

Proof. Set

H_{0} = Ob(L_{0}) = {P ≤ S_{0} | P is F_{0}-centric} and H = {P ≤ S | P ∩ S_{0} ∈ H_{0}}.

To simplify notation, for any P ≤ S, we write P_{0} = P ∩ S_{0}. For g ∈ S and φ ∈ Mor_{L_{0}}(P, Q), we write gφg^{−1} ∈ Mor_{L_{0}}(gP g^{−1}, gQg^{−1}) for the given action of g on φ. By (a), when g ∈ S_{0}, this agrees with the morphism gφg^{−1} already defined.

Step 1: We first define categories L_{1} ⊇ L_{0} and F_{1} ⊇ F_{0}, where Ob(F_{1}) = Ob(F_{0}) and Ob(L_{1}) = H_{0}. Set

Mor(L_{1}) = S ×_{S_{0}} Mor(L_{0}) = (S × Mor(L_{0}))/∼,

where (gg_{0}, φ) ∼ (g, g_{0}φ) for g ∈ S, g_{0} ∈ S_{0}, and φ ∈ Mor(L_{0}). If φ ∈ Mor_{L_{0}}(P, Q), then [[g, φ]] ∈ Mor_{L_{1}}(P, gQg^{−1}) denotes the equivalence class of the pair (g, φ). Composition is defined by

[[g, φ]] ∘ [[h, ψ]] = [[gh, h^{−1}φh ∘ ψ]].

Note that if φ ∈ Mor_{L_{0}}(P, Q), then h^{−1}φh ∈ Mor_{L_{0}}(h^{−1}P h, h^{−1}Qh). To show that this is well defined, we note that for all g, h ∈ S, g_{0}, h_{0} ∈ S_{0}, and φ, ψ ∈ Mor(L_{0}) with appropriate domain and range,

[[gg_{0}, φ]] ∘ [[hh_{0}, ψ]] = [[gg_{0}hh_{0}, h_{0}^{−1}(h^{−1}φh)h_{0} ∘ ψ]] = [[gh·(h^{−1}g_{0}h), (h^{−1}φh) ∘ h_{0}ψ]]

= [[gh, (h^{−1}g_{0}h)·(h^{−1}φh) ∘ h_{0}ψ]] = [[gh, h^{−1}(g_{0}φ)h ∘ h_{0}ψ]] = [[g, g_{0}φ]] ∘ [[h, h_{0}ψ]] .

Here, the second to last equality follows from assumptions (b) and (d).

Let F_{1} be the smallest fusion system over S_{0} which contains F_{0} and Aut_{S}(S_{0}). By assumption, Aut_{S}(S_{0}) ≤ Aut_{fus}(S_{0}, F_{0}). Thus for each g ∈ S, c_{g} normalizes the fusion system F_{0}: for each φ ∈ Mor(F_{0}) there is φ′ ∈ Mor(F_{0}) such that φ ∘c_{g} = c_{g} ∘φ′. Hence each morphism in F_{1} has the form c_{g} ∘ φ for some g ∈ S and φ ∈ Aut(F_{0}). Define

π_{L_{1}} : L_{1} ⟶ F_{1}

by sending π_{L_{1}}([[g, φ]]) = c_{g} ∘ π_{0}(φ), where π_{0} denotes the natural projection from L_{0} to F_{0}. This is a functor by (c).

For all P, Q ∈ H_{0}, define

δ̂_{P,Q}: N_{S}(P, Q) ⟶ Mor_{L_{1}}(P, Q)

by setting δ̂_{P,Q}(g) = [[g, ι_{P}^{g^{−1}Qg}]]. This extends the canonical monomorphism δ_{P,Q} defined from N_{S_{0}}(P, Q) to Mor_{L_{0}}(P, Q). To simplify the notation below, we sometimes write x̂ = δ̂_{P,Q}(x) for x ∈ N_{S}(P, Q).

Step 2: We next construct categories L_{2} and F_{2}, both of which have object sets H, and which contain L_{1} and the restriction of F_{1} to H_{0}, respectively. Afterwards, we let F be the fusion system over S generated by F_{2} and restrictions of morphisms.

Before doing this, we need to know that the following holds for each P, Q ∈ H_{0} and each ψ ∈ Mor_{L_{1}}(P, Q):

∀ x ∈ N_{S}(P) there is at most one y ∈ N_{S}(Q) such that ŷ ∘ ψ = ψ ∘ x̂.  (1)

Since ψ is the composite of an isomorphism and an inclusion (and the claim clearly holds if ψ is an isomorphism), it suffices to prove this when P ≤ Q and ψ is the inclusion. We will show that we must have y = x in that case. By definition,

ι_{P}^{Q} ∘ x̂ = [[1, ι_{P}^{Q}]] ∘ [[x, Id_{P}]] = [[x, ι_{P}^{x^{−1}Qx}]] and ŷ ∘ ι_{P}^{Q} = [[y, ι_{P}^{Q}]]

(since conjugation sends inclusions to inclusions). So if ŷ∘ ι_{P}^{Q} = ι_{P}^{Q} ∘ x̂, then there exists g_{0} ∈ S_{0} such that y = xg_{0} and ι_{P}^{Q} = g_{0}^{−1}ι_{P}^{x^{−1}Qx}, and thus

δ_{P,Q}(1) = ι_{P}^{Q} = g_{0}^{−1}ι_{P}^{x^{−1}Qx} = δ_{x^{−1}Qx,Q}(g_{0}^{−1}) ∘ δ_{P,x^{−1}Qx}(1) = δ_{P,Q}(g_{0}^{−1}).

Hence g_{0} = 1 by the injectivity of δ_{P,Q} in Proposition 1.13; and so ŷ ∘ ι_{P}^{Q} = ι_{P}^{Q} ∘ x̂ only if x = y ∈ N_{S}(Q).

Now let L_{2} be the category with Ob(L_{2}) = H, and where for all P, Q ∈ H,

Mor_{L_{2}}(P, Q) = {ψ ∈ Mor_{L_{1}}(P_{0}, Q_{0}) | ∀ x ∈ P ∃ y ∈ Q such that ψ ∘ x̂ = ŷ ∘ ψ}.

Let

δ̂_{P,Q}: N_{S}(P, Q)_{⊆N_{S}(P_{0},Q_{0})} ⟶ Mor_{L_{2}}(P, Q)_{⊆Mor_{L_{1}}(P_{0},Q_{0})}

be the restriction of δ̂_{P_{0},Q_{0}}. Let F_{2} be the category with Ob(F_{2}) = H, and where

Mor_{F_{2}}(P, Q) = \{φ ∈ Hom(P, Q) | ∃ ψ ∈ Mor_{L_{2}}(P, Q) such that ψ ∘ x̂ = { \atop φ(x)} ∘ ψ ∀ x ∈ P \}.

Let π: L_{2} → F_{2} be the functor which sends ψ ∈ Mor_{L_{2}}(P, Q) to the homomorphism π(ψ)(x) = y if ψ ∘ x̂ = ŷ ∘ ψ (uniquely defined by (1)). Let F be the fusion system over S generated by F_{2} and restriction of homomorphisms.

Set Γ = S/S_{0}. Let

θ̂: L_{2} ⟶ B(Γ)

be the functor defined by setting θ̂([[g, φ]]) = gS_{0}. In particular, (θ^{−1}(1))|_{H_{0}} = L_{0}.

Step 3: We next show that each P ∈ H is F-conjugate to a subgroup P′ such that P′_{0} is fully normalized in F_{0}. Moreover, we show that P′ can be chosen so that the following holds:

∀ g ∈ S such that gP_{0}g^{−1} is F_{0}-conjugate to P_{0}, g·S_{0} ∩ N_{S}(P′_{0}) ≠ ∅.  (2)

To see this, let P_{fn} be the set of all S_{0}-conjugacy classes [P′_{0}] of subgroups P′_{0} ≤ S_{0} which are F_{0}-conjugate to P_{0} and fully normalized in F_{0}. (If P′_{0} is fully normalized in F_{0}, then so is every subgroup in [P′_{0}].) Let S′ ⊆ S be the subset of elements g ∈ S such that gP_{0}g^{−1} is F_{0}-conjugate to P_{0}. In particular, S′ ≥ N_{S}(P_{0}) ≥ P. Since each g ∈ S acts on F_{0} — two subgroups Q, Q′ ≤ S_{0} are F_{0}-conjugate if and only if gQg^{−1} and gQ′g^{−1} are F_{0}-conjugate — S′ is a subgroup of S.

For all g ∈ S′ and [P′_{0}] ∈ P_{fn}, gP′_{0}g^{−1} is F_{0}-conjugate to gP_{0}g^{−1} and hence to P_{0}, and is fully normalized since g normalizes S_{0}. Thus S′/S_{0} acts on P_{fn}, and this set has order prime to p by Proposition 1.16. So we can thus choose a subgroup P′_{0} ≤ S_{0} such that [P′_{0}] ∈ P_{fn} and is fixed by S′. In particular, P′_{0} is fully normalized in F_{0}. Also, for each g ∈ S′, some element of g·S_{0} normalizes P′_{0} (since [P′_{0}] = [gP′_{0}g^{−1}]) — and this proves (2).

Now consider the set Rep_{F_{0}}(P_{0}, S_{0}) = Hom_{F_{0}}(P_{0}, S_{0})/Inn(S_{0}). Since P ≤ S′, the group P/P_{0} acts on this set by conjugation (i.e., gP_{0} ∈ P/P_{0} acts on [φ], for φ ∈ Hom_{F_{0}}(P_{0}, S_{0}), by sending it to [c_{g}φc_{g}^{−1}]). In particular, since the S_{0}-conjugacy class [P′_{0}] is invariant under conjugation by P/P_{0}, this group leaves invariant the subset X ⊆ Rep_{F_{0}}(P_{0}, S_{0}) of all conjugacy classes [φ] of homomorphisms such that [Im(φ)] = [P′_{0}]. Fix any φ ∈ Iso_{F_{0}}(P_{0}, P′_{0}) (recall that subgroups in P_{fn} are F_{0}-conjugate to P_{0}). Every element of X has the form [αφ] for some α ∈ Aut_{F_{0}}(P′_{0}), and [αφ] = [βφ] if and only if αβ^{−1} ∈ Aut_{S_{0}}(P′_{0}). Thus |X| = | Aut_{F_{0}}(P′_{0})|/| Aut_{S_{0}}(P′_{0})|, and is prime to p since P′_{0} is fully normalized in F_{0}. We can thus choose φ_{0} ∈ Hom_{F_{0}}(P_{0}, S_{0}) such that φ_{0}(P_{0}) = P′_{0} and [φ_{0}] is invariant under the P/P_{0}-action.

Fix ψ ∈ Iso_{L_{0}}(P_{0}, P′_{0}) such that π(ψ) = φ_{0}. Since [φ_{0}] is P/P_{0}-invariant, for each x ∈ P, there is some y ∈ x·S_{0} such that c_{y} ∘ φ_{0} = φ_{0} ∘ c_{x}. By (A)_{q} (and since ψ is an isomorphism), there is y′ ∈ y·C_{S_{0}}(P′_{0}) such that ŷ′ ∘ ψ = ψ ∘ x̂ in L_{1}. This element y′ is unique by (1); and upon setting φ(x) = y′ we get a homomorphism φ ∈ Hom_{F}(P, S) which extends φ_{0}. Set P′ = φ(P); then P′ is F-conjugate to P and P′_{0} = P′ ∩ S_{0}.

Step 4: In Step 5, we will prove that F is saturated, using [5A1, Theorem 2.2]. Before that theorem can be applied, a certain technical condition must be checked.

Assume that P is F-centric, but not in H. By Step 3, P is F-conjugate to some P′ such that P′_{0} is fully normalized in F_{0}. Thus P′_{0} is fully centralized in F_{0} and not F_{0}-centric, which implies that C_{S_{0}}(P′_{0}) ≰ P′_{0}. Then P′ acts on C_{S_{0}}(P′_{0})·P′_{0}/P′_{0} with fixed subgroup QP′_{0}/P′_{0} ≠ 1 for some Q ≤ C_{S_{0}}(P′_{0}), and [Q, P′] ≤ P′_{0} since P′/P′_{0} centralizes QP′_{0}/P′_{0}. Hence Q ≰ P′_{0}, and Q ≤ N_{S}(P′) since [Q, P′] ≤ P′. For any x ∈ Q∖P′_{0}, [c_{x}] ≠ 1 ∈ Out(P′) (C_{S}(P′) ≤ P′ since P is F-centric), but c_{x} induces the identity on P′_{0} (since Q ≤ C_{S_{0}}(P′_{0})) and on P′/P′_{0} (since x ∈ S_{0}). Hence [c_{x}] ∈ O_{p}(Out_{F}(P′)) by Lemma 1.15. This shows that

P F-centric, P ∉ H ⟹ ∃ P′ F-conjugate to P, Out_{S}(P′)∩O_{p}(Out_{F}(P′)) ≠ 1. (3)

Step 5: We next show that F is saturated, and also (since it will be needed in the proof of (II)) that axiom (A)_{q} holds for L_{2}. By [5A1, Theorem 2.2] (the stronger form of Theorem 1.5(b)), it suffices to prove that the subgroups in H satisfy the axioms for saturation. Note in particular that condition (∗) in [5A1, Theorem 2.2] is precisely what is shown in (3).

Proof of (I): Fix a subgroup P ∈ H which is fully normalized in F. Let S′/S_{0} ≤ S/S_{0} be the stabilizer of the F_{0}-conjugacy class of P_{0}. By Step 3, P is F-conjugate to a subgroup P′ such that P′_{0} is fully normalized in F_{0}; and such that for each g ∈ S′, some element of g·S_{0} normalizes P′_{0} (see (2)). Hence there are short exact sequences

1 ⟶ Aut_{L_{0}}(P′_{0}) ⟶ Aut_{L_{1}}(P′_{0}) ⟶ S′/S_{0} ⟶ 1
1 ⟶ N_{S_{0}}(P′_{0}) ⟶ N_{S}(P′_{0}) ⟶ S′/S_{0} ⟶ 1.

We consider P′ ≤ N_{S}(P′_{0}) as subgroups of Aut_{L_{1}}(P′_{0}) via δ̂_{P′_{0},P′_{0}}. Then

[Aut_{L_{1}}(P′_{0}) : N_{S}(P′_{0})] = [Aut_{L_{0}}(P′_{0}) : N_{S_{0}}(P′_{0})]

is prime to p since P′_{0} is fully normalized in F_{0}, and hence N_{S}(P′_{0}) ∈ Syl_{p}(Aut_{L_{1}}(P′_{0})). Fix ψ ∈ Aut_{L_{1}}(P′_{0}) such that ψ^{−1}N_{S}(P′_{0})ψ contains a Sylow p-subgroup of the group N_{Aut_{L_{1}}(P′_{0})}(P′) (in particular, P′ ≤ ψ^{−1}N_{S}(P′_{0})ψ), and set P′′ = ψP′ψ^{−1} ≤ N_{S}(P′_{0}). Then ψ ∈ Iso_{L_{2}}(P′, P′′) by definition of L_{2}. In particular, P′′ is F-conjugate to P, and P′′_{0} = P′_{0}. Also, N_{S}(P′_{0}) contains a Sylow p-subgroup of N_{Aut_{L_{1}}(P′_{0})}(P′′),

Aut_{L_{2}}(P′′) = N_{Aut_{L_{1}}(P′_{0})}(P′′) and N_{S}(P′′) = N_{N_{S}(P′_{0})}(P′′),

and it follows that N_{S}(P′′) ∈ Syl_{p}(Aut_{L_{2}}(P′′)).

