EXTENSIONS OF LINKING SYSTEMS AND FUSION SYSTEMS

BOB OLIVER

Abstract. We correct two errors in the statement and proof of a theorem in [BCGLO2], and at the same time extend that result to a more general theorem about extensions of p-local finite groups. Other special cases of this theorem have already been shown in two later papers, so we feel it will be useful to have this more general result in the literature.

This paper has two purposes: to correct some errors in the statement and proof of a theorem in the earlier paper [BCGLO2], and also to prove a more general version of this theorem, describing (very roughly) how to construct extensions of fusion and linking systems by groups of outer automorphisms. Special cases of this construction have been used in at least two papers written since [BCGLO2].

When G is a finite group and S ∈ Syl_{p}(G), the fusion category of G is the category F_{S}(G) whose objects consist of all subgroups of S, and where

Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q) =^{def} {c_{g} ∈ Hom(P, Q) | g ∈ G, gP g^{−1} ≤ Q} .

This gives a means of encoding the p-local structure of G: the conjugacy relations among the p-subgroups of G. The centric linking category of G is a closely related category which (among other things) provides a link between the fusion in G and the homotopy type of its p-completed classifying space. These categories motivated the definition by Puig [Pg] of abstract fusion systems, and by Broto, Levi, and Oliver [BLO] of abstract linking systems.

The main theorem in this note (Theorem 9) describes how to construct certain types of extensions of abstract fusion and linking systems. The special case shown in [BCGLO2, Theorem 4.6] shows how to extend a linking system by a p-group of outer automorphisms. Other special cases were used by Castellana and Libman [CL] to construct wreath products of linking systems, and by Andersen, Oliver, and Ventura [AOV] to construct exotic fusion and linking systems under certain hypotheses. Since all three of these constructions have very similar proofs, it should be useful to have one reference which covers all of these cases, and hopefully any others which might be needed in the future.

There was an omission in the statement of [BCGLO2, Theorem 4.6], in that the group S must be assumed to act on L_{0} via isotypical automorphisms (Definition 5). Without this assumption, it need not induce an action on the fusion system F_{0}. The error in the proof of the theorem occurs in Step 4. In that step, a certain property of subgroups not in a family H was proven using a result shown in Step 3 — a result shown there only for subgroups which are in the family H.

In most cases, our main interest is to construct extensions of fusion systems. However, when trying to do this, one quickly discovers that a fusion system alone does not contain enough information to construct extensions, at least not in a straightforward way. This is why the results in [BCGLO2], [CL], and [AOV] are all stated in terms of linking systems. Furthermore, the extensions L_{0} ◁ L of linking systems which we construct are such that the geometric realization |L_{0}| has the homotopy type of a finite covering space of |L| — with covering group the group of outer automorphisms by which we extended. This is not in general the case for extensions of the fusion systems.

2000 Mathematics Subject Classification. Primary 55R35. Secondary 20D20, 20E22.

Key words and phrases. Classifying spaces, Sylow subgroups, fusion, extensions.

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

We now recall the definitions of abstract fusion and linking systems, and their basic properties which will be needed later.

An abstract fusion system over a finite p-group S is a category F such that Ob(F) is the set of all subgroups of S, and such that 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 F-isomorphism followed by an inclusion.

Some additional conditions are needed to make this very useful.

Definition 1 ([Pg], [BLO]). Let F be a fusion system over a p-group S.

• Two subgroups P, Q ≤ S are F-conjugate if they are isomorphic as objects of the category F.

• 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 saturated 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) For all P ≤ S and φ ∈ Hom_{F}(P, S) such that φP is fully centralized, 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.

More generally, when H is a set of subgroups of S closed under F-conjugacy, F is Hsaturated if axioms (I) and (II) hold for all P ∈ H.

The main objects of interest here are the saturated fusion systems. For example, F_{S}(G) is saturated for any finite group G and any S ∈ Syl_{p}(G). Axioms (I) and (II) follow mostly as consequences of the Sylow theorems (cf. [BLO, Proposition 1.3]).

We will need to refer frequently to the following classes of subgroups in a fusion system.

Definition 2. Let F be a fusion system over a p-group S.

• A subgroup P ≤ S is F-centric if it is fully centralized and C_{S}(P) = Z(P);

• P ≤ S is F-radical if O_{p}(Aut_{F}(P)) = Inn(P); and

• P ≤ S is F-quasicentric if for each P′ ≤ S which is fully centralized and F-conjugate to P, each P′ ≤ Q ≤ P′·C_{S}(P′), and each α ∈ Aut_{F}(Q) such that α|_{P′} = Id, α has p-power order.

We now turn to abstract linking systems. As explained above, they seem to be the most natural structures for describing extensions of the type which we are looking at in this paper. For any finite group G and any set H of subgroups of G, let T_{H}(G) denote the transporter category of G: the category with Ob(T_{H}(G)) = H, and where for each P, Q ∈ H,

Mor_{T_{H}(G)}(P, Q) = N_{G}(P, Q) =^{def} {g ∈ G | gP g^{−1} ≤ Q}

(the transporter set).

Definition 3 ([BLO, Definition 1.7] & [BCGLO1, Definition 3.3]). Let F be a fusion system over a p-group S. A linking system associated to F is a finite category L, together with a pair of functors

T_{Ob(L)}(S) ⟶^{δ} L ⟶^{π} F ,

satisfying the following conditions:

(A) Ob(L) is a set of subgroups of S closed under F-conjugacy and overgroups, and includes all subgroups which are F-centric and F-radical. Also, δ is the identity on objects and π is the inclusion on objects. For each P, Q ∈ Ob(L) such that P is fully centralized in F, C_{S}(P) acts freely on Mor_{L}(P, Q) via δ_{P,P} and right composition, and π_{P,Q} induces a bijection

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

(B) For each P, Q ∈ Ob(L) and each g ∈ N_{S}(P, Q), π_{P,Q} sends δ_{P,Q}(g) ∈ Mor_{L}(P, Q) to c_{g} ∈ Hom_{F}(P, Q).

(C) For all φ ∈ Mor_{L}(P, Q) and all g ∈ P, the diagram

\xymatrix{P \ar[r]^{φ} \ar[d]_{δ_{P,P} (g)} & Q \ar[d]^{δ_{Q,Q}(π(φ)(g))} \\ P \ar[r]^{φ} & Q}

commutes in L.

The main differences between this definition and those in [BLO] and [BCGLO1] are that it is more flexible on the set of objects in L, and that we define here δ as a functor on the transporter category of S. That δ can be defined on T_{Ob(L)}(S) follows as a consequence of the earlier definitions (see [BLO, Proposition 1.11] and [BCGLO1, Lemma 3.7]), and including it in the definition allows us to drop axiom (D)_{q} in [BCGLO1, Definition 3.3].

Note that the functor δ in the above definition must be injective on all morphism sets. For any pair of objects P and Q, π ∘ δ_{P,Q} sends N_{S}(P, Q) = Mor_{T (S)}(P, Q) onto Hom_{S}(P, Q) ≅ N_{S}(P, Q)/C_{S}(P), C_{S}(P) acts freely on π^{−1}(Hom_{S}(P, Q)) by axiom (A), and thus δ_{P,Q} is injective.

The condition that L be closed under overgroups could perhaps be dropped. But it simplifies the proof of Proposition 4 below, and is needed in any case in the hypotheses of the main theorem. Also, it is difficult to imagine a situation where we might need a linking system which is not closed under overgroups.

One assumption which was made in [BCGLO1] and not repeated here was that all objects in a linking system associated to F are F-quasicentric. We will see in the next lemma that this follows as a consequence of the above definition.

The reason for assuming L contains all subgroups which are F-centric and F-radical originates with [BCGLO1, Theorem 3.5], which says that if L′ ⊆ L are two linking systems associated to the same fusion system, such that Ob(L′) contains all F-centric F-radical subgroups and Ob(L) is contained in the set of all F-quasicentric subgroups, then the geometric realizations of these two categories are homotopy equivalent. In other words, this seems to be the minimal set of objects needed to get the information which one needs from a linking system. But as we will see shortly, this also plays an important role in the proof of Theorem 9 below.

