FINITE LOCALITIES II

Andrew Chermak

Kansas State University

June 2020

Introduction

This is Part II of the series that began with [Ch1] - and the reader is assumed to be familiar with Part I. References to results in Part I will be made by prefixing a “I” to the number of the cited result. Thus, for example, lemma I.3.12 is Stellmacher’s Splitting Lemma.

Let (L, Δ, S) be a locality. There is then a category F = F_{S}(L) - the “fusion system” of L - whose objects are the subgroups of S, and whose morphisms are compositions of conjugation maps c_{g} : X → Y from one subgroup of S into another, induced by elements g ∈ L. We say that L is a locality on F.

There is an extensive theory of abstract fusion systems and, more particularly, of “saturated” fusion systems. The references by Craven [Cr], and by Aschbacher, Kessar, and Oliver [AKO] provide far more material than will be needed here. It is possible to provide a completely self-contained treatment of the fusion systems associated with localities; and such a treatment will in fact be necessary when generalizing the results here (where all subgroups of L are finite) to discrete localities (where all subgroups of L are locally finite and countable). Our approach here is to compromise, by presenting elementary material on fusion systems in a self-contained way in section 1, and to avail ourselves of the literature on saturated fusion systems in section 6. Readers who are already familiar with fusion systems, and who may be tempted to simply skip over section 1, are advised to familiarize themselves with the definition in 1.8 of radical subgroups (not the usual one, but equivalent to the usual definition if the ambient fusion system is saturated), and of subcentric subgroups.

Throughout this paper, (L, Δ, S) will be a proper locality on F (to be defined in 2.4). In brief, what this means is that Δ is “not too small” (Δ contains the set F^{cr} of all subgroups of S which are both centric and radical in F), and that the groups N_{L}(P) for P ∈ Δ are of characteristic p - where one says that a finite group G is of characteristic p if C_{G}(O_{p}(G)) ≤ O_{p}(G). It will be shown in 2.9 that if L is an arbitrary locality for which

Typeset by AMS-TEX

Δ is not too small in the above sense, and also not “too large”, then L has a canonical homomorphic image which is a proper locality on F.

A non-empty collection Γ of subgroups of S is F-closed if Q ∈ Γ whenever there exists an F-homomorphism ϕ : P → Q for some P ∈ Γ. For example, the set F^{s} of F-subcentric subgroups is F-closed, and (see 2.8) Δ is an F-closed subset of F^{s}. It is a straightforward exercise with the definitions (see 2.11) to show that if Δ_{0} is an F-closed subset of Δ containing F^{cr}, then there is a unique proper locality (L_{0}, Δ_{0}, S) on F such that the partial group L_{0} is a subset of L and such that the inclusion map L_{o} → L is a homomorphism of partial groups. We shall call L_{0} the restriction of L. The core of this paper, consisting of sections 3 through 5, concerns the opposite sort of operation, by which one expands, rather than restricts, the set of objects.

Theorem A1. Let (L, Δ, S) be a proper locality on F and let Δ^{+} be an F-closed collection of subgroups of S such that Δ ⊆ Δ^{+} ⊆ F^{s}.

(a) There exists a proper locality (L^{+}, Δ^{+}, S) on F such that L is the restriction L^{+} |_{Δ} of L^{+} to Δ. Moreover, L^{+} is generated by L as a partial group.

(b) For any proper locality (L̃, Δ^{+}, S) on F whose restriction to Δ is L, there is a unique isomorphism L^{+} → L̃ which restricts to the identity map on L.

Recall from Part I that a partial subgroup N of a partial group L is normal in L (or is a partial normal subgroup of L, denoted N ⊴ L) if x^{g} := g^{−1}xg ∈ N for all x ∈ N and g ∈ L for which the product g^{−1}xg is defined. Recall also: for any partial group L and any subset X of L, ⟨X⟩ is defined to be the intersection of the set of partial subgroups of L containing X. The intersection of partial subgroups is again a partial subgroup by I.1.8, and ⟨X⟩ is called the partial subgroup generated by X.

Theorem A2. Let the hypothesis and notation be as in Theorem A1, let N be a partial normal subgroup of L, and set T = S ∩ N. Then there exists a unique partial normal subgroup N^{+} of L^{+} such that N = N^{+} ∩ L. Moreover, the mapping N ↦ N^{+} is an inclusion-preserving bijection from the set of partial normal subgroups of L to the set of partial normal subgroups of L^{+}.

As mentioned, section 6 concerns saturated fusion systems, and relies on results from the literature. We shall obtain the following result.

Theorem B. Let F be the fusion system of a proper locality (L, Δ, S). Then F is saturated.

Section 6 contains also a treatment of the set F^{s} of subcentric subgroups of S, taken from Rfrom Henke’s work [He2] on this subject. Section 7 concerns the notions of O_{L}^{p}(N) and O_{L}^{p′}(N) for N a partial normal subgroup of a proper locality L.

Section 1: Fusion systems

We begin this section by providing a brief summary of some of the terminology and some of the basic results pertaining to general fusion systems. Some of the definitions to be given here will be non-standard, but will turn out to be equivalent to the standard definitions in the case of the fusion system of a proper locality.

Definition 1.1. Let S be a finite p-group. A fusion system F on S is a category, whose set of objects is the set of subgroups of S, and whose morphisms satisfy the following conditions (in which P and Q are subgroups of S).

(1) Each F-morphism P → Q is an injective homomorphism of groups.

(2) If g ∈ S and P^{g} ≤ Q then the conjugation map c_{g} : P → Q is an F-morphism.

(3) If ϕ : P → Q is an F-morphism then the bijection P → Im(ϕ) defined by ϕ is an F-isomorphism.

One most often refers to F-morphisms as F-homomorphisms, in order to emphasize condition (1). Notice that (2) implies that all inclusion maps between subgroups of S are F-homomorphisms, and hence the restriction of an F-homomorphism P → Q to a subgroup of P is again an F-homomorphism.

Let G be a finite group and let S be a p-subgroup of G. There is then a fusion system F = F_{S}(G) on S in which the F-homomorphisms P → Q are the maps c_{g} : P → Q given by conjugation by those elements g ∈ G for which P^{g} ≤ Q.

Definition 1.2. Let F be a fusion system on S, and let F′ be a fusion system on S′. A homomorphism α : S → S′ is a fusion-preseving (relative to F and F′) if, for each F-homomorphism ϕ : P → Q, there exists an F′-homomorphism ψ : P α → Qα such that α |_{P} ∘ψ = ϕ ∘ α |_{Q}.

Notice that each of the F′-homomorphisms ψ in the preceding definition is uniquely determined, since all F′-homomorphisms are injective. Thus, if α : S → S′ is a fusionpreserving homomorphism then α induces a mapping Hom_{F}(P, Q) → Hom_{F′}(P α, Qα), for each pair (P, Q) of subgroups of S. The following result is then easily verifed.

Lemma 1.3. Let F be a fusion system on S, let F′ be a fusion system on S′, and let α : S → S′ be a fusion-preserving homomorphism. Then the mapping P ↦ P α from objects of F to objects of F′, together with the set of mappings

α_{P,Q} : Hom_{F}(P, Q) → Hom_{F′}(P α, Qα) (P, Q ≤ S)

defines a functor α^{∗} : F → F′. □

In view of the preceding result, a fusion-preserving homomorphism α may also be called a homomorphism of fusion systems. Notice that the inverse of a fusion-preserving isomorphism is fusion-preserving, and is therefore an isomorphism of fusion systems.

In the special case where S ≤ S′ and the inclusion map S → S′ is fusion-preserving, we say that F is a fusion subsystem of F′. Thus, F_{S}(S) is a fusion subsystem of F for each fusion system F on S, by 1.1(2). We refer to F_{S}(S) as the trivial fusion system on S.

If F is a fusion system on S and P is a subgroup of S, write P^{F} for the set of subgroups of S of the form P ϕ, ϕ ∈ Hom_{F}(P, S). The elements of P^{F} are the F-conjugates of P.

For the remainder of this section let F be a fixed fusion system on the finite p-group S.

Definition 1.4. Let P ≤ S be a subgroup of S. Then P is fully normalized in F if |N_{S}(P)| ≥ |N_{S}(P′)| for all P′ ∈ P^{F}. Similarly, P is fully centralized in F if |C_{S}(P)| ≥ |C_{S}(P′)| for all P′ ∈ P^{F}.

Let U ≤ S be a subgroup of S. The normalizer N_{F}(U) of U in F is the category whose objects are the subgroups of N_{S}(U), and whose morphisms P → Q (P and Q subgroups of N_{S}(U)) are restrictions of F-homomorphisms ϕ : P U → QU such that Uϕ = U. Similarly, the centralizer C_{F}(U) of U in F is the category whose objects are the subgroups of C_{S}(U) and whose morphisms ϕ : P → Q are restrictions of Fhomomorphisms ϕ : P U → QU such that ϕ induces the identity map on U. One observes that N_{F}(U) is a fusion system on N_{S}(U) and that C_{F}(U) is a fusion system on C_{S}(U). The following result is immediate from 1.3.

Lemma 1.5. Let U, V ≤ S be subgroups of S, and suppose that there exists an Fisomorphism α : N_{S}(U) → N_{S}(V) such that Uα = V. Then α is an isomorphism N_{F}(U) → N_{F}(V). Similarly, if β : C_{S}(U)U → C_{S}(V)V is an F-isomorphism such that Uβ = V, then the restriction of β to C_{S}(U) is an isomorphism C_{F}(U) → C_{F}(V). □

Definition 1.6. Let T be a subgroup of S. Then T is weakly closed in F if T^{F} = {T }, strongly closed in F if X^{F} is a set of subgroups of T for each subgroup X of T, and normal in F if F = N_{F}(T).

The following result is immediate from the definitions.

Lemma 1.7. If U and V are subgroups of S which are normal in F then also UV is normal in F. Thus, there is a largest subgroup O_{p}(F) of S which is normal in F. □

A set Δ of subgroups of S is F-invariant if X ∈ Δ ⟹ X^{F} ⊆ Δ. An F-invariant set Δ of subgroups of S is F-closed if Δ is non-empty and is closed with respect to overgroups in S (P ∈ Δ and P ≤ Q ≤ S ⟹ Q ∈ Δ).

Definition 1.8. Let F be a fusion system on S, and let P ≤ S be a subgroup of S. (1) P is centric in F (or P is F-centric) if C_{S}(Q) ≤ Q for all Q ∈ P^{F}.

(2) P is radical in F (or P is F-radical) if there exists Q ∈ P^{F} such that Q is fully normalized in F and such that Q = O_{p}(N_{F}(Q)).

(3) P is quasicentric in F (or P is F-quasicentric) if there exists Q ∈ P^{F} such that Q is fully centralized in F and such that C_{F}(Q) is the trivial fusion system on C_{S}(Q).

(4) P is subcentric in F (or P is F-subcentric) if there exists Q ∈ P^{F} such that Q is

fully normalized in F and such that O_{p}(N_{F}(Q)) is centric in F. Write F^{c}, F^{q}, and F^{s}, respectively, for the set of subgroups of S which are F-centric, F-quasicentric, and F-subcentric. Write F^{cr} for the set of subgroups of S which are both centric and radical in F.

Remark. The above definition of F-radical subgroup is different from the standard one (which is that P is F-radical if Inn(P) = O_{p}(Aut_{F}(P))). But it will turn out to be equivalent to the standard definition in the case that F is the fusion system of a proper locality.

Lemma 1.9. Let F be a fusion system on S. Then F^{cr} is F-invariant, and F^{c} is F-closed.

Proof. Both F^{c} and F^{cr} are F-invariant by definition. Let P ∈ F^{c}, let P ≤ Q ≤ S, and let ϕ : Q → S be an F-homomorphism. Then C_{S}(Qϕ) ≤ C_{S}(P ϕ) ≤ P ϕ ≤ Qϕ, and so Q ∈ F^{c}. Thus F^{c} is closed with respect to overgroups in S. As S ∈ F^{c} it follows that F^{c} is F-closed. □

Lemma 1.10. Let P ≤ S be a subgroup of S and let Q ∈ P^{F} such that Q is fully centralized in F. Then P ∈ F^{c} if and only if C_{S}(Q) ≤ Q.

Proof. Suppose that C_{S}(Q) ≤ Q and let R ∈ Q^{F}. Then

|C_{S}(R)| ≤ |C_{S}(Q)| = |Z(Q)| = |Z(R)|,

and so C_{S}(R) = Z(R). That is, C_{S}(R) ≤ R, and thus Q is F-centric. As F^{c} is Finvariant by 1.9, P is then F-centric. That is:

C_{S}(Q) ≤ Q ⟹ P ∈ F^{c}.

The reverse implication is given by the definition of F^{c}. □

Let Ψ be a non-empty set of F-isomorphisms. Then F is generated by Ψ if every F-isomorphism can be expressed as a composition of restrictions of members of Ψ, and write F = ⟨Ψ⟩ in that case. If Γ is a non-empty set of subgroups of S and

(*)  Ψ = ⟨⋃{Aut_{F}(R) | R ∈ Γ⟩.

then we say that F is Γ-generated if F = ⟨Ψ⟩. An important special case is that in which Γ is the set of all P F^{cr} such that P is fully normalized in F, where we we shall say that F is (cr)-generated if F is Γ-generated.

Definition 1.11. Let F be a fusion system on S, and let Γ be an F-closed set of subgroups of S. Then F is Γ-inductive if:

(*) For each U ∈ Γ, and each V ∈ U^{F} such that V is fully normalized in F, there exists an F-homomorphism ϕ : N_{S}(U) → N_{S}(V) with Uϕ = V.

If F is Γ-inductive where Γ is the set of all subgroups of S, we shall simply say that F is inductive.

Lemma 1.12. Let G be a finite group, let S be a Sylow p-subgroup of G, and set F = F_{S}(G). Then F is inductive, and F is (cr)-generated.

Proof. Let V be fully normalized in F. Equivalently: N_{S}(V) ∈ Syl_{p}(N_{G}(V)). Let U ∈ V^{F}, and let g ∈ L with U^{g} = V. Then N_{S}(U)^{g} ≤ N_{G}(V), and there then exists h ∈ N_{G}(V) with N_{S}(U)^{gh} ≤ N_{S}(V). Thus F is inductive. That F is (cr)-generated is a well-known consequence of the Alperin-Goldschidt fusion theorem [Gold]. □

The next few results provide information about inductive fusion systems. All of these results can be re-stated (and proved) in an obvious way, so as to yield corresponding results about Γ-inductive fusion systems.

Lemma 1.13. Assume that F is inductive, and let V ≤ S be fully normalized in F. Then V is fully centralized in F.

Proof. Let U ∈ V^{F}, and let ϕ : N_{S}(U) → N_{S}(V) be an F-homomorphism such that V = Uϕ. Then ϕ maps C_{S}(U) into C_{S}(V), and thus |C_{S}(U)| ≤ |C_{S}(V)|. □

Lemma 1.14. Assume that F is inductive, and let U and V be F-conjugate subgroups of S.

(a) If U and V are fully normalized in F then N_{F}(U) ≅ N_{F}(V).

(b) If U and V are fully centralized in F then C_{F}(U) ≅ C_{F}(V).

Proof. Suppose that U and V are fully normalized in F. Then (FL1) implies that there exisits an F-isomorphism ϕ : N_{S}(U) → N_{S}(V) with Uϕ = V. By 1.5, ϕ is then an isomorphism N_{F}(U) → N_{F}(V). Thus (a) holds. Now suppose instead that U and V are fully centralized in F, and let X be a fully normalized F-conjugate of U (and hence also of V). There are then F-homomorphisms ρ : N_{S}(U) → N_{S}(X) and σ : N_{S}(V) → N_{S}(X) with Uρ = X = V σ. The restriction of ρ to C_{S}(U)U is then an isomorphism with C_{S}(X)X, and similarly σ restricts to an isomorphism C_{S}(V)V → C_{S}(X)X. A further application of 1.5 now yields (b). □

Lemma 1.15. Assume that F is inductive, let T be strongly closed in F, let Q ≤ S be a subgroup of S, and set V = Q ∩ T. Suppose that V is fully normalized in F and that Q is fully normalized in N_{F}(V). Then Q is fully normalized in F.

Proof. Let P ∈ Q^{F} such that P is fully normalized in F, let ϕ : N_{S}(Q) → N_{S}(P) be an F-homomorphism with Qϕ = P, and set U = V ϕ. Then U = P ∩ T as T is strongly closed in F, and then also N_{S}(P) ≤ N_{S}(U). By (FL1) there exists an F-homomorphism ψ : N_{S}(U) → N_{S}(V) with Uψ = V. Set Q′ = P ψ. Then Q′ = (Qϕ)ψ is an N_{F}(V)- conjugate of Q. Since N_{S}(Q) = N_{N_{S}(V)}(Q) (and similarly for Q′), and since Q is fully normalized in N_{F}(V), we have |N_{S}(Q)| ≥ |N_{S}(Q′)|. The sequence

N_{S}(Q) ⟶^{ϕ} N_{S}(P) ⟶^{ψ} N_{S}(Q′)

of injective homomorphisms then shows that |N_{S}(P)| = |N_{S}(Q)|, and so Q is fully normalized in F. □

Lemma 1.16. Assume that F is inductive, let V ≤ S be a subgroup of S, and let Q ≤ N_{S}(V) be a subgroup of N_{S}(V) containing V. Then each of the following conditions implies that Q is fully centralized in F.

(1) V is fully normalized in F and Q is fully centralized in N_{F}(V).

(2) V is fully centralized in F, Q = V Y for some subgroup Y of C_{S}(V) such that Y is fully centralized in N_{F}(V), and N_{F}(V′) is inductive for each fully normalized F-conjugate V′ of V.

Proof. Assume first that (1) holds. As in the proof of 1.15 let P ∈ Q^{F} such that P is fully normalized in F, and let ϕ : N_{S}(Q) → N_{S}(P) be an F-homomorphism with Qϕ = P. Set U = V ϕ and let ψ : N_{S}(U) → N_{S}(V) be an F-homomorphism with Uψ = V. Set Q′ = P ψ. Then Q′ is an N_{F}(V)-conjugate of Q. As Q is fully centralized in N_{F}(V), and since C_{S}(Q) = C_{C_{S}(V)}(Q) (and similarly for Q′), we then have |C_{S}(Q)| ≥ |C_{S}(Q′)|. The sequence

C_{S}(Q) ⟶^{ϕ} C_{S}(P) ⟶^{ψ} C_{S}(Q′)

then shows that |C_{S}(P)| = |C_{S}(Q)|. On the other hand, as P is fully normalized in F, P is also fully centralized in F by 1.13. Thus |C_{S}(P)| ≥ |C_{S}(Q)|, and Q is fully centralized in F.

Assume next that (2) holds. Let V′ ∈ V^{F} with V′ fully normalized in F, and let α : C_{S}(V)V → C_{S}(V′)V′ be an F-homomorphism with V′ = V α. Then α is an isomorphism, and induces an isomorphism C_{F}(V) → C_{F}(V′). Thus Y′ is fully centralized in C_{F}(V′). Here ϕ^{−1} maps C_{S}(V′Y′) into C_{S}(V Y), so if V′Y′ is fully centralized in F then also V Y is fully centralized in F. Thus we may assume from the beginning that V is fully normalized in F. By hypothesis, N_{F}(V) is then inductive.

Let P ∈ Q^{F} be fully centralized in F, and let ϕ : C_{S}(Q)Q → C_{S}(P)P be an Fhomomorphism with P = Qϕ. Set U = V ϕ and X = Y ϕ. As V is fully centralized in F there exists an F-homomorphism ψ : C_{S}(P)P → C_{S}(V)V with Uψ = V. Set R = P ψ. Then R is an N_{F}(V)-conjugate of Q and R is fully centralized in F, whence also R is fully centralized in N_{F}(V). We have R = V Z where Z = (Y)ϕ∘ψ is an N_{F}(V)- conjugate of Y. As Y is fully centralized in N_{F}(V) there exists an N_{F}(V)-homomorphism ρ : C_{S}(V Z)V Z → C_{S}(V Y)V Y with Zρ = V. Thus C_{S}(P)P is mapped by ψ ∘ ρ into C_{S}(Q)Q, and so Q is fully centralized in F. □

Lemma 1.17. Assume that F is inductive, let T ≤ S be strongly closed in F, and let E be a fusion subsystem of F on T. Let U ≤ T be a subgroup of T such that U is fully normalized in F. Then U is fully normalized in E.

Proof. Let V ∈ U^{E} such that V is fully normalized in E. Then V ∈ U^{F}. As F is inductive there exists an F-homomorphism ϕ : N_{S}(V) → N_{S}(U) with V ϕ = U. Then N_{T} (V)ϕ ≤ N_{T} (U) as T is strongly closed in F. As V is fully normalized in E it follows that N_{T} (V)ϕ = N_{T} (U), and hence U is fully normalized in E. □

Lemma 1.18. Assume that F is inductive, and let U ≤ S be a subgroup of S. Then there exists V ∈ U^{F} such that both V and O_{p}(N_{F}(V)) are fully normalized in F.

Proof. Without loss of generality, U is fully normalized in F. Set P = O_{p}(N_{F}(U)) and let Q ∈ P^{F} with Q fully normalized in F. Let ϕ : N_{S}(P) → N_{S}(Q) be an F-homomorphism which maps P to Q, and set V = Uϕ. As N_{S}(U) ≤ N_{S}(P) we have N_{S}(U)ϕ ≤ N_{S}(V). But |N_{S}(U)| ≥ |N_{S}(V)| as U is fully normalized, so ϕ induces an isomorphism N_{S}(U) → N_{S}(V). Thus V is fully normalized in F, and then Q = O_{p}(N_{F}(V)) by 1.14(a). □

Lemma 1.19. Let F be an inductive fusion system on S, and let V ≤ S be a subgroup of S. Assume that V is fully normalized in F, and let P be a subgroup of N_{S}(V) containing V. Then P is centric in F if and only if P is centric in N_{F}(V).