Now, Aut_{L_{2}}(P) ≅ Aut_{L_{2}}(P′′) since they are F-conjugate, and |N_{S}(P)| ≥ |N_{S}(P′′)| since P is fully normalized. Thus N_{S}(P) ∈ Syl_{p}(Aut_{L_{2}}(P)), and hence Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)). Also, N_{S}(P) contains the kernel of the projection from Aut_{L_{2}}(P) to Aut_{F}(P); i.e., C_{S}(P) is isomorphic to this kernel. For all Q which is F-conjugate to P, C_{S}(Q) is isomorphic to a subgroup of the same kernel, so |C_{S}(Q)| ≤ |C_{S}(P)|, and thus P is fully centralized in F.

Proof of (A)_{q} for L_{2}: It is clear from the construction that for any P, Q ∈ H, C_{S}(P) acts freely on Mor_{L_{2}}(P, Q) via δ̂_{P,P}. So it remains to show that when P is fully centralized in F, then for all ψ, ψ′ ∈ Mor_{L_{2}}(P, Q) such that π(ψ) = π(ψ′), there is some x ∈ C_{S}(P) such that ψ′ = ψ ∘ x̂. Since every morphism in L_{2} is the composite of an isomorphism followed by an inclusion, it suffices to show this when ψ and ψ′ are isomorphisms. But in this case, ψ^{−1}ψ′ ∈ Aut_{L_{2}}(P) lies in the kernel of the map to Aut_{F}(P). We have just seen, in the proof of (I), that this implies there is some x ∈ C_{S}(P) such that x̂ = ψ^{−1}ψ′, so ψ′ = ψ ∘ x̂, and this is what we wanted to prove.

Proof of (II): Fix φ ∈ Hom_{F}(P, S), where φ(P) is fully centralized in F. Set P′ = φ(P). By definition of F_{2} ⊆ F and of L_{2}, there is some ψ ∈ Iso_{L_{2}}(P, P′) ⊆ Iso_{L_{1}}(P_{0}, P′_{0}) such that ψ ∘ ĝ = φ̂(g) ∘ ψ for all g ∈ P. Upon replacing ψ by x̂ ∘ ψ for some appropriate x ∈ S (and replacing φ by c_{x} ∘ φ and P′ by xP′x^{−1}), we can assume that ψ ∈ Iso_{L_{0}}(P_{0}, P′_{0}) and φ|_{P_{0}} ∈ Hom_{F_{0}}(P_{0}, S_{0}).

Consider the subgroups

N_{φ} = {x ∈ N_{S}(P) | φc_{x}φ^{−1} ∈ Aut_{S}(P′)}
N′ = N_{φ|P_{0}} ∩ N_{φ} = {x ∈ N_{φ} ∩ S_{0} | (φc_{x}φ^{−1})|_{P_{0}} ∈ Aut_{S_{0}}(P′_{0})}.

We will see shortly that N′ = N_{φ} ∩ S_{0}. By (II) applied to the saturated fusion system F_{0}, there is φ_{0} ∈ Hom_{F_{0}}(N′, S) which extends φ|_{P_{0}}, and it lifts to ψ ∈ Hom_{L_{0}}(N′, S). By (A) (applied to L_{0}), there is z ∈ Z(P_{0}) such that ψ = (ψ|_{P_{0}}) ∘ ẑ. Upon replacing ψ by ψ ∘ ẑ (and φ_{0} by φ_{0} ∘ c_{z}), we can assume that ψ = ψ|_{P_{0}}.

For any x ∈ N_{φ} ∩ S_{0}, φc_{x}φ^{−1} = c_{y} ∈ Aut_{F}(P′) for some y ∈ N_{S}(P′). Hence by (A)_{q} (for L_{2}), ψx̂ψ^{−1} = ŷz for some unique z ∈ C_{S}(P′). Thus ŷz ∈ Aut_{L_{0}}(P′_{0}), and by definition of the distinguished monomorphisms for L_{1}, this is possible only if yz ∈ S_{0}. Thus φc_{x}φ^{−1} = c_{yz} where yz ∈ N_{S_{0}}(P′), and so x ∈ N′. This shows that N′ = N_{φ} ∩S_{0}.

Define φ ∈ Hom(N_{φ}, S) by the relation φ(x) = y if ψx̂ψ^{−1} = ŷ. In particular, this implies that { \atop yx^{−1}} = ψ∘(xψx^{−1})^{−1} is a morphism in L_{0}, and hence that yx^{−1} ∈ S_{0}. Also, φ|_{P} = φ by the original assumption on ψ. It remains to show that φ ∈ Hom_{F}(N_{φ}, S). To do this, it suffices to show that ψ ∘ x̂ = ŷ ∘ ψ in Mor_{L_{2}}(N′, S) for all x, where y = φ(x). Equivalently, we must show that

ψ = { \atop yx^{−1}} ∘ (xψx^{−1}).  (4)

Since yx^{−1} ∈ S_{0}, both sides in (4) are in L_{0}, and they are equal after restriction to P_{0}. Hence they are equal as morphisms defined on N_{φ} ∩ S_{0} by [5A1, Lemma 3.9], and this finishes the proof.

Step 6: We next check that F_{0} has p-power index in F. For any P ≤ S and any α ∈ Aut_{F}(P) of order prime to P, α induces the identity on P/P_{0} by construction, and hence x^{−1}α(x) ∈ S_{0} for all x ∈ P. This shows that S_{0} ≥ O_{F}^{p}(S) (see Definition 2.1). Also, by construction, for all P ≤ S_{0}, Aut_{F_{0}}(P) is normal of p-power index in Aut_{F}(P), and thus contains O^{p}(Aut_{F}(P)). This proves that the fusion subsystem F_{0} has p-power index in F in the sense of Definition 3.1.

Step 7: It remains to construct a quasicentric linking system L^{q} which contains L_{2} as a full subcategory, and which is associated to F. Note first that the axioms of Definition 1.9 are all satisfied by L_{2}: axiom (A)_{q} holds by Step 5, while axioms (B)_{q}, (C)_{q}, and (D)_{q} follow directly from the construction in Steps 1 and 2.

Let L_{2}^{c} ⊆ L_{2} be the full subcategory whose objects are the set H^{c} ⊆ H of subgroups in H which are F-centric. We first construct a centric linking system L ⊇ L_{2}^{c} associated to F. For any set K of F-centric subgroups of S, let O^{c}(F) be the orbit category of F, let O^{K}(F) ⊆ O^{c}(F) be the full subcategory with object set K, and let Z_{F}^{K} be the functor P ↦ Z(P) on O^{K}(F) which sends (see Definition 1.7 for more detail). By [BLO2, Proposition 3.1], when K is closed under F-conjugacy and overgroups, the obstruction to the existence of a linking system with object set K lies in \underleftarrow{lim}^{3}(Z_{F}^{K}), and the obstruction to its uniqueness lies in \underleftarrow{lim}^{2}(Z_{F}^{K}). Furthermore, by [BLO2, Proposition 3.2], if P is an F-conjugacy class of F-centric subgroups maximal among those not in K, and K′ = K ∪ P, then the inclusion of functors induces an isomorphism between the higher limits of Z_{F}^{K} and Z_{F}^{K′} if certain groups Λ^{∗}(Out_{F}(P); Z(P)) vanish for P ∈ P. By (3), O_{p}(Out_{F}(P)) ≠ 1 for any F_{0}-centric subgroup P ∉ H^{c}, and hence Λ^{∗}(Out_{F}(P); Z(P)) = 0 for such P by [JMO, Proposition 6.1(ii)]. So these obstructions all vanish, and there is a centric linking system L ⊇ L_{2}^{c} associated to F.

Now let L^{q} be the quasicentric linking system associated to (S, F, L). For each P ∈ H = Ob(L_{2}), P ∩ S_{0} is F_{0}-centric by definition, hence is F-quasicentric by Theorem 4.3(a), and thus P is also F-quasicentric. Also, H contains all F-centric F-radical subgroups by (3), and H is closed under F-conjugacy and overgroups (by definition of F and H). Hence by [5A1, Proposition 3.12], since L_{2}^{c} is a full subcategory of L by construction, L_{2} is isomorphic to a full subcategory of L^{q} in a way which preserves the projection functors and distinguished monomorphisms. So L_{0} can also be identified with a linking subsystem of L^{q}, and this finishes the proof. □

We now prove a topological version of Theorem 4.6. By Theorem 4.4, if (S_{0}, F_{0}, L_{0}) ⊆ (S, F, L) is an inclusion of p-local finite groups of p-power index, where S_{0} ◁ S and Γ = S/S_{0}, then there is a fibration sequence |L_{0}|_{p}^{∧} ⟶ |L|_{p}^{∧} ⟶ BΓ. So it is natural to ask whether the opposite is true: given a fibration sequence whose base is the classifying space of a finite p-group, and whose fiber is the classifying space of a p-local finite group, is the total space also the classifying space of a p-local finite group? The next proposition shows that this is, in fact, the case.

Before stating the proposition, we first define some categories which will be needed in its proof. Fix a space Y, a p-group S, and a map f: BS ⟶ Y. For P ≤ S, we regard BP as a subspace of BS; all of these subspaces contain the basepoint ∗ ∈ BS. We define three categories in this situation, F_{S,f}(Y), L_{S,f}(Y), and M_{S,f}(Y), all of which have as objects the subgroups of S. Of these, the first two are discrete categories, while M_{S,f}(Y) has a topology on its morphism sets. Morphisms in F_{S,f}(Y) are defined by setting

Mor_{F_{S,f} (Y)}(P, Q) = {φ ∈ Hom(P, Q) | f|_{BP} ≃ f|_{BQ} ∘ Bφ};

we think of this as the fusion category of Y (with respect to S and f).

Next define

Mor_{M_{S,f} (Y)}(P, Q) = {(φ, H) | φ ∈ Hom(P, Q), H: BP × [0, t] → Y,

t ≥ 0, H|_{BP ×0} = f|_{BP}, H|_{BP ×t} = f|_{BQ} ∘ Bφ} .

Thus a morphism in M_{S,f}(Y) has the form (φ, H) where H is a Moore homotopy in Y. Composition is defined by

(ψ, K) ∘ (φ, H) = (ψφ, (K ∘ (Bφ × Id))·H),

where if H and K are homotopies parameterized by [0, t] and [0, s], respectively, then (K ∘ (Bφ × Id))·H is the composite homotopy parameterized by [0, t + s].

Let P(Y) be the category of Moore paths in Y, and let Res_{∗}: M_{S,f}(Y) → P(Y) be the functor which sends each object to f(∗), and sends a morphism (φ, H) to the path obtained by restricting H to the basepoint ∗ ∈ BS. Define a map ev from |P(Y)| to Y as follows. For any n-simplex Δ^{n} in |P(Y)|, indexed by a composable sequence of paths ϕ_{1}, …, ϕ_{n} where ϕ_{i} is defined on the interval [0, t_{i}], let ev|_{Δ^{n}} be the composite

Δ^{n} ⟶^{λ(t_{1},...,t_{n})} [0, t_{1} + … + t_{n}] ⟶^{ϕ_{n}···ϕ_{1}} Y,

where λ(t_{0}, …, t_{n}) is the affine map which sends the i-th vertex to t_{1} + … + t_{i}. The category M_{S,f}(Y) thus comes equipped with an “evaluation function”

eval: |M_{S,f}(Y)| ⟶^{| Res_{∗} |} |P(Y)| ⟶^{ev} Y.

Now set

Mor_{L_{S,f} (Y)}(P, Q) = π_{0}(Mor_{M_{S,f} (Y)}(P, Q)).

We think of L_{S,f}(Y) as the linking category of Y. Also, for any set H of subgroups of S, we let L_{S,f}^{H}(Y) and M_{S,f}^{H}(Y) denote the full subcategories of L_{S,f}(Y) and M_{S,f}(Y) with object set H. For more about the fusion and linking categories of a space, see [BLO2, §7].

Theorem 4.7. Fix a p-local finite group (S_{0}, F_{0}, L_{0}), a p-group Γ, and a fibration X ⟶^{v} BΓ with fiber X_{0} ≃ |L_{0}|_{p}^{∧}. Then there is a p-local finite group (S, F, L) such that S_{0} ◁ S, F_{0} ⊆ F is a fusion subsystem of p-power index, S/S_{0} ≅ Γ, and X ≃ |L|_{p}^{∧}.

Proof. Let ∗ denote the base point of BΓ, and assume X_{0} = v^{−1}(∗). Fix a homotopy equivalence f: |L_{0}|_{p}^{∧} ⟶ X_{0}, regard BS_{0} as a subspace of |L_{0}|, and set f_{0} = f|_{BS_{0}} : BS_{0} ⟶ X_{0}, also regarded as a map to X. Let H_{0} be the set of F_{0}-centric subgroups of S_{0}.

Step 1: By [BLO2, Proposition 7.3],

F_{0} ≅ F_{S_{0},f_{0}}(X_{0}) and L_{0} ≅ L_{S_{0},f_{0}}^{H_{0}}(X_{0}).

We choose the inclusions ι_{P}^{Q} ∈ Mor_{L_{0}}(P, Q) (for P ≤ Q in H_{0}) to correspond to the morphisms (incl_{P}^{Q}, [c]) in L_{S_{0},f_{0}}(X_{0}), where c is the constant homotopy f_{0}|_{BP}. Set

F_{1} = F_{S_{0},f_{0}}(X) and L_{1} = L_{S_{0},f_{0}}^{H_{0}}(X),

where f_{0} is now being regarded as a map BS_{0} ⟶ X. The inclusion X_{0} ⊆ X makes F_{0} into a subcategory of F_{1} and L_{0} into a subcategory of L_{1}.

For all P ≤ S_{0} which is fully centralized in F_{0}, Map(BP, X_{0})_{f_{0}|BP} ≃ |C_{L_{0}}(P)|_{p}^{∧} by [BLO2, Theorem 6.3], where C_{L_{0}}(P) is a linking system over the centralizer C_{S}(P).

Since P ∈ H_{0} (i.e., P is F_{0}-centric) if and only if it is fully centralized and C_{S_{0}}(P) = Z(P), this shows that

P ∈ H_{0} ⟺ Map(BP, X_{0})_{f_{0}|_{BP}} ≃ BZ(P).  (1)

If P and P′ are F_{1}-conjugate, and φ ∈ Iso_{F_{1}}(P, P′), then the homotopy between f_{0}|_{BP} and f_{0}|_{BP′} ∘Bφ as maps from BP to X induces, using the homotopy lifting property for the fibration v, a homotopy between w∘f_{0}|_{BP} and f_{0}|_{BP′} ∘Bφ (as maps from BP to X_{0}), where w: X_{0} ^{≃}→ X_{0} is the homotopy equivalence induced by lifting some loop in BΓ. Since w is a homotopy equivalence and Bφ is a homeomorphism, this shows that the mapping spaces Map(BP, X_{0})_{f_{0}|_{BP}} and Map(BP′, X_{0})_{f_{0}|_{BP′}} are homotopy equivalent, and hence (by (1)) that P′ ∈ H_{0} if P′ ∈ H_{0}. Thus for all P, P′ ≤ S_{0},

P F_{1}-conjugate to P′ and P ∈ H_{0} ⟹ P′ ∈ H_{0}.  (2)

Fix S ∈ Syl_{p}(Aut_{L_{1}}(S_{0})). We identify S_{0} as a subgroup of S via the distinguished monomorphism δ_{S_{0}} from S_{0} to Aut_{L_{0}}(S_{0}) ≤ Aut_{L_{1}}(S_{0}).

Step 2: For all P ≤ S, and for Y = X_{0} or X, we define

Map(BP, Y)_{Φ} = {f: BP → Y | f ≃ f_{0} ∘ Bφ, φ ∈ Hom(P, S), φ(P) ∈ H_{0}}.