In general, when L is a linking system over S, and P ≤ Q are both objects in L, we define ι_{P}^{Q} = δ_{P,Q}(1), where δ: T_{Ob(L)}(S) ⟶ L is the functor in the definition of L. We regard these morphisms as the inclusion morphisms in L.

Proposition 4. The following hold for any linking system L associated to a saturated fusion system F over a p-group S.

(a) If P ∈ Ob(L) is fully normalized in F, then δ_{P} (N_{S}(P)) ∈ Syl_{p}(Aut_{L}(P)).

(b) For every morphism ψ ∈ Mor_{L}(P, Q), and every P_{0}, Q_{0} ∈ Ob(L) such that P_{0} ≤ P, Q_{0} ≤ Q, and π(ψ)(P_{0}) ≤ Q_{0}, there is a unique morphism ψ|_{P_{0},Q_{0}} ∈ Mor_{L}(P_{0}, Q_{0}) (the “restriction” of ψ) such that ψ ∘ ι_{P_{0},P} = ι_{Q_{0},Q} ∘ ψ|_{P_{0},Q_{0}}.

(c) Let P, Q, P , Q ∈ Ob(L) and ψ ∈ Iso_{L}(P, Q) be such that P ◁ P, Q ◁ Q, and ψδ_{P} (P)ψ^{−1} ≤ δ_{Q}(Q). Then there is a unique morphism ψ ∈ Mor_{L}(P , Q) such that ψ|_{P,Q} = ψ.

(d) All morphisms in L are monomorphisms and epimorphisms in the categorical sense.

(e) All objects in L are F-quasicentric. In other words, if P ∈ Ob(L) is fully centralized in F, and P ≤ Q ≤ P C_{S}(P), then each automorphism α ∈ Aut_{F}(Q) such that α|_{P} = Id_{P} has p-power order.

Proof. (a) If P ∈ Ob(L) is fully normalized in F, then Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)),

Aut_{F}(P) ≅ Aut_{L}(P)/δ_{P} (C_{S}(P)) and Aut_{S}(P) ≅ N_{S}(P)/C_{S}(P),

and hence δ_{P} (N_{S}(P)) ∈ Syl_{p}(Aut_{L}(P)).

(b) Fix ψ ∈ Mor_{L}(P, Q), and P_{0} ≤ P, Q_{0} ≤ Q as above, and set φ = π(ψ)|_{P_{0},Q_{0}} ∈ Hom_{F}(P_{0}, Q_{0}). Let ψ_{1} ∈ Mor_{L}(P_{0}, Q_{0}) be any morphism such that π(ψ_{1}) = φ. Then π(ι_{Q_{0}}^{Q} ∘ ψ_{1}) = π(ψ ∘ ι_{P_{0}}^{P}), so by (A), there is g ∈ C_{S}(P_{0}) such that ι_{Q_{0}}^{Q} ∘ ψ_{1} ∘ δ_{P_{0}}(g) = ψ ∘ ι_{P_{0}}^{P}. We can thus take ψ|_{P_{0},Q_{0}} = ψ_{1} ∘ δ_{P_{0}}(g).

Now assume ψ_{0} and ψ′_{0} are two such restrictions; thus ι_{Q_{0}}^{Q} ∘ ψ_{0} = ι_{Q_{0}}^{Q} ∘ ψ′_{0}. By axiom (A) again, there is h ∈ C_{S}(P_{0}) such that ψ′_{0} = ψ_{0} ∘ δ_{P_{0}}(h). But then h = 1 since C_{S}(P_{0}) acts freely on Mor_{L}(P_{0}, Q), and thus ψ_{0} = ψ′_{0}.

(c) Let ψ ∈ Iso_{L}(P, Q), and P ◁ P ≤ S and Q ◁ Q ≤ S, be such that ψ ∘ δ_{P} (P) ∘ ψ^{−1} ≤ δ_{Q}(Q). If Q is fully centralized in F, then axiom (II) for the saturated fusion system F implies that φ = π(ψ) extends to a homomorphism φ ∈ Hom_{F}(P , S). Choose ψ̂ ∈ Mor_{L}(P , S) such that π(ψ̂) = φ. Then π(ψ̂|_{P,Q}) = π(ψ), and since Q is fully centralized, there is g ∈ C_{S}(Q) such that ψ = δ_{Q}(g) ∘ ψ̂|_{P,Q}. Set ψ = δ_{S}(g) ∘ ψ̂; then ψ|_{P,Q} = ψ. Also, π(ψ)(P) ≤ Q by (C) and the original assumption on ψ, and so ψ restricts to a morphism in Mor_{L}(P , Q) by (b). Note that this is an isomorphism if ψ ∘ δ_{P} (P) ∘ ψ^{−1} = δ_{Q}(Q).

Now assume Q is not fully centralized. Choose R which is F-conjugate to P and Q and fully normalized in F. Fix an isomorphism φ ∈ Iso_{L}(Q, R). Then φδ_{Q}(Q)φ^{−1} is a p-subgroup of Aut_{L}(R), so by (a), there is χ ∈ Aut_{L}(R) such that (χφ)δ_{Q}(Q)(χφ)^{−1} = δ_{R}(R) for some R ≤ N_{S}(R). We just showed that χφ and χφψ extend to morphisms φ ∈ Iso_{L}(Q, R) and ψ ∈ Mor_{L}(P , R), and so φ^{−1} ∘ ψ ∈ Mor_{L}(P , Q) extends ψ. This proves (c), except for the uniqueness which will follow from (d).

(d) We claim that L is a transporter system in the sense of [OV1, Definition 3.1]. Once this has been shown, then (d) follows from [OV1, Lemma 3.2(b,d)].

Points (A1), (B), and (C) in [OV1, Definition 3.1] hold for L by definition, while (II) holds by (c) (it requires only the existence of ψ and not uniqueness), and (I) by (a). Upon comparing axiom (A) for a linking system with (A2) for a transporter system, we see that the only thing left to check is that for each P, Q ∈ Ob(L), δ_{Q}(C_{S}(Q)) acts freely on Mor_{L}(P, Q).

To see this, fix ψ ∈ Mor_{L}(P, Q) and g ∈ C_{S}(Q) such that δ_{Q}(g)∘ψ = ψ. Set Q_{0} = π(ψ)(P), and set ψ_{0} = ψ|_{P,Q_{0}}. Thus ψ = ι_{Q_{0}}^{Q} ∘ ψ_{0} where ψ_{0} is an isomorphism, so δ_{Q}(g) ∘ ι_{Q_{0}}^{Q} = ι_{Q_{0}}^{Q}. In other words, δ_{Q_{0},Q}(g) = δ_{Q_{0},Q}(1). Since g ∈ C_{S}(Q_{0}), and this group acts freely on Mor_{L}(Q_{0}, Q) by (A), we conclude that g = 1.

(e) Fix P ∈ Ob(L); we claim P is F-quasicentric. It suffices to show this when P is fully centralized. If P is not F-quasicentric, then by definition, there is some Q and some Id ≠ α ∈ Aut_{F}(Q) such that P ≤ Q ≤ P C_{S}(P), α|_{P} = Id_{P}, and α has order prime to p. Assume this is the case, and choose ψ ∈ Aut_{L}(Q) such that π(ψ) = α. We can assume ψ also has order prime to p; otherwise replace it by ψ^{k} for some appropriate k. Then π(ψ|_{P,P}) = Id_{P}, so ψ|_{P,P} = δ(g) for some g ∈ C_{S}(P). Thus ψ|_{P,P} has p-power order and order prime to p, so ψ|_{P,P} = Id in Aut_{L}(P). But this means that ψ ∘ ι_{P}^{Q} = Id_{Q} ∘ ι_{P}^{Q}, so ψ = Id since ι_{P}^{Q} is an epimorphism by (d). Hence α = Id_{Q}, which contradicts the original assumption on α. □

Since we want to construct extensions of fusion and linking systems, we must say what we mean by automorphisms of linking systems and by normal linking subsystems. We first look at automorphisms.

Definition 5. For any linking system L, an automorphism of categories α: L ⟶^{≅} L is isotypical if for each P ∈ Ob(L), α(δ_{P} (P)) = δ_{α(P)}(α(P)). Let Aut_{typ}^{I}(L) be the group of isotypical automorphisms of L which send inclusions to inclusions.