Proof. We are free to replace P by any N_{F}(V)-conjugate of P, so we may assume that P is fully normalized in N_{F}(V). Then P is fully centralized in F by 1.16, and so P ∈ F^{c} if and only if C_{S}(P) ≤ P by 1.10. As V ≤ P we have C_{S}(P) = C_{N_{S}(V)}(P), and the lemma follows. □

We end with this section with a result on quotients of fusion systems. The hypothesis will be shown to be fulfilled in 2.12, in the context of projections of localities.

Lemma 1.20. Let F and \overline{F} be fusion systems on p-groups S and \overline{S}, and let λ : S → \overline{S} be a fusion-preserving homomomorphism. Denote also by λ the corresponding homomorphism F → \overline{F} of fusion systems given by 1.3, and write \overline{X} for the image under λ of any subgroup or set of subgroups X of S. Assume that λ is surjective, and that each of the mappings

λ_{X,Y} : Hom_{F}(X, Y) → Hom\overline{F}(\overline{X}, \overline{Y}) (Ker(λ) ≤ X ∩ Y)

is surjective. Let \overline{P} ≤ \overline{S} and let P be the λ-preimage of \overline{P}. Then the following hold. (a) P is fully normalized in F if and only if \overline{P} is fully normalized in \overline{F}.

(b) O_{p}(N_{F}(P))λ ≤ O_{p}(N\overline{F}(\overline{P})).

(c) For each of the symbols ⋆ ∈ \{cr, c, q, s\} we have \overline{P} ∈ \overline{F}^{∗} if P ∈ F^{∗}.

Proof. For any subgroup or element \overline{U} of \overline{S} write U for the preimage of \overline{U} in S. For any such U we have \overline{N_{S}(U)} = N\overline{S}(\overline{U}), and it is this observation that yields (a).

Set Q = O_{p}(N_{F}(P)) and set \overline{R} = O_{p}(N\overline{F}(\overline{P}). Let \overline{ϕ} : \overline{X} → \overline{Y} be a N\overline{F}(\overline{P})- homomorphism. By hypothesis there exists an N_{F}(P)-homomorphism ϕ : X → Y with \overline{ϕ} = (ϕ)λ. Then ϕ extends to an N_{F}(P)-homomorphism ψ : QX → QY which fixes Q, and then (ψ)λ is an extension of \overline{ϕ} to an N\overline{F}(\overline{P})-homomorphism \overline{Q}\overline{X} → \overline{Q}\overline{Y} which fixes \overline{Q}. This shows that \overline{Q} ⊴ N\overline{F}(\overline{P}), and so \overline{Q} ≤ \overline{R}. A similar argument shows that R ≤ Q, and establishes (b).

The statement in (c) concerning \overline{F}^{c} and F^{c} is immediate from the observation that \overline{C_{S}(U)} ≤ C\overline{S}(\overline{U}) for any subgroup U ≤ S. For any fusion system E on a p-group T write E^{r} for the set of E-radical subgroups of T (cf. 1.8). Suppose that \overline{P} ∈ \overline{F}^{r}. By definition 1.8 there exists an N\overline{F}(\overline{P})-conjugate \overline{P}_{1} of \overline{P} such that \overline{P}_{1} is fully normalized in \overline{F} and such that \overline{P}_{1} = O_{p}(N\overline{F}(\overline{P}_{1})). Then P_{1} is an F-conjugate of P, P_{1} is fully normalized in F by (a), and P_{1} = O_{p}(N_{F}(P_{1})) by (b). Thus P ∈ F^{r}, and this establishes the statement in (c) concerning “cr”. The statement concerning “s” follows from (a) and from the statement concerning “c”. Finally, if C\overline{F}(\overline{P}) is the trivial fusion system on C\overline{S}(\overline{P}) then C_{F}(P) is the trivial fusion system on C_{S}(P), and then (a) completes the proof of the statement in (c) concerning “q”. □

Section 2: Fusion systems of localities

Fix a locality (L, Δ, S) throughout this section. For each g ∈ L there is then an isomorphism

c_{g} : S_{g} → S_{g^{−1}}

given by conjugation by g. Define F_{S}(L) to be the fusion system F on S which is generated by these isomorphisms. We then say that (L, Δ, S) is a locality on F.

It will be useful to have the following explicit description of the set of F-isomorphisms. For w = (g_{1}, cdots, g_{n}) ∈ W(L) and for U ≤ S_{w} define V = U^{w} to be the subgroup of S_{w^{−1}} obtained by successively applying the conjugation maps c_{g_{i}} to U:

U = U_{0} ⟶^{c_{g_{1}}} U_{1} ⟶^{c_{g_{2}}} ⋯ ⟶^{c_{gn}} U_{n} = V.

The set of such mappings c_{w} : U → V (U, V ≤ S) is then the set of F-isomorphisms.

Lemma 2.1. Let (L, Δ, S) be a locality on F, and let P ∈ Δ. Then

(*)  P^{F} = {P^{g} | g ∈ L, P ≤ S_{g}}.

Moreover, P is fully normalized in F if and only if N_{S}(P) ∈ Syl_{p}(N_{L}(P)), and P is fully centralized in F if and only if C_{S}(P) ∈ Syl_{p}(C_{L}(P)).

Proof. Let ϕ : P → Q be an F-isomorphism. As noted above, ϕ = c_{w} for some w ∈ W(L) with P ≤ S_{w}. Then w ∈ D, and c_{w} = c_{Π(w)} by I.2.3(c). This yields (*).

As P is in Δ, N_{L}(P) is a subgroup of L, and C_{L}(P) is a normal subgroup of N_{L}(P). Let X be a Sylow p-subgroup of N_{L}(P) containing N_{S}(P). By I.2.11 there exists g ∈ L with X^{g} ≤ S, and conjugation by g induces an isomorphism N_{L}(P) → N_{L}(P^{g}) by I.2.3(b). If P is fully normalized in F then |N_{S}(P)| ≥ |N_{S}(P^{g})| ≥ |X^{g}| = |X|, so N_{S}(P) = X, and N_{S}(P) ∈ Syl_{p}(N_{L}(P) in that case. Conversely, suppose that N_{S}(P) is a Sylow subgroup of N_{L}(P) and let Q ∈ P^{F}. We have P = Q^{h} for some h ∈ L by (*), so N_{S}(Q)^{h} ≤ N_{L}(P). By Sylow’s theorem there exists f ∈ N_{L}(P) with (N_{S}(Q)^{g})^{f} ≤ N_{S}(P), and thus |N_{S}(P)| ≥ |N_{S}(Q)|. This establishes the first of the two “if and only ifs” of the lemma. The proof of the second “if and only if” is obtained in similar fashion. □

Proposition 2.2. Let (L, Δ, S) be a locality on F. Then F is Δ-inductive. Moreover, for each P ∈ Δ such that P is fully normalized in F:

(a) N_{F}(P) = F_{N_{S}(P)}(N_{L}(P)) and C_{F}(P) = F_{C_{S}(P)}(C_{L}(P)).

(b) N_{F}(P) and C_{F}(P) are (cr)-generated.

Proof. That F is Δ-inductive is immediate from the preceding lemma and from I.2.10. Let P ∈ Δ with P fully normalized in F, and let ϕ : X → Y be an N_{F}(P)-isomorphism between two subgroups X and Y of N_{S}(P) containing P. As in the proof of (*) in 2.1, we find that ϕ = c_{g} for some g ∈ N_{L}(P), and this shows that N_{F}(P) = F_{N_{S}(P)}(N_{L}(P)). Then N_{F}(P) is N_{F}(P)^{cr}-generated by 1.12.

By 1.13 P is fully centralized in F. Let ϕ : X → Y be a C_{F}(P)-isomorphism between two subgroups X and Y of C_{S}(P). By definition of C_{F}(P), ϕ extends to an F-isomorphism ψ : XP → Y P such that ψ restricts to the identity map on P. Then ψ = c_{g} for some g ∈ C_{L}(P), and thus C_{F}(P) = F_{C_{S}(P)}(C_{L}(P)). We again appeal to 1.12, obtaining (cr)-generation for C_{F}(P). □

A finite group G is of characteristic p if C_{G}(O_{p}(G)) ≤ O_{p}(G).

Lemma 2.3. Let (L, Δ, S) be a locality on F, and assume that N_{L}(P) is of characteristic p for all P ∈ Δ. Then O_{p}(F) = O_{p}(L).

Proof. As F is generated by the conjugation maps c_{g} : S_{g} → S for g ∈ L, the inclusion O_{p}(L) ≤ O_{p}(F) is immediate.

Set R = O_{p}(F) and assume by way of contradiction that R is not normal in L. Among all elements of L not in N_{L}(R), choose g so that |S_{g}| is as large as possible. Then R ≰ S_{g}, since c_{g} : S_{g} → S is an F-homomorphism. In particular, S_{g} ≠ S, and S_{g} is a proper subgroup of N_{S}(S_{g}).

Set P = S_{g}, P′ = P^{g}, and let Q ∈ P^{F} be fully normalized in F. As L is Δ-inductive by 2.2, there exists x ∈ L such that P^{x} = Q and such that N_{S}(P) ≤ S_{x}. Then also R ≤ S_{x} by the maximality of |P | in the choice of g. Since Q ∈ (P′)^{F} there exists also y ∈ L such that (P′)^{y} = Q and such that N_{S}(P′)R ≤ S_{y}.

Note that (x^{−1}, g, y) ∈ D via Q, and that f := Π(x^{−1}, g, y) ∈ N_{L}(Q). Note also that (x, x^{−1}, g, y, y^{−1}) ∈ D via P, and that

Π(x, f, y^{−1}) = Π(x, x^{−1}, g, y, y^{−1}) = Π(g)

by D-associativity (I.1.4). If R ≤ S_{f} then R ≤ S_{(x,f,y^{−1})}, and then R ≤ S_{g}. Thus R ≰ S_{f}, and we may therefore replace g with f, and P with Q. That is, we may assume that P is fully normalized in F and g ∈ N_{L}(P).

Set M = N_{L}(P). Then M is a subgroup of L and, by hypothesis, M is of characteristic p. Set D = N_{R}(P), and set E = F_{N_{S}(P)}(M). Then D ⊴ E, and so the conjugation map c_{g} : P → P extends to an E-automorphism of P D. Thus, there exists h ∈ N_{M}(P D) such that c_{h} : P D → P D restricts to c_{g} on P. Then gh^{−1} ∈ C_{M}(P) ≤ P, and so gh^{−1} ∈ N_{M}(D). This yields D^{g} = D, so D ≤ P, and then R ≤ P. Then R^{g} = R. This result is contrary to the choice of g, and completes the proof. □

Definition 2.4. Let (L, Δ, S) be a locality on F. Then L is proper if: (PL1) F^{cr} ⊆ Δ, and

(PL2) N_{L}(P) is of characteristic p for each P ∈ Δ.

The next two results are well known, and are important for an understanding of the structure of finite groups of characteristic p.

Lemma 2.5. Let G be a finite group, and let A and B be subgroups of G such that |A| is relatively prime to |B|. Suppose that [A, B] ≤ C_{B}(A). Then [A, B] = 1.

Proof. Let a ∈ A and b ∈ B. Then a^{−1}a^{b} = [a, b] commutes with a, by hypothesis, and so a^{b} commutes with ⟨a⟩. As |a| = |a^{b}| is relatively prime to |B|, the same is then true of a^{−1}a^{b}. As [a, b] ∈ B by hypothesis, we conclude that [a, b] = 1, and thus [A, B] = 1. □

Lemma 2.6 (Thompson’s A × P Lemma). Let G be a finite group and let H ≤ G be a subgroup of G such that:

(i) H = P A, where P is a normal p-subgroup of H, and where A is a p′-subgroup of H.

(ii) There exists a subgroup B of C_{P} (A) such that [C_{P} (B), A] = 1. Then [P, A] = 1 and H is isomorphic to the direct product A × P.

Proof. We have A ∩ P = 1 since |A| and |P | are relatively prime, and so it suffices to show that P = C_{P} (A). Suppose false, so that C_{P} (A) is a proper subgroup of P. Set Q = N_{P} (C_{P} (A)). Thus B ≤ C_{P} (A) ⊴ Q, and so [Q, B] ≤ C_{P} (A). One may express this in the standard way as [Q, B, A] = 1. Also [B, A, Q] = 1 since B ≤ C_{P} (A). The Three Subgroups Lemma (2.2.3 in [Gor]) then yields [A, Q, B] = 1. That is, we have [A, Q] ≤ C_{Q}(B), and hence [A, Q] ≤ C_{Q}(A) by (ii). Then Q ≤ C_{P} (A) by 2.5. That is, N_{P} (C_{P} (A)) ≤ C_{P} (A), and hence P = C_{P} (A), as required. □

Lemma 2.7. Let G be a finite group of characteristic p.

(a) Every normal subgroup of G is of characteristic p.

(b) Every p-local subgroup of G is of characteristic p.

(c) Let V be a normal p-subgroup of G, and let X be the set of elements x ∈ C_{G}(V) such that [O_{p}(G), x] ≤ V. Then X is a normal p-subgroup of G.

(d) Let N ⊴ G be a normal subgroup such that C_{G}(N) ≤ N. Then C_{G}(O_{p}(N)) ≤ O_{p}(G), and every overgroup of N in G is of characteristic p.

Proof. Let K ⊴ G be a normal subgroup of G, and set R = O_{p}(K). Then [O_{p}(G), K] ≤ R, and so [O_{p}(G), C_{K}(R), C_{K}(R)] = 1. Then O^{p}(C_{K}(R)) ≤ C_{G}(O_{p}(G)) by 2.5, and thus O^{p}(C_{K}(R)) ≤ O_{p}(G). Then C_{K}(R) is a normal p-subgroup of K, and so C_{K}(R) ≤ R. This establishes point (a).

Next, let U be a p-subgroup of G, and set H = N_{G}(U), P = O_{p}(H), and Q = O_{p}(G). Then N_{Q}(U) ≤ P. Let A be a p′-subgroup of C_{H}(P). Then [N_{Q}(U), A] = 1, and so [Q, A] = 1 by 2.6. Then A ≤ Q, and thus A = 1, proving (b).

For the proof of (c), notice that C_{G}(V) ⊴ G, and that X is the intersection of C_{G}(V) with the preimage in G of the normal subgroup C_{G/V} (O_{p}(G)/V) of G/V. Thus X ⊴ G. Each p′-subgroup of X centralizes O_{p}(G) by 2.5, so X is a p-group.

Finally, let N ⊴ G with C_{G}(N) ≤ N, and set Q = O_{p}(N). Then [N, C_{G}(Q)] ≤ C_{N}(Q) ≤ Q, and so [N, C_{G}(Q), C_{G}(Q)] = 1. Set Z = [C_{G}(Q), C_{G}(Q)]. The Three Subgroups Lemma then yields Z ≤ C_{G}(N). As C_{G}(N) ≤ N by hypothesis, we conclude that Z is a p-group and that Z ≤ Z(C_{G}(Q)). Then O^{p}(C_{G}(Q)) is a p′-group. As C_{G}(Q) ⊴ G and G is of characteristic p, we conclude that O^{p}(C_{G}(Q)) = 1. Thus C_{G}(Q) ≤ O_{p}(G), as desired. Now let H be a subgroup of G containing N. Then C_{H}(O_{p}(H)) ≤ C_{H}(Q) ≤ O_{p}(G) ∩ H ≤ O_{p}(H), and thus H is of characteristic p. This completes the proof of (d). □

The next result refers to the terminology and notation of 1.8.

Lemma 2.8. Let (L, Δ, S) be a proper locality on F and let P ∈ Δ. Then P is subcentric in F, and the following hold.

(a) P ∈ F^{cr} if and only if P = O_{p}(N_{L}(P)).

(b) P is centric in F if and only if C_{L}(P) = Z(P).

(c) P is quasicentric in F if and only if C_{L}(P) ≤ O_{p}(N_{L}(P)).

Proof. Set M = N_{L}(P), and let Q ∈ P^{F} with Q fully normalized in F. Then Q = P^{g} for some g ∈ L by 2.1, and then the conjugation map c_{g} : M → M^{g} is an isomorphism by I.2.3(b). As F^{cr}, F^{c}, F^{q}, and F^{s} are F-invariant by 1.9, it follows that it suffices to establish the lemma under the assumption (which we now make) that P itself is fully normalized in F. Since M may be regarded as a proper locality whose set of objects is the set of all subgroups of N_{S}(P), it follows from 2.2 and 2.3 that O_{p}(N_{F}(P)) = O_{p}(M). As F is Δ-inductive by 2.3, and M is of characteristic p, it follows from 1.10 and 1.13 that O_{p}(M) is centric in F. Thus P ∈ F^{s}.

Set K = C_{L}(P). As L is proper, M is of characteristic p, and then K is of characteristic p by 2.7(a). Further, we have C_{F}(P) = F_{C_{S}(P)}(K) by 2.2. If P ∈ F^{c} then C_{S}(P) = Z(P) ≤ Z(K), so O^{p}(K) is a normal p′-subgroup of K. Then O^{p}(K) = 1, and K = Z(P). Conversely, if K = Z(P) then C_{S}(P) ≤ P. As P is fully normalized in F, P is also fully centralized by 1.13, and we then conclude from 1.10 that P ∈ F^{c}. This establishes (b).

Suppose next that P ∈ F^{q}, so that C_{F}(P) is the trivial fusion system on C_{S}(P). Then N_{K}(U)/C_{K}(U) is a p-group for every subgroup U of C_{S}(P). Take U = O_{p}(K). Thus K/Z(U) is a p-group, so K is a p-group, and then K = C_{S}(P) is a normal p-subgroup of M. Conversely, if K ≤ O_{p}(M) then C_{F}(P) is trivial, so we have (c).

Set R = O_{p}(M) and let Γ be the set of all overgroups of R in N_{S}(P). As M is of characteristic p, R is centric in N_{F}(P), and then Γ ⊆ N_{F}(P)^{c}. We may view M as a locality (M, Γ, N_{S}(P)) which happens to be a group, and this locality is proper by 2.7(b). Then R = O_{p}(N_{F}(P)) by 2.3. This completes the proof of (a). □

The next result shows that if (L, Δ, S) is a locality on F, such that the set Δ of objects is not “too small” (F^{cr} ⊆ Δ) and not “too large” (Δ ⊆ F^{q}), then L has a canonical homomorphic image which is a proper locality on F.

Proposition 2.9. Let (L, Δ, S) be a locality on F, with F^{cr} ⊆ Δ ⊆ F^{q}. Let Θ be the union of the groups Θ(P) := O_{p′}(C_{L}(P)) over all P ∈ Δ. Then Θ is a partial normal subgroup of L, and L/Θ is a proper locality on F.

Proof. Let P ∈ Δ and let Q ∈ P^{F} be fully normalized in F. Set M = N_{L}(Q) and K = C_{L}(Q). Then C_{F}(Q) = F_{C_{S}(Q)}(K) by 2.2. As Q is F-quasicentric by hypothesis, C_{F}(Q)^{cr} is the trivial fusion system on C_{S}(Q), and a classical theorem of Frobenius [Theorem 7.4.5 in Gor] then implies that K has a normal p-complement. That is, we have K = Θ(Q)C_{S}(Q).

Set \overline{M} = M/Θ(Q), set \overline{R} = O_{p}(\overline{M}), and let R be the preimage of \overline{R} in N_{S}(Q). Let \overline{X} be a p′-subgroup of C\overline{M}(\overline{R}), and let X be the preimage of \overline{X} in M. Since M = N_{M}(R)Θ(Q) by the Frattini argument, we obtain X = N_{X}(R)Θ(Q). But

[R, N_{X}(R)] ≤ R ∩ [R, X] ≤ R ∩ Θ(Q) = 1,

so

N_{X}(R) ≤ C_{X}(R) ≤ O^{p}(C_{K}(Q)) = Θ(Q),

and thus X = Θ(Q). This shows that C\overline{M}(\overline{R}) is a p-group, and thus \overline{M} is of characteristic p. In this way the hypothesis (*) of I.4.12 is fulfilled, and we conclude that Θ ⊴ L, (L/Θ, Δ, S) is a locality, F = F_{S}(L/Θ), and N_{L/Θ}(P) is of characteristic p for all P ∈ Δ. As F^{cr} ⊆ Δ, L/Θ is a proper locality on F. □

There are proper localities (L, Δ, S), with fusion system F, such that Δ is strictly larger than F^{q}. For example, if G is a finite group of Lie type, defined over a field of characteristic p, and Δ is the set of all non-identity subgroups of a Sylow p-subgroup S of G, then a theorem of Borel and Tits shows that N_{G}(P) is of characteristic p for all P ∈ Δ. Then (G |_{Δ}, Δ, S) is a proper locality on F_{S}(G), whereas in general there are non-identity subgroups of S which are not quasicentric in F.

Lemma 2.10. Let (L, Δ, S) be a locality on F, and suppose that F^{cr} ⊆ Δ. Then F is (cr)-generated.

Proof. Let Γ be the set of all R ∈ F^{cr} and such that R is fully normalized in F, and let F_{0} be the fusion system on S generated by the union of the groups Aut_{F}(R), for R ∈ Γ. Assuming the lemma to be false, there exists an F-isomorphism ϕ : P → P′ such that ϕ is not an F_{0}-isomorphism.