Using (2), we see that the fibration sequence X_{0} ⟶ X ⟶ BΓ induces a fibration sequence of mapping spaces

Map(BP, X_{0})_{Φ} ⟶ Map(BP, X)_{Φ} ⟶ Map(BP, BΓ)_{ct} ≃ BΓ,  (3)

where Map(BP, BΓ)_{ct} is the space of null homotopic maps, and the last equivalence is induced by evaluation at the basepoint. By (1), each connected component of Map(BP, X_{0})_{Φ} is homotopy equivalent to BZ(P), and hence the connected components of Map(BP, X)_{Φ} are also aspherical.

For any morphism (φ, [H]) ∈ Mor_{L_{1}}(P, Q), where φ ∈ Hom_{F_{1}}(P, Q) and [H] is the homotopy class of the path H in Map(BP, X)_{Φ}, restricting v ∘ H to the basepoint of BP defines a loop in BΓ, and thus an element of Γ. This defines a map from Mor(L_{1}) to Γ which sends composites to products, and thus a functor

θ̂: L_{1} ⟶ B(Γ).

By construction (and the fibration sequence (3)), for any ψ ∈ Mor(L_{1}), we have ψ ∈ Mor(L_{0}) if and only if θ̂(ψ) = 1.

Using the homotopy lifting property in (3), we see that θ̂ restricts to a surjection of Aut_{L_{1}}(S_{0}) onto Γ, with kernel Aut_{L_{0}}(S_{0}). Since S_{0} ∈ Syl_{p}(Aut_{L_{0}}(S_{0})), this shows that θ̂ induces an isomorphism S/S_{0} ≅ Γ, where S is a Sylow p-subgroup of Aut_{L_{1}}(S_{0}) which contains S_{0}. Hence for any ψ ∈ Mor(L_{1}), there is g ∈ S such that θ̂(ψ) = θ̂(g); and for any such g there is a unique morphism ψ_{0} ∈ Mor(L_{0}) such that ψ = δ(g) ∘ ψ_{0} where δ(g) ∈ Mor(L_{1}) denotes the appropriate restriction of g ∈ Aut_{L_{1}}(S_{0}). In other words,

Mor(L_{1}) ≅ S ×_{S_{0}} Mor(L_{0}).  (4)

Step 3: The conjugation action of S on Mor(L_{0}) ⊆ Mor(L_{1}) defines an action of S on L_{0}, which satisfies the hypotheses of Theorem 4.6. (Note in particular that this action sends inclusions to inclusions, since they are assumed to be represented by constant homotopies.) So we can now apply that theorem to construct a p-local finite group (S, F, L) which contains (S_{0}, F_{0}, L_{0}) with p-power index. Let L^{q} be the quasicentric linking system associated to (S, F, L). By (4), the category L_{1} defined here is equal to the category L_{1} defined in the proof of Theorem 4.6; i.e., the full subcategory of L^{q} with object set H_{0}.

Let H be the set of subgroups P ≤ S such that P ∩ S_{0} ∈ H_{0}, and let L_{2} ⊆ L^{q} be the full subcategory with object set H. By Step 4 in the proof of Theorem 4.6, all F-centric F-radical subgroups of S lie in H, and so |L| ≃ |L^{q}| ≃ |L_{2}| by Proposition 1.12(a). Also, |L_{1}| is a deformation retract of |L_{2}|, where the retraction is defined by sending P ∈ H to P ∩ S_{0} ∈ H_{0} (and morphisms are sent to their restrictions, uniquely defined by Proposition 1.12(b)). Thus |L| ≃ |L_{1}|. So by Theorem 4.4, we have a homotopy fibration sequence |L_{0}|_{p}^{∧} → |L_{1}|_{p}^{∧} → BΓ.

Step 4: It remains to construct a homotopy equivalence |L_{1}|_{p}^{∧} ⟶ X, which extends to a homotopy equivalence between the fibration sequences. This is where we need to use the topological linking categories defined above. Set

M_{0} = M_{S_{0},f_{0}}^{H_{0}}(X_{0}) and M_{1} = M_{S_{0},f_{0}}^{H_{0}}(X)

for short, and consider the following commutative diagram:

|L_{0}|_{p}^{∧} ←\begin{matrix}proj & _{0} \\ ≃ \end{matrix} |M_{0}|_{p}^{∧}  \overline{^{eval_{0}} →} X_{0}

|L_{1}|_{p}^{∧} ↓ ←\begin{matrix}proj & _{1} \\ ≃ \end{matrix} |M_{1}|_{p}^{∧} ↓  \overline{^{eval_{1}} →} X. ↓

The vertical maps in the diagram are all inclusions. Also, X_{0} is p-complete by definition and X by [BK, II.5.2(iv)], so the evaluation maps defined above extend to the pcompleted nerves |M_{i}|_{p}^{∧}. The maps proj_{1} and eval_{1} both commute up to homotopy with the projections to BΓ. The projection maps proj_{0} and proj_{1} are both homotopy equivalences: the connected components of the morphism spaces in M_{i} are contractible since the connected components of the fiber and total space in (3) are aspherical (see Step 2).

We claim that eval_{0} is homotopic to f ∘proj_{0} as maps to X_{0}. By naturality, it suffices to check this on the uncompleted nerve |M_{0}|, and in the case where X_{0} = |L|_{p}^{∧} and f = Id. But in this case, the only real difference between the maps is that eval_{0} sends all vertices of |M_{0}| to the base point of X_{0} = |L|_{p}^{∧}, while proj_{0} sends the vertex for a subgroup P ≤ S to the corresponding vertex in |L|_{p}^{∧}. So the maps are homotopic via a homotopy which sends vertices to the base point along the edges of |L| corresponding to the inclusion morphisms. In particular, this shows that eval_{0} is also a homotopy equivalence. We thus have an equivalence between the fibration sequences

\begin{aligned}|L_{0}|_{p}^{∧} & \overline{→} & |L_{1}|_{p}^{∧} & \overline{→} & BΓ \\ |L_{0}|_{p}^{∧} & ↓^{≃} & \overline{→} & X & ↓ & \overline{→} & BΓ , & ‖ \end{aligned}

and this proves that the two sequences are equivalent. □

Recall (Definition 4.5) that for any saturated fusion system F over a p-group S, Aut_{fus}(S, F) denotes the group of all fusion preserving automorphisms of S. The following corollary to Proposition 4.6 describes how “exotic” fusion systems could potentially arise as extensions of p-power index; we still do not know whether the situation it describes can occur.

Corollary 4.8. Fix a finite group G, with Sylow subgroup S ∈ Syl_{p}(G). Assume there is an automorphism α ∈ Aut_{fus}(S, F_{S}(G)) of p-power order, which is not the restriction to S of an automorphism of G, and which moreover is not the restriction of an automorphism of G′ for any finite group G′ with S ∈ Syl_{p}(G′) and F_{S}(G′) = F_{S}(G). Then there is a saturated fusion system (Ŝ, F̂) ⊇ (S, F_{S}(G)), such that F_{S}(G) has ppower index in F̂, and such that F̂ is not the fusion system of any finite group.

Proof. By [BLO1, Theorem E], together with [O1, Theorem A] and [O2, Theorem A], there is a short exact sequence

0 ⟶ \underleftarrow{lim}^{1}_{O_{S}^{c}(G)}(Z_{G}) ⟶ Out_{typ}(L_{S}^{c}(G)) ⟶ Out_{fus}(S, F_{S}(G)) ⟶ 0

where Out_{fus}(S, F) = Aut_{fus}(S, F)/ Aut_{F}(S), where O_{S}^{c}(G) and Z_{G} are the category and functor of Definition 1.7(b), and where Out_{typ}(L_{S}^{c}(G)) is the group of “isotypical” automorphisms of L_{S}^{c}(G) modulo natural isomorphism (see the introduction of [BLO1], or [BLO1, Definition 3.2], for the definition). Let [α] be the class of α in Out_{fus}(S, F_{S}(G)); then [α] lifts to an automorphism α of the linking system L_{S}^{c}(G). Upon replacing α by some appropriate power, we can assume that it still has p-power order. We can also assume, upon replacing α by another automorphism in the same conjugacy class if necessary, that the α-action on Aut_{L}(S) leaves invariant the subgroup δ_{S}(S); i.e., that the action of α on L_{S}^{c}(G) restricts to an action on S.

Set Ŝ = S⋊⟨x⟩, where |x| = |α| and x acts on S via α. Then Ŝ has an action on L_{S}^{c}(G) induced by the actions of S and of α, and this action satisfies conditions (a)–(d) in Theorem 4.6. Let F̂ ⊇ F_{S}(G) be the saturated fusion system over Ŝ constructed by the theorem.

We claim that F̂ is not the fusion system of any finite group. Assume otherwise: assume F̂ is the fusion system of a group Ĝ. Since F has p-power index in F̂, S ≥ Op ; F̂(Ŝ), and so S ≥ Ŝ ∩ O^{p}(Ĝ) by the hyperfocal subgroup theorem (Lemma 2.2). Set G′ = S·O^{p}(Ĝ). Then G′ ◁ Ĝ since S ◁ Ŝ; Ĝ/G′ ≅ Ŝ/S′, and thus G′ ◁ Ĝ has p-power index and S ∈ Syl_{p}(G′). By Theorem 4.3, there is a unique saturated fusion subsystem over S of p-power index in F̂, and thus F = F_{S}(G′). Also, x ∈ Ŝ ≤ Ĝ acts on G′ via an automorphism whose restriction to S is α, and this contradicts the original assumption about α. □

5. Fusion subsystems and extensions of index prime to p.

In this section, we classify all saturated fusion subsystems of index prime to p in a given saturated fusion system F, and show that there is a unique minimal subsystem O^{p′}(F) of this type. More precisely, we show that there is a certain finite group of order prime to p associated to F, denoted below Γ_{p′}(F), and a one-to-one correspondence between subgroups T ≤ Γ_{p′}(F) and fusion subsystems F_{T} of index prime to p in F. The index of F_{T} in F can then be defined to be the index of T in Γ_{p′}(F).

Conversely, we also describe extensions of saturated fusion systems of index prime to p. Once more the terminology requires motivation. Roughly speaking, an extension of index prime to p of a given saturated fusion system F is a saturated fusion system F′ over the same p-group S, but where the morphism set has been “extended” by an action of a group of automorphisms whose order is prime to p. Here again, a one-to-one correspondence statement is obtained, thus providing a full classification.

\underline{5.1 Subsystems of index prime to p} We first classify all saturated fusion subsystems of index prime to p in a given saturated fusion system F. The fusion subsystems and associated linking systems will be constructed using Proposition 3.8 and Theorem 3.9, respectively. More precisely, Theorem 3.9 has already told us that for any p-local finite group (S, F, L), any surjection θ of π_{1}(|L|) onto a finite p′-group Γ, and any subgroup H ≤ Γ, there is a p-local finite subgroup (S′, F′, L′) such that |L′| is homotopy equivalent to the covering space of |L| with fundamental group θ^{−1}(H). What is new in this section is first, that we describe the “universal” p′-group quotient of π_{1}(|L|) as a certain quotient group of Out_{F}(S); and second, we show that all fusion subsystems of index prime to p in F (in the sense of Definition 3.1) are obtained in this way.

When applying Proposition 3.8 to this situation, we need to consider fusion mapping triples (Γ, θ, Θ) on F^{q}, where Γ is finite of order prime to p. Since θ ∈ Hom(S, Γ), it must then be the trivial homomorphism. In this case, conditions (i)–(iii) in Definition 3.6 are equivalent to requiring that there is some functor Θ̂: F^{q} → B(Γ) such that Θ(φ) = {Θ̂(φ)} for each φ ∈ Mor(F^{q}) (and condition (iv) is redundant). So instead of explicitly constructing fusion mapping triples, we instead construct functors of this form.

We start with some definitions. For a finite group G, one defines O^{p′}(G) to be the smallest normal subgroup of G of index prime to p, or equivalently the subgroup generated by elements of p-power order in G. These two definitions are not, in general, equivalent in the case of an infinite group, (the case G = ℤ being an obvious example). We do, however, need to deal with such subgroups here. The following definition is most suitable for our purposes, and it is a generalization of the finite case.

Definition 5.1. For any group G (possibly infinite), let O^{p′}(G) be the intersection of all normal subgroups in G of finite index prime to p.

In particular, under this definition, an epimorphism α: G −↠ H with Ker(α) ≤ O^{p′}(G) induces an isomorphism G/O^{p′}(G) ≅ H/O^{p′}(H). Thus by Proposition 2.6, for any p-local finite group (S, F, L), the projections of |L| ≃ |L^{q}| onto |F^{c}| and |F^{q}| induce isomorphisms

π_{1}(|L|)/O^{p′}(π_{1}(|L|)) ≅ π_{1}(|F^{c}|)/O^{p′}(π_{1}(|F^{c}|)) ≅ π_{1}(|F^{q}|)/O^{p′}(π_{1}(|F^{q}|)).

Fix a saturated fusion system F over a p-group S, and define

Γ_{p′}(F) = π_{1}(|F^{c}|)/O^{p′}(π_{1}(|F^{c}|)).

We will show that the natural functor

ε_{F^{c}}: F^{c} ⟶ B(Γ_{p′}(F))

induces a bijective correspondence between subgroups of Γ_{p′}(F) and fusion subsystems of F of index prime to p.

Recall (Definition 3.3) that for any saturated fusion system F over a p-group S, O_{∗}^{p′}(F) ⊆ F is the smallest fusion subsystem which contains the groups O^{p′}(Aut_{F}(P)) for all P ≤ S; i.e., the smallest fusion subsystem which contains all automorphisms in F of p-power order. Define

Out_{F}^{0}(S) = ⟨α ∈ Out_{F}(S) | α|_{P} ∈ Mor_{O_{∗}^{p′}(F)}(P, S), some F-centric P ≤ S⟩.

Then Out_{F}^{0}(S) ◁ Out_{F}(S), since O_{∗}^{p′}(F) is normalized by Aut_{F}(S) (Lemma 3.4(a)).

Proposition 5.2. There is a unique functor

θ̂: F^{c} ⟶ B(Out_{F}(S)/ Out_{F}^{0}(S))

with the following properties:

(a) θ̂(α) = α (modulo Out_{F}^{0}(S)) for all α ∈ Aut_{F}(S).

(b) θ̂(φ) = 1 if φ ∈ Mor(O_{∗}^{p′}(F)^{c}). In particular, θ̂ sends inclusion morphisms to the identity.

Furthermore, there is an isomorphism

θ: Γ_{p′}(F) = π_{1}(|F^{c}|)/O^{p′}(π_{1}(|F^{c}|)) ⟶^{≅} Out_{F}(S)/ Out_{F}^{0}(S)

such that θ̂ = Bθ ∘ ε_{F}^{c}.

Proof. By Lemma 3.4(c), each morphism in F^{c} factors as the composite of the restriction of a morphism in Aut_{F}(S) followed by a morphism in O_{∗}^{p′}(F)^{c}. If

φ = φ_{1} ∘ α_{1}|_{P} = φ_{2} ∘ α_{2}|_{P} ,

where φ ∈ Hom_{F}(P, Q), α_{i} ∈ Aut_{F}(S), and φ_{i} ∈ Hom_{O_{∗}^{p′}(F)^{c}}(α_{i}(P), Q), then we can assume (after factoring out inclusions) that all of these are isomorphisms, and hence

(α_{2} ∘ α_{1}^{−1})|_{P} = φ_{2}^{−1} ∘ φ_{1} ∈ Iso_{O_{∗}^{p′}(F)^{c}}(α_{1}(P), α_{2}(P)).