Let F: L ⟶ Grp be the functor π: L ⟶ F followed by the forgetful functor from F to groups. An automorphism α of a linking system L is isotypical if and only if there is a natural isomorphism of functors F ∘ α ≅ F. This was shown for centric linking systems in [BLO, Lemma 8.2], and the same proof applies in this more general setting.

The next proposition shows that each α ∈ Aut_{typ}^{I}(L) induces an automorphism of F in a natural way.

Proposition 6. Let L be a linking system associated to a fusion system F over a p-group S, with structure functors T_{Ob(L)}(S) ⟶^{δ} L ⟶^{π} F. Fix α ∈ Aut_{typ}^{I}(L). Let β ∈ Aut(S) be such that α(δ_{S}(g)) = δ_{S}(β(g)) for all g ∈ S. Then β is “fusion preserving” in the following sense: there is an automorphism α̂ of the category F which sends P ≤ S to β(P) and sends φ ∈ Hom_{F}(P, Q) to βφβ^{−1} ∈ Hom_{F}(β(P), β(Q)). Furthermore, π ∘ α = α̂ ∘ π.

Proof. Clearly, α(S) = S, and hence α sends δ_{S}(S) to itself. Thus β is well defined. Since α sends inclusions to inclusions, it commutes with restrictions. So for P ∈ Ob(L), since α_{P} sends δ_{P} (P) to δ_{α(P)}(α(P)), α_{S} sends δ_{S}(P) to δ_{S}(α(P)), and thus α(P) = β(P). Furthermore, for each g ∈ P, α sends δ_{P} (g) to δ_{β(P)}(β(g)).

Now fix P, Q ∈ Ob(L) and ψ ∈ Mor_{L}(P, Q), and set φ = π(ψ) ∈ Hom_{F}(P, Q). For each g ∈ P, α sends

\xymatrix{P \ar[r]^{ψ} \ar[d]_{δ_{P} (g)} & Q \ar[d]^{δ_{Q}(φ(g))} \\ P \ar[r]^{ψ} & Q}  to  \xymatrix{β(P) \ar[r]^{α(ψ)} \ar[d]_{δ_{β(P)}(β(g))} & β(Q) \ar[d]^{δ_{β(Q)}(β(φ(g)))} \\ β(P) \ar[r]^{α(ψ)} & β(Q)} .

By axiom (C) in Definition 3, these squares commute, and also (since morphisms in L are epimorphisms) β(φ(g)) = π(α(ψ))(β(g)). Thus

π(α(ψ)) = βπ(ψ)β^{−1}.  (1)

In particular, βφβ^{−1} ∈ Hom_{F}(β(P), β(Q)) for each P, Q ∈ Ob(L) and φ ∈ Hom_{F}(P, Q). Since Ob(L) includes all subgroups which are F-centric and F-radical, all morphisms in F are composites of restrictions of morphisms between objects of L (cf. [BLO, Theorem A.10]). Hence there is a well defined functor α̂ from F to itself which sends each P ≤ S to β(P) and sends each φ ∈ Hom_{F}(P, Q) to βφβ^{−1}. This is an automorphism of the category F by the same argument applied to α^{−1}. By (1), π ∘ α = α̂ ∘ π. □

The following definition of a normal fusion subsystem is the most convenient for our purposes. When F is a fusion system over S and S_{0} ◁ S, then S_{0} is strongly closed if no element of S_{0} is F-conjugate to any element of S∖S_{0}.

Definition 7. Let F be a saturated fusion system over a p-group S, and let F_{0} ⊆ F be a subcategory which is a saturated fusion subsystem over a subgroup S_{0} ≤ S. Then F_{0} is normal in F (F_{0} ◁ F) if

(i) S_{0} is strongly closed in F;

(ii) for all P, Q ≤ S_{0} and φ ∈ Hom_{F}(P, Q), there are morphisms α ∈ Aut_{F}(S_{0}) and φ_{0} ∈ Hom_{F_{0}}(α(P), Q) such that φ = φ_{0} ∘ α|_{P,α(P)}; and

(iii) for each P, Q ≤ S_{0}, each φ ∈ Hom_{F_{0}}(P, Q), and each β ∈ Aut_{F}(S_{0}), βφβ^{−1} ∈ Hom_{F_{0}}(β(P), β(Q)).

This is equivalent to Puig’s definition of a normal subsystem [Pg, § 6.4]. It is also equivalent to Aschbacher’s definition [Asch, § 3] of an F-invariant subsystem, except that he does not require the subsystem to be saturated. See [Asch, Theorem 3.3] for a proof of the equivalence of these definitions.

For example, when G_{0} ◁ G are finite groups, S ∈ Syl_{p}(G), and S_{0} = S ∩ G_{0} ∈ Syl_{p}(G_{0}), then F_{S_{0}}(G_{0}) is a normal subsystem of F_{S}(G) under the above definition. The first and last conditions clearly hold, and (ii) holds by the Frattini argument: G = G_{0}·N_{G}(S_{0}), since any subgroup G-conjugate to S_{0} is also G_{0}-conjugate.

There is now an obvious analogous definition of a normal linking subsystem.

Definition 8. Fix a pair of saturated fusion systems F_{0} ◁ F over p-groups S_{0} ◁ S such that F_{0} is normal in F, and let L_{0} ⊆ L be associated linking systems. Then L_{0} is normal in L (L_{0} ◁ L) if

(i) Ob(L) = {P ≤ S | P ∩ S_{0} ∈ Ob(L_{0})};

(ii) for all P, Q ∈ Ob(L_{0}) and ψ ∈ Mor_{L}(P, Q), there are morphisms γ ∈ Aut_{L}(S_{0}) and ψ_{0} ∈ Mor_{L_{0}}(γ(P), Q) such that ψ = ψ_{0} ∘ γ|_{P,γ(P)}; and

(iii) for all γ ∈ Aut_{L}(S_{0}), P, Q ∈ Ob(L_{0}), and ψ ∈ Mor_{L_{0}}(P, Q), γ|_{Q,γ(Q)} ∘ ψ ∘ γ|_{P,γ(P)}^{−1} is in Mor_{L_{0}}(γ(P), γ(Q)).

Here, in (ii) and (iii), we write γ(P) = π(γ)(P) and γ(Q) = π(γ)(Q) for short.

In fact, condition (ii) in Definition 8 follows from other conditions in that definition, together with condition (ii) in Definition 7. But we include it here anyway to make the analogy between the two definitions clearer.

Whenever L_{0} ◁ L is a pair of linking systems over p-groups S_{0} ◁ S, then the geometric realization |L_{0}| has the homotopy type of a covering space over |L| with covering group the quotient Aut_{L}(S_{0})/Aut_{L_{0}}(S_{0}). We don’t prove this here since it isn’t used, but it follows by essentially the same proof as that of [BCGLO2, Theorem 3.9] or [OV1, Proposition 4.1(d)].

We are now ready to describe the procedure for constructing extensions of linking systems. To help motivate the hypotheses, we first describe an analogous construction with groups. Assume we are given three groups H_{0}, H, and G_{0}, where H_{0} ≤ G_{0} and H_{0} ◁ H, together with an action τ of H on G_{0} which leaves H_{0} invariant. Regard τ as a homomorphism from H to Aut(G_{0}:H_{0}), where Aut(G_{0}:H_{0}) is the group of automorphisms of G_{0} which leave H_{0} invariant. We want to construct a group G = G_{0}H, where G_{0} ◁ G, and where the conjugation action of H on G_{0} is that defined by τ. The obvious way to do this is to start with the semidirect product G_{0} ⋊ H defined by τ, and then set G = (G_{0} ⋊ H)/{(g, g^{−1}) | g ∈ H_{0}}. In order to do this, the set {(g, g^{−1})} must be a normal subgroup, and one quickly discovers that the necessary and sufficient condition for this to be the case is for the following diagram to commute:

\xymatrix{H_{0} \ar[r]^{conj} \ar[d]_{incl} & Aut(G_{0}:H_{0}) \ar[d]^{restr.} \\ H \ar[r]^{conj} \ar[ur]^{τ} & Aut(H_{0}) .}

The hypotheses in Theorem 9 are similar, except that G_{0} ◁ G are replaced by a pair of linking systems L_{0} ◁ L.