By definition, F is generated by the conjugation maps c_{g} : S_{g} → S_{g^{−1}}, so we may take ϕ to be such a c_{g}, with P = S_{g} and P′ = P^{g} (and where P and P′ are in Δ). Among all such obstructions g to the lemma, choose g so that |P | is as large as possible. As S ∈ Γ, we have P ≠ S, so P is a proper subgroup of N_{S}(P).

Let Q ∈ P^{F} with Q fully normalized in F. As F is Δ-inductive by 2.2, there exist elements x, y ∈ L such that N_{S}(P)^{x} ≤ N_{S}(Q) ≥ N_{S}(P′)^{y}, and such that P^{x} = Q = (P′)^{y}. As P is a proper subgroup of N_{S}(P), c_{x} and c_{y} are F_{0}-isomorphisms. If also c_{x}^{−1} ∘ c_{g} ∘ c_{y} is an F_{0}-isomorphism, then so is c_{g}, which is contrary to the case. We may therefore replace g with x^{−1}gy, so that g ∈ N_{L}(Q). Points (a) and (b) of 2.2 then imply that N_{F}(Q) is (cr)-generated and that N_{F}(Q) is the fusion system of N_{L}(Q) over N_{S}(Q). In this way the maximality of Q reduces the problem to the case where Q ∈ N_{F}(Q)^{cr}. Then Q = O_{p}(N_{F}(Q)) and Q ∈ F^{c}, and so Q ∈ F^{cr}. Thus ϕ is in F_{0}, and this contradiction proves the lemma. □

Lemma 2.11. Let (L, Δ, S) be a locality on F and let N ⊴ L be a partial subgroup. Set T = S ∩ N and set H = N_{L}(T). Assume that x^{h} is defined for all x ∈ N and all h ∈ H. Let ϕ : X → Y be an F-isomorphism. Then there exists u ∈ W(N) with X ≤ S_{v}, and there exists v ∈ W(H) with X^{u} ≤ S_{v}, such that:

(a) For each entry h of v and each x ∈ N we have S_{(h,x)} = S_{(x^{h^{−1}} ,h)}, and

(b) ϕ = c_{u} ∘ c_{v}.

Proof. Among all counterexamples choose ϕ : X → Y so that the length ℓ(w) of a shortest word w ∈ W(L) with ϕ = c_{w} is as small as possible. Then ϕ is not an identity map, and so w is not the empty word. Write w = w_{0} ∘ (g) with g ∈ L, and let ϕ_{0} : X → X^{w_{0}} be the conjugation map induced by w_{0}. The minimality of ℓ(w) implies that there exist u_{0} ∈ W(N) and v_{0} ∈ W(H) such that (a) and (b) hold with respect to ϕ_{0}, u_{0}, and v_{0}. The splitting lemma (I.3.12) implies that g = xh with x ∈ N and h ∈ H, and such that (a) holds. Write v_{0} = (h_{1}, ⋯, h_{n}), set x_{0} = x, and recursively define x_{k+1} = (x_{k})^{h_{n−k}^{−1}} for 0 ≤ k < n. Set y = x_{n}, and set u = u_{0} ∘ (y) and v = v_{0} ∘ (h). Then u and v satisfy the conclusion of the lemma, and hence ϕ is not a counterexample. □

Lemma 2.12. Let (L, Δ, S) and (L′, Δ′, S′) be localities, with fusion systems F = F_{S}(L) and F′ = F_{S′}(L′). Let λ : L → L′ be a homomorphism of partial groups such that Sλ ≤ S′. Then the restriction λ |_{S} of λ to S is fusion-preserving with respect to F and F′. That is, λ |_{S} is a morphism F → F′ in the category of fusion systems. Moreover, if λ is a projection (I.4.4) then λ |_{S} maps Ob(F) onto Ob(F′), and maps Mor(F) onto Mor(F′).

Proof. For any subgroup X ≤ S and any g ∈ L write \overline{X} for Xλ and \overline{g} for gλ. Also, for X ≤ S write λ_{X} for λ | X. Thus, λ_{S} is fusion-preserving if for each F-homomorphism α : X → Y there exists an F′-homomorphism β : \overline{X} → \overline{Y} such that

(*)  λ_{X} ∘ β = α ∘ λ_{Y} .

The uniqueness of β then follows from the observation that \overline{x}β = (xα)λ_{Y} for all x ∈ X. In the case that α = c_{g} for some g ∈ L the formula (*) is immediate from the definition of homomorphism of partial groups, and β = c\overline{g} in that case. Since F is generated by such homomorphisms c_{g}, (*) then follows from the further observation that if also h ∈ L and Y^{h} ≤ Z ≤ S, then

c_{g} ∘ c_{h} ∘ λ = λ |_{X} ∘c\overline{g} ∘ c\overline{h}.

Thus λ |_{S} is fusion-preserving.

Suppose that λ is a projection. Then λ maps S onto S′ by I.4.3(c), and so λ |_{S} maps Ob(F) onto Ob(F′). Let \overline{w} ∈ W(L′). By I.3.13(e) there exists w ∈ W(L) such that that the λ |_{S}-preimage of S′ ; \overline{w} in S is a subgroup of S_{w}. The mapping λ^{∗} : W(L) → W(L′) of free monoids, induced by λ, sends w to \overline{w}, and then λ |_{S} sends c_{w} to c\overline{w}. Thus, λ |_{S} maps Mor(F) onto Mor(F′). □

Lemma 2.13. Let (L, Δ, S) be a locality on F, and let D ≤ S be a subgroup of S such that every F-isomorphism X → Y extends to an F-isomorphism XD → Y D. Then D ≤ O_{p}(F).

Proof. The set of F-isomorphisms is the set of all conjugation maps c_{w} : X → X^{w} with w ∈ W(L) and with X ≤ S_{w}. Set E = ⋂ S_{w} | w ∈ W(L). Then E^{v} = E for all v ∈ W(L) since W(L) is closed under concatenation of words. Thus D ≤ E ≤ O_{p}(L). □

Section 3: Restrictions and expansions

This section is based closely on [section 5 in Ch1].

Definition 3.1. Let (L, Δ, S) be a locality and let H be a partial subgroup of L. Set R = S ∩ H and let Γ be a non-empty collection of subgroups of R, closed with respect to overgroups in R. Set

H |_{Γ}= {g ∈ H | S_{g} ∩ R ∈ Γ}.

Then (H, Γ) is a localizable pair in L if the following conditions hold.

(1) w ∈ W(H) with S_{w} ∩ R ∈ Γ ⟹ w ∈ D(L). In particular, N_{H}(P) is a subgroup of H for all P ∈ Γ.

(2) X^{g} ∈ Γ for all (X, g) ∈ Γ × H such that X ≤ S_{g}.

(3) R is a maximal p-subgroup of H. Define F_{Γ}(H) to be the fusion system E on R which is generated by the conjugation maps c_{g} : X → Y where g ∈ H and where X, Y ∈ Γ. If E^{cr} ⊆ Γ and each of the groups N_{H}(P) for P ∈ Γ is of characteristic p then we say that (H, Γ) is properly localizable.

Lemma 3.2. Let (H, Γ) be a localizable pair in the locality L, and set E := F_{Γ}(H). Then (H |_{Γ}, Γ, S ∩ H) is a locality on E, with

D(H |_{Γ}) = {w ∈ W(H) | S_{w} ∩ R ∈ Γ},

and the inclusion map H |_{Γ}→ L is a homomorphism of partial groups. Moreover, if (H, Γ) is properly localizable then (H |_{Γ}, Γ, S ∩ H) is a proper locality.

Proof. By 3.1(1) and (2) H |_{Γ} is an objective partial group. As Γ is a set of subgroups of R, 3.1(3) then implies that (H |_{Γ}, Γ, S ∩ H) is a locality, and plainly a locality on E. Also, 3.1(1) implies that the inclusion H |_{Γ}→ L is a homomorphism of partial groups. It is immediate from definition 2.4 that H |_{Γ} is proper if (H, Γ) is properly localizable. □

Example 3.3. Take H = L and let Γ be an F-closed set of subgroups of S. The conditions (1) through (3) in definition 3.1 are fulfilled in an obvious way, so (L, Γ) is a localizable pair. The locality (L |_{Γ}, Γ, S) will be called the restriction of L to Γ.

Example 3.4. Let R ≤ S be fully normalized in F and assume that N_{S}(R) ∈ Δ. Set Δ_{R} = {P ∈ Δ | R ⊴ P }. By I.2.12(a) N_{L}(R) is a partial subgroup of L. Then (N_{L}(R), Δ_{R}) is a localizable pair (conditions (1) and (2) of definition 3.1 are obvious, while condition (3) is given by I.2.11(b)). Set F_{R} = F_{Γ}(N_{L}(R). We write L_{R} for the locality N_{L}(R) |_{Δ_{R}}.

Along with the notion of restriction of a locality there is a corresponding notion of expansion.

Definition 3.5. Let (L, Δ, S) be a locality. An expansion of L consists of a locality (L′, Δ′, S′) together with an injective homomorphism η : L → L′ such that η restricts to an isomorphism S → S′, and such that Im(η) is a restriction of L′. Such an expansion is elementary if there exists R ∈ Δ′ such that Δ′ = Δη ⋃ R^{F}.

Of course, one always has trivial expansions of L, for which η is an isomorphism. Our goal in the remainder of this section is to construct a non-trivial elementary expansion of L, under conditions (see Hypothesis 3.7 below) which will be found to hold if L is proper, and provided that Δ is not already “as large as possible”. The following lemma prepares the way for such a construction.

Lemma 3.6. Let (L, Δ, S) be a locality on F, let R ≤ S with R fully normalized in F, and suppose that ⟨U, V ⟩ ∈ Δ for every pair U ≠ V of distinct F-conjugates of R. Set Δ_{R} = {P ∈ Δ | R ⊴ P }. Then the following hold.

(a) Δ_{R} = {N_{Q}(R) | R ≤ Q ∈ Δ}, and (N_{L}(R), Δ_{R}, N_{S}(R)) is a locality. Moreover, if every strict overgroup of R in S is in Δ, then F_{N_{S}(R)}(N_{L}(R) = N_{F}(R)).

(b) R^{F} = {R^{g} | g ∈ L, R ≤ S_{g}}.

(c) F is Δ ∪ R^{F}-inductive.

Proof. Let Q ∈ Δ with R ≤ Q. If R ⊴ Q then N_{Q}(R) = Q ∈ Δ, while otherwise N_{Q}(R) contains a pair of distinct Q-conjugates of R. Thus N_{Q}(R) ∈ Δ in either case, and thus Δ_{R} = {N_{Q}(R) | R ≤ Q ∈ Δ}.

For any g ∈ N_{L}(R) we have N_{S_{g}}(R) ∈ Δ, so (N_{L}(R), Δ_{R}, N_{S}(R)) is a locality by 3.4. Set F_{R} = F_{N_{S}(R)}(N_{L}(R)). Then F_{R} is a fusion subsystem of N_{F}(R), and it remains to show that F_{R} = N_{F}(R) in order to complete the proof of (a). The proof of this last step in point (a) will require (c), which we now prove together with (b).

Let V ∈ R^{F}, and let Y_{V} be the set of elements y ∈ L such that R^{y} = V and such that N_{S}(V) ≤ S_{y^{−1}}. By definition, each F-isomorphism ϕ : V → R can be factored as a composition of conjugation maps

(*)  V ⟶^{c_{x_{1}}} V_{1} ⟶ ⋯ ⟶ V_{k−1} ⟶^{c_{x_{k}}} R

for some word w = (x_{1}, ⋯, x_{k}) ∈ W(L). Write V^{w} = R to indicate this. Assume now that there exists V ∈ R^{F} such that Y_{V} = ∅. Among all such V, choose V so that the minimum length k, taken over all words w for which V^{w} = R, is as small as possible. Then, subject to this condition, choose w so that |N_{S_{w}}(V)| is as large as possible. We evidently have k > 0, and R ≠ S.

Suppose that k = 1. Thus w = (x) for some x ∈ L with V^{x} = R. Set P = N_{S_{x}}(V) and set P̃ = N_{N_{S}(V)}(P). Then P ∈ Δ, and conjugation by x then induces an isomorphism N_{L}(P) → N_{L}(P^{x}) by I.2.3(b). Thus P̃^{x} is a p-subgroup of the locality N_{L}(R). By I.2.11 there exists z ∈ N_{L}(R) with (P̃^{x})^{z} ≤ N_{S}(R), and then (x, z) ∈ D via P. Now P̃^{xz} ≤ N_{S}(R), and the maximality of |N_{S_{w}}(V)| in the choice of V and w yields P = P̃. Then P = N_{S}(V), so x^{−1} ∈ Y_{V}. This shows that k > 1.

The minimality of k now yields Y_{V_{1}} ≠ ∅, where V_{1} is defined by (*). Thus V^{(c,d)} = R, where d may be chosen in Y_{V_{1}}. Set A = N_{S_{c}}(V). Then A ∈ Δ as S_{c} ∈ Δ, and A^{c} ≤ N_{S}(V_{1}). As b ∈ Y_{V_{1}} we then have w ∈ D via A. Thus k = 1, contrary to the result of the preceding paragraph. We conclude that Y_{V} = ∅ for V ∈ R^{F}, and this proves (b) and (c).

We now return to the final step in the proof of (a). Set F_{R} = F_{N_{S}(R)}(N_{L}(R)), and assume that Δ_{R} contains the set of all strict overgroups of R in S. Assume also, by way of contradiction, that F_{R} is a proper subsystem of N_{F}(R). Then there exists a subgroup Y of N_{S}(R), containing R and fully normalized in N_{F}(R), such that Aut_{F_{R}}(Y) is a proper subgroup of Aut_{N_{F} (R)}(Y). If R is a proper subgroup of Y then Y ∈ Δ, and then every N_{F}(R)-automorphism of Y is induced by conjugation by an element of N_{L}(Y) which fixes R. That is, we have Aut_{F_{R}}(Y) = Aut_{N_{F} (R)}(Y) as is contrary to the case. Thus Y = R. Now let α ∈ Aut_{F}(R) and let w ∈ W(L) with c_{w} = α. Write w = (g_{1}, ⋯, g_{n}), set R_{0} = R, and recursively define R_{k} for 0 < k ≤ n by R_{k} = R^{g_{k}}. Thus R_{n} = R. Set a_{0} = a_{n} = 1. As a consequence of (b) we may choose and for each k with 0 < k < n an element a_{k} ∈ L such that (R_{k})^{a_{k}} = R and such that N_{S}(R_{k}) ≤ S_{a_{k}}. Then the word u_{k} = (a_{k−1}, g_{k}, a_{k}) (with 0 < k ≤ n) is in Δ via N_{S_{a_{k−1}}} (R_{k−1})^{a_{k−1}}. Setting f_{k} = Π(u_{k}) and w′ = (f_{1}, ⋯, f_{n}), we obtain f_{k} ∈ N_{L}(R) and α = c_{w′}. Thus α is an F_{R}- automorphism of R, which is again contrary to the case. We conclude that F_{R} = N_{F}(R), completing the proof of (a) and of the lemma. □

We assume the following setup for the remainder of this section.

Hypothesis 3.7. (L, Δ, S) is a locality on F, and R ≤ S is a subgroup of S such that:

(1) Each strict overgroup of R in S is in Δ.

(2) Both R and O_{p}(N_{F}(R)) are fully normalized in F.

(3) N_{L}(R) is a subgroup of L, and N_{F}(R) = F_{N_{S}(R)}(N_{L}(R)).

For each V ∈ R^{F} let Y_{V} be the set of elements y ∈ L such that R^{y} = V and such that N_{S}(V) ≤ S_{y^{−1}}. Define Y to be the union of the sets Y_{V}, taken over all V ∈ R^{F}. Notice that points (b) and (c) of 3.6 are equivalent to the condition that Y_{V} be non-empty for each V ∈ R^{F}.

Theorem 3.8. Assume Hypothesis 3.7, and set Δ^{+} = Δ ⋃ R^{F}. Then there exists a locality (L^{+}, Δ^{+}, S) such that L is the restriction of L^{+} to Δ, and such that N_{L^{+}}(R) = N_{L}(R). Moreover, we then have F_{S}(L^{+}) = F, and the following hold.

(a) For any locality (L̃, Δ^{+}, S) such that L̃ |_{Δ}= L, and such that NL̃(R) = N_{L}(R), we have F_{S}(L̃) = F, and there is a unique isomorphism β : L^{+} → L̃ which restricts to the identity map on L.

(b) Let L_{0}^{+} be the set of all g ∈ L^{+} such that S_{g} contains an F-conjugate of R. Then L_{0}^{+} is a partial subgroup of L, and L^{+} is the pushout in the category of partial groups of the diagram

L_{0}^{+} ⟵ L ∩ L_{0} ⟶ L

of inclusion homomorphisms.

Along the way to proving Theorem 3.8, we shall explicitly determine the partial group structure of L^{+} in 3.14 through 3.18. These results will then play an important computational role in section 4.

Set X = {x^{−1} | x ∈ Y} and set

Φ = X × N_{L}(R) × Y.

For any ϕ = (x^{−1}, h, y) ∈ Φ set U_{ϕ} = R^{x} and V_{ϕ} = R^{y}. Thus:

U_{ϕ} ⟶^{x^{−1}} R ⟶^{h} R ⟶^{y} V_{ϕ}, and N_{S}(U_{ϕ}) ⟶^{x^{−1}} N_{S}(R) ⟵^{y^{−1}} N_{S}(V_{ϕ})

are diagrams of conjugation maps, labelled by the conjugating elements. (In the first of these diagrams the conjugation maps are isomorphisms, and in the second they are homomorphisms into N_{S}(R).)

(3.9). Define a relation ∼ on Φ as follows. For ϕ = (x^{−1}, h, y) and \overline{ϕ} = (\overline{x}^{−1}, \overline{h}, \overline{y}) in Φ, write ϕ ∼ \overline{ϕ} if

(i) U_{ϕ} = U\overline{ϕ}, V_{ϕ} = V\overline{ϕ}, and (ii) (\overline{x}x^{−1})h = \overline{h}(\overline{y}y^{−1}).

The products in 3.9(ii) are well-defined. Namely, by 3.9(i), (\overline{x}, x^{−1}) ∈ D via N_{S}(U)^{\overline{x}^{−1}}, \overline{x}x^{−1} ∈ N_{L}(R), and then (\overline{x}x^{−1}, h) ∈ D since N_{L}(R) is a group (3.7(3)). The same considerations apply to (\overline{y}, y) and (\overline{h}, \overline{y}y^{−1}).

One may depict the relation ∼ by means of a commutative diagram, as follows.

U ⟶^{x^{−1}} R ⟶^{g} R ⟶^{y} V
∥  _{\overline{x}x^{−1}}↑  ↑^{\overline{y}y^{−1}}  ∥
U ⟶_{\overline{x}^{−1}} R ⟶_{\overline{g}} R ⟶_{\overline{y}} V

Lemma 3.10. ∼ is an equivalence relation on Φ.

Proof. Evidently ∼ is reflexive and symmetric. Let ϕ_{i} = (x_{i}^{−1}, g_{i}, y_{i}) ∈ Φ (1 ≤ i ≤ 3) with ϕ_{1} ∼ ϕ_{2} ∼ ϕ_{3}. Then R^{x_{1}} = R^{x_{3}} and R^{y_{1}} = R^{y_{3}}. Notice that

(x_{3}, x_{2}^{−1}, x_{2}, x_{1}^{−1}) ∈ D via N_{S}(U)^{x_{3}^{−1}} and,
(y_{3}, y_{2}^{−1}, y_{2}, y_{1}^{−1}) ∈ D via N_{S}(V)^{y_{3}^{−1}}.

Computation in the group N_{L}(R) then yields

(x_{3}x_{1}^{−1})g_{1} = (x_{3}x_{2}^{−1})(x_{2}x_{1}^{−1})g_{1} = (x_{3}x_{2}^{−1})g_{2}(y_{2}y_{1}^{−1})
= g_{3}(y_{3}y_{2}^{−1})(y_{2}y_{1}^{−1}) = g_{3}(y_{3}y_{1}^{−1}),

which completes the proof of transitivity. □

Lemma 3.11. Let ψ ∈ Φ and set U = U_{ψ} and V = V_{ϕ}. Let x ∈ Y_{U} and y ∈ Y_{V}. Then there exists a unique h ∈ N_{L}(V) such that ψ ∼ (x^{−1}, h, y).

Proof. Write ψ = (\overline{x}^{−1}, \overline{h}, \overline{y}). Then (\overline{x}, x^{−1}) ∈ D via N_{S}(U)^{\overline{x}^{−1}}, and \overline{x}x^{−1} ∈ N_{L}(R). Similarly (\overline{y}, y^{−1}) ∈ D and \overline{y}y^{−1} ∈ N_{L}(R). As N_{L}(R) is a subgroup of L we may form the product h := (x\overline{x}^{−1})\overline{h}(\overline{y}y^{−1}) and obtain (\overline{x}x^{−1})h = \overline{h}(\overline{y}y^{−1}). Setting ϕ = (x^{−1}, h, y), we thus have ϕ ∼ ψ. If h′ ∈ N_{L}(R) with also (x^{−1}, h′, y) ∼ ψ then h′ = (x\overline{x}^{−1})\overline{h}(\overline{y}y^{−1}) = h. □

For ease of reference we record the following observation, even though it is simply part of the definition of the relation ∼.