Thus α_{2} ∘ α_{1}^{−1} ∈ Out_{F}^{0}(S); and so we can define

θ̂(φ) = [α_{1}] = [α_{2}] ∈ Out_{F}(S)/ Out_{F}^{0}(S).

This shows that θ̂ is well defined on morphisms (and sends each object in F^{c} to the unique object of B(Out_{F}(S)/ Out_{F}^{0}(S))). By Lemma 3.4(c) again, θ̂ preserves compositions, and hence is a well defined functor. It satisfies conditions (a) and (b) above by construction. The uniqueness of θ̂ is clear.

It remains to prove the last statement. Since Out_{F}(S)/ Out_{F}^{0}(S) is finite of order prime to p, π_{1}(|θ̂|) factors through a homomorphism

θ̄: π_{1}(|F^{c}|)/O^{p′}(π_{1}(|F^{c}|)) ⟶ Out_{F}(S)/ Out_{F}^{0}(S).

The inclusion of B Aut_{F}(S) into |F^{c}| (as the subcomplex with one vertex S) induces a homomorphism

τ: Out_{F}(S) ⟶ π_{1}(|F^{c}|)/O^{p′}(π_{1}(|F^{c}|)).

Furthermore, τ is surjective since F = ⟨O_{∗}^{p′}(F), Aut_{F}(S)⟩ (Lemma 3.4(b)), and since any automorphism in O_{∗}^{p′}(F) is a composite of restrictions of automorphisms of p-power order. By (a), and since θ restricted to Aut_{F}(S) is the projection onto Out_{F}(S), the composite θ∘τ is the projection of Out_{F}(S) onto the quotient group Out_{F}(S)/ Out_{F}^{0}(S). Finally, Out_{F}^{0}(S) ≤ Ker(τ) by definition of Out_{F}^{0}(S), and this shows that θ is an isomorphism. □

The following lemma shows that any fusion mapping triple on F^{c} can be extended uniquely to F^{q}. This will allow us later to apply Proposition 3.8 in order to produce saturated fusion subsystems of index prime to p.

Lemma 5.3. Let F be a saturated fusion system over a p-group S, and let (Γ, θ, Θ_{0}) be a fusion mapping triple on F^{c}. Then there is a unique extension

Θ: Mor(F^{q}) → 𝔖𝔲𝔟(Γ)

of Θ_{0}, such that (Γ, θ, Θ) is a fusion mapping triple on F^{q}.

Proof. We construct the extension Θ one F-conjugacy class at a time. Thus, assume Θ has been defined on a set H_{0} of F-quasicentric subgroups of S which is a union of F-conjugacy classes, contains all F-centric subgroups, and is closed under overgroups. Let P be maximal among F-conjugacy classes of F-quasicentric subgroups not in H_{0}; we show that Θ can be extended to H = H_{0} ∪ P.

Fix P ∈ P which is fully normalized in F. For each α ∈ Aut_{F}(P), there is an extension α ∈ Aut_{F}(P ·C_{S}(P)) (axiom (II)), and we define a map

Θ_{P} : Aut_{F}(P) → 𝔖𝔲𝔟(N_{Γ}(θ(C_{S}(P))))

by Θ_{P} (α) = Θ(α)·θ(C_{S}(P)). By Definition 3.6 ((i) and (ii)) Θ(α) is a left coset of θ(C_{S}(P ·C_{S}(P))), and by (iv), it is also a right coset (where the right coset representative can be taken to be the same as the one representing the left coset); hence Θ_{P} (α) is a left and a right coset of θ(C_{S}(P)) (again with the same coset representative on both sides). If α′ ∈ Aut_{F}(P ·C_{S}(P)) is any other extension of α, then by [5A1, Lemma 3.8], there is some g ∈ C_{S}(P) such that α′ = c_{g} ∘ α. Thus, by Definition 3.6 again, Θ(α′) = Θ(c_{g} ∘ α) = θ(g)Θ(α), and

Θ(α′) · θ(C_{S}(P)) = θ(g)Θ(α) · θ(C_{S}(P)) = Θ(α)θ(α(g)) · θ(C_{S}(P)) = Θ(α) · θ(C_{S}(P)),

and so the definition of Θ_{P} (α) is independent of the choice of the extension α. This shows that Θ_{P} is well defined.

Notice also that Θ_{P} clearly respects compositions, and since Θ_{P} (α) = x·θ(C_{S}(P)) = θ(C_{S}(P)) · x, for some x ∈ Γ, we conclude that x ∈ N_{Γ}(θ(C_{S}(P))). Thus Θ_{P} induces a homomorphism

Θ_{P} : Aut_{F}(P) → N_{Γ}(θ(C_{S}(P)))/θ(C_{S}(P)).

We now make use of Lemma 4.1, which gives sufficient conditions to the existence of an extension of a fusion mapping triple.

If α ∈ Aut_{F}(P), and x ∈ Θ_{P} (α), then x = y ·θ(h) for some h ∈ C_{S}(P) and y ∈ Θ(α), where α is an extension of α to P · C_{S}(P). Hence for any g ∈ P,

xθ(g)x^{−1} = y · θ(hgh^{−1})y^{−1} = yθ(g)y^{−1} = θ(α(g)) = θ(α(g)).

This shows that point (+) of Lemma 4.1 holds, and so it remains to check (∗).

Assume P ≨ Q ≤ S, P ◁ Q, and let α ∈ Aut_{F}(P) and β ∈ Aut_{F}(Q) be such that α = β|_{P}. Then Q·C_{S}(P) ≤ N_{α} in the terminology of axiom (II), so α extends to another automorphism γ ∈ Aut_{F}(Q·C_{S}(P)), and Θ_{P} (α) = Θ(γ)·θ(C_{S}(P)) by definition of Θ_{P}. By [5A1, Lemma 3.8] again, γ|_{Q} = c_{g} ∘ β for some g ∈ C_{S}(P). Hence, by Definition 3.6, Θ(γ) = Θ(c_{g} ∘ β) = θ(g) · Θ(β), and so

Θ_{P} (α) = θ(g) · Θ(β) · θ(C_{S}(P)) = Θ(β)θ(β(g)) · θ(C_{S}(P)) = Θ(β) · θ(C_{S}(P)).

In particular, Θ_{P} (α) ⊇ Θ(β). This shows that point (∗) of Lemma 4.1 is satisfied as well, and thus, by the lemma, Θ can be extended to a fusion mapping triple on F^{H}. □

Recall that a fusion subsystem of index prime to p in a saturated fusion system F over S is a saturated subsystem F_{0} ⊆ F over the same p-group S, such that Aut_{F_{0}}(P) ≥ O^{p′}(Aut_{F}(P)) for all P ≤ S. Equivalently, F_{0} ⊆ F has index prime to p if and only if it is saturated and contains the subcategory O_{∗}^{p′}(F) of Definition 3.3. We are now ready to prove our main result about these subsystems.

Theorem 5.4. For any saturated fusion system F over a p-group S, there is a bijective correspondence between subgroups

H ≤ Γ_{p′}(F) = Out_{F}(S)/ Out_{F}^{0}(S),

and saturated fusion subsystems F_{H} of F over S of index prime to p in F. The correspondence is given by associating to H the fusion system generated by θ̂^{−1}(B(H)), where θ̂ is the functor of Proposition 5.2.

Proof. Let F_{0} ⊆ F be any saturated fusion subsystem over S which contains O_{∗}^{p′}(F). Then Out_{F}^{0}(S) ◁ Out_{F_{0}}(S), and one can set H = Out_{F_{0}}(S)/ Out_{F}^{0}(S). We first show that a morphism φ of F^{c} is in F_{0} if and only if θ̂(φ) ∈ H. Clearly it suffices to prove this for isomorphisms in F^{c}.

Let P, Q ≤ S be F-centric, F-conjugate subgroups, and fix an isomorphism φ ∈ Iso_{F}(P, Q). By Lemma 3.4, we can write φ = ψ ∘ (α|_{P}), where α ∈ Aut_{F}(S) and ψ ∈ Iso_{O_{∗}^{p′}(F)}(α(P), Q). Then φ is in F_{0} if and only if α|_{P} is in F_{0}. Also, by definition

of θ̂ (and of H), θ̂(φ) ∈ H if and only if α ∈ Aut_{F_{0}}(S). So it remains to prove that α|_{P} ∈ Mor(F_{0}) if and only if α ∈ Aut_{F_{0}}(S).

If α ∈ Aut_{F_{0}}(S), then α|_{P} is also in F_{0} by definition of a fusion system. So it remains to prove the converse. Assume α|_{P} is in F_{0}. The same argument as that used to prove Proposition 3.8(c) shows that α(P) is F_{0}-centric, and hence fully centralized in F_{0}. Since α|_{P} extends to an (abstract) automorphism of S, axiom (II) implies that it extends to some α_{1} ∈ Hom_{F_{0}}(N_{S}(P), S). Since P is F-centric, [BLO2, Proposition A.8] applies to show that α_{1} = (α|_{N_{S}(P)})∘c_{g} for some g ∈ Z(P), and thus that α|_{N_{S}(P)} ∈ Hom_{F_{0}}(N_{S}(P), S). Also, N_{S}(P) ≩ P whenever P ≨ S, and so we can continue this process to show that α ∈ Aut_{F_{0}}(S). This finishes the proof that F_{0} = θ̂^{−1}(H).

Now fix a subgroup H ≤ Out_{F}(S)/ Out_{F}^{0}(S), and let F_{H} be the smallest fusion system over S which contains θ̂^{−1}(B(H)). We must show that F_{H} is a saturated fusion subsystem of index prime to p in F. For F-centric subgroups P, Q ≤ S, Hom_{F_{H}}(P, Q) is the set of all morphisms φ in Hom_{F}(P, Q) such that θ̂(φ) ∈ H. In particular, F_{H} ⊇ O_{∗}^{p′}(F), since morphisms in O_{∗}^{p′}(F) are sent by θ̂ to the identity element.

Define Θ: Mor(F^{c}) → 𝔖𝔲𝔟(Γ_{p′}(F)) by setting Θ(φ) = {θ̂(φ)}; i.e., the image consists of subsets with one element. Let θ ∈ Hom(S, Γ_{p′}(F)) be the trivial (and unique) homomorphism. Then (Γ_{p′}(F), θ, Θ) is a fusion mapping triple on F^{c}. By Lemma 5.3, this can be extended to a fusion mapping triple on F^{q}; and hence F_{H} is saturated by Proposition 3.8.

By Theorem 1.5(a) (Alperin’s fusion theorem), F_{H} is the unique saturated fusion subsystem of F with the property that a morphism φ ∈ Hom_{F}(P, Q) between F-centric subgroups of S lies in F_{H} if and only if θ̂(φ) ∈ H. This shows that the correspondence is indeed bijective. □

The next theorem describes the relationship between subgroups of index prime to p in a p-local finite group (S, F, L) and certain covering spaces of |L|.

Theorem 5.5. Fix a p-local finite group (S, F, L). Then for each subgroup H ≤ Out_{F}(S) containing Out_{F}^{0}(S), there is a unique p-local finite subgroup (S, F_{H}, L_{H}), such that F_{H} has index prime to p in F, Out_{F_{H}}(S) = H, and L_{H} = π^{−1}(F_{H}) (where π is the usual functor from L to F). Furthermore, |L_{H}| is homotopy equivalent, via its inclusion into |L|, to the covering space of |L| with fundamental group H̃: where H̃ ≤ π_{1}(|L|) is the subgroup such that θ̄(H̃/O^{p′}(π_{1}(|L|))) corresponds to H/ Out_{F}^{0}(S) under the isomorphism

π_{1}(|L|)/O^{p′}(π_{1}(|L|)) ≅ π_{1}(|F^{c}|)/O^{p′}(π_{1}(|F^{c}|)) ⟶_{≅}^{θ} Out_{F}(S)/ Out_{F}^{0}(S)

of Proposition 5.2.

Proof. By Theorem 3.9, applied to the composite functor

L ⟶^{π} F^{c} \underrightarrow{θ̂} B(Out_{F}(S)/ Out_{F}^{0}(S)),

(S, F_{H}, L_{H}) is a p-local finite group, and |L_{H}| is homotopy equivalent to the covering space of |L| with fundamental group H̃ defined above. The uniqueness follows from Theorem 5.4. □

Theorem 5.4 shows, in particular, that any saturated fusion system F over S contains a unique minimal saturated subsystem O^{p′}(F) of index prime to p: the subsystem F_{0} ⊆ F with Out_{F_{0}}(S) = Out_{F}^{0}(S). Furthermore, if F has an associated centric linking system L, then Theorem 5.5 shows that O^{p′}(F) has an associated linking system O^{p′}(L), whose geometric realization |O^{p′}(L)| is homotopy equivalent to the covering space of |L| with fundamental group O^{p′}(π_{1}(|L|)).

\underline{5.2 Extensions of index prime to p} It remains to consider the opposite problem: describing the extensions of a given saturated fusion system of index prime to p. As before, for a saturated fusion system F over a p-group S, Aut_{fus}(S, F) denotes the group of fusion preserving automorphisms of S (Definition 4.5). Theorem 5.7 below states that each subgroup of Out_{fus}(S, F) = Aut_{fus}(S, F)/ Aut_{F}(S) of order prime to p gives rise to an extension of F.

The following lemma will be needed to compare the obstructions to the existence and uniqueness of linking systems, in a fusion system and in a fusion subsystem of index prime to p. Recall the definition of the orbit category of a fusion system in Definition 1.7.

Lemma 5.6. Fix a saturated fusion system F over a p-group S, and let F′ ⊆ F be a saturated subsystem of index prime to p. Assume Out_{F′}(S) ◁ Out_{F}(S), and set π = Out_{F}(S)/ Out_{F′}(S). Then for any F : O^{c}(F) → ℤ_{(p)}-mod, there is a natural action of π on the higher limits of F |_{O^{c}(F′)}, and

\underleftarrow{lim}^{∗}_{O^{c}(F)}(F) ≅ [ \underleftarrow{lim}^{∗}_{O^{c}(F′)}(F |_{O^{c}(F′)})]^{π}.

Proof. Let O^{c}(F)-mod denote the category of functors O^{c}(F)^{op} → Ab. For any F in O^{c}(F)-mod, Out_{F}(S) acts on ∏_{P ∈Ob(O^{c}(F))} F (P) by letting α ∈ Out_{F}(S) send

F (α(P)) to F (P) via the induced map α^{∗}. This restricts to an action of π on λ(F) =^{def} \underleftarrow{lim}_{O^{c}(F′)}(F |_{O^{c}(F′)}); and by definition of inverse limits,

\underleftarrow{lim}_{O^{c}(F)}(F) ≅ [ \underleftarrow{lim}_{O^{c}(F′)}(F |_{O^{c}(F′)})]^{π} = [λ(F)]^{π}.

Since F′ is a subsystem of index prime to p, Out_{F′}(S) is a subgroup of Out_{F}^{0}(S). Hence, by a slight abuse of notation, one has a functor θ̂: F → B(π) given as the composite of the functor θ̂ of Proposition 5.2 with projection to π. For any ℤ[π]- module M, regarded as a functor on B(π), we let β(M) denote the composite functor M ∘ θ̂. Then ℤ[π]-mod ^{β}→ O^{c}(F)-mod is an exact functor, and a left adjoint to O^{c}(F)-mod ^{λ}→ ℤ[π]-mod. In particular, the existence of a left adjoint shows that λ sends injective objects in O^{c}(F)-mod to injective ℤ[π]-modules.