The following theorem generalizes [BCGLO2, Theorem 4.6], and also generalizes a related result in [CL]. Recall that by definition, the set of objects in a linking system L associated to a fusion system F over S is closed under F-conjugacy and overgroups, and must contain among its objects all subgroups which are F-centric and F-radical. By the conjugation action of ψ ∈ Aut_{L}(S) on L is meant the action which sends P ≤ S to π(ψ)(P), and which sends α ∈ Mor(L) to ψαψ^{−1} (after replacing each ψ by the appropriate restriction).

Theorem 9. Fix a saturated fusion system F_{0} over a finite p-group S_{0}, and let L_{0} be a linking system associated to F_{0}. Set H_{0} = Ob(L_{0}), and assume it is closed under overgroups. Set Γ_{0} = Aut_{L_{0}}(S_{0}), and regard S_{0} as a subgroup of Γ_{0} via the inclusion of T_{H_{0}}(S_{0}) into L_{0}. Fix a finite group Γ such that Γ_{0} ◁ Γ, and a homomorphism τ: Γ ⟶ Aut_{typ}^{I}(L_{0}) which makes both triangles in the following diagram commute:

[figure: Aut_{L_{0}}(S_{0}) = Γ_{0}; conj; incl; (1a); Aut_{typ}^{I}(L_{0}); (α↦α_{S_{0}}); (1b); Γ; conj; τ; Aut(Γ_{0}) ,]  (1)

Let F_{1} be the smallest fusion system over S_{0} (not necessarily saturated) which contains F_{0} and Aut_{Γ}(S_{0}), where Γ acts on S_{0} via conjugation on S_{0} = O_{p}(Γ_{0}) ◁ Γ. Fix S ∈ Syl_{p}(Γ). Then there is a saturated fusion system F over S which contains F_{1} as full subcategory, and such that F_{0} ◁ F.

Assume, in addition, that

C_{Γ}(S_{0}) is a p-group;  (2)

and also that

Γ/Γ_{0} is a p-group, or each P ∈ H_{0} is F_{0}-centric,  (3′)

or more generally

P ∈ H_{0}, P ≤ Q ≤ P ·C_{S_{0}}(P), α ∈ Aut_{F_{1}}(Q), α|_{P} = Id_{P} ⟹ α has p-power order. (3)

Then F can be chosen so as to have an associated linking system L for which L_{0} ◁ L,

Ob(L) = H =^{def} {P ≤ S | P ∩ S_{0} ∈ H_{0}} ,

and Aut_{L}(S_{0}) = Γ with the given action on L_{0}. If L′ is another linking system, associated to a saturated fusion system F′ over S, such that L_{0} ◁ L′, F_{0} ◁ F′, and Aut_{L′}(S_{0}) = Γ with the given action on L_{0} and the given inclusion S ≤ Γ, then F′ = F and L′ ≅ L.

Proof. The categories L and F will be constructed in Steps 1 and 2. We then show F is H-saturated in Steps 3 and 4, and finish the proof that F is saturated in Step 5. Step 5 is essentially a corrected version of Step 4 in the proof of [BCGLO2, Theorem 4.6]. We prove (3′) implies (3) at the beginning of Step 6. Afterwards, we assume (2) and (3) throughout the rest of the proof, show L is a linking system in Steps 6 and 7, and prove its uniqueness in Step 8. The normality of F_{0} in F and of L_{0} in L are shown at the end of Steps 5 and 7, respectively.

We first fix some notation. For all P ≤ S, we write P_{0} = P ∩ S_{0}. Let

T_{H_{0}}(S_{0}) ⟶^{δ_{0}} L_{0} ⟶^{π_{0}} F_{0}

be the structure functors for the linking system L_{0}. For P ≤ Q, set ι_{P}^{Q} = (δ_{0})_{P,Q}(1). Recall we regard S_{0} as a subgroup of Γ_{0} = Aut_{L_{0}}(S_{0}) via (δ_{0})_{S_{0}}.

For each γ ∈ Γ, let c_{γ} ∈ Aut(S_{0}) denote conjugation by γ on S_{0} = O_{p}(Γ_{0}) ◁ Γ. By the commutativity of (1b), this is the restriction to S_{0} of τ(γ)_{S_{0}} ∈ Aut(Γ_{0}). Hence by Proposition 6, c_{γ} is fusion preserving (induces an automorphism of the category F_{0}), and τ(γ)(P) = c_{γ}(P) for all P ∈ H_{0}. To simplify notation below, we write γ(P) = τ(γ)(P) to denote this action of γ on H_{0}.

For each γ_{0} ∈ Γ_{0} = Aut_{L_{0}}(S_{0}), γ_{0} acts on the set Mor(L_{0}) by composing on the left or right with γ_{0} and its restrictions. Thus for any φ ∈ Mor_{L_{0}}(P, Q), we set

γ_{0}φ = γ_{0}|_{Q,γ_{0}(Q)} ∘ φ ∈ Mor_{L_{0}}(P, γ_{0}(Q))

and

φγ_{0} = φ ∘ γ_{0}|_{γ_{0}^{−1}(P),P} ∈ Mor_{L_{0}}(γ_{0}^{−1}(P), Q).

Here, we write γ_{0}(P) = π(γ_{0})(P) for short. This defines natural left and right actions of Γ_{0} on the set Mor(L_{0}). By the commutativity of (1a), the conjugation action ψ ↦ γ_{0}ψγ_{0}^{−1} on Mor(L_{0}) is the restriction to Γ_{0} of τ; in particular, γ_{0}(P) = τ(γ_{0})(P) as in the last paragraph.

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}) = Mor(L_{0}) ×_{Γ_{0}} Γ = (Mor(L_{0}) × Γ)/∼,

where (φ, γ) ∼ (φ′, γ′) if and only if there is γ_{0} ∈ Γ_{0} such that φ′ = φγ_{0} and γ′ = γ_{0}^{−1}γ. Thus (φγ_{0}, γ) ∼ (φ, γ_{0}γ) for all φ ∈ Mor(L_{0}), γ_{0} ∈ Γ_{0}, and γ ∈ Γ. If φ ∈ Mor_{L_{0}}(P, Q), then [[φ, γ]] ∈ Mor_{L_{1}}(γ^{−1}(P), Q) denotes the equivalence class of the pair (φ, γ). Composition is defined by

[[φ, γ]] ∘ [[ψ, η]] = [[φ ∘ τ(γ)(ψ), γη]].

Here, τ(γ)(ψ) ∈ Mor_{L_{0}}(γ(P), γ(Q)) for ψ ∈ Mor_{L_{0}}(P, Q) (recall γ(P) = τ(γ)(P), etc.).

To show composition is well defined, we note that for all φ, ψ ∈ Mor(L_{0}), γ_{0}, η_{0} ∈ Γ_{0}, and γ, η ∈ Γ with appropriate domain and range,

[[φγ_{0}, γ]] ∘ [[ψη_{0}, η]] = [[φγ_{0} ∘ τ(γ)(ψη_{0}), γη]] = [[φγ_{0} ∘ τ(γ)(ψ), (γη_{0}γ^{−1})γη]]

= [[φγ_{0} ∘ τ(γ)(ψ)γ_{0}^{−1}, γ_{0}γη_{0}η]] = [[φ ∘ τ(γ_{0}γ)(ψ), γ_{0}γη_{0}η]] = [[φ, γ_{0}γ]] ∘ [[ψ, η_{0}η]]

The second equality follows from the commutativity of (1b), and the fourth from that of (1a).

We claim that

all morphisms in L_{1} are monomorphisms and epimorphisms.  (4)

For any [[φ, γ]], [[ψ, η]], and [[ψ′, η′]] with appropriate domain and range,

[[φ, γ]] ∘ [[ψ, η]] = [[φ, γ]] ∘ [[ψ′, η′]] ⟹ γη = γη′ and φ ∘ τ(γ)(ψ) = φ ∘ τ(γ)(ψ′)
⟹ η = η′ and τ(γ)(ψ) = τ(γ)(ψ′) ⟹ ψ = ψ′

since morphisms in L_{0} are monomorphisms (Proposition 4(d)); and thus [[φ, γ]] is a monomorphism. The proof that morphisms are epimorphisms is similar.