Lemma 3.12. Let C be a ∼-class of Φ, let ϕ = (x^{−1}, g, y) ∈ C, and set U = R^{x} and V = R^{y}. Then the pair (U, V) depends only on C, and not on the choice of representative ϕ. □

Lemma 3.13. Let L_{0} be the set of all g ∈ L such that S_{g} contains an F-conjugate U of R. For each g ∈ L_{0} set

Φ_{g} = {ϕ ∈ Φ ∩ D | Π(ϕ) = g},

and set

U_{g} = {U ∈ R^{F} | U ≤ S_{g}}.

(a) U_{g} is the set of all U_{ϕ} such that ϕ ∈ Φ_{g}.

(b) Φ_{g} is a union of ∼-classes.

(c) Let U ∈ U_{g}, set V = U^{g}, and let x ∈ X_{U} and y ∈ X_{V}. Then (x, g, y^{−1}) ∈ D, h := xgy^{−1} ∈ N_{L}(U), and ϕ := (x^{−1}, h, y) ∈ Φ_{g}. If also (x^{−1}, h′, y) ∈ Φ_{g} then h = h′.

(d) Let ϕ and ψ in Φ_{g}. Then

ϕ ∼ ψ ⟺ U_{ϕ} = U_{ψ} ⟺ V_{ϕ} = V_{ψ}.

Proof. Let g ∈ L_{0}, let U ∈ U_{g}, and set V = U^{g}. Let (x^{−1}, y) ∈ X_{U} × Y_{V} and set w = (x, g, y^{−1}). Then w ∈ D via

(N_{S_{g}}(U)^{x^{−1}}, N_{S_{g}}(U), N_{S_{g^{−1}}} (V), N_{S_{g^{−1}}} (V)^{y^{−1}}).

Set h = Π(w). Then h ∈ N_{L}(R) since R^{x} = U and R^{y} = V. Set w′ = (x^{−1}, x, g, y^{−1}, y). Then w′ ∈ D via N_{S_{w}}(U), and g = Π(w′) = Π(x^{−1}, h, y) by D-associativity (I.1.4(b)).

If also (x^{−1}, h′, y) ∈ D with Π(x^{−1}, h′, y) = g then h = h′ by the cancellation rule (I.1.4(e)). This establishes (c), and shows that U_{g} ⊆ {U_{ϕ} | ϕ ∈ Φ_{g}}. The opposite inclusion is immediate (cf. I.2.3(c)), so also (a) is established.

Let ϕ ∈ Φ_{g} and let \overline{ϕ} ∈ Φ with ϕ ∼ \overline{ϕ}. Write ϕ = (x^{−1}, h, y) and \overline{ϕ} = (\overline{x}^{−1}, \overline{h}, \overline{y}), and set

(*)  w′ = (\overline{x}^{−1}, \overline{x}, x^{−1}, h, y, \overline{y}^{−1}, \overline{y}).

As ϕ ∈ D we have N_{S_{ϕ}}(U) ∈ Δ by 3.7(1), and then w′ ∈ D via N_{S_{ϕ}}(U). Now Π(w′) = Π(ϕ) = g, while also

(**)  Π(w′) = Π(\overline{x}^{−1}, Π((\overline{x}x^{−1}), h, (y\overline{y}^{−1})), \overline{y}) = Π(\overline{x}^{−1}, \overline{h}, \overline{y}) = Π(\overline{ϕ}),

and thus \overline{ϕ} ∈ Φ_{g}. This proves (b).

It remains to prove (d). So, let ϕ, ψ ∈ Φ_{g}. If ϕ ∼ ψ then U_{ϕ} = U_{ψ} and V_{ϕ} = V_{ψ} by 3.12. On the other hand, assume that U_{ϕ} = U_{ψ} or that V_{ϕ} = V_{ψ}. Then both equalities obtain, since V_{ϕ} = (U_{ϕ})^{g} and V_{ψ} = (U_{ψ})^{g}. Write ϕ = (x^{−1}, h, y) and ψ = (\overline{x}^{−1}, \overline{h}, \overline{y}) in the usual way, and define w′ as in (*). Then w′ ∈ D via N_{S_{ϕ}}(V_{ϕ}), and Π(w′) = Π(ϕ) = g. Set

h′ = Π((\overline{x}x^{−1})h(y\overline{y}^{−1})).

Then g = Π(w′) = Π(\overline{x}^{−1}, h′, \overline{y}), so (\overline{x}^{−1}, h′, \overline{y}) ∈ Φ_{g}. Then 3.11 yields h′ = \overline{h}, and thus (\overline{x}^{−1}, h′, \overline{y}) = \overline{ϕ}. This shows that ϕ ∼ \overline{ϕ}, completing the proof of (d). □

(3.14). We now have a partition of the disjoint union L ⨆ Φ (and a corresponding equivalence relation ≈ on L ⨆ Φ) by means of three types of ≈-classes, as follows.

· Singletons {f}, where f ∈ L and where S_{f} contains no F-conjugate of R (classes whose intersection with Φ is empty). · ∼-classes [ϕ] such that [ϕ] ∩ D = ∅ (classes whose intersection with L is empty). · Classes Φ_{g} ∪{g}, where g ∈ L and where S_{g} contains an F-conjugate of R (classes having a non-empty intersection with both L and Φ).

For any element E ∈ L ∪ Φ, write [E] for the ≈-class of E. (Thus [E] is also a ∼-class if and only if [E] ∩ D = ∅.) Let L^{+} be the set of all ≈-classes. Let L_{0}^{+} be the subset of L^{+}, consisting of those ≈-classes whose intersection with Φ is non-empty. That is, the members of L_{0}^{+} are the ≈-classes of the form [ϕ] for some ϕ ∈ Φ.

Recall from Part I that there is an inversion map w ↦ w^{−1} on W(L), given by (g_{1}, ⋯, g_{n})^{−1} = (g_{n}^{−1}, ⋯, g_{1}^{−1}). The following result is then a straightforward consequence of the definitions of Φ, ∼, and ≈.

Lemma 3.15. The inversion map on W(L) preserves Φ. Further, for each E ∈ L ∪ Φ we have E^{−1} ∈ L ⋃ Φ, and [E^{−1}] is the set [E]^{−1} of inverses of members of [E]. □

Definition 3.16. For any γ = (ϕ_{1}, ⋯, ϕ_{n}) ∈ W(Φ) let w_{γ} be the word ϕ_{1} ∘ ⋯ ∘ ϕ_{n} in W(L). Let Γ be the set of all γ ∈ W(Φ) such that S_{w_{γ}} contains an F-conjugate of R. Let D_{0}^{+} be the set of all sequences w = ([ϕ_{1}], ⋯, [ϕ_{n}]) ∈ W(L_{0}^{+}) for which there exists a sequence γ of representatives for w with γ ∈ Γ. We shall say that γ is a Γ-form of w.

The following lemma shows how to define a product Π_{0}^{+} : D_{0}^{+} → L_{0}^{+}.

Lemma 3.17. Let w = ([ϕ_{1}], ⋯, [ϕ_{n}]) ∈ D_{0}^{+}, and let γ = (ϕ_{1}, ⋯, ϕ_{n}) be a Γ-form of w. Write ϕ_{i} = (x_{i}^{−1}, h_{i}, y_{i}).

(a) (y_{i}, x_{i+1}^{−1}) ∈ D and y_{i}x_{i+1}^{−1} ∈ N_{L}(R) for each i with 1 ≤ i < n.

(b) Set

w_{0} = (h_{1}, y_{1}x_{2}^{−1}, ⋯, y_{n−1}x_{n}^{−1}, h_{n}).

Then w_{0} ∈ W(N_{L}(R)) and (x_{1}^{−1}, Π(w_{0}), y_{n}) ∈ Φ. Moreover:

(c) The ≈-class [x_{1}^{−1}, Π(w_{0}), y_{n}] depends only on w, and not on the choice of Γ-form of w.

Proof. Let U ∈ R^{F} and let x, y ∈ Y_{U}. Then (y, x^{−1}) ∈ D via N_{S}(U)^{y^{−1}}, and then yx^{−1} ∈ N_{L}(R). This proves (a), and shows that w_{0} ∈ W(N_{L}(R)). As N_{L}(R)) is a subgroup of L, (b) follows.

Let \overline{γ} = (\overline{ϕ}_{1}, ⋯, \overline{ϕ}_{n}) be any Γ-form of w, write \overline{ϕ}_{i} = (\overline{x}_{i}^{−1}, \overline{h}_{i}, \overline{y}_{i}), and define \overline{w}_{0} in analogy with w_{0}. Set U_{0} = U_{ϕ_{1}} and \overline{U}_{0} = U_{\overline{ϕ}_{1}}. For each i with 1 ≤ i ≤ n set U_{i} = V_{ϕ_{i}} and \overline{V}_{i} = U_{\overline{ϕ}_{i}}. Thus:

U_{i−1} ⟶^{x_{i}^{−1}} R ⟶^{h_{i}} R ⟶^{y_{i}} U_{i} and \overline{U}_{i−1} ⟶^{\overline{x}_{i}^{−1}} R ⟶^{\overline{h}_{i}} R ⟶^{\overline{y}_{i}} \overline{U}_{i}.

Suppose that there exists an index j with U_{j} ≠ \overline{U}_{j}. As ϕ_{j} ≈ \overline{ϕ}_{j} it follows from 3.10 and 3.11 that ϕ_{j} and \overline{ϕ}_{j} are in D, and that there is an element g_{j} ∈ L_{0} such that ϕ_{j} and \overline{ϕ}_{j} are in Φ_{g}. If j < n then

U_{j+1} = (U_{j})^{g} ≠ (\overline{U}_{j})^{g} = \overline{U}_{j+1},

and if j > 0 one obtains U_{j−1} ≠ \overline{U}_{j−1} in similar fashion, by consideration of w^{−1} via 3.14. Thus, for each index i we have U_{i} ≠ \overline{U}_{i}. Then ϕ_{i} and \overline{ϕ} lie in distinct ∼-classes by 3.10. As ϕ_{i} ≈ \overline{ϕ}_{i} it follows that ϕ_{i} and \overline{ϕ}_{i} are members of Φ ∩ D, and that there is a word v = (g_{1}, ⋯, g_{n}) ∈ W(L) with g_{i} = Π(ϕ_{i}) = Π(\overline{ϕ}_{i}). As ⟨U_{0}, \overline{U}_{0}⟩ ≤ S_{v}, we have v ∈ D.

Set P = S_{v} and set P_{0} = N_{P} (U_{0}). Then P_{0} ∈ Δ by 3.7(1). Set P_{i} = (P_{i−1})^{g_{i}} for 1 ≤ i ≤ n. Then

P_{i−1}^{x_{i}^{−1}} ≤ N_{S}(R) and (P_{i})^{y_{i}^{−1}} ≤ N_{S}(R).

Thus conjugation by g_{i} maps P_{i−1}^{x_{i}^{−1}} to (P_{i})^{y_{i}^{−1}}, and this shows:

(*) Set w_{γ} = γ_{1} ∘ ⋯ ∘ γ_{n}. Then w_{γ} ∈ D via P_{0}.

Similarly, one obtains w\overline{γ} ∈ D via N_{P} (\overline{U}_{0}), where Now D-associativity yields:

Π(x_{1}^{−1}, Π(w_{0}), y_{n}) = Π(w_{γ}) = Π(v) = Π(w\overline{γ}) = Π(\overline{x}_{1}^{−1}, Π(\overline{w}_{0}), \overline{y}_{n}).

Thus Π(x_{1}^{−1}, Π(w_{0}), y_{n}) and (\overline{x}_{1}^{−1}, Π(\overline{w}_{0}), \overline{y}_{n}) lie in the same fiber of Π : Φ ∩ D → L_{0}, and thus (x_{1}^{−1}, Π(w_{0}), y_{n}) ≈ (\overline{x}_{1}^{−1}, Π(\overline{w}_{0}), \overline{y}_{n}). This reduces (c) to the following claim.

(**) Suppose that U_{i} = \overline{U}_{i} for all i. Then (x_{1}^{−1}, Π(w_{0}), y_{n}) ∼ (\overline{x}_{1}^{−1}, Π(\overline{w}_{0}), \overline{y}_{n}). Among all counter-examples to (**), let w be chosen so that n is as small as possible. Then n ≠ 0 (i.e. w and \overline{w} are non-empty words), as there is otherwise nothing to verify. If n = 1 (so that w = (ϕ_{1}) and \overline{w} = (\overline{ϕ}_{1})) then w_{0} = (h), \overline{w}_{0} = (\overline{h}), and (**) follows since ϕ ∼ \overline{ϕ}. Thus, n ≥ 2.

Set ψ = (x_{1}^{−1}, h_{1}(y_{1}x_{2}^{−1})h_{2}, y_{2}), and similarly define \overline{ψ}. Then ψ and \overline{ψ} are in Φ. As ϕ_{i} ∼ \overline{ϕ}_{i} for i = 1, 2, one has the commutative diagram

U_{0} ⟶^{x_{1}^{−1}} R ⟶^{h_{1}} R ⟶^{y_{1}} U_{1} ⟶^{x_{2}^{−1}} R ⟶^{h_{2}} R ⟶^{y_{2}} U_{2}
∥  ^{\overline{x}_{1}x_{1}^{−1}}↑  ↑^{\overline{y}_{1}y_{1}^{−1}} A∥ ^{\overline{x}_{2}x_{2}^{−1}}↑  ↑^{\overline{y}_{2}y_{2}^{−1}}  ∥
U_{0} ⟶^{\overline{x}_{1}^{−1}} R ⟶^{\overline{h}_{1}} R ⟶^{\overline{y}_{1}} U_{1} ⟶^{\overline{x}_{2}^{−1}} R ⟶^{\overline{h}_{2}} R ⟶^{\overline{y}_{2}} U_{2}

of conjugation maps. This diagram then collapses to the commutative diagram

U_{0} ⟶^{x_{1}^{−1}} R ⟶^{h_{1}(y_{1}x_{2}^{−1})h_{2}} R ⟶^{y_{2}} U_{2}
∥  ^{\overline{x}_{1}x_{1}^{−1}}↑  ↑^{\overline{y}_{2}y_{2}^{−1}}  ∥
U_{0} ⟶^{\overline{x}_{1}^{−1}} R ⟶^{\overline{h}_{1}(\overline{y}_{1}\overline{x}_{2}^{−1})\overline{h}_{2}} R ⟶^{\overline{y}_{2}} U_{2}

which shows that ψ ∼ \overline{ψ}. Here (ψ, ϕ_{3}, ⋯, ϕ_{n}) and (\overline{ψ}, \overline{ϕ}_{3}, ⋯, \overline{ϕ}_{n}) are Γ-forms of the word u = ([ψ], [ϕ_{3}], ⋯, [ϕ_{n}]). Set

u_{0} = (h_{1}(y_{1}x_{2}^{−1})h_{2}, y_{2}x_{3}^{−1}, ⋯, y_{n−1}x_{n}^{−1}, h_{n}),

and similarly define \overline{u}_{0}. Then Π(u_{0}) = Π(w_{0}) and Π(\overline{u}_{0}) = Π(\overline{w}_{0}). The minimality of n then yields

(x^{−1}, Π(w_{0}), y_{n}) ≈ (\overline{x}_{1}^{−1}, Π(\overline{w}_{0}), \overline{y}_{n}).

This proves (**), and thereby completes the proof of (c). □

Proposition 3.18. There is a mapping Π^{+} : D_{0}^{+} → L_{0}^{+}, given by

Π_{0}^{+}(∅) = [1, 1, 1],

and by

(*)  Π_{0}^{+}(w) = [x_{1}^{−1}, Π(w_{0}), y_{n}],

on non-empty words w ∈ D_{0}^{+}; where w_{0} is given by a Γ-form of w as in 3.15(b). Further, there is an involutory bijection on L_{0}^{+} given by

(**)  [x^{−1}, g, y]^{−1} = [y^{−1}, g^{−1}, x].

With these structures, L_{0}^{+} is a partial group.

Proof. The reader may refer to I.1.1 for the conditions (1) through (4) defining the notion of partial group. Condition (1) requires that D_{0}^{+} contain all words of length 1 in the alphabet L_{0}^{+}, and that D_{0}^{+} be closed with respect to decomposition. (That is, if u and v are two words in the free monoid W(L_{0}^{+}), and the concatenation u ∘ v is in D_{0}^{+}, then u and v are in D_{0}^{+}.) Both of these conditions are immediate consequences of the definition of D_{0}^{+}.

That Π_{0}^{+} is a well-defined mapping is given by 3.15. The proof that Π_{0}^{+} satisfies the conditions I.1.1(2) (Π_{0}^{+} restricts to the identity map on words of length 1) and I.1.1(3):

u ∘ v ∘ ˜∈D_{0}^{+} ⟹ Π_{0}^{+}(u ∘ v ∘ v) = Π_{0}^{+}(u ∘ Π_{0}^{+}(v) ∘ w)

are then straightforward, and may safely be omitted.

The inversion map [x^{−1}, h, y] ↦ [y^{−1}, h^{−1}, x] is well-defined by 3.13. Evidently this mapping is an involutory bijection, and it extends to an involutory bijection

(C_{1}, ⋯, C_{n})^{−1} = (C_{n}^{−1}, ⋯, C_{1}^{−1})

on W(L_{0}^{+}). It thus remains to show I.1.1(4). That is, we must check that

w ∈ D_{0}^{+} ⟹ w^{−1} ∘ w ∈ D_{0}^{+} and Π_{0}^{+}(w^{−1} ∘ w) = [1, 1, 1].

In detail: take w = (C_{1}, ⋯, C_{n}) and let γ = (ϕ_{1}, ⋯, ϕ_{n}) be a Γ-form of w, where ϕ_{i} is written as [x_{i}^{−1}, h_{i}, y_{i}). One easily verifies that γ^{−1} ∘ γ ∈ Γ, and hence w^{−1} ∘ w ∈ D_{0}^{+}. Now

Π_{0} + (w^{−1} ∘ w) = [y_{n}^{−1}, Π(u_{0}), y_{n}],

where

u_{0} = (g_{n}^{−1}, x_{n}y_{n−1}^{−1}, ⋯, x_{2}y_{1}^{−1}, g_{1}^{−1}, x_{1}x_{1}^{−1}, g_{1}, y_{1}x_{2}^{−1}), ⋯, y_{n−1}x_{n}^{−1}, g_{n}).

One observes that Π(u_{0}) = 1, and so Π_{0} + (w^{−1} ∘ w) = [y_{n}^{−1}, 1, y_{n}]. Now observe that (y_{n}^{−1}, 1, y_{n}) ≡ (1, 1, 1), since (y_{n}^{−1}, 1, y_{n}) and (1, 1, 1) are in D and since Π(y_{n}^{−1}, 1, y_{n}) = 1 = Π(1, 1, 1). Thus Π_{0} + (w^{−1} ∘ w) = [1, 1, 1] = Π_{0}^{+}(∅). Thus I.1.1(4) holds, and the proof is complete. □

Lemma 3.19. Let D_{0} be the set of all w ∈ D such that S_{w} contains an F-conjugate of R, and let L_{0} be the set of words of length 1 in D_{0}, regarded as a subset of L. Let Π_{0} : D_{0} → L_{0} be the restriction of Π to D_{0}. Then:

(a) L_{0}, with Π_{0} and the restriction of the inversion map on L to L_{0}, is a partial group.

(b) Let ι_{0} : L_{0} → L be the inclusion map, and let λ_{0} : L_{0} → L_{0}^{+} be the mapping g ↦ Φ_{g} ∪ {g}. Then ι_{0} and λ_{0} are homomorphisms of partial groups.

Proof. The verification of (a) is straightforward, and is left to the reader (see I.1.1). Moreover, since the product in L_{0} is inherited from L, it is immediate that ι_{0} is a homomorphism of partial groups.

Let v = (g_{1}, ⋯, g_{n}) ∈ D_{0}, set w = ([g_{1}], ⋯, [g_{n}]), and let U ∈ R^{F} be chosen so that U ≤ S_{v}. By 3.17 there exists a word γ = (ϕ_{1}, ⋯, ϕ_{n}) ∈ W(Φ) such that ϕ_{i} ∈ [g_{i}] and such that U ≤ S_{w_{γ}}, where w_{γ} is the word ϕ_{1} ∘ ⋯ ∘ ϕ_{n} ∈ W(L). Then γ is a Γ-form of w, and so w ∈ D_{0}^{+}. Set P = S_{w}. The proof of the intermediary result (*) in the proof of 3.15 shows that w_{γ} ∈ D via N_{P} (U). Write ϕ_{i} = (x_{i}^{−1}, h_{i}, y_{i}), and form the word w_{0} ∈ W(N_{L}(R)) as in 3.15(b). Set g = Π(v). Then g = Π(w_{γ}) = Π(x_{1}^{−1}, w_{0}, y_{n}) by D-associativity, and thus (x_{1}^{−1}, w_{0}, y_{n}) ∈ Φ_{g}. That is, we have [g] = [x_{1}^{−1}, w_{0}, y_{n}]. This shows that λ_{0} is a homomorphism of partial groups, completing the proof of (b). □

We remark that it is easily verified that Im(λ_{0}) is in fact a partial subgroup of L_{0}^{+}. But there is no reason to suppose that Im(ι_{0}) is a partial subgroup of L, as it may be the case that W(L_{0}) ∩ D is not contained in D_{0}.

For any w = (g_{1}, ⋯, g_{n}) ∈ W(L) set [w] = ([g_{1}], ⋯, [g_{n}]). For any subset W of W(L) set [X] = {[w] | w ∈ W }. Since each ≈-class [g_{i}] intersects L in {g_{i}} it follows that the product Π : D → L may be regarded as a mapping [D] → L.

Recall from Theorem I.1.17 that the category of partial groups has all colimits (and all limits). In particular, pushouts are available.

Proposition 3.20. Set D^{+} = [D] ∪ D_{0}^{+}.