Thus, if 0 → F → I_{0} → I_{1} → ⋯ is an injective resolution of F in O^{c}(F)-mod, and F takes values in ℤ_{(p)}-modules, then

\underleftarrow{lim}^{∗}_{O^{c}(F)}(F) ≅ H^{∗}(0 → λ(I_{0})_{(p)}^{π} → λ(I_{1})_{(p)}^{π} → ⋯)
≅ [H^{∗}(0 → λ(I_{0})_{(p)} → λ(I_{1})_{(p)} → ⋯)]^{π} ≅ [ \underleftarrow{lim}^{∗}_{O^{c}(F′)}(F |_{O^{c}(F′)})]^{π} . □

We are now ready to examine extensions of index prime to p.

Theorem 5.7. Fix a saturated fusion system F over a p-group S. Let

π ≤ Out_{fus}(S, F) =^{def} Aut_{fus}(S, F)/ Aut_{F}(S)

be any subgroup of order prime to p, and let π̃ denote the inverse image of π in Aut_{fus}(S, F).

(a) Let F.π be the fusion system over S generated (as a category) by F together with restrictions of automorphisms in π̃. Then F.π is a saturated fusion system, which contains F as a fusion subsystem of index prime to p.

(b) If F has an associated centric linking system, then so does F.π.

(c) Let L be a centric linking system associated to F. Assume that for each α ∈ π, the action of α on F lifts to an action on L. Then there is a unique centric linking system L.π associated to F.π whose restriction to F is L.

Proof. (a) By definition, every morphism in F.π is the composite of morphisms in F and restrictions of automorphisms of S which normalize F. If ψ ∈ Hom_{F}(P, Q) and φ ∈ Aut_{fus}(S, F), then one has

φ|_{Q} ∘ ψ = φ|_{Q} ∘ ψ ∘ φ^{−1}|_{φ(P)} ∘ φ|_{P} = ψ′ ∘ φ|_{P} ,  (1)

where ψ′ ∈ Hom_{F}(φ(P), φ(Q)) (since φ is fusion preserving). Hence each morphism in F.π is the composite of the restriction of a morphism in π̃ followed by a morphism in F.

We next claim F is a fusion subsystem of F.π of index prime to p. More precisely, we will show, for all P ≤ S, that Aut_{F}(P) ◁ Aut_{F.π}(P) with index prime to p. To see this, let π_{1} ⊆ π̃ be the set of automorphisms φ in π̃ ≤ Aut_{fus}(S, F) such that φ(P) is F-conjugate to P, and let π_{0} ⊆ π_{1} be the set of classes φ such that φ|_{P} ∈ Hom_{F}(P, S). If φ(P) and ψ(P) are both F-conjugate to P, then ψ(φ(P)) is F-conjugate to ψ(P) since ψ is fusion preserving, and thus is F-conjugate to P. This shows that π_{1} is a subgroup of π̃, and an argument using (1) shows that π_{0} is also a subgroup. By definition, π_{0} ≥ Aut_{F}(S), so π_{1}/π_{0} has order prime to p since π does. Using (1), define

θ_{P} : Aut_{F.π}(P) ⟶ π_{1}/π_{0}

by setting θ_{P} (α) = [ψ] if α = ψ|_{β(P)}∘β for some β ∈ Hom_{F}(P, S) and some ψ· Aut_{F}(S) ∈ π. By definition of π_{0}, this is well defined, and Ker(θ_{P}) = Aut_{F}(P). Since π_{1}/π_{0} has order prime to p, this shows that Aut_{F}(P) ◁ Aut_{F.π}(P) with index prime to p.

We next claim that F and F.π have the same fully centralized, fully normalized and centric subgroups (compare with the proof of Proposition 3.8(c)). By definition, F and F.π are fusion systems over the same p-group S. Since each F-conjugacy class is contained in some F.π-conjugacy class, any subgroup P ≤ S which is fully centralized (fully normalized) in F.π is fully centralized (fully normalized) in F. By the same argument, any F.π-centric subgroup is also F-centric.

Conversely, assume that P is not fully centralized in F.π, and let P′ ≤ S be a subgroup F.π-conjugate to P such that |C_{S}(P′)| > |C_{S}(P)|. Let ψ: P′ → P be an F.π-isomorphism between them. Then, by the argument above ψ = ψ′ ∘ φ, where ψ′ is a morphism in F and φ ∈ π̃ is the restriction to P′ of an automorphism of S. Hence |C_{S}(φ(P′))| = |C_{S}(P′)| > |C_{S}(P)|, and since φ(P′) is F-conjugate to P, this shows that P is not fully centralized in F. A similar argument shows that if P is not fully normalized (or not centric) in F.π, then it is not fully normalized (or not centric) in F.

We prove that F.π is saturated using Theorem 1.5(b). Thus, we must show that conditions (I) and (II) of Definition 1.2 are satisfied for all F.π-centric subgroups, and that F.π is generated by restrictions of morphisms between its centric subgroups. Let P ≤ S be a subgroup which is fully normalized in F.π. Then it is fully normalized in F, and since F is saturated, it is fully centralized there and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) = Syl_{p}(Aut_{F.π}). Hence, condition (I) holds for any P in (F.π)^{c}.

Let ψ: P → Q be a morphism in F.π, and write ψ = ψ′ ∘ φ, as before. Set

N_{ψ} = {g ∈ N_{S}(P) | ψ ∘ c_{g} ∘ ψ^{−1} ∈ Aut_{S}(ψ(P))};

then φ(N_{ψ}) = N_{ψ′} since φc_{g}φ^{−1} = c_{φ(g)} for all g ∈ S. Since condition (II) holds for F, the morphism ψ′ can be extended to N_{ψ′}. Hence ψ = ψ′ ∘ φ can be extended to N_{ψ} = φ(N_{ψ′}), and so condition (II) holds for F.π.

That all morphisms in F.π are composites of restrictions of F.π-morphisms between F.π-centric subgroups holds by construction. Thus Theorem 1.5(b)applies, and F.π is saturated.

(b) Let Z_{F}: O^{c}(F) → Ab be the functor Z_{F}(P) = Z(P), and similarly for Z_{F.π} (see Definition 1.7). By Lemma 5.6, restriction of categories induces a monomorphism

\underleftarrow{lim}^{∗}_{O^{c}(F.π)}(Z_{F.π}) ⟶ \underleftarrow{lim}^{∗}_{O^{c}(F)}(Z_{F})  (2)

whose image is the subgroup of π-invariant elements.

By [BLO2, Proposition 3.1], the obstruction η(F.π) to the existence of a centric linking system associated to F.π lies in \underleftarrow{lim}^{3}(Z_{F.π}). From the construction in [BLO2] of these obstructions (and the fact that F.π and F have the same centric subgroups), it is clear that the restriction map (2) sends η(F.π) to η(F). So if there is a linking system L associated to F, then η(F) = 0, so η(F.π) = 0 by Lemma 5.6, and there is a linking system L.π associated to F.π.

(c) Let L.π be a centric linking system associated to F.π, as constructed in (b), and let L′ be its restriction to F (i.e., the inverse image of F under the the projection L.π → (F.π)^{c}). By [BLO2, Proposition 3.1] again, the group \underleftarrow{lim}^{2}(Z_{F}) acts freely and transitively on the set of all centric linking systems associated to F. If the action on F of each α ∈ π lifts to an action on L, then the element of \underleftarrow{lim}^{2}(Z_{F}) which measures the difference between L and L′ is π-invariant, and hence (by Lemma 5.6 again) is the restriction of an element of \underleftarrow{lim}^{2}(Z_{F.π}). Upon modifying the equivalence class of L.π by this element, if necessary, we get a centric linking system associated to F.π whose restriction to F is L. □

As was done in the last section for extensions with p-group quotient, we now translate this last result to a theorem stated in terms of fibration sequences.

Theorem 5.8. Fix a p-local finite group (S, F, L), a finite group Γ of order prime to p, and a fibration E ⟶^{v} BΓ with fiber X ≃ |L|_{p}^{∧}. Then there is a p-local finite group (S, F′, L′) such that F ⊆ F′ is normal of index prime to p, Aut_{F′}(S)/ Aut_{F}(S) is a quotient group of Γ, and E_{p}^{∧} ≃ |L′|_{p}^{∧}.

Proof. For any space Y, let Aut(Y) denote the topological monoid of homotopy equivalences Y ⟶^{≃} Y. Fibrations with fiber Y and base B are classified by homotopy classes of maps B → B Aut(Y). This follows, for example, as a special case of the main theorem in [DKS].

We are thus interested in the classifying space B Aut(|L|_{p}^{∧}), whose homotopy groups were determined in [BLO2, §8]. To describe these, let Aut_{typ}(L) be the monoid of isotypical self equivalences of the category L; i.e., the monoid of all equivalences of categories ψ ∈ Aut(L) such that for all P ∈ Ob(L), ψ_{P,P} (Im(δ_{P})) = Im(δ_{ψ(P)}). Let Out_{typ}(L) be the group of all isotypical self equivalences modulo natural isomorphisms of functors. By [BLO2, Theorem 8.1], π_{i}(B Aut(|L|_{p}^{∧})) is a finite p-group for i = 2 and vanishes for i > 2, and

π_{1}(B Aut(|L|_{p}^{∧})) ≅ Out_{typ}(L).

Each isotypical self equivalence of L is naturally isomorphic to one which sends inclusions to inclusions (this was shown in [BLO1, Lemma 5.1] for linking systems of a group, and the general case follows by the same argument). Thus each element of Out_{typ}(L) is represented by some β which sends inclusions to inclusions. This in turn implies that β_{S} — the restriction of β_{S,S} to δ_{S}(S) ≤ Aut_{L}(S) — lies in Aut_{fus}(S, F), and that for every F-centric subgroup P of S, the functor β sends P to the subgroup β_{S}(P) ≤ S. In particular, β is an automorphism of L, since it induces a bijection on the set of objects.

Now let BΓ ^{f_{v}}→ B Aut(|L|_{p}^{∧}) be the map which classifies the fibration E ⟶^{v} BΓ. Since Γ is a p′-group, the fibration E ⟶^{v} BΓ is uniquely determined by the action of Γ on |L|_{p}^{∧}; more precisely, by the map induced by f_{v} on fundamental groups:

π_{1}(f_{v}): Γ ⟶ Out_{typ}(L) .

By an argument identical to that used to prove [BLO1, Theorem 6.2], there is an exact sequence

0 ⟶ \underleftarrow{lim}^{1}_{O^{c}(F)}(Z_{F}) ⟶ Out_{typ}(L) ⟶^{μ_{L}} Out_{fus}(S, F),

where μ_{L} is defined by restricting a functor L → L to δ_{S}(S) ≤ Aut_{L}(S). Also, O^{c}(F) and Z_{F} are as in Definition 1.7, but all that we need to know here is that Ker(μ_{L}) is a p-group. Set

ψ_{v} = μ_{L} ∘ π_{1}(f_{v}): Γ ⟶ Out_{fus}(S, F).

Set π = Im(ψ_{v}) ≤ Out_{fus}(S, F). By its definition, ψ_{v} comes equipped with a lift to Out_{typ}(L). So by the above remarks, every element of π lifts to an automorphism of L. Let (S, F.π, L.π) be the p-local finite group constructed in Theorem 5.7(a,c) as an extension of (S, F, L). This induces a fibration

|L| ⟶ |L.π| ⟶ Bπ.

By [BK, Corollary I.8.3], there is a fiberwise completion E_{w} of |L.π| which sits in a fibration sequence

|L|_{p}^{∧} ⟶ E_{w} ⟶^{w} Bπ,

and this fibration is classified by a map

f_{w}: Bπ ⟶ B Aut(|L|_{p}^{∧}).

As was the case for f_{v}, the induced map between fundamental groups π_{1}(f_{w}) determines the fibration.

Consider the diagram

Γ  \overline{^{ρ} →} π
[figure: π_{1}(f_{w})]
Out_{typ}(L) ↓^{π_{1}(f_{v})}  \overline{→^{μ_{L}}} Out_{fus}(S, F) , ↓^{incl}

(1)

where ρ: Γ → π is the restriction of ψ_{v} to its image, and where the square commutes since each composite is equal to ψ_{v}. The composite μ_{L} ∘π_{1}(f_{w}) ∈ Hom(π, Out_{fus}(S, F)) is the homomorphism induced by the homotopy action of π = π_{1}(Bπ) on the fiber |L|_{p}^{∧}. By the construction in Theorem 5.7, this is just the inclusion map. Hence the lower right triangle in the above diagram commutes. Since Γ has order prime to p and Ker(μ_{L}) ≅ \underleftarrow{lim}^{1}(Z_{F}) is a p-group, any homomorphism from Γ to Out_{fus}(S, F) has a unique conjugacy class of lifting to Out_{typ}(L) by the Schur-Zassenhaus theorem (cf. [Go, Theorem 6.2.1]). In particular, π_{1}(f_{w}) ∘ ρ and π_{1}(f_{v}) are conjugate as homomorphisms from Γ to Out_{typ}(L). Thus, the upper left triangle in (1) commutes up to conjugacy.

Since π_{2}(B Aut(|L|_{p}^{∧})) is a finite p-group and the higher homotopy groups all vanish, we have shown that f_{w} ∘ Bρ is homotopic to f_{v}. In other words, we have a map of fibrations

\begin{aligned}|L|_{p}^{∧} & \overline{→} & E & \overline{^{v} →} & BΓ \\ |L|_{p}^{∧} & ‖ & \overline{→} & E_{w} & ↓^{λ} & \overline{^{w} →} & Bπ, & ↓^{Bρ} \end{aligned}

A comparison of the spectral sequences for these two fibrations shows that λ is an 𝔽_{p}- homology equivalence. Since |L| is p-good [BLO2, Proposition 1.12], the natural map from |L.π| to its fiberwise completion E_{w} is also an 𝔽_{p}-homology equivalence. Hence these maps induce homotopy equivalences E_{p}^{∧} ≃ (E_{w})_{p}^{∧} ≃ |L.π|_{p}^{∧}. This finishes the proof of the theorem, with F′ = F.π and L′ = L.π. □

Note that we are not assuming that Γ injects into Out_{fus}(S, F) in the hypotheses of Theorem 5.8. Thus, for example, if we start with a product fibration E = |L|_{p}^{∧} × BΓ, then we end up with (S, F′, L′) = (S, F, L) (and E_{p}^{∧} ≃ |L|_{p}^{∧}).

We do not know whether the index prime to p analog of Corollary 4.8 holds. The key point in the proof of Corollary 4.8 is the observation that if G is a finite group, whose fusion system at p contains a normal subsystem of index p^{m}, then G contains a normal subgroup of index p^{m}. This is not true for subsystems of index prime to p. For example, the fusion system at the prime 3 of the simple groups J_{4} and Ru has a subsystem of index 2.

6. Central extensions of fusion systems and linking systems

Let F be a fusion system over a p-group S. We say that a subgroup A ≤ S is central in F if C_{F}(A) = F, where C_{F}(A) is the centralizer fusion system defined in [BLO2, Definition A.3]. Thus A is central in F if A ≤ Z(S), and each morphism φ ∈ Hom_{F}(P, Q) in F extends to a morphism φ ∈ Hom_{F}(P A, QA) such that φ|_{A} = Id_{A}.

In this section, we first study quotients of fusion systems, and of p-local finite groups, by central subgroups. Afterwards, we will invert this procedure, and study central extensions of fusion systems and p-local finite groups.

\underline{6.1 Central quotients of fusion and linking systems} We first note that every saturated fusion system F contains a unique maximal central subgroup, which we regard as the center of F.

Proposition 6.1. For any saturated fusion system F over a p-group S, define

Z_{F}(S) = \{x ∈ Z(S) | φ(x) = x, all φ ∈ Mor(F^{c})\} = \underleftarrow{lim}_{F^{c}} Z(−) :

the inverse limit of the centers of all F-centric subgroups of S. Then Z_{F}(S) is the center of F: it is central in F, and contains all other central subgroups.