Set Aut_{Γ}(S_{0}) = {c_{γ} ∈ Aut(S_{0}) | γ ∈ Γ}. Let F_{1} be the smallest fusion system over S_{0} which contains F_{0} and Aut_{Γ}(S_{0}). Define

π_{1} : L_{1} ⟶ F_{1}

by setting π_{1}([[φ, γ]]) = π_{0}(φ) ∘ c_{γ}. To show this is a functor (that it preserves composition), we must show the following square commutes

\xymatrix{P \ar[r]^{π_{0}(φ)} \ar[d]_{c_{γ}} & Q \ar[d]^{c_{γ}} \\ γ(P) \ar[r]^{π_{0}(τ(γ)(φ))} & γ(Q)}

for each φ ∈ Mor_{L_{0}}(P, Q) and each γ ∈ Γ; and this follows from the commutativity of (1b) together with the last statement in Proposition 6 (applied with α = τ(γ) and β = c_{γ}). Since π_{1}(L_{1}) contains F_{0}|_{H_{0}} and Aut_{Γ}(S_{0}), and is closed under restrictions of morphisms to subgroups in H_{0} (Proposition 4(b)), π_{1} maps onto F_{1}|_{H_{0}}.

We regard L_{0} as a subcategory of L_{1} by identifying φ ∈ Mor_{L_{0}}(P, Q) with [[φ, 1]] ∈ Mor_{L_{1}}(P, Q). By construction, π_{0} = π_{1}|_{L_{0}}. For P ≤ Q in H_{0}, the inclusion morphism ι_{P}^{Q} = (δ_{0})_{P,Q}(1) for L_{0} is also considered as an inclusion morphism in L_{1}. The existence of restriction morphisms in L_{0} (Proposition 4(b)) carries over easily to the existence of restriction morphisms in L_{1}, and they are unique by (4).

For all P, Q ∈ H_{0}, define

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

by setting δ_{P,Q}(s) = [[ι_{s(P)}^{Q}, s]]. When s ∈ S_{0}, [[ι_{s(P)}^{Q}, s]] = [[(δ_{0})_{P,Q}(s), 1]]; and thus δ_{P,Q} extends the monomorphism (δ_{0})_{P,Q} from N_{S_{0}}(P, Q) to Mor_{L_{0}}(P, Q) defined for L_{0}. To simplify the notation, we write δ(x) = δ_{P,Q}(x) when P and Q are understood.

We claim that for all P, Q ∈ H_{0}, ψ ∈ Mor_{L_{1}}(P, Q), and x ∈ P,

δ(π_{1}(ψ)(x)) ∘ ψ = ψ ∘ δ(x).

Set ψ = [[φ, γ]], where γ ∈ Γ and φ ∈ Mor_{L_{0}}(γ(P), Q). Then

ψ ∘ δ(x) = [[φ, γ]] ∘ [[Id_{P} , x]] = [[φ, γx]] = [[φ, c_{γ}(x)γ]] = [[φ ∘ δ_{0}(c_{γ}(x)), γ]]
= [[δ_{0}(π_{0}(φ)(c_{γ}(x))) ∘ φ, γ]] = [[δ_{0}(π_{1}(ψ)(x)) ∘ φ, γ]]
= [[δ_{0}(π_{1}(ψ)(x)), 1]] ∘ [[φ, γ]] = δ(π_{1}(ψ)(x)) ∘ ψ ,

where the fifth equality holds by axiom (C) for the linking system L_{0}.

We next show that morphisms in L_{1} have the following extension property:

∀ P, Q ∈ H_{0}, ψ ∈ Iso_{L_{1}}(P, Q), and P , Q ≤ S_{0} for which P ◁ P, Q ◁ Q,
and ψδ(P)ψ^{−1} ≤ δ(Q), ∃! ψ ∈ Mor_{L_{1}}(P , Q) such that ψ|_{P,Q} = ψ.

(5)

Set ψ = [[φ, γ]] where φ ∈ Mor_{L_{0}}(γ(P), Q). For all x ∈ P,

[[φ, γ]] ∘ [[δ_{0}(x), 1]] ∘ [[φ, γ]]^{−1} = [[φ ∘ τ(γ)(δ_{0}(x)) ∘ φ^{−1}, 1]] = [[φ ∘ δ_{0}(c_{γ}(x)) ∘ φ^{−1}, 1]] ∈ δ(Q) ,

where τ(γ)(δ_{0}(x)) = δ_{0}(c_{γ}(x)) by the commutativity of (1b). Thus φδ_{0}(γ(P))φ^{−1} ≤ δ_{0}(Q), so φ extends to φ ∈ Mor_{L_{0}}(γ(P), Q) by Proposition 4(c). Set ψ = [[φ, γ]]. Then ψ|_{P,Q} = ψ

since τ(γ)(ι_{P}^{P}) = ι_{γ(P)}^{γ(P)} (τ(γ) sends inclusions to inclusions), and this proves (5).

Step 2: We next construct categories L 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.

Now let L be the category with Ob(L) = H, and where for all P, Q ∈ H,

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

Let

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

be the restriction of δ_{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_{1}}(P_{0}, Q_{0}) where ψ ∘ δ(x) = δ(φ(x)) ∘ ψ ∀ x ∈ P }.

Define π: L ⟶ F_{2} to be the identity on objects, and to send ψ ∈ Mor_{L}(P, Q) to the homomorphism π(ψ)(x) = y if ψ ∘ δ(x) = δ(y) ∘ ψ (uniquely defined by (4)). This is clearly a functor: it is seen to preserve composition by juxtaposing the commutative squares which define π on morphisms.

Let F be the fusion system over S generated by F_{2} and restriction of homomorphisms. Since H = Ob(F_{2}) is closed under overgroups, F_{2} is a full subcategory of F. Since L_{1} is a full subcategory of L, Hom_{F_{1}}(P, Q) = Hom_{F_{2}}(P, Q) for all P, Q ∈ H_{0}. If P, Q ≤ S_{0} are any subgroups and φ ∈ Hom_{F}(P, Q), then φ is a composite of restrictions of morphisms in F_{2}, and hence (since P ∈ Ob(F_{2}) = H implies P_{0} ∈ H_{0}) a composite of restrictions of morphisms in F_{2} (equivalently F_{1}) between subgroups in H_{0}. Thus φ ∈ Hom_{F_{1}}(P, Q); and we conclude that F_{1} is also a full subcategory of F.

Step 3: We next prove that

each P ∈ H is F-conjugate to some P′ such that δ(N_{S}(P′_{0})) ∈ Syl_{p}(Aut_{L}(P′_{0})). (6)

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}].) If γ ∈ Γ and Q_{0}, Q′_{0} ∈ H_{0}, then since γ acts on L_{0} and hence on F_{0} as a group of automorphisms (Proposition 6), Q_{0} is F_{0}-conjugate to Q′_{0} if and only if γ(Q_{0}) is F_{0}-conjugate to γ(Q′_{0}).

Let Γ′ ⊆ Γ be the subset of those γ ∈ Γ such that γ(P_{0}) is F_{0}-conjugate to P_{0}. For γ_{1}, γ_{2} ∈ Γ′, γ_{1}γ_{2}(P_{0}) is F_{0}-conjugate to γ_{1}(P_{0}) since γ_{2}(P_{0}) is F_{0}-conjugate to P_{0}, and hence γ_{1}γ_{2} ∈ Γ′. Thus Γ′ is a subgroup of Γ. Since S_{0} ◁ Γ, Q_{0} and Q′_{0} are S_{0}-conjugate if and only if γ(Q_{0}) and γ(Q′_{0}) are. Since each γ ∈ Γ acts on S_{0} via the fusion preserving automorphism c_{γ} ∈ Aut(S_{0}) as shown above, γ permutes the subgroups fully normalized in S_{0}. This proves that Γ′ permutes the set P_{fn}.

Fix S^{∗} ∈ Syl_{p}(Γ′). Let η ∈ Γ be such that S′ =^{def} ηS^{∗}η^{−1} ≤ S. Since P_{fn} has order prime to p by [BCGLO1, Proposition 1.16], there is some [P_{0}^{∗}] ∈ P_{fn} fixed by S^{∗}. In other words, for each γ ∈ S^{∗}, γ(P_{0}^{∗}) is S_{0}-conjugate to P_{0}^{∗}. So for each s = ηγη^{−1} ∈ S′ (where γ ∈ S^{∗}), s(η(P_{0}^{∗})) is S_{0}-conjugate to η(P_{0}^{∗}). Set Q_{0} = η(P_{0}^{∗}). Then each coset in S′/S_{0} contains some element s which normalizes Q_{0}, and hence

|N_{S}(Q_{0})| ≥ |N_{S_{0}}(Q_{0})|·|S′/S_{0}| = |N_{S_{0}}(P_{0}^{∗})|·|S^{∗}/S_{0}| .