(a) The products Π : D → L and Π_{0}^{+} : D_{0}^{+} → L_{0}^{+} agree on [D] ∩ D_{0}^{+}, and Π ∪ Π_{0}^{+} is

a mapping

Π^{+} : D^{+} → L^{+}.

(b) L^{+} is a partial group via the product Π^{+} and the involutory bijection given by 3.13.

(c) Let λ : L → L^{+} be the mapping g ↦ [g], and let ι : L_{0}^{+} → L^{+} be the inclusion map. Then λ and ι are injective homomorphisms of partial groups, and

L_{0}^{+} ⟶^{ι} L^{+}
λ_{0}↑  ↑^{λ}
L_{0} ⟶_{ι_{0}} L

is a pushout diagram in the category of partial groups.

Proof. That Π and Π_{0}^{+} agree on [D] ∩ D_{0}^{+} is one way of interpreting the fact (3.17(b)) that λ_{0} and ι_{0} are homomorphisms of partial groups. Thus (a) holds, and point (b) is then a straightforward exercise with definition I.1.1.

Let L^{∗} (with the appropriate diagram of homomorphisms) be a pushout for

(*)  L_{0}^{+} ⟵^{λ_{0}} L_{0} ⟶^{ι_{0}} L.

By I.1.17 we may in fact take the underlying set of L^{∗} to be the standard pushout of (*) as a diagram of mappings of sets. That is, we may take L^{∗} to be the disjoint union L_{0}^{+} ⨆ L modulo the relation ≡ which identifies g ∈ L_{0} with gλ_{0}. Here gλ_{0} = Φ_{g} ∪ {g}, and the elements of L which are not in L_{0} are by definition the singletons {f} such that S_{f} contains no F-conjugate of R. By identifying such a singleton {f} with its unique element we thereby obtain L^{∗} = L^{+} as sets.

The domain D^{∗} of the product in L^{∗} is obtained by from the disjoint union D_{0}^{+} ⨆ D by imposing the ≡-relation componentwise. That is, we have D^{∗} = D^{+}. The product Π^{∗} : D^{∗} → L^{∗} is then the union of the products Π_{0}^{∗} and Π; which is to say that Π^{∗} = Π^{+}. Similarly, the inversion maps on L^{∗} and on L^{+} coincide, and so L^{∗} = L^{+} as partial groups. That λ and ι are the homomorphisms which give the required pushout diagram in then immediate.

Now let f, g ∈ L with fλ = gλ. Then 3.12 yields

{f} = [f] ∩ L = [g] ∩ L = {g}

and so λ is injective. The inclusion map ι is of course injective, so the proof is complete. □

Let Δ^{+} be the union of Δ with the set of all subgroups P ≤ S such that P contains an F-conjugate of R. The following lemma prepares the way for showing that (L^{+}, Δ^{+}) is an objective partial group.

Lemma 3.21. Let λ : L → L^{+} be the homomorphism of partial groups given by g ↦ [g]. Let [ϕ] ∈ L_{0}^{+}, and let S_{[ϕ]} be the set of all a ∈ S such that [a]^{[ϕ]} is defined in L^{+}, and such that [a]^{[ϕ]} ∈ [S]. Then S_{[ϕ]} = S_{ϕ}.

Proof. Let a ∈ S such that [a]^{[ϕ]} = [b] for some b ∈ S. Suppose first that [ϕ] ∩ L is non-empty, and let g be the unique element of [ϕ] ∩ L. The equality [a]^{[ϕ]} = [b] then simply means that (a^{g})λ = bλ, and the injectivity of λ (3.20(c)) yields a^{g} = b. Thus a ∈ S_{g}. Conversely, for any x ∈ S_{g} we have xλ = [x] ∈ S_{[ϕ]} = gλ, and thus the lemma holds in this case.

Assume now that [ϕ] ∩ L = ∅. Then [ϕ] is a ∼-class by 3.14, and 3.12 shows that the pair (U, V) := (U_{ϕ}, V_{ϕ}) is constant over all ϕ ∈ [ϕ]. Set w = ([ϕ]^{−1}, [a], [ϕ]). Then w ∈ D_{0}^{+} by hypothesis, so there is a Γ-form γ = (ϕ^{−1}, ψ, \overline{ϕ}) of w. This means that, upon setting wγ = ϕ^{−1} ∘ ψ ∘ \overline{ϕ}, we have V ≤ S_{w_{γ}}. The uniqueness of (U, V) for [ϕ] (and of (V, U) for [ϕ]^{−1}) yields U = U_{ψ} = V_{ψ}, and then a ∈ N_{S}(U) since a = Π(ψ). Since [b]^{[ϕ]^{−1}} = [a] we similarly obtain b ∈ N_{S}(V). Notice also that since (U, V) is independant of the choice of representative for [ϕ] we may take \overline{ϕ} to be ϕ. As x ∈ Y_{U} and y ∈ Y_{V} we obtain:

(*) a^{x^{−1}}, b^{y^{−1}} ∈ N_{S}(R).

Write ϕ = (x^{−1}, h, y), and set θ = (a^{−1}x^{−1}, 1, xa^{2}). As a ∈ N_{S}(U) we get xa^{2} ∈ Y_{U} and a^{−1}x^{−1} ∈ X_{U}. Thus θ ∈ Φ. Moreover, we have θ ∈ D via N_{S}(U), and Π(θ) = a. Thus θ ∈ Φ_{a}, and (ϕ^{−1}, θ, ϕ) is a Γ-form of w. Then

Π^{+}(w) = [y^{−1}, h^{−1}(x(a^{−1}x^{−1}))((xa^{2})x^{−1})h, y]

by the definition of Π^{+} in 3.20. Observing now that

(x, a^{−1}, x^{−1}, x, a^{2}, x^{−1}) ∈ D,

via N_{S}(U)^{x^{−1}}, we obtain

[b] = Π^{+}(w) = [y^{−1}, h^{−1}(xax^{−1})h, y] = [y^{−1}, (a^{x^{−1}})^{h}, y].

We now claim that (y^{−1}, b^{y^{−1}}, y) ∈ [b]. In order to see this, one observes first of all that (y^{−1}, b^{y^{−1}}, y) ∈ Φ. Also, since b^{y^{−1}} normalizes N_{S}(V)^{y^{−1}} we have (y^{−1}, y, b, y^{−1}, y) ∈ D via N_{S}(V). Then Π(y^{−1}, b^{y^{−1}}, y) = b, and the claim is proved. Thus:

(**)  [y^{−1}, (a^{x^{−1}})^{h}, y] = [y^{−1}, b^{y^{−1}}, y].

Application of Π to both sides of (**) then yields

((a^{x^{−1}})^{h})^{y} = (b^{y^{−1}})^{y},

and then (a^{x^{−1}})^{h} = b^{y^{−1}} by the cancellation rule in L. As a^{x^{−1}}, b^{y^{−1}} ∈ S by (*), we conclude that a ∈ S_{ϕ}. This completes the proof of (a), and thereby completes the proof. □

At this point it will be convenient (and need cause no confusion) to view λ_{0} and λ as inclusion maps. Then L^{+} = L ∪ L_{1}^{+}, where

L_{1}^{+} = {[ϕ] | ϕ ∈ Φ, ϕ ∉ D}.

Proposition 3.22. (L^{+}, Δ^{+}, S) is a locality. Moreover:

(a) L is the restriction L^{+} |_{Δ} of L^{+} to Δ.

(b) N_{L^{+}}(R) = N_{L}(R).

(c) F_{S}(L^{+}) = F.

Proof. We have first to show that (L^{+}, Δ^{+}) satisfies the conditions (O1) and (O2) in the definition I.2.1 of objectivity. Condition (O2) is the requirement that Δ^{+} be F-closed (i.e. that Δ^{+} be preserved by F-homomorphisms). Since Δ is F-closed, and Δ^{+} is given by attaching to Δ an F-conjugacy class R^{F} and all overgroups in S of members of R^{F}, (O2) holds for (L^{+}, Δ^{+}).

Condition (O1) requires that D^{+} be equal to D_{Δ^{+}}. This means:

(*) The word w = (C_{1}, ⋯, C_{n}) ∈ W(L^{+}) is in D^{+} ⟺ there exists a sequence (X_{0}, ⋯, X_{n}) ∈ W(Δ^{+}) such that X_{i−1}^{C_{i}} = X_{i} for all i, (1 ≤ i ≤ n).

Here we need only be concerned with the case w ∈ W(L_{0}^{+}) since D^{+} = D_{0}^{+} ∪ [D], and since D = D_{Δ}. The implication ⟹ in (*) is then given by the definition of D_{0}^{+}, with X_{0} ∈ R^{F}. The reverse implication is given by 3.21, and thus (L^{+}, Δ^{+}) is objective. We note that L^{+} is finite since L and Φ are finite.

Let S̃ be a p-subgroup of L^{+} containing S, and let a ∈ NS̃(S). As S ∈ Δ we get a ∉ L_{1}^{+}, so a ∈ N_{L}(S), and then a ∈ S since S is a maximal p-subgroup of L. Thus S̃ = S, S is a maximal p-subgroup of L^{+}, and (L^{+}, Δ^{+}, S) is a locality. The restriction of L^{+} to Δ is by definition the partial group whose product is the restriction of Π^{+} to D_{Δ}, whose underlying set is the image of Π^{+} | D_{Δ}, and whose inversion map is inherited from L^{+}. That is, (a) holds.

Let [ϕ] ∈ L_{1}^{+}, let ϕ = (x^{−1}, h, y) ∈ [ϕ], and set U = Uϕ. Then U ⊴ S_{[ϕ]} = S_{ϕ} by 4.17(a), and the conjugation map c_{[ϕ]} : S_{ϕ} → S is then the composite c_{x}^{−1} ∘ c_{h} ∘ c_{y} applied to S_{ϕ}. Thus c_{[ϕ]} is an F-homomorphism, and this yields (c). Suppose now that [ϕ] ∈ N_{L^{+}}(R). Then x, h, and y are in N_{L}(R), and then ϕ ∈ D as N_{L}(R) is a subgroup of L. Then [ϕ] ∉ L_{1}^{+}, and so (b) holds. □

Proposition 3.23. Let (L̃, Δ^{+}, S) be a locality having the same set Δ^{+} of objects as L^{+}. Assume that L̃ |_{Δ}= L and that NL̃(R) = N_{L}(R). Then the identity map on L extends in a unique way to an isomorphism L^{+} → L̃.

Proof. Write Π̃ : D̃ → L̃ for the product in L̃. (It isn’t necessary to distinguish the inversion map on Π̃ in any way, since by I.1.13 it restricts to the inversion map on L.) Let ϕ = (x^{−1}, h, y) and \overline{ϕ} = (\overline{x}^{−1}, \overline{h}, \overline{y}) be members of Φ such that ϕ ∼ \overline{ϕ}. Set U = U_{ϕ}. Then U = U\overline{ϕ}, and (\overline{x}, x^{−1}) ∈ D via N_{S}(U)^{\overline{x}^{−1}}. Then Π̃(\overline{x}, x^{−1}) = Π(\overline{x}, x^{−1}) as L is the restriction of L̃ to Δ. Similarly, we obtain Π̃(\overline{y}, y^{−1}) = Π(\overline{y}, y^{−1}). Then:

(1)  Π̃(\overline{x}, x^{−1}, h) = Π̃(\overline{x}x^{−1}, h) = Π(\overline{x}x^{−1}, h) = Π(\overline{h}, \overline{y}y^{−1}) = Π̃(\overline{h}, \overline{y}, y^{−1}).

Observe that both (\overline{x}^{−1}, \overline{x}, x^{−1}, h, y) and (\overline{x}^{−1}, \overline{h}, \overline{y}, y^{−1}, y) are in D^{+}∩D̃ (via the obvious conjugates of U). Then (1) yields

Π̃(ϕ) = Π̃(\overline{x}^{−1}, \overline{x}, x^{−1}, h, y) = Π̃(\overline{x}^{−1}, \overline{h}, \overline{y}, y^{−1}, y) = Π̃(\overline{ϕ}).

If ϕ ∈ D then Π̃(ϕ) = Π(ϕ), so we have shown that there is a well-defined mapping

β : L^{+} → L̃

such that β restricts to the identity map on L and sends the element [ϕ] ∈ L_{0}^{+} to Π̃(ϕ).

We now show that β is a homomorphism of partial groups. As L^{+} is a pushout, in the manner described in 3.20(c), and since the inclusion L → L̃ is a homomorphism, it suffices to show that the restriction β_{0} of β to L_{0}^{+} is a homomorphism.

So, let w = ([ϕ_{1}], ⋯, [ϕ_{n}]) ∈ D^{+}, let γ = (ϕ_{1}, ⋯, ϕ_{n}) be a Γ-form of w, and set w_{γ} = (ϕ_{1}, ⋯, ϕ_{n}). Write ϕ_{i} = (x_{i}^{−1}, h_{i}, y_{i}). Then Π^{+}(w) = [x_{1}^{−1}, Π(w_{0}), y_{n}], where w_{0} ∈ W(N_{L}(R)) is the word

w_{0} = (h_{1}, y_{1}x_{2}^{−1}, ⋯, y_{n−1}x_{n}^{−1}, h_{n}).

given by 3.17(b). Let β∗ be the induced mapping W(L^{+}) → W(L̃) of free monoids. Then

Π̃(wβ^{∗}) = Π̃([ϕ_{1}]β, ⋯, [ϕ_{n}]β) = Π̃(Π̃(ϕ_{1}), ⋯, Π̃(ϕ_{n})) = Π̃(w_{γ})

by D̃-associativity, and

(Π^{+}(w))β = [x_{1}^{−1}, Π(w_{0}), y_{n}]β = Π̃(x_{1}^{−1}, Π(w_{0}), y_{n}).

Set w′_{0} = (h_{1}, y_{1}, x_{2}^{−1}, ⋯, y_{n−1}, x_{n}^{−1}, h_{n}). Then w′_{0} ∈ D̃ via U_{ϕ_{1}}, and Π̃(w′_{0}) = Π̃(w_{0}) = Π(w_{0}). Then

(Π^{+}(w))β = Π̃((x_{1}^{−1}) ∘ Π̃(w′_{0}) ∘ (y_{n})) = Π̃(w_{γ}) = Π̃(wβ^{∗}),

and so β_{0} is a homomorphism. As already mentioned, this result implies that β is a homomorphism.

Let f ∈ L̃ with f ∉ L. Then S_{f} ∉ Δ (as L = L̃ |_{Δ}), and so S_{f} contains a unique U ∈ R^{F}. Set V = U^{f}, and recall that the inversion map on L is induced from the inversion map on L̃. If V ∈ Δ then f^{−1} ∈ L̃, and then f ∈ L. Thus V ∉ Δ, and hence V ∈ R^{F}. By 3.6 there exist elements x, y ∈ L such that U^{x} = R = V^{y} and such that N_{S}(U)^{x^{−1}} ≤ N_{S}(R) ≥ N_{S}(V)^{y^{−1}}. We find that (x, f, y^{−1}) ∈ D̃ via R and that h := Π̃(x, f, y^{−1}) ∈ NL̃(R). Then h ∈ N_{L}(R) by hypothesis, and f = Π̃(x^{−1}, h, y). In particular, we have shown:

(*) Every element of L̃ is a product of elements of L.

We may now show that β^{∗} maps D_{0}^{+} onto D̃. Thus, let w̃ = (f_{1}, ⋯, f_{n}) ∈ D̃ and set X = Sw̃. If X ∈ Δ then w̃ ∈ D and w̃ = w̃β^{∗}. So assume that X ∉ D. Then X contains a unique F-conjugate U_{0} of R, and there is a sequence (U_{1}, ⋯, U_{n}) ∈ W(Δ^{+}) given by U_{i} = (U_{i−1})_{i}^{f}. As seen in the preceding paragraph, we then have U_{i} ∈ R^{F} for all i, and there exists a sequence γ = (ϕ_{1}, ⋯, ϕ_{n}) ∈ W(Φ) such that Π̃(ϕ_{i}) = f_{i}. Set w_{γ} = [ϕ_{1}] ∘ ⋯ ∘ [ϕ_{n}]. Then w_{γ} ∈ D^{+} and (w_{γ})β^{∗} = w̃. Thus β^{∗} : D_{0}^{+} → D̃ is surjective. That is, β is a projection, as defined in I.4.5.

Let g ∈ Ker(β). If g ∈ L then g = 1 since β |_{L} is the inclusion map. So assume that g ∉ L. Then g = [ϕ] for some ϕ = (x^{−1}, h, y) ∈ Φ, and [ϕ] is a ∼-class. Set U = U_{ϕ} and V = V_{ϕ}. As 1 = gβ = Π̃(ϕ) it follows that U = V = R. But then ϕ ∈ D as N_{L}(R) is a subgroup of L, and so [ϕ] is not a ∼-class. Thus Ker(β) = 1. As β is a projection, β is then an isomorphism by I.4.3(d). It follows from (*) that β is the unique isomorphism L^{+} → L̃ which restricts to the identity on L. □

Notice that Propositions 3.22 and 3.23 complete the proof of Theorem 3.8.

Section 4: Elementary expansions and partial normal subgroups

We continue to assume Hypothesis 3.7 throughout this section. Our aim is to prove the following result.

Theorem 4.1. Assume Hypothesis 3.7, and let (L^{+}, Δ^{+}, S) be the expansion of (L, Δ, S) given by Theorem 3.8. Let N ⊴ L be a partial normal subgroup of L, and let N^{L^{+}} be the set of all L^{+}-conjugates of elements of N (the set of all f^{g} such that f ∈ N, g ∈ L^{+} and (g^{−1}, f, g) ∈ D^{+}). Set X = N^{L^{+}} and let N^{+} be partial subgroup of L^{+} generated by X (cf. I.1.9). Then:

(a) N^{+} ⊴ L^{+}.

(b) N^{+} ∩ L = N.

(c) If M ⊴ L^{+} is a partial normal subgroup of L^{+} such that M ∩ L = N then M = N^{+}.

We will employ all of the notation from section 3. Thus, the reader will need to have in mind the meanings of L_{0}, L_{0}^{+}, L^{+}, Φ, ∼, and ≈. Set Ω = N^{L^{+}}.

Lemma 4.2. Let w ∈ D^{+}, and suppose that w ∉ W(L). Then S_{w} is an F-conjugate of R, and S_{w} = S_{Π^{+}(w)} if Π^{+}(w) ∉ L.

Proof. As w ∉ L it follows that S_{w} ∉ Δ. Then, as w ∈ L^{+}, it follows from 3.5(1) that S_{w} is an F-conjugate of R. Write w = (g_{1}, ⋯, g_{n}) with g_{i} ∈ L^{+}, set g = Π^{+}(w), and suppose that g ∉ L. Then S_{g} is an F-conjugate, and since S_{w} ≤ S_{g} we obtain S_{w} = S_{g}, as required. □

Lemma 4.3. Assume that 4.1(b) holds. Then:

(a) Theorem 4.1 holds in its entirety.

(b) Let ρ : L → L/N and ρ^{+} : L^{+}/N^{+} be the canonical projections, and let ι : L → L^{+} be the inclusion map. Then there is a unique homomorphism η : L/N → L^{+}/N^{+} such that ι ∘ ρ^{+} = ρ ∘ η. Moreover, η is injective, and L^{+}/N^{+} is an elementary expansion of L/N via η (cf. 3.1). Further, if S ∩ N ≰ R then η is an isomorphism, while if S ∩ N ≤ R then N_{L/N} (Rη) = N_{L^{+}/N^{+}}(Rη).

Proof. We assume that L ∩ N^{+} = N (4.1(b)). Set Ω_{0} = Ω, and recursively define Ω_{n} for n > 0 by

Ω_{n} = {Π^{+}(w) | w ∈ W(Ω_{n−1}) ∩ D^{+}}.

Then N^{+} = ⟨Ω⟩ is the union of the sets X_{n}, by I.1.9. In order to show that N^{+} ⊴ L^{+} it then suffices to show that each Ω_{n} is closed with respect to conjugation (whenever defined) in L^{+}.

Let g ∈ Ω_{0}. Then there exists f ∈ N and a ∈ L^{+} such that (a^{−1}, f, a) ∈ D^{+} and with g = f^{a}. Now let b ∈ L^{+} such that (b^{−1}, g, b) ∈ D^{+}. If g ∈ L then g ∈ N by 4.1(b), and then g^{b} ∈ X_{0}. On the other hand, assume g ∉ L. Then S_{g} ∈ R^{F}, and 4.2 yields S_{g} = S_{(a^{−1},f,a)}. Set u = (b^{−1}, g, b) and v = (b^{−1}, a^{−1}, f, a, b). Then S_{u} = S_{v} = (S_{g})^{b}, so v ∈ D^{+}, and g^{b} = Π^{+}(u) = Π^{+}(v) = f^{ab}. Thus g^{b} ∈ X_{0} in any case, and so X_{0} is closed with respect to conjugation in L^{+}.

Assume now that there exists a largest integer k such that Ω_{k} is closed with respect to conjugation in L^{+}, and let now f ∈ Ω_{k+1} and b ∈ L^{+} such that f^{b} is defined in L^{+} and is not in Ω_{k+1}. By definition there exists w = (f_{1}, ⋯, f_{n}) ∈ W(Ω_{k}) ∩ D^{+} such that f = Π^{+}(w). Suppose f ∈ L. Then f ∈ N by 4.1(b), and then f^{b} ∈ Ω. As Ω ⊆ Ω_{m} for all m, we have f^{b} ∈ Ω_{k+1} in this case. So assume that f ∉ L. Then S_{f} = S_{w} ∈ R^{F} by 4.2. Set w′ = (b^{−1}, f_{1}, b, ⋯, b^{−1}, f_{n}, b). Then w′ ∈ D^{+} via (S_{w})^{b}, and we obtain f^{b} = Π^{+}(w′) = Π^{+}(f_{1}^{b}, ⋯, f_{n}^{b}). As Ω_{k} is closed with respect to conjugation in L^{+} we conclude that f^{b} ∈ Ω_{k+1}, contrary to the maximality of k; and this contradiction completes the proof that N^{+} ⊴ L^{+}. That is, 4.1(a) holds.