Proof. By definition, if A is central in F, then A ≤ Z(S), and any morphism in F between subgroups containing A restricts to the identity on A. Since all F-centric subgroups contain Z(S), this shows that A ≤ Z_{F}(S).

By Alperin’s fusion theorem (Theorem 1.5(a)), each morphism in F is a composite of restrictions of morphisms between F-centric subgroups. In particular, each morphism is a restriction of a morphism between subgroups containing Z_{F}(S) which is the identity on Z_{F}(S), and thus Z_{F}(S) is central in F. □

The center of a fusion system F has already appeared when studying mapping spaces of classifying spaces associated to F. By [BLO2, Theorem 8.1], for any p-local finite group (S, F, L),

Map(|L|_{p}^{∧}, |L|_{p}^{∧})_{Id} ≃ BZ_{F}(S).

We next define the quotient of a fusion system by a central subgroup.

Definition 6.2. Let F be a fusion system over a p-group S, and let A be a central subgroup. Define F/A to be the fusion system over S/A with morphism sets

Hom_{F/A}(P/A, Q/A) = Im[Hom_{F}(P, Q) ⟶ Hom(P/A, Q/A)].

By [BLO2, Lemma 5.6], if F is a saturated fusion system over a p-group S, and A ≤ Z_{F}(S), then F/A is also saturated as a fusion system over S/A. We now want to study the opposite question: if F/A is saturated, is F also saturated? The following example shows that it is very easy to construct counterexamples to this. In fact, under a mild hypothesis on S, if a fusion system over S/A has the form F/A for any fusion system F over S, then it has that form for a fusion system F which is not saturated.

Example 6.3. Fix a p-group S and a central subgroup A ≤ Z(S). Assume there is α ∈ Aut(S)∖Inn(S) such that α|_{A} = Id_{A} and α induces the identity on S/A. Let F be a saturated fusion system over S/A, and assume there is some fusion system F_{0} over S such that F_{0}/A = F. Let F ⊇ F_{0} be the fusion system over S defined by setting

Hom_{F}(P, Q) = {φ ∈ Hom(P, Q) | ∃ φ ∈ Hom(P A, QA),

φ|_{P} = φ, φ|_{A} = Id_{A}, φ/A ∈ Hom_{F}(P A/A, QA/A)}.

Then A is a central subgroup of F, and F/A = F. But F is not saturated, since Aut_{F}(S) contains as normal subgroup the group of automorphisms of S which are the identity on A and on S/A (a p-group by Lemma 1.15), and this subgroup is by assumption not contained in Inn(S). Hence Aut_{S}(S) is not a Sylow subgroup of Aut_{F}(S).

The hypotheses of the above example are satisfied, for example, by any pair of pgroups 1 ≠ A ≨ S with A ≤ Z(S), such that S is abelian, or more generally such that A ∩ [S, S] = 1.

We will now describe conditions which allow us to say when F is saturated. Before doing so, in the next two lemmas, we first compare properties of subgroups in F with those of subgroups of F/A, when F is a fusion system with central subgroup A. This is done under varying assumptions as to whether F or F/A is saturated.

Lemma 6.4. The following hold for any fusion system F over a p-group S, and any subgroup A ≤ Z(S) central in F.

(a) If A ≤ P ≤ S and P/A is F/A-centric, then P is F-centric.

(b) If F/A is saturated, and P ≤ S is F-quasicentric, then P A/A is F/A-quasicentric.

(c) If F and F/A are both saturated and P/A ≤ S/A is F/A-quasicentric, then P is F-quasicentric.

Proof. For each P ≤ S containing A, let C̃_{S}(P) ≤ S be the subgroup such that C̃_{S}(P)/A = C_{S/A}(P/A). Let

η_{P} : C̃_{S}(P) ⟶ Hom(P, A)

be the homomorphism η_{P} (x)(g) = [x, g] for x ∈ C̃_{S}(P) and g ∈ P. Thus Ker(η_{P}) = C_{S}(P).

For each P ≤ S containing A, set

Γ_{P} = Ker[Aut_{F}(P) ⟶ Aut_{F/A}(P/A)].

Since every α ∈ Aut_{F}(P) is the identity on A, Γ_{P} is a p-group by Lemma 1.15.

(a) If g, g′ commute in S, then their images commute in S/A. Thus for all Q ≤ S, C_{S}(Q)/A ≤ C_{S/A}(Q/A).

Assume P/A is F/A-centric. Then for each P′ which is F-conjugate to P, P′/A is F/A-conjugate to P/A, and hence C_{S}(P′)/A ≤ C_{S/A}(P′/A) ≤ P′/A. Thus P is F-centric.

(b) Fix P ≤ S such that P is F-quasicentric. Since C_{S}(P′A) = C_{S}(P′) for all P′ which is F-conjugate to P, P′A is fully centralized in F if and only if P′ is. Also, C_{F}(P′) = C_{F}(P′A) for such P′, and this shows that P A is also F-quasicentric. So after replacing P by P A, if necessary, we can assume P ≥ A.

Choose P′/A which is fully centralized in F/A and F/A-conjugate to P/A. Then P′ is F-conjugate to P, hence still F-quasicentric. So upon replacing P by P′, we are reduced to showing that P/A is F/A-quasicentric when P ≥ A, P is F-quasicentric, and P/A is fully centralized.

For any P′ which is F-conjugate to P, there is a morphism

φ/A ∈ Hom_{F/A}((P′·C̃_{S}(P′))/A, (P ·C̃_{S}(P))/A)

which sends P′/A to P/A (by axiom (II) for the saturated fusion system F/A). Then φ restricts to a morphism from C_{S}(P′) to C_{S}(P). Thus |C_{S}(P′)| ≤ |C_{S}(P)| for all P′ which is F-conjugate to P, and this proves that P is fully centralized in F.

If P/A is not F/A-quasicentric, then by Lemma 1.6 (and since F/A is saturated), there is some Q/A ≤ P/A·C_{S/A}(P/A) containing P/A, and some α ∈ Aut_{F}(Q), such that Id ≠ α/A ∈ Aut_{F/A}(Q/A) has order prime to p and α/A is the identity on P/A. Since α is also the identity on A, Lemma 1.15 implies that α|_{P} = Id_{P}. Set Q′ = Q ∩ C̃_{S}(P). Then α(Q′) = Q′, and η_{P} ∘ α = η_{P} since α is the identity on P ≥ A. Thus α induces the identity on Q′/C_{Q′}(P) since Ker(η_{P}) = C_{S}(P). Since α has order prime to p, Lemma 1.15 now implies that α|_{C_{Q′}(P)} ≠ Id. Thus C_{Q′}(P) ≤ C_{S}(P) and α|_{P ·C_{Q′}(P)} is a nontrivial automorphism in C_{F}(P) of order prime to p. Since P is fully centralized in F, this implies (by definition) that C_{F}(P) is not the fusion system of C_{S}(P), and hence that P is not F-quasicentric.

(c) Now assume that F and F/A are both saturated, and that P/A ≤ S/A is F/A-quasicentric. If P is not F-quasicentric, and P′ is F-conjugate to P and fully centralized in F, then by Lemma 1.6(b), there is some P′ ≤ Q ≤ P′·C_{S}(P′) and some Id ≠ α ∈ Aut_{F}(Q) such that α|_{P′} = Id_{P′} and α has order prime to p. Then Q/A ≤ (P′/A)·C_{S/A}(P′/A), α/A ∈ Aut_{F/A}(Q/A) also has order prime to p, and so α/A ≠ Id by Lemma 1.15 again. But by Lemma 1.6(a), this contradicts the assumption that P/A is F/A-quasicentric. □

In the next lemma, we compare conditions for being fully normalized in F and in F/A.

Lemma 6.5. The following hold for any fusion system F over a p-group S, and any subgroup A ≤ Z(S) central in F.

(a) Assume F is saturated. Then for all P, Q ≤ S containing A such that P is fully normalized in F, if φ, φ′ ∈ Hom_{F}(P, Q) are such that φ/A = φ′/A, then φ′ = φ ∘c_{x} for some x ∈ N_{S}(P) such that xA ∈ C_{S/A}(P/A).

(b) Assume F/A is saturated, and let P, P′ ≤ S be F-conjugate subgroups which contain A. Then P is fully normalized in F if and only if P/A is fully normalized in F/A. Moreover, if P is fully normalized in F, then there is ψ ∈ Hom_{F}(N_{S}(P′), N_{S}(P)) such that ψ(P′) = P.

Proof. (a) Assume F is saturated, and fix P, Q ≤ S containing A such that P is fully normalized in F. Let φ, φ′ ∈ Hom_{F}(P, Q) be such that φ/A = φ′/A. Then Im(φ) = Im(φ′). Set α = φ^{−1} ∘ φ′ ∈ Aut_{F}(P); then φ′ = φ ∘ α, and α/A = Id_{P/A}. Thus

α ∈ Ker[Aut_{F}(P) ⟶ Aut_{F/A}(P/A)],

which is a normal p-subgroup by Lemma 1.15. Since P is fully normalized, Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) and so α ∈ Aut_{S}(P). Thus α ∈ c_{x} for some x ∈ N_{S}(P), and xA ∈ C_{S/A}(P/A) since α/A = Id_{P/A}.

(b) We are now assuming that F/A is saturated. Assume first that P/A is fully normalized in F/A. Then by Lemma 1.3, there is a morphism

φ_{0} ∈ Hom_{F/A}(N_{S/A}(P′/A), N_{S/A}(P/A))  (1)

such that φ_{0}(P′/A) = P/A, and this lifts to φ ∈ Hom_{F}(N_{S}(P′), N_{S}(P)) such that φ(P′) = P. Therefore |N_{S}(P′)| ≤ |N_{S}(P)| for any P′ which is F-conjugate to P and P is fully normalized in F. This also proves that the last statement in (b) holds, once we know that P/A is fully normalized.

Assume now that P is fully normalized in F; we want to show that P/A is fully normalized in F/A. Fix P′ which is F-conjugate to P and such that P′/A is fully normalized in F/A. By Lemma 1.3 again, there exists

φ_{0} ∈ Hom_{F/A}(N_{S/A}(P/A), N_{S/A}(P′/A))

such that φ_{0}(P/A) = P′/A. Then φ_{0} = φ/A for some φ ∈ Hom_{F}(N_{S}(P), N_{S}(P′)), and φ is an isomorphism since P is fully normalized in F. Thus φ_{0} is an isomorphism, and hence P/A is fully normalized in F/A.

Thus if P is fully normalized in F, then P/A is also fully normalized, and we have already seen that the last statement in (b) holds in this case. □

We are now ready to give conditions under which we show that F is saturated if F/A is saturated. As in [5A1, §2], for any fusion system F over a p-group S, and any set H of subgroups of S, we say that F is H-generated if each morphism in F is a composite of restrictions of morphisms between subgroups in H.

Proposition 6.6. Let A be a central subgroup of a fusion system F over a p-group S, such that F/A is a saturated fusion system. Let H be any set of subgroups of S, closed under F-conjugacy and overgroups, which contains all F-centric subgroups of S. Assume

(a) Ker[Aut_{F}(P) ⟶ Aut_{F/A}(P/A)] ≤ Aut_{S}(P) for each P ∈ H which is fully normalized in F; and

(b) F is H-generated.

Then F is saturated.

Proof. Let H_{0} be the set of all P ∈ H such that P ≥ A. Since A is central, each morphism φ ∈ Hom_{F}(P, Q) extends to some φ ∈ Hom_{F}(P A, QA) such that φ|_{A} = Id_{A}. Thus F is H_{0}-generated if it is H-generated. So upon replacing H by H_{0}, we can assume all subgroups in H contain A.

By assumption, H contains all F-centric subgroups of S. So by Theorem 1.5(b), it suffices to check that axioms (I) and (II) hold for all P ∈ H. For all P ≤ S containing A, we write

Γ_{P} =^{def} Ker[Aut_{F}(P) ⟶ Aut_{F/A}(P/A)].

(I) Assume P ∈ H is fully normalized in F. Then P/A is fully normalized in F/A by Lemma 6.5(b). By assumption, Γ_{P} ≤ Aut_{S}(P). Hence

[Aut_{F}(P) : Aut_{S}(P)] = [Aut_{F/A}(P/A) : Aut_{S/A}(P/A)],

and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) since Aut_{S/A}(P/A) ∈ Syl_{p}(Aut_{F/A}(P/A)).

Assume P′ ≤ S is F-conjugate to P and fully centralized in F. By Lemma 6.5(b) again, there is ψ ∈ Hom_{F}(N_{S}(P′), N_{S}(P)) such that ψ(P′) = P. Then ψ(C_{S}(P′)) ≤ C_{S}(P), so P is also fully centralized.

(II) Assume φ ∈ Hom_{F}(P, S) is such that P ∈ H and φ(P) is fully centralized. Set

N_{φ} = {g ∈ N_{S}(P) | φc_{g}φ^{−1} ∈ Aut_{S}(φ(P))},

as usual. Then N_{φ}/A ≤ N_{φ/A}.

Assume first that φ(P) is fully normalized in F. Then φ(P)/A is fully normalized in F/A by Lemma 6.5(b), and hence also fully centralized. So by (II), applied to the saturated fusion system F/A, there is φ̂ ∈ Hom_{F}(N_{φ/A}, S/A) such that φ̂|_{(P/A)} = φ/A. Let φ ∈ Hom_{F}(N_{φ}, S) be a lift of φ̂, then (φ|_{P})/A = φ/A and φ(P) = φ(P). So there is α = φ ∘ (φ|_{P})^{−1} ∈ Γ_{φ(P)} ≤ Aut_{F}(φ(P)) such that α ∘ φ|_{P} = φ. By (a), there is x ∈ N_{S}(φ(P)) such that α = c_{x}; then c_{x} ∘ φ lies in Hom_{F}(N_{φ}, S) and extends φ.

It remains to prove the general case. Choose P′ which is fully normalized in F and F-conjugate to P. By Lemma 6.5(b), there is ψ ∈ Hom_{F}(N_{S}(φ(P)), N_{S}(P′)) such that ψ(φ(P)) = P′. Then N_{φ} ≤ N_{ψφ}. Since ψφ(P) = P′ is fully normalized, ψφ extends to some ψ ∈ Hom_{F}(N_{φ}, N_{S}(P′)). We will show that Im(ψ) ≤ Im(ψ), and thus there is φ ∈ Hom_{F}(N_{φ}, N_{S}(φ(P))) such that φ|_{P} = φ.

For each g ∈ N_{φ}, choose x ∈ N_{S}(φ(P)) such that φc_{g}φ^{−1} = c_{x}; then we obtain c_{ψ(x)} = ψc_{x}ψ^{−1} = c_{ψ(g)}, and so ψ(x)ψ(g)^{−1} ∈ C_{S}(P′). Since φ(P) is fully centralized

in F, ψ(C_{S}(φ(P))) = C_{S}(P′), and thus ψ(g) ∈ Im(ψ). □

For example, one can take as the set H in Proposition 6.6 either the set of subgroups P ≤ S containing A such that P/A is F/A-quasicentric (by Lemma 6.4(b)), or the set of F-centric subgroups of S.

Now let (S, F, L) be a p-local finite group. One can also define a centralizer linking system C_{L}(A) when A ≤ S is fully centralized [BLO2, Definition 2.4]. Since this is always a linking system associated to C_{F}(A), A is central in L (i.e., C_{L}(A) = L) if and only if A is central in F. So from now on, by a central subgroup of (S, F, L), we just mean a subgroup A ≤ Z(S) which is central in F.