Since Γ′ is the subgroup of elements of Γ which send P_{0} to a subgroup in its F_{0}-conjugacy class,

|Aut_{L}(Q_{0})| = |Aut_{L}(P_{0})| = |Aut_{L_{0}}(P_{0})|·|Γ′/Γ_{0}| = |Aut_{L_{0}}(P_{0}^{∗})|·|Γ′/Γ_{0}| .

Since P_{0}^{∗} is fully normalized in F_{0}, S^{∗} ∈ Syl_{p}(Γ′), and S_{0} ∈ Syl_{p}(Γ_{0}), this shows that δ(N_{S}(Q_{0})) is a Sylow p-subgroup of Aut_{L}(Q_{0}).

Choose any ψ ∈ Iso_{L}(P_{0}, Q_{0}). Then ψδ(N_{S}(P_{0}))ψ^{−1} is a p-subgroup of Aut_{L}(Q_{0}). Choose χ ∈ Aut_{L}(Q_{0}) such that (χψ)δ(N_{S}(P_{0}))(χψ)^{−1} ≤ δ(N_{S}(Q_{0})). By definition of the category L, χψ extends to a morphism ψ ∈ Mor_{L}(P, N_{S}(Q_{0})). Set P′ = π(ψ)(P). Then P′_{0} = Q_{0}, P′ is F-conjugate to P, and this finishes the proof of (6).

Step 4: We are now ready to show that F is H-saturated. For each P ∈ H, set

E(P) = Ker[Aut_{L}(P) ⟶^{π_{P}} Aut_{F}(P)] .

Fix P ∈ H such that δ(N_{S}(P_{0})) ∈ Syl_{p}(Aut_{L}(P_{0})). By (6), every subgroup in H is Fconjugate to some such P. Write G = Aut_{L}(P_{0}), T = δ(N_{S}(P_{0})), and P̂ = δ(P) for short, where δ = δ_{P_{0}}. Thus P̂ ≤ T ∈ Syl_{p}(G). Fix R ∈ Syl_{p}(N_{G}(P̂)), and choose α ∈ G such that αRα^{−1} ≤ T. Then

αRα^{−1} ∈ Syl_{p}(N_{G}(αP̂α^{−1})) , αRα^{−1} ≤ T ⟹ αRα^{−1} = N_{T} (αP̂α^{−1}) .

Set P′ = π(α)(P) ≤ S. Then αP̂α^{−1} = δ(P′) by definition of π, and

N_{T} (αP̂α^{−1}) = N_{δ(N_{S}(P_{0}))}(δ(P′)) = δ(N_{S}(P′)) ,
N_{G}(αP̂α^{−1}) = N_{Aut_{L}(P_{0})}(δ(P′)) = Aut_{L}(P′) .

We have thus found P′ F-conjugate to P such that δ(N_{S}(P′)) ∈ Syl_{p}(Aut_{L}(P′)). This in turn implies

Aut_{S}(P′) ∈ Syl_{p}(Aut_{F}(P′)) and δ(C_{S}(P′)) ∈ Syl_{p}(E(P′)).  (7)

Since F_{2} is a full subcategory of F, all F-morphisms between subgroups in H lift to morphisms in L by definition of F_{2}. Since P′ ∈ H, |Aut_{F}(P′)| and |E(P′)| depend only on the F-conjugacy class (= L-isomorphism class) of P′. So by (7), the subgroup P′ must be fully normalized and fully centralized in F. If P′′ ∈ H is any other subgroup F-conjugate to P′ and fully normalized, then since |N_{S}(P′′)| = |Aut_{S}(P′′)|·|C_{S}(P′′)|, P′′ must be fully centralized and Aut_{S}(P′′) ∈ Syl_{p}(Aut_{F}(P′′)). This finishes the proof of axiom (I) for the fusion system F, and also shows that a subgroup P ∈ H is fully centralized if and only if δ_{P} (C_{S}(P)) ∈ Syl_{p}(E(P)).

Now assume P ∈ H is fully centralized in F. Thus δ(C_{S}(P)) ∈ Syl_{p}(E(P)). Fix Q ∈ H and φ ∈ Iso_{F}(Q, P), and set

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

Let ψ ∈ Iso_{L}(Q, P) be any morphism in π^{−1}(φ). Then ψδ_{Q}(N)ψ^{−1} ≤ δ_{P} (N_{S}(P))·E(P). Since δ_{P} (C_{S}(P)) ∈ Syl_{p}(E(P)), ψδ_{Q}(N)ψ^{−1} is conjugate by an element of E(P) to a subgroup of δ_{P} (N_{S}(P)). So upon replacing ψ by χψ for some appropriate χ ∈ E(P), we can assume ψδ_{Q}(N)ψ^{−1} ≤ δ_{P} (N_{S}(P)).

Set ψ_{0} = ψ, regarded as an element ψ_{0} ∈ Iso_{L_{1}}(Q_{0}, P_{0}). By definition of L, we have the inclusion

ψ_{0}δ_{Q_{0}}(N)ψ_{0}^{−1} ≤ δ_{P_{0}}(N_{S}(P))

in Aut_{L_{1}}(P_{0}). By (4), for each g ∈ N, there is a unique element φ(g) ∈ N_{S}(P) such that ψ_{0}δ_{Q_{0}}(g)ψ_{0}^{−1} = δ_{P_{0}}(φ(g)). Then φ ∈ Hom(N, N_{S}(P)). We claim φ is a morphism in F. By (5), there is ψ_{0} ∈ Mor_{L_{1}}(N_{0}, N_{S_{0}}(P)) such that ψ_{0}|_{Q_{0},P_{0}} = ψ_{0}. By the uniqueness of extensions (4), for all g ∈ N,

ψ_{0} ∘ δ_{N_{0}}(g) = δ_{N_{S_{0}}(P)}(φ(g)) ∘ ψ_{0}

in L_{1}, and hence φ ∈ Hom_{F}(N, N_{S}(P)) by the definition of F in Step 2.

Step 5: By Step 4, F is H-saturated; i.e., it satisfies the saturation axioms for subgroups in H. It is also H-generated by definition: each morphism in F is a composite of restrictions of morphisms between subgroups in H. So by [BCGLO1, Theorem A], to prove F is saturated, it suffices to show the following holds for all P ≤ S:

P F-centric, P ∉ H ⟹ ∃ P′ F-conj. to P, Aut_{S}(P′) ∩ O_{p}(Aut_{F}(P′)) ≩ Inn(P′). (8)

Let K be the set of all P ≤ S such that the saturation axioms hold for subgroups Fconjugate to P and all of their overgroups. Since H_{0} is closed under overgroups and F_{0}- conjugacy by assumption, H is closed under overgroups and F-conjugacy by construction, and thus K ⊇ H. Let K^{∗} be the set of subgroups of S not in K, and let K_{0}^{∗} be the set of subgroups of S_{0} not in K. We must show K^{∗} = ∅. This will be done by first proving that for all P ≤ S,

P ∈ K or P_{0} maximal in K_{0}^{∗} ⟹ (8) holds for P  (9)

Fix P as in (9). We first show there exists P^{∗} F-conjugate to P such that P_{0}^{∗} is fully normalized in F_{0}. If P_{0} is fully normalized, we are done, so assume otherwise. Let P′_{0} be F_{0}-conjugate to P_{0} and fully normalized in F_{0}. By [BLO, Proposition A.2(b)], there is ρ ∈ Hom_{F_{0}}(N_{S_{0}}(P_{0}), N_{S_{0}}(P′_{0})) such that ρ(P_{0}) = P′_{0}. Clearly, P_{0} ≨ S_{0}, so N_{S_{0}}(P_{0}) ≩ P_{0}. Hence by the maximality assumption, the saturation axioms hold for N_{S_{0}}(P_{0}), N_{S_{0}}(P′_{0}), and all subgroups of S which contain them.