Next: let M ⊴ L^{+} be any partial normal subgroup of L^{+} such that M ∩ L = N. Let ϕ be the composite of the inclusion map L → L^{+} followed by the quotient map ρ : L^{+} → L^{+}/M. Then Ker(ρ) = M (cf. I.4.3), and so Ker(ϕ) = M ∩ L = N. Let ρ_{S} be the restriction of ρ (and hence also of ϕ) to S.

Set \overline{Δ} = \{P ϕ | P ∈ Δ\}, and write \overline{P} for P ϕ. Let \overline{w} ∈ W(L^{+}/M), and set \overline{X} = \overline{S}\overline{w}. That is (and in the usual way) \overline{X} is the largest subgroup of \overline{S} which is conjugated successively into \overline{S} by the entries of \overline{w}. Write \overline{w} = (\overline{g}_{1}, ⋯, \overline{g}_{n}), and for each i let g_{i} ∈ L^{+} be a representative of \overline{g}_{i} such that g_{i} is ↑-maximal with respect to M. Set w = (g_{1}, ⋯, g_{n}), and let X be the preimage of \overline{X} in S via ρ_{S}. Then X ≤ S_{w} by I.3.14(e). If \overline{X} ∈ \overline{Δ} then X ∈ Δ and w ∈ D(L^{+} |_{Δ}). As L^{+} | Δ = L, we have w ∈ D(L) in this case, and then \overline{w} ∈ D((L^{+}/M) |\overline{D}). On the other hand, if \overline{w} ∈ D((L^{+}/M) |\overline{D}) then \overline{X} ∈ \overline{Δ}. Thus:

Im(ϕ) = (L^{+}/M) |\overline{D} .

In particular, Im(ϕ) is a partial group (and indeed a locality) in this way, and the mapping

ϕ_{0} : L → Im(ϕ)

given by ϕ is then a homomorphism of partial groups. One observes that the induced mapping

ϕ_{0}^{∗} : W(L) → W(Im(ϕ))

of free monoids carries D(L) onto W(Im(ϕ)). That is, ϕ_{0} is a projection. Then, by I.4.6 there is an isomorphism \overline{ϕ} : L/N → Im(ϕ) induced by ϕ.

Let η : L/N → L^{+}/M be the composition of \overline{ϕ} followed by the inclusion of Im(ϕ) into L^{+}/M. Our aim now is to show that L^{+}/M is an expansion of L/N via η. For this, what is required (cf. definition 3.1) is: (A) η is injective (obvious, since η is a composite of injective mappings); (B) Im(η) is the restriction of L^{+}/M to Δϕ (which has been verified above); and (C) Δϕ is closed in the fusion system F′ = F\overline{S}(L^{+}/M). In order to verify (C) we note first of all that F = F_{S}(L^{+}) by 3.4(a). Since ρ and ϕ are projections, (C) then follows from 1.20. Thus η is an expansion, as claimed. Moreover, η is an elementary expansion, since 1.20 shows that the set \{\overline{P} | P ∈ Δ^{+}\} of objects of L^{+}/M is the union of \overline{D} with the set of F\overline{S}(L^{+}/M)-conjugates of \overline{R}.

We have now to verify that Hypothesis 3.3 holds with respect to L/M and \overline{R}, in order to apply Theorem 3.4 to the expansion L^{+}/M of Im(ϕ). Set T = S ∩ N, and suppose first that T ≰ R. Then T R ∈ Δ, so \overline{R} ∈ \overline{Δ}, and then 3.4 holds trivially. On the other hand, suppose that T ≤ R. Then every subgroup \overline{Q} of \overline{S} which properly contains an F\overline{S}(L^{+}/M)-conjugate of \overline{R} is the image under ϕ of a subgroup Q of S which properly contains an F-conjugate of R. That is, 3.3(1) obtains. Now let \overline{g} ∈ N_{L^{+}/M}(\overline{R}), and let g ∈ L^{+} be a ρ-preimage of g such that g is ↑-maximal with respect to M. Then g ∈ N_{L^{+}}(R) by I.3.14(e). Here N_{L^{+}}(R) = N_{L}(R) by 3.4 applied to L^{+}, so we have shown that ρ maps N_{L}(R) onto N_{L^{+}/M}(\overline{R}). This verifies 3.3(2).

The preceding analysis applies to N^{+} in the role of M. We now observe that N^{+} ≤ M since N ⊆ M, and since N^{+} = ⟨N^{L^{+}}⟩. Let η_{0} : L/N → L^{+}/N^{+} be the homomorphism which defines the expansion of L/N to L^{+}/N^{+}. By Theorem 3.4(b) there is then a unique homomorphism β : L/N^{+} → L/M such that η_{0} ∘ β = η, and β is in fact an isomorphism. On the other hand, let γ : L/N^{+} → L/M be the mapping which sends a maximal coset [g]_{N^{+}} of N^{+} in L^{+} to the unique maximal coset [g]_{M} containing it. One easily verifies that γ is a homomorphism and that η_{0} ∘ γ = η, and thus γ is an isomorphism. For g ∈ M we then have [g]_{N^{+}} = M, and this shows that M = N^{+}. That is, 4.1(c) holds, and we have obtained point (a) of the lemma. Point (b) was obtained in the course of proving (a). □

(4.4) Notation. Set M = N_{L}(R), and K = M ∩ N. As in the preceding section, for each g ∈ L^{+} set

U_{g} = {U ∈ R^{F} | U ≤ S_{g}},

and for each U ∈ R^{F} set

Y_{U} = {y ∈ L | R^{y} = U, N_{S}(U)^{y^{−1}} ≤ N_{S}(R)}.

The notation (4.5) will remain fixed until the proof of Theorem 4.1 is complete. Note that since M is a subgroup of L by 3.3, K is a normal subgroup of M by I.1.8(c).

Lemma 4.6. Suppose T ≤ R. Then N^{∗} = Ω, and Ω ∩ L = N.

Proof. Let f ∈ N, and suppose that S_{f} contains an F-conjugate of R. Then T ≤ S_{f} since, by I.3.1(a) T is weakly closed in F. Then I.3.1(b) yields the following result.

(1) Let f ∈ N such that S_{f} contains an F-conjugate U of R. Then P^{f} = P for each subgroup P of S_{f} containing U. In particular, U^{f} = U.

Let f′ ∈ Ω, and let f ∈ N and g ∈ L^{+} such that f′ = f^{g}. That is, assume that v := (g^{−1}, f, g) ∈ D^{+} and that f′ = Π^{+}(v). If v ∈ D then f′ = Π(v) ∈ N. On the other hand, suppose that v ∉ D. Then S_{v} is an F-conjugate of R by 4.1(1). Set U = (S_{v})^{g^{−1}}. Then U = U^{f} by (1). Now choose a ∈ Y_{U}, and set v′ = (g^{−1}, a^{−1}, a, f, a^{−1}, a, g). Then v′ ∈ D^{+} via U^{g}, and Π^{+}(v) = Π^{+}(v′). Notice that (1) implies that (a, f, a^{−1}) ∈ D via P := N_{S_{f}} (U), and that

T ⟶^{a} U ⟶^{f} U ⟶^{a^{−1}} T,

so that f^{a^{−1}} ∈ M. Thus f^{a^{−1}} ∈ K, and Π^{+}(v) = Π^{+}(g^{−1}a^{−1}, f^{a^{−1}}, ag). This shows:

(2) Ω is the union of N with the set of all Π^{+}(g^{−1}, f, g) such that f ∈ K and such that (g^{−1}, f, g) ∈ D^{+}. Moreover, for any such v = (g^{−1}, f, g), Π^{+}(v) normalizes each V ∈ R^{F} such that V ≤ S_{v}.

Assume now that we have v = (g^{−1}, f, g) as in (2) (so that f ∈ K), and let A be an F-conjugate of R contained in S_{v}. In order to analyze these things further, we shall need to be able to compute products in L^{+} in the manner described in 4.13 and 4.14. To that end, note first of all that since A^{g} = T there exists a unique h ∈ M and y ∈ Y_{A} such that g ≈ (1, h, y) ∈ Φ, by 4.7. Set ϕ = (1, h, y) and set ψ = (1, f, 1). Then (ϕ^{−1}, ψ, ϕ) is a Γ-form of (g^{−1}, f, g), as defined in 4.12. We then compute via 4.13 that

f′ = Π^{+}(g^{−1}, f, g) = [y^{−1}, h^{−1}, 1][1, f, 1][1, h, y] = [y^{−1}, f^{h}, y].

(3) Let f′ ∈ Ω, such that S_{f′} contains an F-conjugate of R. Then f′ is an ≈-class

[y^{−1}, k, y] with k ∈ K. If f′ ∈ L then (y^{−1}, k, y) ∈ D by 4.10, and so f′ ∈ N. Thus:

(4) Ω ∩ L = N.

Now let w = (f′_{1}, ⋯, f′_{n}) ∈ W(Ω) ∩ D^{+}, and set B = S_{w}. Suppose that Π^{+}(w) ∉ Ω. Then Π^{+}(w) ∉ N, so Π^{+}(w) ∉ L by (4). Thus w ∉ D, so B ∈ R^{F}, and then (2) shows that each f′_{i} normalizes B. Fix b ∈ Y_{B}. Then (3) implies that there exist elements k_{i} ∈ K such that f′_{i} = [b^{−1}, k_{i}, b]. One observes that the sequence of elements (b^{−1}, k_{i}, b) of Φ is a Γ-form for w, and then 4.13 yields Π^{+}(w) = [b^{−1}, k, b] where k = Π(k_{1}, ⋯, k_{n}) ∈ K. This simply means that Π^{+}(w) = k^{b}, since b^{−1} = [b^{−1}, 1, 1], k = [1, k, 1], and b = [1, 1, b]. Thus Π^{+}(w) ∈ Ω. Since Ω is closed under inversion, we have thus shown that Ω is a partial subgroup of L.

Finally, let c ∈ L^{+} be given so that (c^{−1}, f′, c) ∈ D^{+} (and where f′ ∈ Ω). Suppose that Π^{+}(c^{−1}, f′, c) ∉ Ω. Then f′ ∉ N, so f′ ∉ L by (4), and then f′ = Π^{+}(g^{−1}, f, g) for some f ∈ N and some g ∈ L^{+}, and where S_{f′} = S_{(g^{−1},f,g)}. Set u = (c^{−1}, g^{−1}, f, g, c). Then u ∈ D^{+} via (S_{f′})^{c}, and Π^{+}(u) = Π^{+}(c^{−1}g^{−1}, f, gc). Thus Π^{+}(c^{−1}, f′, c) ∈ Ω after all, and Ω ⊴ L^{+}. □

Let \overline{L} be the quotient locality L/N (cf. 4.4), and let ρ : L → \overline{L} be the quotient map and let ρ^{∗} be the induced homomorphism W(L) → W(\overline{L}) of free monoids. For any subset or element X of L, \overline{X} shall denote the image of X under ρ. We extend this convention to subsets and elements of W(\overline{L}) in the obvious way. Set \overline{Δ} = \{\overline{P} | P ∈ Δ\}.

Lemma 4.7. Assume that T ≰ R, and set \overline{F} = F\overline{S}(\overline{L}). Then \overline{R}\overline{F} ⊆ \overline{Δ}.

Proof. Let U ∈ R^{F}. As T ≰ R we then have T ≰ U by I.3.1(a). Then U is a proper subgroup of UT, so UT ∈ Δ by 3.3(1). Then \overline{U} = \overline{UT} ∈ \overline{Δ} by 4.3. □

Lemma 4.8. Assume that T ≰ R. There is then a homomorphism σ : L^{+} → \overline{L} such that the restriction of σ to L is the quotient map ρ.

Proof. Set \overline{D} = D(\overline{L}) and let \overline{Π} : \overline{D} → \overline{L} be the product in \overline{L}. As \overline{R}\overline{F} ⊆ \overline{Δ} by 4.7, we have \overline{Φ} ⊆ \overline{D}, and so there is a mapping λ : Φ → \overline{L} given by ϕλ = \overline{Π}(ϕρ^{∗}), where ρ^{∗} : W(L) → W(\overline{L}) is the homomorphism of free monoids induced by ρ. That is: ϕλ = \overline{Π}(\overline{ϕ}).

Let ϕ_{1}, ϕ_{2} ∈ Φ with ϕ_{1} ∼ ϕ_{2}, and write ϕ_{i} = (x_{i}^{−1}, h_{i}, y_{i}). Then (x_{2}x_{1}^{−1})h_{1} = h_{2}(y_{2}y_{1}^{−1}) (in M), and so

(1)  (\overline{x}_{2}\overline{x}_{1}^{−1})\overline{h}_{1} = \overline{h}_{2}(\overline{y}_{2}\overline{y}_{1}^{−1})

in \overline{M}. As (x_{2}, x_{1}^{−1}, h_{1}) and (h_{2}, y_{2}, y_{1}^{−1}) are in D^{+} via the appropriate conjugates of R, we have (\overline{x}_{2}, \overline{x}_{1}^{−1}, \overline{h}_{1}) and (\overline{h}_{2}, \overline{y}_{2}, \overline{y}_{1}^{−1}) in \overline{D}, and then

\overline{Π}(\overline{x}_{2}, \overline{x}_{1}^{−1}, \overline{h}_{1}) = \overline{Π}(\overline{h}_{2}, \overline{y}_{2}, \overline{y}_{1}^{−1})

by (1) and \overline{D}-associativity. A standard cancellation argument (cf. I.2.4(a)) then yields \overline{Π}(\overline{ϕ}_{1}) = \overline{Π}(\overline{ϕ}_{2}), and thus λ is constant on ∼-classes.

Now suppose that ϕ ∈ Φ_{g} for some g ∈ L. That is, suppose that ϕ ∈ Φ ∩ D and that g = Π(ϕ). Then ϕλ = \overline{g} since ρ is a homomorphism of partial groups, and so λ is constant on ≈-classes. This shows that there is a (well-defined) mapping σ : L^{+} → \overline{L} given by ρ on L and by λ on L_{0}^{+}. It remains only to check that σ is a homomorphism.

Let w = (f_{1}, ⋯, f_{n}) ∈ D^{+}. If w ∈ D then wσ^{∗} = wρ^{∗} ∈ \overline{D} and \overline{Π}(wσ^{∗}) = (Π(w))σ. On the other hand, suppose that w ∉ D. Then f_{i} = [ϕ_{i}] for some ϕ_{i} ∈ Φ, and there is a Γ-form γ = (ϕ_{1}, ⋯, ϕ_{n}) of w. Thus the word w_{γ} = ϕ_{1} ∘ ⋯ ∘ ϕ_{n} has the property that S_{w_{γ}} is an F-conjugate of R, and hence \overline{w_{γ}} ∈ \overline{D}. Then

\overline{Π}((w_{γ})σ^{∗}) = \overline{Π}(\overline{w_{γ}}).

Write ϕ_{i} = (x_{i}^{−1}, h_{i}, y_{i}). Then

(Π^{+}(w_{γ}))σ = [x_{1}^{−1}, Π(w_{0}), y_{n}]σ

where w_{0} ∈ D(M) is given by the formula in 4.13(b). One then obtains \overline{Π}((w_{γ})σ^{∗}) = (Π^{+}(w_{γ}))σ via \overline{D}-associativity, and so σ is a homomorphism, as desired. □

Lemma 4.9. N^{∗} ∩ L = N.

Proof. Let σ : L^{+} → \overline{L} be a homomorphism, as in 4.8, whose restriction to L is the quotient map ρ : L → \overline{L}. Then Ker(σ) ∩ L = Ker(ρ), where Ker(ρ) = N, by 4.4. As Ker(σ) ⊴ L^{+}, by 1.14, the lemma follows. □

Theorem 4.1 follows at once from the union of lemmas 4.3, 4.6, and 4.9.

Section 5: Theorem A

Recall from 1.8 that for any fusion system F on S, F^{s} denotes the set of F-subcentric subgroups of S. These are the subgroups U ≤ S such that there exists an F-conjugate V of U with V fully normalized in F and with O_{p}(N_{F}(V)) ∈ F^{c}.

Lemma 5.1. Let (L, Δ, S) be a proper locality on F, and let R be a subgroup of S which satisfies 3.3(1) and 3.3(2). Assume that R ∈ F^{s}. Then R satisfies 3.3(3). That is, N_{L}(R) is a subgroup of L of characteristic p, and N_{F}(R) = F_{N_{S}(R)}(N_{L}(R)).

Proof. Set L_{R} = N_{L}(R) and set Δ_{R} = {P ∈ Δ | R ⊴ P }. By 3.2 (L_{R}, Δ_{R}, N_{S}(R)) is a locality, and we set F_{R} = F_{N_{S}(R)}(L_{R}). Evidently F_{R} is a fusion subsystem of N_{F}(R).

If R ∈ Δ then L_{R} is a subgroup of L, of characteristic p since L is proper; and N_{F}(R) = F_{N_{S}(R)}(L_{R}) by 2.2. Thus there is nothing to show in this case, and so we may assume that R ∉ Δ. As R ∈ F^{s} there exists an F-conjugate R′ of R such that O_{p}(N_{F}(R′)) is centric in F. As F is Δ ∪ R^{F}-inductive by 3.2(c), it follows from 1.14(a) that Q := O_{p}(N_{F}(R)) is centric in F. If R = Q then R ∈ F^{cr}, contrary to R ∉ Δ. Thus R ≠ Q and so Q ∈ Δ by 3.3(1). As F_{R} is a fusion subsystem of N_{F}(R), on N_{S}(R), we have Q ⊴ F_{R}, and then Q is contained in every member of (F_{R})^{cr}. Thus:

(*) (F_{R})^{cr} ⊆ Δ_{R}. For P ∈ Δ_{R} write L_{P} for the subgroup N_{L}(P) of L. Then R ≤ L_{P}, and and N_{L_{P}} (R) = N_{L_{R}}(P). As L is proper, L_{P} is of characteristic p, and N_{L_{R}}(P) is then of characteristic p by II.2.7(b). This result, along with (*), shows that L_{R} is a proper locality on F_{R}. Then Q ⊴ L_{R} by 2.3, and thus L_{R} = N_{L_{Q}}(R) is a subgroup of L of characteristic p.

It now remains only to show that F_{R} = N_{F}(R), in order to complete the proof. Observe that N_{F}(R) is a fusion subsystem of N_{F}(Q), and that N_{F}(Q) is the fusion system F_{N_{S}(Q)}(L_{Q}) of a finite group, by 2.2(a). Then each N_{F}(R)-isomorphism is a conjugation map by an element of N_{L_{Q}}(R), and so N_{F}(R) is a fusion subsystem of F_{R}. That F_{R} is a subsystem of N_{F}(R) has already been noted, so the required equality of fusion systems obtains. □

5.2 (Theorem A1). Let (L, Δ, S) be a proper locality on F and let Δ^{+} be an F-closed collection of subgroups of S such that Δ ⊆ Δ^{+} ⊆ F^{s}.

(a) There exists a proper locality (L^{+}, Δ^{+}, S) on F such that L is the restriction L^{+} |_{Δ} of L^{+} to Δ. Moreover, L^{+} is generated by L as a partial group.

(b) For any proper locality (L̃, Δ^{+}, S) on F whose restriction to Δ is L, there is a unique isomorphism L^{+} → L̃ which restricts to the identity map on L.

Proof. Suppose false, and among all counterexamples choose (L, Δ, S) and Δ^{+} so that the set U = Δ^{+} − Δ has the smallest possible cardinality. We may choose R ∈ U so that the set Δ_{1} = Δ ⋃ R^{F} is F-closed, and by 1.15 R may be chosen so that both R and O_{p}(N_{F}(R)) are fully normalized in F. Thus the conditions (1) and (2) in Hypothesis 3.3 hold, with Δ_{1} in the role of Δ^{+}. Then 3.3 holds in its entirety, by 5.1. Theorem 3.4 then yields a proper locality (L_{1}, Δ_{1}, S) on F, such that L_{1} |_{Δ}= L, and such that N_{L_{1}}(R) = N_{L}(R). The minimality of |U| yields the existence of a proper locality (L^{+}, Δ^{+}, S) such that L^{+} |_{Δ_{1}}= L_{1} and with F_{S}(L^{+}) = F. Then L^{+} |_{Δ}= L. The explicit construction of L^{+} in section 3 shows that every element of L^{+} is a Π^{+}-product of elements of L, so we have (a). Point (b) is then immediate from the corresponding uniqueness result for L_{1} (given by Theorem 3.4) and for L^{+} with respect to L_{1}. □

5.3 (Theorem A2). Let the hypothesis and notation be as in Theorem A1. Let 𝔑 be the set of partial normal subgroups of L, and let 𝔑^{+} be the set of partial normal subgroups of L^{+}. For each N ∈ 𝔑 let N^{L^{+}} be the set of all elements of L^{+} of the form Π^{+}(g^{−1}, f, g), such that f ∈ N and such that (g^{−1}, f, g) ∈ D^{+}. Then there is an inclusion-preserving bijection

𝔑^{+} → 𝔑 (N^{+} ↦ N^{+} ∩ L).