Definition 6.7. Let (S, F, L) be a p-local finite group with a central subgroup A. Define L/A to be the category with objects the subgroups P/A for P ∈ Ob(L) (i.e., such that P is F-centric), and with morphism sets

Mor_{L/A}(P/A, Q/A) = Mor_{L}(P, Q)/δ_{P} (A).

Let (L/A)^{c} ⊆ L/A be the full subcategory with object set the F/A-centric subgroups in S/A.

Similarly, if L^{q} is the associated quasicentric linking system, then define L^{q}/A to be the category with objects the F/A-quasicentric subgroups of S/A — equivalently, the subgroups P A/A for P ∈ Ob(L^{q}) — and with morphisms

Mor_{L^{q}/A}(P/A, Q/A) = Mor_{L^{q}}(P, Q)/δ_{P} (A).

Note that by Lemma 6.4(a,c), for any P/A ≤ S/A, if P/A is F/A-centric or F/Aquasicentric, then P is F-centric or F-quasicentric, respectively. Thus the categories (L/A)^{c} ⊆ L/A and L^{q}/A are well defined.

We are now ready to prove our main theorem about quotient fusion and linking systems.

Theorem 6.8. Let A be a central subgroup of a p-local finite group (S, F, L) with associated quasicentric linking system L^{q}. Let L_{≥A}^{q} ⊆ L^{q} be the full subcategory with objects those F-quasicentric subgroups of S which contain A. Then the following hold.

(a) (S/A, F/A, (L/A)^{c}) is again a p-local finite group, and L^{q}/A is a quasicentric linking system associated to F/A.

(b) The inclusions of categories induce homotopy equivalences |L| ≃ |L_{≥A}^{q}| ≃ |L^{q}| and |(L/A)^{c}| ≃ |L/A| ≃ |L^{q}/A|.

(c) The functor τ: L^{q} → L^{q}/A, defined by τ(P) = P A/A and with the obvious maps on morphisms, induces principal fibration sequences

BA ⟶ |L| ⟶^{|τ|} |L/A|  and  BA ⟶ |L_{≥A}^{q}| ⟶^{|τ|} |L^{q}/A|

which remain principal fibration sequences after p-completion.

Proof. (a) The first statement is shown in [BLO2, Lemma 5.6] when |A| = p, and the general case follows by iteration. So we need only prove that L^{q}/A is a quasicentric linking system associated to F/A. Axioms (B)_{q}, (C)_{q}, and (D)_{q} for L^{q}/A follow immediately from those axioms applied to L^{q}, so it remains only to prove (A)_{q}.

Let H be the set of subgroups P ≤ S such that A ≤ P and P/A is F/A-quasicentric and fix P, Q ∈ H. By construction, C_{S/A}(P/A) acts freely on Mor_{L^{q}/A}(P/A, Q/A) and induces a surjection

Mor_{L^{q}/A}(P/A, Q/A)/C_{S/A}(P/A) −↠ Hom_{F/A}(P/A, Q/A).

We must show that this is a bijection whenever P/A is fully centralized in F/A. Since any other fully centralized subgroup in the same F-conjugacy class has centralizer of the same order, it suffices to show this when P/A is fully normalized in F/A; or equivalently (by Lemma 6.5(b)) when P is fully normalized in F.

Let C̃_{S}(P) ≤ S be the subgroup such that C̃_{S}(P)/A = C_{S/A}(P/A). Fix any F/Aquasicentric subgroup Q/A, and consider the following sequence of maps

Mor_{L^{q}}(P, Q) ⟶ Hom_{F}(P, Q) ⟶ Hom_{F/A}(P/A, Q/A).

Since L^{q} is the quasicentric linking system of F, by property (A)_{q} the first map is the orbit map of the action of C_{S}(P). The second is the orbit map for the action of Aut_{C̃_{S}(P)}(P) by Lemma 6.5(a). Thus the composite is the orbit map for the free action of C̃_{S}(P). It now follows that C̃_{S}(P)/A ≅ C_{S/A}(P/A) acts freely on Mor_{L^{q}/A}(P/A, Q/A) = Mor_{L^{q}}(P, Q)/A with orbit set Hom_{F/A}(P/A, Q/A).

(b) These homotopy equivalences are all special cases of Proposition 1.12(a).

(c) For any category C and any n ≥ 0, let C_{n} denote the set of n-simplices in the nerve of C; i.e., the set of composable n-tuples of morphisms. For each n ≥ 0, the group B(A)_{n} acts freely on (L_{≥A}^{q})_{n}: an element (a_{1}, …, a_{n}) acts by composing the i-th component with δ_{P} (a_{i}) for appropriate P. This action commutes with the face and degeneracy maps, and its orbit set is (L^{q}/A)_{n}. It follows that the projection of |L_{≥A}^{q}| onto |L^{q}/A| (and of |L| onto |L/A|) is a principal fibration with fiber the topological group BA = |B(A)| (see, e.g., [May, §§18–20] or [GJ, Corollary V.2.7]).

By the principal fibration lemma of Bousfield and Kan [BK, II.2.2], these sequences are still principal fibration sequences after p-completion. □

\underline{6.2 Central extensions of p-local finite groups} We first make it more precise what we mean by this.

Definition 6.9. A central extension of a (saturated) fusion system F over a p-group S by an abelian p-group A consists of a (saturated) fusion system F̃ over a p-group S̃, together with an inclusion A ≤ Z(S̃), such that A is a central subgroup of F̃, S̃/A ≅ S, and F̃/A ≅ F as fusion systems over S.

Similarly, a central extension of a p-local finite group (S, F, L) by an abelian group A consists of a p-local finite group (S̃, F̃, L̃), together with an inclusion A ≤ ZF̃(S̃), such that (S̃/A, F̃/A, (L̃/A)^{c}) ≅ (S, F, L).

Extensions of categories were defined and studied by Georges Hoff in [Hf], where he proves that they are classified by certain Ext-groups. We will deal with one particular case of this. Hoff’s theorem implies that an extension of categories of the type B(A) ⟶ L̃_{≥A}^{q} ⟶^{τ} L^{q} is classified by an element in \underleftarrow{lim}_{L^{q}}^{2}(A). What this extension

really means is that L^{q} is a quotient category of L̃_{≥A}^{q}, where each morphism set in L̃_{≥A}^{q} admits a free action of A with orbit set the corresponding morphism set in L^{q}. Also, \underleftarrow{lim}^{2}(A) means the second derived functor of the limit of the constant functor which sends each object of L^{q} to A and each morphism to Id_{A}.

We regard A as an additive group. Fix an element [ω] ∈ \underleftarrow{lim}_{L^{q}}^{2}(A), where ω is a

(reduced) 2-cocyle. Thus ω is a function from pairs of composable morphisms in L^{q} to A such that ω(f, g) = 0 if f or g is an identity morphism, and such that for any triple f, g, h of composable morphisms, the cocycle condition is satisfied

dω(f, g, h) =^{def} ω(g, h) − ω(gf, h) + ω(f, hg) − ω(f, g) = 0.

Consider the composite i ∘ δ_{S} : BS → L^{q}, where δ_{S} is induced by the distinguished morphism S → Aut_{L^{q}}(S) and i is induced by the inclusion of Aut_{L^{q}}(S) in L^{q} as the full subcategory with one object S. Then ω_{S} = (i ∘ δ_{S})^{∗}(ω) is a 2-cocycle defined on S which classifies a central extension of p-groups

1 ⟶ A ⟶ S̃ ⟶^{τ} S ⟶ 1 .

Thus S̃ = S × A, with group multiplication defined by

(h, b)·(g, a) = (hg, b + a + ω_{S}(g, h)),

and τ(g, a) = g. For each F-quasicentric subgroup P ≤ S, set P̃ = τ^{−1}(P) ≤ S̃.

Using this cocycle ω, we can define a new category L̃_{0} as follows. There is one object P̃ = τ^{−1}(P) of L̃_{0} for each object P of L^{q}. Morphism sets in L̃_{0} are defined by setting

MorL̃_{0}(P̃, Q̃) = Mor_{L^{q}}(P, Q) × A.

Composition in L̃_{0} is defined by

(g, b) ∘ (f, a) = (gf, a + b + ω(f, g)).

The associativity of this composition law follows since dω = 0. Furthermore, if we chose another representative ω + dμ where μ is a 1-cochain, the categories obtained are isomorphic. This construction comes together with a projection functor τ: L̃_{0} → L^{q}.

Finally, we define

δS̃: S ⟶ AutL̃_{0}(S)

by setting δS̃(g, a) = (δ_{S}(g), a). This is a group homomorphism by definition of ω_{S}.

Proposition 6.10. Let (S, F, L) be a p-local finite group, and let L^{q} be the associated quasicentric linking system. Fix an abelian group A and a reduced 2-cocycle ω on L^{q} with coefficients in A. Let

L̃_{0} ⟶^{τ} L^{q}  and  S̃ ⟶^{τ} S

be the induced extensions of categories and of groups, with distinguished monomorphism δS̃ as defined above.

Then there is a unique saturated fusion system F̃ over S̃ and a functor π̃: L̃_{0} → F̃, together with distinguished monomorphisms δP̃: P̃ → AutL̃_{0}(P).

If L̃ ⊆ L̃_{0} denotes the full subcategory whose objects are the F̃-centric subgroups of S̃ then (S̃, F̃, L̃) is a p-local finite group with central subgroup A ≤ Z(S̃), such that

(S̃/A, F̃/A, L̃_{0}/A) ≅ (S, F, L^{q}).  (1)

Also, L̃_{0} extends to a quasicentric linking system L̃^{q} associated to F̃.

Proof. Assume that inclusion morphisms ι_{P} have been chosen in the quasicentric linking system L^{q} associated to (S, F, L). For each F-quasicentric P ≨ S, define

ιP̃ = (ι_{P} , 0) ∈ MorL̃_{0}(P̃, S̃).

Next, define the distinguished monomorphism

δP̃: P̃·CS̃(P̃) ⟶ AutL̃_{0}(P̃)

to be the unique monomorphism such that the following square commutes for each P̃ and each q ∈ P̃·CS̃(P̃):

\begin{matrix}P̃ & \overline{^{ιP̃} →} & S̃ \\ P̃ & ↓^{δP̃(q,a)} & \overline{^{ιP̃} →} & S̃. & ↓^{δS̃(q,a)} \end{matrix}  (2)

More precisely, since δS̃(q, a) = (δ_{S}(q), a) by definition, this means that

δP̃(q, a) = (δ_{P} (q), a + ω(ι_{P} , δ_{S}(q)) − ω(δ_{P} (q), ι_{P})).

For each morphism (f, a) ∈ MorL̃_{0}(P̃, Q̃), and each element (q, b) ∈ P̃, there is a unique element c ∈ A such that the following square commutes:

\begin{matrix}P̃ & \overline{^{(f,a)} →} & Q̃ \\ P̃ & ↓^{δP̃(q,b)} & \overline{^{(f,a)} →} & Q̃. & ↓^{δQ̃(π(f)(q),c)} \end{matrix}  (3)

In this situation, we set

π̃(f, a)(q, b) = (π(f)(q), c) ∈ Q̃.

By juxtaposing squares of the form (3), we see that π̃(f, a) ∈ Hom(P̃, Q̃), and that this defines a functor π̃ from L̃_{0} to the category of subgroups of S̃ with monomorphisms.

Define F̃ to be the fusion system over S̃ generated by the image of π̃ and restrictions. By construction, the surjection τ: S̃ ⟶ S induces a functor τ_{∗}: F̃ ⟶ F between the fusion systems, which is surjective since F is generated by restrictions of morphisms between F-quasicentric subgroups (Theorem 1.5(a)). So we can identify F with F̃/A. By Lemma 6.4(b), for each F̃-quasicentric subgroup P ≤ S̃, P A/A is F-quasicentric. So we can extend L̃_{0} to a category L̃^{q} defined on all F̃-quasicentric subgroups, by setting

Mor_{L^{q}}(P, Q) = {f ∈ MorL̃_{0}(P A, QA) | π̃(f)(P) ≤ Q} and δ_{P} = δ_{P A}

for each pair P, Q of F̃-quasicentric subgroups. We extend π̃ to L̃^{q} in the obvious way.

It remains to prove that F̃ is saturated, and that L̃^{q} is a quasicentric linking system associated to F̃. In this process of doing this, we will also prove the isomorphism (1).

Let H be the set of subgroups P̃ = τ^{−1}(P) ≤ S̃ for all F-quasicentric subgroups P ≤ S.

F̃ is saturated: We want to apply Proposition 6.6 to prove that F̃ is saturated. By Lemma 6.4(b), H contains all F̃-quasicentric subgroups of S̃ which contain A, and in particular, all F̃-centric subgroups (since every F̃-centric subgroup of S̃ must contain A). Since F̃ is H-generated by construction, it remains only to prove condition (a) in Proposition 6.6.

Fix some P̃ = τ^{−1}(P) in H, and let φ ∈ AutF̃(P̃) be such that τ_{∗}(φ) = Id_{P}. Choose (f, a) ∈ π̃^{−1}(φ); thus

φ = π̃(f, a) ∈ Ker[AutF̃(P̃) ⟶ Aut_{F}(P)].

Then π(f) = Id_{P}, so f = δ_{P} (q) for some q ∈ C_{S}(P), and (f, a) = δP̃(q, c) for some c ∈ A. Since δP̃ is a homomorphism, the definition of φ = π̃(f, a) via (3) shows that φ = π̃(f, a) = δ̃P̃(q, c) = c_{(q,c)}, and thus that φ ∈ AutS̃(P̃). Thus condition (a) in Proposition 6.6 holds, and this finishes the proof that F̃ is saturated.

L̃^{q} is a quasicentric linking system associated to F̃: The distinguished monomorphisms δP̃, for P̃ ∈ Ob(L̃^{q}), were chosen so as to satisfy (D)_{q}, and this was independent of the choice of inclusion morphisms which lift the chosen inclusion morphisms in L^{q}. Once the δ_{P} were determined, then π̃ was defined to satisfy (C)_{q}, and F̃ was defined as the category generated by Im(π̃) and restrictions. Thus all of these structures were uniquely determined by the starting data. Axiom (B)_{q} follows from (C)_{q} by Lemma 1.10.

It remains only to prove (A)_{q}. We have already seen that the functor π̃ is surjective on all morphism sets. Also, since CS̃(P) = CS̃(P A) for all P ≤ S̃, it suffices to prove (A)_{q} for morphisms between subgroups P̃ = τ^{−1}(P) and Q̃ = τ^{−1}(Q) containing A. By construction, CS̃(P̃) acts freely on each morphism set MorL̃^{q}(P̃, Q̃), and it remains to show that if P̃ is fully centralized, then π̃_{P̃,Q̃} is the orbit map of this action. As in the proof of Theorem 6.8(a), it suffices to do this when P̃ and P are fully normalized.

Fix two morphisms (f, a) and (g, b) from P̃ to Q̃ such that π̃(f, a) = π̃(g, b). Then π(f) = π(g), so g = f ∘ δ_{P} (x) for some x ∈ C_{S}(P), and

(g, b) = (f, a) ∘ (δ_{P} (x), b − a − ω(δ_{P} (x), f)) = (f, a) ∘ δP̃(x, c),

where c = b − a − ω(δ_{P} (x), f) − (ω(ι, δ_{S}(x)) − ω(δ_{S}(x), ι_{P})). Also, π̃(δP̃(x, c)) = IdP̃ implies (via (3)) that (x, c) commutes with all elements of P̃, so (x, c) ∈ CS̃(P̃), and this finishes the argument. □

We now want to relate the obstruction theory for central extensions of linking systems with kernel A to those for central extensions of p-groups, and to those for principal fibrations with fiber BA. As a consequence of this, we will show (in Theorem 6.13) that when appropriate restrictions are added, these three types of extensions are equivalent.