Set R = N_{S_{0}}(P_{0}) and R′ = ρ(R). Choose R′′ ≤ S_{0} and τ ∈ Iso_{F}(R, R′′) such that R′′ is fully normalized in F, and set P′′_{0} = τ(P_{0}). In general, for a pair of subgroups Q_{1} ≤ Q ≤ S, we write Aut_{F}(Q:Q_{1}) for the group of elements in Aut_{F}(Q) which leave Q_{1} invariant, and similarly for Aut_{S}(Q:Q_{1}) and N_{S}(Q:Q_{1}) = N_{S}(Q)∩N_{S}(Q_{1}). Since Aut_{S}(R′′) ∈ Syl_{p}(Aut_{F}(R′′)), there is ω ∈ Aut_{F}(R′′) such that Aut_{S}(R′′:ω(P′′_{0})) ∈ Syl_{p}(Aut_{F}(R′′:ω(P′′_{0}))). So upon replacing τ by ωτ, we can assume Aut_{S}(R′′:P′′_{0}) ∈ Syl_{p}(Aut_{F}(R′′:P′′_{0})).

In particular, there is χ ∈ Aut_{F}(R′′:P′′_{0}) such that (χτ)Aut_{S}(R:P_{0})(χτ)^{−1} ≤ Aut_{S}(R′′:P′′_{0}). Since F is H-saturated and R′′ ∈ H (and is fully centralized), χτ extends to some τ̄ ∈ Hom_{F}(N_{S}(R:P_{0}), N_{S}(R′′:P′′_{0})). Note that P ≤ N_{S}(R:P_{0}), and τ̄(P_{0}) = τ(P_{0}) = P′′_{0}. Set P′′ = τ̄(P). By a similar argument, there is χ′ ∈ Aut_{F}(R′′:P′′_{0}) such that χ′τρ^{−1} ∈ Iso_{F}(R′, R′′) extends to some ρ̄ ∈ Hom_{F}(N_{S}(R′:P′_{0}), N_{S}(R′′:P′′_{0})), where ρ̄(P′_{0}) = τρ^{−1}(P′_{0}) = P′′_{0}.

We claim that

|N_{S_{0}}(P_{0})| < |N_{S_{0}}(R′:P′_{0})| ≤ |N_{S_{0}}(R′′:P′′_{0})| ≤ |N_{S_{0}}(P′′_{0})| .  (10)

Since P_{0} is not fully normalized in F_{0}, R′ = ρ(N_{S_{0}}(P_{0})) ≨ N_{S_{0}}(P′_{0}), and hence

R′ ≨ N_{N_{S_{0}}(P′_{0})}(R′) = N_{S_{0}}(R′:P′_{0}) .

This proves the first inequality in (10). The next one holds since ρ̄ sends N_{S_{0}}(R′:P′_{0}) into N_{S_{0}}(R′′:P′′_{0}), and the last since all elements of N_{S_{0}}(R′′:P′′_{0}) normalize P′′_{0}. Thus P′′ is Fconjugate to P and |N_{S_{0}}(P′′_{0})| > |N_{S_{0}}(P_{0})|. If P′′_{0} is not fully normalized in F_{0}, then we repeat this procedure, until we do find a subgroup P^{∗} which is F-conjugate to P and such that P_{0}^{∗} is fully normalized in F_{0}.

We are now ready to prove (9). Assume P is F-centric and P ∉ H; otherwise the statement is empty. Thus P_{0} ∉ H_{0} = Ob(L_{0}). By definition of a linking system, either P_{0} and P_{0}^{∗} are not F_{0}-centric or they are not F_{0}-radical. If P_{0}^{∗} is not F_{0}-centric, then there is g ∈ C_{S_{0}}(P_{0}^{∗})∖P_{0}^{∗} (since P_{0}^{∗} is fully centralized). If P_{0} is not F_{0}-radical, then O_{p}(Aut_{F_{0}}(P_{0}^{∗})) ≩ Inn(P_{0}^{∗}) and is contained in the Sylow subgroup Aut_{S_{0}}(P_{0}^{∗}) (P_{0}^{∗} is fully normalized), and thus there is g ∈ N_{S_{0}}(P_{0}^{∗})∖P_{0}^{∗} such that c_{g} ∈ O_{p}(Aut_{F_{0}}(P_{0}^{∗})). In either case,

g ∈ Q^{∗} =^{def} {g ∈ N_{S_{0}}(P_{0}^{∗}) | c_{g} ∈ O_{p}(Aut_{F_{0}}(P_{0}^{∗}))} and g ∉ P_{0}^{∗} ,

and hence Q^{∗} ≩ P_{0}^{∗}. Also, P^{∗} normalizes Q^{∗} and P^{∗}Q^{∗} ≩ P^{∗}, so N_{P^{∗}Q^{∗}}(P^{∗}) ≩ P^{∗}, and there is x ∈ Q^{∗}∖P^{∗} such that x ∈ N_{S_{0}}(P^{∗}). For any such x, c_{x} ∉ Inn(P^{∗}) (C_{S}(P^{∗}) ≤ P^{∗} since P is F-centric), but c_{x}|_{P_{0}^{∗}} is in O_{p}(Aut_{F_{0}}(P_{0}^{∗})) and c_{x} induces the identity on P^{∗}/P_{0}^{∗} (since x ∈ S_{0}). Hence c_{x} ∈ O_{p}(Aut_{F}(P^{∗})), since the group of all α ∈ Aut(P^{∗}) which induce the identity on P_{0}^{∗} and on P^{∗}/P_{0}^{∗} is a p-group (cf. [G, Corollary 5.3.3]). Thus (8) holds for P, and this finishes the proof of (9).

We want to show that F is saturated; i.e., that K^{∗} = ∅. Assume otherwise; then K_{0}^{∗} ≠ ∅ since P ∈ K^{∗} implies P_{0} ∈ K_{0}^{∗}. Choose Q to be maximal in K_{0}^{∗}, and choose P to be maximal among those P ∈ K^{∗} such that P_{0} = Q. Then P is also maximal in K^{∗}. By [BCGLO1, Lemmas 2.4 & 2.5], this maximality of P among subgroups not satisfying the saturation axioms implies (8) does not hold for P. Since this contradicts (9), we now conclude that K^{∗} = ∅, and hence that F is saturated and (8) holds for all P.

Now that we know F is saturated, (8) is equivalent to the statement that H contains all subgroups which are F-centric and F-radical. Also, F_{0} is normal in F (Definition 7).

Step 6: We show here that (3′) implies (3), and that (2) and (3) imply E(P) is a p-group for all P ∈ H. Assume first (3′) holds. Fix P ∈ H_{0}, P ≤ Q ≤ P ·C_{S_{0}}(P), and α ∈ Aut_{F}(Q) such that α|_{P} = Id. If P is F_{0}-centric, then Q = P and α = Id. If Γ/Γ_{0} is a p-group, then Aut_{L_{0}}(Q) has p-power index in Aut_{L}(Q) (there is a homomorphism from Aut_{L}(Q) to Γ/Γ_{0} sending [[φ, γ]] to γΓ_{0} with kernel Aut_{L_{0}}(Q)); so Aut_{F_{0}}(Q) has p-power index in Aut_{F}(Q), and hence α^{p^{k}} ∈ Aut_{F_{0}}(Q) for some k. By Proposition 4(e), P is F_{0}-quasicentric, and so α^{p^{k}} has p-power order. Thus α has p-power order in both cases, and this proves (3).

For the rest of the proof, we assume (2) and (3) hold. We next show E(P) is a p-group when P ∈ H_{0}; i.e., when P ≤ S_{0}. Assume otherwise, and let P be maximal among those P ∈ H_{0} for which E(P) is not a p-group. Since this depends only on the F_{0}-conjugacy class of P, we can assume P is fully normalized in F_{0}. Since E(S_{0}) = C_{Γ}(S_{0}) is a p-group by (2), we have P ≨ S_{0}.