In particular, N^{+} is the unique partial normal subgroup of L^{+} whose intersection with L is equal to N, and S ∩ N^{+} = S ∩ N for each N ∈ 𝔑.

Proof. As in the proof of Theorem A1, assume that L is a counter-example with |Δ^{+}−Δ| as small as possible. Define R, Δ_{1}, and L_{1} as in the preceding proof. Let N ⊴ L be a partial normal subgroup, let N_{1} be the partial subgroup of L_{1} generated by the set N^{L_{1}} of L_{1}-conjugates of elements of N. Theorem 4.1 then shows N_{1} ⊴ L_{1} and N = L ∩ N_{1}. The minimality of |Δ^{+} − Δ| yields the existence of a unique partial normal subgroup N^{+} ⊴ L^{+} with L_{1} ∩ N^{+} = N_{1}, and then L ∩ N^{+} = L ∩ N_{1} = N. Thus, the map

μ : 𝔑^{+} → 𝔑 (N^{+} ↦ N^{+} ∩ L)

is surjective.

Now let Ñ ⊴ L^{+} with L ∩ Ñ = N. Set Ñ_{1} = L_{1} ∩ Ñ. Then Ñ_{1} = N_{1} by 4.1, and then Ñ = N^{+} by the minimality of |Δ^{+} − Δ|. This shows that μ is injective, and so it only remains to show that μ is inclusion-preserving.

Let M ⊴ L with M ≤ N, and set M_{1} = ⟨M^{L_{1}}⟩. Then M_{1} ≤ ⟨N^{L_{1}}⟩ = N_{1}. The minimality of |Δ^{+} − Δ| then gives M^{+} ≤ N^{+}, completing the proof. □

Here is an application. Note that by [He1] the product of partial normal subgroups of a locality is again a partial normal subgroup.

Corollary 5.3. Let M and N be partial normal subgroups of the proper locality (L, Δ, S), and let (L^{+}, Δ^{+}, S) be a proper expansion of L. Then (MN)^{+} = M^{+}N^{+}.

Proof. Both M^{+}N^{+} and (MN)^{+} are partial normal subgroups of L^{+}, and (MN)^{+} contains both M^{+} and N^{+}, so M^{+}N^{+} ≤ (MN)^{+}. Then

L ∩ M^{+}N^{+} ≤ L ∩ (MN)^{+} = MN .

As M ≤ M^{+} and N ≤ N^{+} we obtain

L ∩ (MN)^{+} = L ∩ M^{+}N^{+},

and hence the desired result. □

We wish also to obtain a version of Theorem A for localities which are homomorphic images of proper localities. Thus, for the remainder of this section (L, Δ, S) is a locality (not necessarily proper) on F, and N ⊴ L is a partial normal subgroup of L. Set T = S ∩ N.

For any g ∈ L, N g denotes the set of all products xg such that x ∈ N and (x, g) ∈ D, and we say that N g is a right coset of N in L. The analogous notion of left coset is obvious. The set of all cosets (left or right) of N in L is partially ordered by inclusion, and one thus has the notion of the maximal cosets of N in L.

By I.3.14 the maximal left cosets of N are the maximal right cosets, and these maximal cosets form a partition \overline{L} = L/N of L. Let ρ : L → \overline{L} be the map which sends g ∈ L to the unique maximal coset of N containing g. By I.3.16 there is a unique partial group structure on \overline{L} which makes ρ into a homomorphism of partial groups; and then the induced map

ρ^{∗} : D(L) → D(\overline{L})

is surjective. A homomorphism from a locality to a partial group is by definition a projection (cf. I.4.4) if the induced map between the domains of their products is surjective. Thus, ρ is a projection, and it is shown in I.4.3 that \overline{L} thereby has the structure of a locality

(\overline{L}, \overline{Δ}, \overline{S}),

where \overline{D} = \{P ρ | P ∈ Δ\}, and where \overline{S} = Sρ ≅ S/T. An important (and elementary) property of any homomorphism of partial groups (I.1.15) is that it sends subgroups to subgroups.

The following lemma clarifies the relationship between the fusion system F of L and the fusion system of a homomorphic image of L.

Lemma 5.4. Let (L, Δ, S) be a locality on F, and let N ⊴ L be a partial normal subgroup. Let (\overline{L}, \overline{Δ}, \overline{S}) be the corresponding quotient locality, and let ρ : L → \overline{L} be the canonical projection. Set T = S ∩ N, and set \overline{F} := F\overline{S}(\overline{L}).

(a) ρ restricts to a surjective homomorphism σ : S → \overline{S}, and σ is fusion-preserving relative to F and \overline{F}. That is, σ is a homomorphism F → \overline{F} of fusion systems. Moreover, if X and Y are subgroups of S containing T then the induced map

σ_{X,Y} : Hom_{F}(X, Y) → Hom\overline{F}(\overline{X}, \overline{Y})

is surjective.

(b) Let X ≤ S be a subgroup of S containing T. Then X is fully normalized in F if and only if \overline{X} is fully normalized in \overline{F}.

(c) Let \overline{X} ∈ \overline{F}^{c}, and let X be the preimage of \overline{X} in S. Then X ∈ F^{c}.

(d) Let \overline{X} ∈ \overline{F}^{cr}, and let X be the preimage of \overline{X} in S. Then X ∈ F^{cr}.

(e) If F^{cr} ⊆ Δ then \overline{F}^{cr} ⊆ \overline{Δ}.

Proof. Point (a) is igiven by 1.20. For any subgroup H of L write \overline{H} for the image of H under ρ. Point (b) is given by the observation that if X is a subgroup of S containing T then \overline{N_{S}(X)} = N\overline{S}(\overline{X}).

Now let \overline{X} ∈ \overline{F}^{c}, let X be the preimage of \overline{X} in S, and let Y be an F-conjugate of X. Then \overline{Y} is an \overline{F}-conjugate of \overline{X}, so C\overline{S}(\overline{Y}) ≤ \overline{Y}, and hence C_{S}(Y) ≤ Y. This proves (c). Now assume further that \overline{X} ∈ \overline{F}^{cr}, and let Y ∈ X^{F} with Y fully normalized in F. Then \overline{Y} is fully normalized in \overline{F} by (b). Set Q = O_{p}(N_{F}(Y)). Then \overline{Q} ≤ O_{p}(N\overline{F}(\overline{Y})) by (a), so \overline{Q} = \overline{Y}, and hence Q = Y. This shows that X ∈ F^{cr} (as defined in 1.1), and proves (d). Point (e) is immediate from (d). □

If (L, Δ, S) is a proper locality on F, then the quotient locality L/N (see I.4.5) need not be proper. For example, let (N , Γ, T) be a direct product of pair-wise isomorphic proper localities (N_{i}, Γ_{i}, T_{i}) with 1 ≤ i ≤ k. This means (cf. I.1.17) that the underlying set of N is the direct product of the sets N_{i}, D(N) is the direct product of the domains D(N_{i}) (with the obvious product and inversion), Γ is the direct product of the collections Γ_{i}, and T is the direct product of the groups T_{i}. Then, let L be the partial group obtained as the natural semi-direct product N ⋊ H, where H is the symmetric group on k letters. (The reader should have no difficulty, at this stage, in defining the partial group L.) Let S be a maximal p-subgroup of L containing T, and let Δ be the set of all P ≤ S such that P ∩ T ∈ Γ. Then (L, Δ, S) is a proper locality, while L/N is isomorphic to the group H, which is a proper locality if and only if k = 1, or (k, p) is (2, 2), (3, 3), or (4, 2). Thus, Theorems A1 and A2 are not directly applicable to homomorphic images of proper localities.

Theorem 5.5. Let (L, Δ, S) be a proper locality on F, let N ⊴ L, let \overline{L} = L/N be the quotient locality, and let ρ : L → \overline{L} be the canonical projection. Let Δ^{+} be an F-closed set of subgroups of S such that Δ ⊆ Δ^{+} ⊆ F^{s}. Let (L^{+}, Δ^{+}, S) be the expansion of L to Δ^{+}, set \overline{Δ}^{+} = \{P ρ | P ∈ Δ^{+}\}, and set \overline{F} = F\overline{S}(\overline{L}).

(a) There is a locality (\overline{L}^{+}, \overline{Δ}^{+}, \overline{S}) on \overline{F} whose restriction to \overline{Δ} is \overline{L}, and \overline{L}^{+} is unique up to a unique isomorphism which restricts to the identity map on \overline{L}.

(b) There is a unique projection ρ^{+} : L^{+} → \overline{L}^{+} whose restriction to L is ρ. Moreover, Ker(ρ^{+}) = N^{+}, where N^{+} is the partial normal subgroup of L^{+} which intersects L in N.

Proof. Note first of all that since F^{cr} ⊆ Δ we have \overline{F}^{cr} ⊆ \overline{Δ} by 5.4(e). Notice also that if Δ = Δ^{+} then there is nothing to prove, so we may assume that Δ is properly contained in Δ^{+}.

Among all F-closed sets Δ_{1} with Δ ⊆ Δ_{1} ⊆ Δ^{+}, let Δ_{1} be maximal subject to the condition that (a) and (b) hold with Δ_{1} in the role of Δ^{+}. Then Δ_{1} is a proper subset of Δ^{+}, and by replacing Δ with Δ_{1} we reduce (as in the proof of Theorem A1) to the case where Δ^{+} = Δ ∪ R^{F} for some R ≤ S.

Take R to be fully normalized in F, and suppose that T ≰ R. Then RT ∈ Δ, and then Δ^{+}ρ = Δρ = \overline{Δ}. Then (\overline{L}, \overline{Δ}, \overline{S}) is the unique locality on \overline{F} whose restriction to \overline{Δ} (namely, itself) is \overline{L}, and thus (a) holds in this case. In order to verify (b) in this case recall that, by Theorem A1, L^{+} is the “free amalgamated product” in the category of partial groups of the “amalgam” given by the inclusion maps L_{0} → L and L_{0} → L_{0}^{+}. Point (b) will follow if:

(1) There is a unique homomorphism ρ_{0}^{+} : L_{0}^{+} → \overline{L} such that ρ_{0}^{+} agrees with ρ on L_{0}.

(2) ρ_{0}+ induces a surjection D(L_{0}^{+}) → D(\overline{L}_{0}), where \overline{L}_{0} is the set of all \overline{f} ∈ \overline{L} such that \overline{S}\overline{f} contains a \overline{F}-conjugate of \overline{R}.

(3) Ker(ρ^{+}) = N^{+}. Versions of the same three points will need to be verified in the case where T ≤ R, and our approach will be to merely sketch the proofs in each case, leaving some of the entirely mechanical details to the reader. Thus, let Φ be the set of triples (x^{−1}, g, y) defined following 3.4, let ≈ be the relation on Φ defined in 3.10, and let ρ^{∗} : W(L) → W(\overline{L}) be the ρ-induced homomorphism of free monoids. Then ρ^{∗} maps Φ into D(\overline{L}), and it may be routinely verified that the composition

Φ ⟶^{ρ^{∗}} D(\overline{L}) ⟶^{\overline{Π}} \overline{L}

is constant on ≈-classes. Thus, there is a mapping ρ_{0}^{+} : L_{0}^{+} → \overline{L} given by [x^{−1}, g, y] ↦ \overline{Π}(\overline{x}^{−1}, \overline{g}, \overline{y}). The verification that ρ_{0}^{+} is a homomorphism is also routine, and yields (1). Since ρ^{∗} maps D(L) onto D(\overline{L}), (2) is then immediate. As Ker(ρ^{+}) ∩ L = N, (3) is immediate from Theorem A2. Thus, the theorem holds if T ≰ R.

Assume henceforth that T ≤ R. Then N\overline{L}(\overline{R}) = \overline{N_{L}(R))}, since each element of \overline{L} has a ρ-preimage in N_{L}(T) by I.3.11. Then N\overline{L}(\overline{R}) is a subgroup of \overline{L} by I.1.15. Thus Hypothesis 3.3 is satisfied, with \overline{L} and \overline{R} in the roles of L and R. Theorem 3.4 then yields (a). For the same reasons as in the preceding case (and because the verification of (3) did not in fact make use of the hypothesis that T ≰ R) it now suffices to verify:

(4) There is a unique homomorphism ρ_{0}^{+} : L_{0}^{+} → \overline{L}^{+} such that ρ_{0}^{+} agrees with ρ on L_{0}.

(5) ρ_{0}+ induces a surjection D(L_{0}^{+}) → D(\overline{L}_{0}^{+}), where \overline{L}_{0}^{+} is the set of all \overline{f} ∈ \overline{L}^{+} such that \overline{S}\overline{f} contains a \overline{F}-conjugate of \overline{R}.

Let Φ and ρ^{∗} be as above, and define \overline{Φ} to be the set of triples \overline{ϕ} = (\overline{x}^{−1}, \overline{g}, \overline{y}) such that one has

\overline{U} ⟶^{\overline{x}^{−1}} \overline{R} ⟶^{\overline{g}} \overline{R} ⟶^{\overline{y}} \overline{V}

(a sequence of conjugation isomorphisms between subgroups of \overline{S}, labeled by the conjugating elements), with N\overline{S}(\overline{U}) ≤ \overline{S}\overline{x}^{−1}, and with N\overline{S}(\overline{V}) ≤ \overline{S}\overline{y}^{−1}. Then ρ^{∗} maps Φ onto \overline{Φ} by 5.4(a). It need cause no confusion to denote also by ∼ and ≈ the two equivalence relations on \overline{Φ} given by direct analogy with 3.5 and 3.10. Again by means of 5.4(a), one verifies that the restriction of ρ^{∗} to Φ preserves these equivalence relations, and hence induces a surjective mapping ρ_{0}^{+}L_{0}^{+} → \overline{L}_{0}^{+}. Here L_{0} is by definition the set of elements f ∈ L such that S_{f} contains an F-conjugate of R, and any such f is identified with its ≈-class [f], consisting of all those ϕ ∈ Φ ∩ D(L) such that Π(ϕ) = f. The analogous definition of \overline{L}_{0} leads to the conclusion that the restriction of ρ_{0}^{+} to L_{0} is the (surjective) homorphism ρ_{0} : L_{0} → \overline{L}_{0} induced by ρ. The verification that ρ_{0}^{+} is a homomorphism, and hence that (4) holds, is then straightforward.

Set D_{0}^{+} = D(L_{0}^{+}) and \overline{D}_{0}^{+} = D(\overline{L_{0}}^{+}), let \overline{w} ∈ \overline{D}_{0}^{+}, and write \overline{w} = ([\overline{ϕ}_{1}], ⋯, [\overline{ϕ}_{n}]). Here [\overline{ϕ}_{i}] is the ≈-class of an element \overline{ϕ}_{i} = (\overline{x}_{i}^{−1}, \overline{g}_{i}, \overline{y}_{i}) in \overline{Φ}, and the representatives \overline{ϕ}_{i} may be chosen so that the sequence \overline{γ} = (\overline{ϕ}_{1}, ⋯, \overline{ϕ}_{n}) is a \overline{Γ}-form of \overline{w}, as defined in 3.12. That is, the word

\overline{w}\overline{g} = \overline{ϕ}_{1} ∘ ⋯ ∘ \overline{ϕ}_{n}

has the property that \overline{S}_{\overline{w}\overline{γ}} contains a \overline{F}-conjugate of \overline{R}. Let ϕ_{i} be a ρ^{∗}-preimage of \overline{ϕ}_{i} in Φ, set γ = (ϕ_{1}, ⋯, ϕ_{n}), let [ϕ_{i}] be the ≈-class of ϕ_{i} (in L_{0}^{+}), and set w = ([ϕ_{1}], ⋯, [ϕ_{n}]). One verifies that γ is a Γ-form of w, and hence that w ∈ D_{0}^{+}. Since ρ^{∗} maps w_{γ} to \overline{w}\overline{γ}, it follows that ρ_{0}^{+} induces a surjection as required in (5). Thus (5) holds, and the proof is complete. □

The preceding result is essentially a generalization of Theorem A1 to homomorphic images of proper localities. Here is the corresponding version of Theorem A2.

Theorem 5.6. Let the hypotheses and the setup be as in the preceding theorem, let 𝔑 be the set of partial normal subgroups of L containing Ker(ρ), and let 𝔑^{+} be the set of partial normal subgroups of L^{+} containing Ker(ρ^{+}). Also, let \overline{𝔑} be the set of partial normal subgroups of \overline{L}, and let \overline{𝔑}^{+} be the set of partial normal subgroups of \overline{L}^{+}. Then there is a commutative diagram of inclusion-preserving bijections:

𝔑 ⟶^{η} 𝔑^{+}
β↓  ↓^{β^{+}}
\overline{𝔑} ⟶_{\overline{η}} \overline{𝔑}^{+}

where η is the map N ↦ N^{+}, β is the map N ↦ N ρ, and β^{+} is the map N^{+} ↦ N^{+}ρ^{+}. Moreover:

(a) Ker(ρ^{+}) = Ker(ρ)η, and

(b) for each N ∈ 𝔑, \overline{N}^{+} is the unique partial normal subgroup of \overline{L}^{+} whose intersection with \overline{L} is \overline{N}.

Proof. By I.4.7 there are inclusion-preserving bijections

β : 𝔑 → \overline{𝔑} and β^{+} : 𝔑^{+} → \overline{𝔑}^{+},

by which a partial normal subgroup is sent to its image in \overline{L} (or in \overline{L}^{+}) under the canonical projection ρ (or ρ^{+}). Theorem A2 yields the inclusion-preserving bijection η : 𝔑 → 𝔑^{+} which sends N ∈ 𝔑 to the unique N^{+} ⊴ L^{+} such that N^{+} intersects L in N. We obtain the above commutative square by taking \overline{η} to be β^{−1} ∘ η ∘ β^{+}, so it remains to prove (a) and (b),

We have Ker(ρ) = L ∩ Ker(ρ^{+}) since ρ is the restriction of ρ^{+} to L. Thus Ker(ρ^{+}) is a partial normal subgroup of L^{+} which intersects L in Ker(ρ). As this condition characterized Ker(ρ)η, we have (a).

Fix N ∈ 𝔑, set \overline{N} = N ρ, and set \overline{N}^{+} = N^{+}ρ^{+}. Thus \overline{N}^{+} = \overline{N} \overline{η}. Identify \overline{L}^{+} with L^{+}/Ker(ρ^{+}), let g ∈ N^{+}, and let [g] be the maximal coset of Ker(ρ^{+}) in L^{+} containing g. Then [g] = Ker(ρ^{+})h for some h ∈ [g] such that h is ↑-maximalin L^{+} with respect to Ker(ρ^{+}). The Splitting Lemma (I.3.12) yields x ∈ Ker(ρ^{+}) such that (x, h) ∈ D(L^{+}), xh = g, and S_{(x,h)} = S_{g}. Then (x^{−1}, x, h) ∈ D(L^{+}), and D-associativity yields h = x^{−1}g. As Ker(ρ) ≤ N, (a) implies that Ker(ρ^{+}) ≤ N^{+}. Thus h ∈ N^{+}, and then [g] ⊆ N^{+}. Now suppose that also [g] ∈ \overline{L}. As \overline{L} is the image of L under ρ^{+}, we may then assume that g was chosen so that g ∈ L. That is, we may take g ∈ L ∩ N^{+}. Then g ∈ N, and this shows that \overline{N}^{+} intersects \overline{L} in \overline{N}. The uniqueness of \overline{N}^{+} for this property is given by the commutativity of the displayed diagram, and so (b) holds. □

Proposition 5.7. Let (L, Δ, S) be a proper locality on F, let (L^{+}, Δ^{+}, S) be a proper expansion of L on F, and let Λ be a subgroup of Aut(L) such that both Δ and Δ^{+} are Λ-invariant. Then each λ ∈ Λ extends in a unique way to an automorphism of L^{+}.

Proof. It suffices to consider the case where Δ^{+} = Δ ∪ R, where R is a set of subgroups of S which is minimal subject to being both F-invariant and Λ-invariant. Thus R is the disjoint union of sets (R_{i})^{F} (1 ≤ i ≤ m), where {R_{1}, ⋯, R_{m}} is a Λ-orbit of subgroups of S. By 3.20(c) L^{+} is a colimit in the category of partial groups (cf. I.1.17):

L^{+} = L_{1}^{+}∐_{L} ⋯ ∐_{L}L_{m}^{+},

where each L_{i}^{+} is an elementary expansion of L whose set Δ_{i}^{+} of objects is Δ ∪ (R_{i})^{F}.

We may assume that R_{1} has been chosen so that both R_{1} and O_{p}(N_{F}(R_{1})) are fully normalized in F. Plainly Λ induces a group of automorphisms of F, so the preceding condition applies also to each R_{i}. We may then form the set Φ_{i} if all triples (x^{−1}, g, y) as in 3.9, with respect to R_{i} in the role of R. Then L_{i}^{+} is itself a colimit of the form

L_{i}^{+} = L∐_{M}M^{+},

where M is the partial subgroup of L consisting of those elements h ∈ L that can be written as a product Π(x^{−1}, g, y) for some (x^{−1}, g, y) ∈ Φ_{i}, and where M^{+} is the partial group consisting of all of the equivalence classes [x^{−1}, g, y] obtained from Φ_{i}.

Let λ ∈ Λ and let an index i be given. Then R_{i}λ = R_{j} for some j. In order to extend λ to a mapping L^{+} → L^{+} it now suffices to show that λ induces a mapping M_{i}^{+} → M_{j}^{+} via

(*)  [x^{−1}, g, y] ↦ [(x^{−1})λ, gλ, yλ].