Given a central extension F̃ of F by the central subgroup A, there is an induced central extension 1 → A → S̃ → S → 1 of Sylow subgroups. Restriction to subgroups P ≤ S produces corresponding central extensions 1 → A → P̃ → P → 1. The homology classes of these central extensions are all compatible with morphisms from the fusion system, and hence define an element in \underleftarrow{lim}_{F} H^{2}(−; A). This, together with

notation already used in [BLO2], motivates the following definition:

Definition 6.11. For any saturated fusion system over a p-group S, and any finite abelian p-group A, define

H^{∗}(F; A) = \underleftarrow{lim}_{F} H^{∗}(−; A) ≅ \underleftarrow{lim}_{F^{c}} H^{∗}(−; A).

The following lemma describes the relation between the cohomology of F and the cohomology of the geometric realization of any linking system associated to F.

Lemma 6.12. For any p-local finite group (S, F, L), and any finite abelian p-group A, the natural homomorphism

H^{∗}(|L|; A) ⟶ H^{∗}(F; A),  (1)

induced by the inclusion of BS into |L|, is an isomorphism. Furthermore, there are natural isomorphisms

\underleftarrow{lim}_{L^{q}}^{2}(A) ≅ H^{2}(|L^{q}|; A) ≅ H^{2}(|L|; A).  (2)

Proof. The second isomorphism in (2) follows from Proposition 1.12(a). The first isomorphism holds for higher limits of any constant functor over any small, discrete category C, since both groups are cohomology groups of the same cochain complex

0 ⟶ ∏_{c} A ⟶ ∏_{c_{0}→c_{1}} A ⟶ ∏_{c_{0}→c_{1}→c_{2}} A ⟶ · · · .

This cochain complex for higher limits is shown in [GZ, Appendix II, Proposition 3.3] (applied with M = Ab^{op}).

To prove the isomorphism (1), it suffices to consider the case where A = ℤ/p^{n} for some n. This was shown in [BLO2, Theorem 5.8] when A = ℤ/p, so we can assume that n ≥ 2, and that the lemma holds when A = ℤ/p^{n−1}. Consider the following diagram of Bockstein exact sequences

\begin{aligned}→ & H^{i+1}(|L|; ℤ/p) & \overline{→} & H^{i}(|L|; ℤ/p^{n−1}) & \overline{→} & H^{i}(|L|; ℤ/p^{n}) & \overline{→} & H^{i}(|L|; ℤ/p) & \overline{→} \\ → & H^{i+1}(BS; ℤ/p) & ↓ & → & H^{i}(BS; ℤ/p^{n−1}) & ↓ & → & H^{i}(BS; ℤ/p^{n}) & ↓ & → & H^{i}(BS; ℤ/p) & ↓ & → . \end{aligned} (1)

We claim that the bottom row restricts to an exact sequence of groups H^{∗}(F; −). Once this is shown, the result follows by the 5-lemma.

By [BLO2, Proposition 5.5], there is a certain (S, S)-biset Ω which induces, via a sum of composites of transfer maps and maps induced by homomorphisms, an idempotent endomorphism of H^{∗}(BS; ℤ/p) whose image is H^{∗}(F; ℤ/p). This biset Ω also induces endomorphisms [Ω] of H^{∗}(BS; ℤ/p^{n}) and H^{∗}(BS; ℤ/p^{n−1}) which commute with the bottom row in (1), since any exact sequence induced by a short exact sequence of coefficient groups will commute with transfer maps and maps induced by homomorphisms. The same argument as that used in the proof of [BLO2, Proposition 5.5] shows that in all of these cases, Im([Ω]) = H^{∗}(F; −), and the restriction of [Ω] to its image is multiplication by |Ω|/|S| ∈ 1 + pℤ. Thus the sequence of the H^{∗}(F; −) splits as a direct summand of the bottom row in (1), and hence is exact. □

We can now collect the results about central extensions of (S, F, L) in the following theorem, which is the analog for p-local finite groups of the classical classification of central extensions of groups.

Theorem 6.13. Let (S, F, L) be a p-local finite group. For each finite abelian p-group A, the following three sets are in one-to-one correspondence:

(a) equivalence classes of central extensions of (S, F, L) by A;

(b) equivalence classes of principal fibrations BA → X → |L|_{p}^{∧}; and

(c) isomorphism classes of central extensions 1 → A → S̃ ^{τ}→ S → 1 for which each morphism φ ∈ Hom_{F}(P, Q) lifts to some φ̃ ∈ Hom(τ^{−1}(P), τ^{−1}(Q)).

The equivalence between the first two is induced by taking classifying spaces, and the equivalence between (a) and (c) is induced by restriction to the underlying p-group. These sets are all in natural one-to-one correspondence with

\underleftarrow{lim}_{L}^{2}(A) ≅ H^{2}(|L|; A) ≅ H^{2}(F; A).  (1)

Proof. The three groups in (1) are isomorphic by Lemma 6.12. By Proposition 6.10, central extensions of L are classified by \underleftarrow{lim}_{L}^{2}(A). Principal fibrations over |L|_{p}^{∧} with

fiber BA are classified by

[|L|_{p}^{∧}, B(BA)] ≅ H^{2}(|L|_{p}^{∧}; A) ≅ H^{2}(|L|; A),

where the second isomorphism holds since |L| is p-good ([BLO2, Proposition 1.12]) and A is an abelian p-group. Central extensions of S by A are classified by H^{2}(S; A), and the central extension satisfies the condition in (c) if and only if the corresponding element of H^{2}(S; A) extends to an element in the inverse limit H^{2}(F; A).

A central extension of p-local finite groups induces a principal fibration of classifying spaces by Theorem 6.8(c), and this principal fibration restricts to the principal fibration over BS of classifying spaces of p-groups. Thus, if the fibration over |L| is classified by χ ∈ H^{2}(|L|; A), then the fibration over BS is classified by the restriction of χ to H^{2}(BS; A). It is well known that the invariant in H^{2}(S; A) = H^{2}(BS; A) for a central extension of S by A is the same as that which describes the principal fibration over BS with fiber BA (see [AM, Lemma IV.1.12]). Since H^{2}(|L|; A) injects into H^{2}(F; A) (Lemma 6.12), this shows that χ is also the class of the 2-cocycle which defines the extension of categories. So the map between the sets in (a) and (b) defined by taking geometric realization is equal to the bijection defined by the obstruction theory.

Since the isomorphism \underleftarrow{lim}_{L}^{2}(A) ≅ H^{2}(F; A) is defined by restriction to S (as a group

of automorphisms in L), the bijection between (a) and (c) induced by the bijection of obstruction groups is the same as that induced by restriction to S. □

The following corollary shows that all minimal examples of “exotic” fusion systems have trivial center.

Corollary 6.14. Let F be a saturated fusion system over a p-group S, and assume there is a nontrivial subgroup 1 ≠ A ≤ Z(S) which is central in F. Then F is the fusion system of some finite group if and only if F/A is.

Proof. Assume F is the fusion system of the finite group G, with S ∈ Syl_{p}(G). By assumption (A is central in F), each morphism in F extends to a morphism between subgroups containing A which is the identity on A. Hence F is also the fusion system of C_{G}(A) over S, and so F/A is the fusion system of C_{G}(A)/A.

It remains to prove the converse. Assume F/A is isomorphic to the fusion system of the finite group G, and identify S/A with a Sylow p-subgroup of G. Since by Lemma 6.12, H^{2}(F/A; A) ≅ H^{2}(BG; A) and A is F-central, the cocycle classifying the extension is in H^{2}(BG; A), and hence there is an extension of finite groups

1 ⟶ A ⟶ G ⟶^{τ} G ⟶ 1

with the same obstruction invariant as the extension F → F/A. In particular, we can identify S = τ^{−1}(S/A) ∈ Syl_{p}(G), and G = G/A.

We will prove the following two statements:

(a) F has an associated centric linking system L; and

(b) L_{S/A}^{c}(G/A) is the unique centric linking system associated to F/A = F_{S/A}(G/A).

Once these have been shown, then they imply that

(S/A, F/A, (L/A)^{c}) ≅ (S/A, F_{S/A}(G/A), L_{S/A}^{c}(G/A))

as p-local finite groups. Hence (S, F, L) ≅ (S, F_{S}(G), L_{S}^{c}(G)) as p-local finite groups by Theorem 6.13, and thus F is the fusion system of G.

It remains to prove (a) and (b). Let

Z_{F}: O^{c}(F) ⟶ ℤ_{(p)}-mod and Z_{G}: O_{S}^{c}(G) ⟶ ℤ_{(p)}-mod

be the categories and functors of Definition 1.7. By [O1, Lemma 2.1], Z_{G} can also be regarded as a functor on O^{c}(F_{S}(G)), and

(c) \underleftarrow{lim}^{∗}_{O_{S}^{c}(G)}(Z_{G}) ≅ \underleftarrow{lim}^{∗}_{O^{c}(F_{S}(G))}(Z_{G}).

By [BLO2, Proposition 3.1], the existence and uniqueness of a centric linking system depends on the vanishing of certain obstruction classes: the obstruction to existence lies in \underleftarrow{lim}^{3}_{O_{S}^{c}(G)}(Z_{F}) and the obstruction to uniqueness in \underleftarrow{lim}^{2}_{O_{S}^{c}(G)}(Z_{F}). Thus (b) follows

from [BLO2, Proposition 3.1] and (c), once we know that \underleftarrow{lim}^{2}(Z_{G/A}) = 0; and this is shown in [O1, Theorem A] (if p is odd) or [O2, Theorem A] (if p = 2).

It remains to prove point (a), and we will do this by showing that

\underleftarrow{lim}^{3}_{O^{c}(F)}(Z_{F}) ≅ \underleftarrow{lim}^{3}_{O_{S}^{c}(G)}(Z_{G}) = 0.  (1)

The last equality follows from [O1, Theorem A] or [O2, Theorem A] again, so it remains only to prove the isomorphism.

Let H be the set of subgroups P ≤ S containing A such that P/A is F/A-centric; or equivalently, p-centric in G/A. Let O^{H}(F) ⊆ O^{c}(F) and O^{H}(F_{S}(G)) ⊆ O^{c}(F_{S}(G)) be the full subcategories with object sets H.

We claim that

\underleftarrow{lim}^{∗}_{O^{H}(F)}(Z_{F}) ≅ \underleftarrow{lim}^{∗}_{O^{c}(F)}(Z_{F}) and  \underleftarrow{lim}^{∗}_{O^{H}(F_{S}(G))}(Z_{G}) ≅ \underleftarrow{lim}^{∗}_{O^{c}(F_{S}(G))}(Z_{G}) .  (2)

If P ≤ S is F-centric but not in H, then there is x ∈ N_{S}(P)∖P such that c_{x} ∈ Aut_{F}(P) is the identity on P/A and on A, but is not in Inn(P). Thus 1 ≠ [c_{x}] ∈ O^{p}(Out_{F}(P)), and so P is not F-radical. By [BLO2, Proposition 3.2], if F: O^{c}(F)^{op} → ℤ_{(p)}-mod is any functor which vanishes except on the conjugacy class of P, then \underleftarrow{lim}^{∗}(F) ≅ Λ^{∗}(Out_{F}(P); F (P)), where Λ^{∗} is a certain functor defined in [JMO, §5]. By [JMO, Proposition 6.1(ii)], Λ^{∗}(Out_{F}(P); F (P)) = 0 for any F, since O_{p}(Out_{F}(P)) ≠ 1. From the long exact sequences of higher limits induced by extension of functors, it now follows that \underleftarrow{lim}^{∗}_{O^{c}(F)}(F) = 0 for any functor F on O^{c}(F) which vanishes on O^{H}(F); and

this proves the first isomorphism in (2). The second isomorphism follows by a similar argument.

The natural surjection of F onto F/A induces isomorphisms of categories

O^{H}(F) ≅ O^{c}(F/A) ≅ O^{H}(F_{S}(G)) .  (3)

By the definition of H, we clearly have bijections between the sets of objects in these categories, so it remains only to compare morphism sets. The result follows from Lemma 6.5(a) for sets of morphisms between pairs of fully normalized subgroups, and the general case follows since every object is isomorphic to one which is fully normalized.

We next claim that for all i > 0,

\underleftarrow{lim}^{i}_{O^{H}(F)}(Z_{F}) ≅ \underleftarrow{lim}^{i}_{O^{c}(F/A)}(Z_{F}/A) and  \underleftarrow{lim}^{i}_{O^{H}(F_{S}(G))}(Z_{G}) ≅ \underleftarrow{lim}^{i}_{O^{c}(F/A)}(Z_{G}/A) . (4)

To show this, by (3), together with the long exact sequence of higher limits induced by the short exact sequence of functors

1 → A ⟶ Z_{F} ⟶ Z_{F}/A → 1,

we need only show that \underleftarrow{lim}^{i}_{O^{c}(F/A)}(A) = 0 for all i > 0. Here, A denotes the constant

functor on O^{c}(F/A) which sends all objects to A and all morphisms to Id_{A}. For each P ∈ H, let F_{A,P} be the functor on O^{c}(F/A) where F_{A,P} (P′/A) = A if P′ is F-conjugate to P and F_{A,P} (P′/A) = 0 otherwise; and which sends isomorphisms between subgroups conjugate to P to Id_{A}. By [BLO2, Proposition 3.2], \underleftarrow{lim}^{∗}(F_{A,P}) ≅ Λ^{∗}(Out_{F/A}(P/A); A).

Also, Λ^{i}(Out_{F/A}(P/A); A) = 0 for i > 0 by [JMO, Proposition 6.1(i,ii)], since the action of Out_{F/A}(P/A) on A is trivial. From the long exact sequences of higher limits induced by extension of functors, we now see that \underleftarrow{lim}^{i}(A) = 0 for all i > 0.

Finally, we claim that

Z_{F}/A ≅ Z_{G}/A  (5)

as functors on O^{c}(F/A) under the identifications in (3). To see this, note that for each P, (Z_{F}/A)(P) = Z(P)/A = (Z_{G}/A)(P); and since these are both identified as subgroups of P/A, any morphism in O^{c}(F/A) from P/A to Q/A induces the same map (under the two functors) from Z(Q)/A to Z(P)/A. (This argument does not apply to prove that Z_{F} ≅ Z_{G}. These two functors send each object to the same subgroup of S, but we do not know that they send each morphism to the same homomorphism between the subgroups.)

Thus by (2), (4), and (5), for all i > 0,

\underleftarrow{lim}^{i}_{O^{c}(F)}(Z_{F}) ≅ \underleftarrow{lim}^{i}_{O^{H}(F)}(Z_{F}) ≅ \underleftarrow{lim}^{i}_{O^{c}(F/A)}(Z_{F}/A)

≅ \underleftarrow{lim}^{i}_{O^{c}(F/A)}(Z_{G}/A) ≅ \underleftarrow{lim}^{i}_{O^{H}(F_{S}(G))}(Z_{G}) ≅ \underleftarrow{lim}^{i}_{O_{S}^{c}(G)}(Z_{G}).

This finishes the proof of (1), and hence finishes the proof of the corollary. □

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66 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER

Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain

E-mail address: broto@mat.uab.es

Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain

E-mail address: natalia@mat.uab.es

Department of Mathematics, University of Chicago, Chicago, IL 60637, USA

E-mail address: jg@math.uchicago.edu

Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, U.K.

E-mail address: ran@maths.abdn.ac.uk

LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

E-mail address: bob@math.univ-paris13.fr