Fix Id ≠ ψ ∈ E(P) of order prime to p. Write ψ = [[φ, γ]], where γ ∈ Γ and φ ∈ Iso_{L_{0}}(γ(P), P). Thus π(φ^{−1}) = π(γ)|_{P,γ(P)} ∈ Hom_{F_{0}}(P, γ(P)). Set N = N_{S_{0}}(P) ≩ P. For each g ∈ N,

π(φ^{−1}) ∘ c_{g} = (π(γ) ∘ c_{g})|_{P,γ(P)} = (c_{γ(g)} ∘ π(γ))|_{P,γ(P)} = c_{γ(g)} ∘ π(φ^{−1}) .

Since P is fully normalized in F_{0}, there is x ∈ C_{S_{0}}(P) ≤ N such that φ^{−1} ∘ δ_{0}(gx) = δ_{0}(γ(g)) ∘ φ^{−1} in L_{0}. Thus φδ_{0}(γ(N))φ^{−1} = δ_{0}(N).

By Proposition 4(c) (applied to L_{0}), φ extends to some φ ∈ Iso_{L_{0}}(γ(N), N), and hence ψ also extends to an automorphism ψ ∈ Aut_{L}(N). By the uniqueness of extension, ψ has the same order as ψ, which is thus prime to p. Set α = π(ψ) ∈ Aut_{F}(N). By (3), α|_{P ·C_{N} (P)} has p-power order, hence is the identity since α has order prime to p. For all g ∈ N and x ∈ P, α(g)xα(g)^{−1} = α(gxg^{−1}) = gxg^{−1} since α|_{P} = Id, and hence g^{−1}α(g) ∈ C_{N}(P). Thus α induces the identity on N/C_{N}(P) and on C_{N}(P), so α has p-power order by [G, Corollary 5.3.3], and hence α = Id. This proves that ψ ∈ E(N), which contradicts the assumption that P was maximal among subgroups in H_{0} with E(P) not a p-group.

Thus E(P) is a p-group for all P ∈ H_{0}. Now assume P ∈ H∖H_{0}. Fix Id ≠ ψ ∈ Aut_{L}(P) of order prime to p such that π_{P} (ψ) = Id_{P}, and set ψ_{0} = ψ|_{P_{0},P_{0}}. Then ψ_{0} ∈ E(P_{0}), E(P_{0}) is a p-group, and hence ψ_{0} = Id. But then ψ, Id ∈ Aut_{L}(P) are two automorphisms with the same restriction to Aut_{L}(P_{0}), which contradicts the definition of Aut_{L}(P) in Step 2.

Step 7: We now prove that L is a linking system associated to F by checking the axioms in Definition 3. We first claim that for each P, Q ∈ H such that P is fully centralized in F,

δ_{P} (C_{S}(P)) acts freely on Mor_{L}(P, Q) and π_{P,Q} is the orbit map.  (11)

Since every morphism in L (and also in F) factors uniquely as the composite of an isomorphism followed by an inclusion, it suffices to prove this when P and Q are F-conjugate. It thus suffices to prove it when P = Q, and this follows from (7) (δ_{P} (C_{S}(P)) ∈ Syl_{p}(E(P))) and Step 6 (E(P) is a p-group).

This proves the last part of axiom (A), and the rest holds by construction and by Step 5 (all objects in L are F-centric and F-radical). Axiom (B) holds by construction, and (C) by definition of the functor π: L ⟶ F.

The pair L_{0} ⊆ L clearly satisfies the conditions in Definition 8, and so L_{0} is normal in L.

Step 8: Now assume L′ is another linking system, associated to a fusion system F′ over S, with L_{0} ◁ L′, and where Aut_{L′}(S_{0}) = Γ with the same conjugation action on L_{0}. Let F′_{1} ⊆ F′ and L′_{1} ⊆ L′ be the full subcategories with Ob(F′_{1}) = {P ≤ S} and Ob(L′_{1}) = H_{0}. Then F′_{1} and F_{1} are both fusion systems over S_{0} (not necessarily saturated). By condition (ii) in the definition of a normal fusion system, F′_{1} and F_{1} are both generated (as fusion systems) by F_{0} and Aut_{Γ}(S_{0}), and hence F′_{1} = F_{1}.

Define Φ_{1} : L_{1} ⟶ L′_{1} to be the identity on objects, and to send [[φ, γ]] ∈ Mor_{L_{1}}(P, Q) to φ∘γ|_{P,γ(P)} ∈ Mor_{L′_{1}}(P, Q). This preserves composition (hence is a functor) by the assumption that L and L′ induce the same Γ-actions on L_{0}. By axiom (A) in the definition of a linking system (applied to L and L′), for each P, Q ∈ H_{0}, C_{S}(P) acts freely on Mor_{L}(P, Q) and on Mor_{L′}(P, Q) with orbit sets Hom_{F}(P, Q) = Hom_{F′}(P, Q). By construction, (Φ_{1})_{P,Q} is equivariant with respect to these actions, and hence is also a bijection. So Φ_{1} is an isomorphism of categories L_{1} ≅ L′_{1}.

For each P, Q ∈ H, consider the restriction homomorphism

Mor_{L′}(P, Q) ⟶^{Res_{P,Q}} Mor_{L′}(P_{0}, Q_{0}).

This is injective by Proposition 4(d) (all morphisms are epimorphisms). By Proposition 4(c),

Im(Res_{P,Q}) = {ψ ∈ Mor_{L′}(P_{0}, Q_{0}) | ∀ g ∈ P ∃ h ∈ Q with δ_{Q}(h) ∘ ψ = ψ ∘ δ_{P} (g)} . (12)

Using this and the definition of L in Step 2, Φ_{1} extends to an isomorphism Φ: L ⟶ L′ of linking systems. By axiom (C), Hom_{F}(P, Q) is determined by (12) for each P, Q, and thus F′ = F. □

In general, the uniqueness of the extension of fusion systems in Theorem 9 does not follow from the information about the fusion systems alone. For example, let F_{0} be the fusion system of A_{6} (over S_{0} ≅ D_{8}). Set S = C_{2} × S_{0}, identified as a Sylow 2-subgroup of C_{2} × A_{6} and also of Σ_{6}. Then F = F_{S}(C_{2} × A_{6}) and F′ = F_{S}(Σ_{6}) are both fusion systems over S containing F_{0} as a normal subsystem, and Aut_{F}(S_{0}) = Aut_{F′}(S_{0}) (= Inn(S_{0})). But these fusion systems are not isomorphic.

Condition (2) in Theorem 9 is clearly necessary to get a linking system, since C_{Γ}(S_{0}) = Ker(π_{S_{0},S_{0}}) must be isomorphic to C_{S}(S_{0}). Condition (3) is necessary since by Proposition 4(e), each P ∈ H = Ob(L) must be F-quasicentric. If Condition (3) in Theorem 9 fails to hold, and all of the other hypotheses do hold, then one can arrange for (3′) (hence (3)) to hold by restricting the objects in L_{0} to those which are F_{0}-centric. In other words, one can always construct some linking system associated to F in this situation, but sometimes only after restricting the set of objects.

As remarked earlier, whenever L_{0} ◁ L and Γ_{0} ◁ Γ are as in Theorem 9, the geometric realisation |L_{0}| has the homotopy type of a covering space of |L| with covering group Γ/Γ_{0}.

The author would like to give profound thanks to Natalia Castellana and Assaf Libman, for having read this paper carefully and sending many suggestions which helped to greatly improve it.

References

[AOV] K. Andersen, B. Oliver, & J. Ventura, Reduced, tame, and exotic fusion systems (in preparation)

[Asch] M. Aschbacher, Normal subsystems of fusion systems, Proc. London Math. Soc. 97 (2008), 239–271 [BLO] C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16

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[BCGLO1] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Subgroup families controlling p-local finite groups, Proc. London Math. Soc. 91 (2005), 325–354

[BCGLO2] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Extensions of p-local finite groups, Trans. Amer. Math. Soc. 359 (2007), 3791–3858

[CL] N. Castellana & A. Libman, Wreath products and representations of p-local finite groups, preprint [G] D. Gorenstein, Finite groups, Harper & Row (1968)

[OV1] B. Oliver & J. Ventura, Extensions of linking systems with p-group kernel, Math. Annalen 338 (2007), 983–1043

[OV2] B. Oliver & J. Ventura, Saturated fusion systems over 2-groups, Trans. Amer. Math. Soc. (to appear)

[Pg] L. Puig, Frobenius categories, J. Algebra 303 (2006), 309–357

LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

  E-mail address: bobol@math.univ-paris13.fr