For this it suffices to verify that λ is equivariant with respect to the equivalence relations ≈_{i} on Φ_{i} and ≈_{j} on Φ_{j} given by 3.10 (with the obvious adjustment needed to accomodate the indices i and j). This verification - that (*) is well defined - is entirely straightforward, and is left to the reader. That the mapping (*) is a homomorphism M_{i}^{+} → M_{j}^{+} is then immediate from the definition of the product in an expansion (see 3.13 and 3.14). One then observes that λ^{−1} induces a homomorphism M_{j}^{+} → M_{i}^{+} which is the inverse of the homomorphism induced by λ. The functoriality of the colimit of partial groups then yields the desired extension of λ to an automorphism λ^{+} of L^{+}. The uniqueness of λ^{+} follows from the observation that [x^{−1}, g, y] can be written as the product x^{−1}gy in M_{i}^{+}. Thus any extension of λ to an endomorphism of L^{+} must necessarily satisfy (*) and will thereby be equal to λ^{+}. □

Section 6: Saturated fusion systems

Throughout this section F is a fusion system on a finite p-group S.

We have been able to proceed, so far, with only very naive notions pertaining to fusion systems in general. The most useful of these has been the notion of inductivity (and Γinductivity) introduced in 1.11. This notion cannot be sufficient for our purposes, for two reasons. First, we have no criteria for deciding if inductivity follows from Γ-inductivity for suitable F-closed sets Γ; and second, inductivity of F is not necessarily inherited by subsystems N_{F}(X), for X fully normalized in F. What is needed is a condition on F which implies inductivity and which resolves these two issues. This is provided by saturation, whose definition we take from [AKO], and which we now review.

First: a subgroup Q of S is defined to be fully automized in F if Aut_{S}(Q) is a Sylow p-subgroup of Aut_{F}(Q). Second: a subgroup Q of S is defined to be receptive in F if every F-isomorphism ϕ : P → Q extends to an F-homomorphism \overline{ϕ} : N_{ϕ} → S, where

N_{ϕ} = {x ∈ N_{S}(P) | ϕ^{−1} ∘ c_{x} ∘ ϕ ∈ Aut_{S}(Q)}.

Definition 6.1. Let Γ be an F-closed set of subgroups of S. Then F is Γ-saturated if for every P ∈ Γ there exists an F-conjugate Q of P such that Q is fully automized and receptive in F. We say F is saturated if F is Γ-saturated for the set Γ of all subgroups of S.

Lemma 6.2. Let F be a fusion system on the finite p-group S, let Γ be an F-closed set of subgroups of S, and assume that F is Γ-saturated.

(a) A subgroup Q ∈ Γ is fully normalized in F if and only if Q is fully automized and receptive in F.

(b) F is Γ-inductive.

Proof. Let P ∈ Γ, let Q be an F-conjugate of P such that Q is fully automized and receptive in F, and let ϕ : P → Q be an F-isomorphism. Then conjugation by ϕ is an isomorphism

c_{ϕ} : Aut_{F}(P) → Aut_{F}(Q).

Set X = Aut_{S}(P) and Y = Aut_{S}(Q). Then Y ∈ Syl_{p}(Aut_{F}(Q)), so there exists α ∈ Aut_{F}(Q) such that the inner automorphism c_{β} of Aut_{F}(Q) carries (X)c_{ϕ} into Y. Set ψ = ϕ ∘ β. Then c_{ψ} carries X into Y, and thus N_{S}(P) ≤ N_{ψ}. There is then an extension of ψ to an F-homomorphism \overline{ψ} : N_{S}(P) → N_{S}(Q). In particular, this shows that (a) implies (b), and so we need only prove (a).

Suppose now that, in the preceding argument, P is fully normalized in F. Then \overline{ψ} is an isomorphism, so \overline{ψ} restricts to an isomorphism C_{S}(P) → C_{S}(Q), and thus c\overline{ψ} is an isomorphism Aut_{S}(P) → Aut_{S}(Q). Thus P is fully automized (and Q is fully normalized) in F.

Now let R ∈ P^{F}, let γ : R → P be an F-isomorphism, and set η = γ ∘ ψ. Then N_{γ} ≤ N_{η}, and η extends to an F-homomorphism η̃: N_{γ} → N_{S}(Q). Set D = (N_{γ})η̃, and let β be the restriction of \overline{ψ}^{−1} to D. Then η̃ ∘ β is an extension of γ to N_{γ}, and thus P is receptive. Thus (a) holds. □

Recall from the discussion preceding 1.11 that if Γ is a non-empty set of subgroups of S then F is defined to be Γ-generated if every F-isomorphism is a composite of restrictions of F-isomorphisms ϕ : X → Y with X, Y ∈ Γ.

Lemma 6.3. Let (L, Δ, S) be a locality on F. Then F is Δ-saturated and Δ-generated.

Proof. By definition, F is generated by the set of conjugation maps c_{g} : S_{g} → S_{g^{−1}}. Since S_{g} ∈ Δ for all g ∈ L, F is thus Δ-generated.

Let Q ∈ Δ with Q fully normalized in F. Then N_{S}(Q) ∈ Syl_{p}(N_{L}(Q)) by I.2.10. Since Aut_{F}(Q) is a homorphic image of N_{L}(Q), Q is then fully automized in F. Now let P ∈ Q^{F} and let ϕ : P → Q be an F-isomorphism. Then ϕ = c_{g} for some g ∈ L, and c_{g} is an isomorphism N_{L}(P) → N_{L}(Q) by I.2.3(b). Here N_{ϕ} is simply N_{S_{g}}(P), so Q is receptive, and thus F is Δ-saturated. □

The following result, due to Broto, Castellana, Grodal, Levi, and Oliver [Theorem 2.2 in 5A1], is an essential tool for determining whether a fusion system is saturated. Since our definition of “centric radical” is not the standard one (cf. 1.8) we provide a few lines of proof to address that point.

Theorem 6.4. Let F be a fusion system on S, and define P(F) to be the set of all subgroups P ≤ S such that P is F-centric and such that for each Q ∈ P^{F} with Q fully normalized in F we have:

(*)  O_{p}(Aut_{F}(Q)) ∩ Aut_{S}(Q) = Inn(Q).

Then P(F) ⊆ F^{cr}. Further, let Γ be an F-closed set of subgroups of S, and assume: (1) Γ is Γ-saturated and Γ-generated.

(2) P(F) ⊆ Γ.

Then F is saturated, and P(F) = F^{cr}.

Proof. As mentioned, the only issue here is that of showing P(F) = F^{cr}. Let P ∈ P(F) be fully normalized in F, and set P̃ = O_{p}(N_{F}(P)). By definition, every N_{F}(P)- isomorphism extends to an N_{F}(P)-isomorphism whose domain contains P̃, so AutP̃(P) ≤ O_{p}(Aut_{F}(P)) ∩ Aut_{S}(P). The condition (*) then yields AutP̃(P) ≤ Inn(P). As P is centric in F we then have P̃ = P, and so P ∈ F^{cr} by definition 1.8. Thus P(F) ⊆ F^{cr}, and it remains to show the reverse inclusion under the assumption (which we now make) that F is saturated.

Let Q ∈ F^{c}, with Q fully normalized in F. Then definition 6.1 implies that every F-automorphism of Q extends to an F-automorphism of the preimage Q̃ in N_{S}(Q) of O_{p}(Aut_{F}(Q)). Thus Q̃ ≤ O_{p}(N_{F}(Q)), and so Q = Q̃ if Q ∈ F^{cr}. Thus P(F) = F^{cr} if F is saturated. □

Corollary 6.5. Let (L, Δ, S) be a locality on F, and suppose that P(F) ⊆ Δ or that F^{cr} ⊆ Δ. Then F is saturated, and P(F) = F^{cr}. In particular, the fusion system of a proper locality is saturated.

Proof. As P(F) ⊆ F^{cr} by 6.3, we have P(F) ⊆ Δ. Let Γ be the overgroup-closure of P(F) in S. Then Γ is F-closed, since P(F) is F-invariant. By 6.2, F is Γ-saturated and Γ-generated. Thus the hypotheses (1) and (2) of 6.4 are fulfilled, hence F is saturated and P(F) = F^{cr}. □

The following result is an immediate consequence of the Alperin-Goldschmidt fusion theorem for fusion systems (cf. I.1.14 in [AKO]). Recall from the discussion preceding 1.11 that F is defined to be (cr)-generated if every F-isomorphism is a composite of restrictions of F-automorphisms of members of F^{cr} which are fully normalized in F.

Lemma 6.6. Let F be a saturated fusion system on S. Then F is (cr)-generated. Thus, if F_{0} is a subsystem of F and Aut_{F}(Q) = Aut_{F_{0}}(Q) for each Q ∈ F^{cr} such that Q is fully normalized in F, then F_{0} = F. □

Recall (from 3.1) the notion of a localizable pair.

Lemma 6.7. Let (L, Δ, S) be a locality and let (H, Γ) be a localizable pair in L. Set R = S ∩ H, set E = F_{R}(H), and assume that E^{cr} ⊆ Γ. Then E is (cr)-generated and E = F_{R}(H_{Γ}). Moreover, if L is proper then (H, Γ, R) is proper if either:

(1) H ⊴ L, or

(2) H = N_{L}(X) for some subgroup X ≤ S.

Proof. Set E_{0} = F_{R}(H_{Γ}). Then E_{0} is a fusion subsystem of E. But also E = ⟨Aut_{E}(P) | P ∈ Γ⟩ is a subsystem of E_{0}since E is (cr)-generated and since E^{cr} ⊆ Γ. Thus E = E_{0}, and H_{Γ} is a locality on E. Now assume that L is proper and let P ∈ Γ. Then N_{H}(P) is a subgroup of L, so there exists Q ∈ Δ with N_{H}(P) ≤ N_{L}(Q) by I.2.11(a). Set G = N_{L}(Q). We may assume that P ≤ Q, and then N_{H}(P) = N_{G∩H}(P). If H ⊴ L then G ∩ H ⊴ G and then N_{H}(P) is of characteristic p by 2.7(a). On the other hand, if H = N_{L}(X) with X ≤ S we may also take X ≤ Q, and then N_{H}(P) = N_{N_{G∩H}(X)}(P)). In this case N_{H}(P) is of characteristic p by 2.7(b). Thus H_{Γ} is a proper locality if L is proper and if (1) or (2) obtains. □

[—- now find all the places where this can be used.]

The following result (and references to several proofs) may be found in [AKO, Theorem I.11].

Proposition 6.8. Let F be a saturated fusion system on the finite p-group S, let X be fully normalized in F, and let Y be fully centralized in F. Then N_{F}(X) and C_{F}(Y) are saturated. □

The next few results concern the set F^{s} of F-subcentric subgroups of the fusion system F of a proper locality, and they are due to Ellen Henke [He2]. We provide proofs for the convenience of the reader. The key insight into F^{s} is given by [Lemma 3.1 in He2], which is as follows.

Lemma 6.9. Let F be the fusion system on S of a proper locality, and let V ≤ S be fully centralized in F. Then V ∈ F^{s} if and only if O_{p}(C_{F}(V)) is centric in C_{F}(V).

Proof. If U ∈ V^{F} is fully normalized in F then 1.14 yields an isomorphism C_{F}(U) ≅ C_{F}(V) of fusion systems. We may therefore assume to begin with that V is fully normalized in F. Set F_{V} = N_{F}(V) and R = O_{p}(F_{V}). Also, set C_{V} = C_{F}(V) and Q = O_{p}(C_{V}).

Suppose first that the lemma holds with F_{V} in place of F. That is, suppose that R is centric in F_{V} if and only if Q is centric in C_{V}. Since R ∈ (F_{V})^{c} if and only if R ∈ F^{c} by 1.19, we then have the lemma in general. Thus, we are reduced to the case where V ⊴ F, and R = O_{p}(F).

Fix a proper locality (L, Δ, S) on F. We may assume that F^{c} ⊆ Δ by Theorem A1. Suppose that V ∈ F^{s}. Then R ∈ F^{c}, so R ∈ Δ, and L is the group N_{L}(R). Notice that [R, C_{S}(V Q)] ≤ C_{R}(V Q) ≤ Q, and that both R and Q are normal subgroups of L. Then C_{S}(V Q) ≤ R by 2.7(c). Then also

C_{C_{S}(V)}(Q) = C_{S}(V Q) ≤ C_{R}(V) ≤ Q,

and thus Q is centric in C_{F}(V). On the other hand, assuming that Q is centric in C_{F}(V), we obtain

C_{S}(R) ≤ C_{S}(V Q) ≤ Q ≤ R,

and so R ∈ F^{c}, as required. □

Corollary 6.10. Let F be a fusion system of a proper locality. Then

F^{cr} ⊆ F^{c} ⊆ F^{q} ⊆ F^{s}.

Proof. We have F^{cr} ⊆ F^{c} by definition. Let (L, Δ, S) be a proper locality on F, and P ≤ S be fully normalized in F. If P ∈ F^{c} then C_{F}(P) is the trivial fusion system on Z(P), and so P ∈ F^{q}. Now suppose instead that we are given P ∈ F^{q}. Then O_{p}(C_{F}(Q)) = C_{S}(Q), and then P ∈ F^{s} by 6.9. □

Corollary 6.11. Let F be the fusion system on S of a proper locality, and let V ≤ S with V fully normalized in F. Then

{Q ∈ N_{F}(V)^{s} | V ≤ Q} ⊆ F^{s}.

Proof. One need only observe that C_{F}(V) = C_{C_{F} (V)}(V), in order to obtain the desired result from 6.9. □

The next result is [lemma 3.2 in He].

Theorem 6.12. Let F be the fusion system of a proper locality. Then F^{s} is F-closed.

Proof. Fix a proper locality (L, Δ, S) on F, with Δ = F^{c}. Let U, V be subgroups of S. By 1.14, N_{F}(U) ≅ N_{F}(V) if U and V are F-conjugate subgroups of S, so F^{s} is F-invariant. Clearly S ∈ F^{s}. Thus, we are reduced to showing that F^{s} is closed with respect to overgroups in S.

Among all V ∈ F^{s} such that some overgroup of V in S is not subcentric in F, choose V so that |V | is as large as possible. Then there exists an overgroup P of V in S such that P ∉ F^{s} and such that V has index p in P. Then P ≤ N_{S}(V). Let U ∈ V^{F} such that U is fully normalized in F. As F is inductive there is an F-homomorphism ϕ : N_{S}(V) → N_{S}(U) such that V ϕ = U. Then P ϕ ∉ F^{s} as F^{s} is F-invariant, and so we may assume to begin with that V is fully normalized in F. Set F_{V} = N_{F}(V). Replacing P with a suitable F_{V} -conjugate of P, we may assume that P is fully normalized in F_{V}.

Suppose that P ∈ (F_{V})^{s}. As C_{F}(P) = C_{F_{V}} (P) we then obtain P ∈ F^{s} from 6.9. Thus P ∉ (F_{V})^{s}, so F_{V} is a counterexample, and we may therefore assume that F_{V} = F. Then O_{p}(F) ∈ F^{c}, so P ∈ Δ, and L is a group of characteristic p. Then N_{L}(P) is of characteristic p by 2.7(b), and since P is fully normalized in F we have also N_{F}(P) = F_{N_{S}(P)}(N_{L}(P)). Then O_{p}(N_{F}(P)) ∈ N_{F}(P)^{c}, and hence O_{p}(N_{F}(P)) ∈ F^{c} by 1.19. Thus P ∈ F^{s}, as required. □

The following result is immediate from 6.10 and Theorem A1.

Corollary 6.13. Let (L, Δ, S) be a proper locality on F. Then there exists an expansion of L to a proper locality (L^{s}, F^{s}, S) on F . □

The following result is [lemma 3.3 in He].

Lemma 6.14. Let F be the fusion system on S of a proper locality, and let P ≤ S with O_{p}(F)P ∈ F^{s}. Then P ∈ F^{s}.

Proof. Among all counter-examples, choose P so that |P | is as large as possible. Set Q = O_{p}(F), and let ϕ : P → P′ be an F-isomorphism such that P′ is fully normalized in F. Then ϕ extends to an F-isomorphism P Q → P′Q, so P′Q ∈ F^{s}. Thus we may assume that P is fully normalized in F.

Set E = N_{F}(P) and set D = N_{Q}(P). Then D ≤ O_{p}(E). If D ≤ P then P = P Q, and P is not a counter-example. Thus D ≰ P, and DP ∈ F^{s} by the maximality of |P |. Since N_{S}(P) ≤ N_{S}(P D) we may assume that P was chosen so that also P D is fully normalized in F. Set R = N_{S}(P D) and set G = N_{L}(DP). We may take Δ to be F^{s} by 6.13, so G is a subgroup of L, and is of characteristic p. Here R ∈ Syl_{p}(G), and F_{R}(G) = N_{F}(DP). Since each E-homomorphism extends to an N_{F}(DP)-homomorphism, E is then the fusion system of N_{G}(P) at N_{S}(P). As G is of characteristic p, so is N_{G}(P) by 2.7(b). Thus O_{p}(E) is centric in E, and hence centric in F by 1.19. Thus P ∈ F^{s}. □

This completes our treatment of F^{s}. We end the section with three further results, which will be needed in Part III.

Lemma 6.15. Let L be a proper locality on F, let P ∈ F^{cr}, let T ≤ S be strongly closed in F, and let E be an inductive fusion system on T such that E is a subsystem of F. Then there exists Q ∈ P^{F} with Q ∩ T ∈ E^{c}.

Proof. Set U = P ∩ T and set A = N_{C_{T} (U)}(P). Then [P, A] ≤ U, and thus A centralizes the chain (P ≥ U ≥ 1) of normal subgroups of the group N_{L}(P). Then A ≤ O_{p}(N_{L}(P)) by 2.7(c), and then A ≤ P by an application of 2.3 to the fusion systen N_{F}(P). Thus C_{T} (U) ≤ P, and so C_{T} (U) ≤ U.

Let V ∈ U^{F} be fully normalized in F. As F is inductive there exists an F-homomorphism ϕ : N_{S}(U) → N_{S}(V) with Uϕ = V. Set Q = P ϕ. Then Q ∈ F^{cr}, and so C_{T} (V) ≤ V by the result of the preceding paragraph. Let V′ ∈ V^{E}. Then V′ ∈ V^{F}, and so there exists an F-homomorphism ψ : N_{S}(V′) → N_{S}(V) with V′ = V ψ. Then N_{T} (V′)ψ ≤ N_{T} (V) as T strongly closed in F, and thus |N_{T} (V′)| ≤ |N_{T} (V)|. This shows that V is fully normalized in E. As E is inductive, V is fully centralized in E by 1.13. As C_{T} (V) ≤ V, V is then centric in E by 1.10. □

Lemma 6.16. Let (L, Δ, S) be a proper locality on F, let X be fully normalized in F, and let X ≤ P ≤ N_{S}(X) with P fully normalized in N_{F}(X). Then there exists an Fhomomorphism ϕ : N_{S}(X) ∩ N_{S}(P) → S such that Q := P ϕ is fully normalized in F and Y := Xϕ is fully normalized in N_{F}(Q). Moreover, ϕ then induces an isomorphism N_{N_{F} (X)}(P) → N_{N_{F} (Q)}(Y) of fusion systems.

Proof. First, choose an F-homomorphism ψ : N_{S}(X) → S so that Q := P ψ is fully normalized in F, and set Y′ = Xψ. By 6.8 N_{F}(Q) is saturated, so N_{F}(Q) is inductive, and so there exists an N_{F}(Q)-homomorphism ξ : N_{S}(Y′) ∩ N_{S}(Q) → N_{S}(Q) such that Y := Y′ξ is fully normalized in N_{F}(Q). Let ψ_{0} be the restriction of ψ to N_{S}(X)∩N_{S}(P). Then ψ_{0} ∘ ξ sends N_{S}(X) ∩ N_{S}(P) into N_{S}(Y) ∩ N_{S}(Q), and maps X to Y and P to Q.

Now let ϕ be any F-homomorphism N_{S}(X) ∩ N_{S}(P) → S such that P ϕ is fully normalized in F and such that Xϕ is fully normalized in N_{F}(P ϕ), and write Q = P ϕ and Y = Xϕ. Set U = N_{S}(X) ∩ N_{S}(P) and V = N_{S}(Y) ∩ N_{S}(Q). Then Uϕ ≤ V. The argument of the preceding paragraph shows that there exists an F-homomorphism V → U, so UϕV, and thus ϕ induces an F-isomorphism α : U → V. It follows from two applications of 1.5 that α is fusion-preserving. □

Lemma 6.17. Let F be a saturated fusion system on S, let X be fully normalized in F, and let P be a subgroup of N_{S}(X) containing X. Set F_{X} = N_{F}(X). Then:

(a) P ∈ (F_{X})^{c} if and only if P ∈ F^{c}.

(b) P ∈ (F_{X})^{s} if and only if P ∈ F^{s}.

Proof. As F_{X} is saturated by 6.8, we may appeal to 1.18 to obtain the existence of an F_{X}-conjugate Q of P such that both Q and R := O_{p}(N_{F_{X}} (Q)) are fully normalized in F_{X}. Then Q and R are fully centralized in F, by 1.13. If P ∈ (F_{X})^{c} then C_{S}(Q) = C_{C_{S}(X)}(Q) ≤ Q, so Q ∈ F^{c}, and then P ∈ F^{c}. Similarly, if R ∈ (F_{X})^{c} then R ∈ F^{c}, whence Q ∈ F^{s} and P ∈ F^{s}. This establishes the forward direction of the “if and only if” in both (a) and (b). As F_{X} is a subsystem of F the reverse direction in each of the two cases is obvious. □

Section 7: O_{L}^{p}(N) and O_{L}^{p′}(N)

Definition 7.1. Let (L, Δ, S) be a locality, let N ⊴ L be a partial normal subgroup of L, and set T = S ∩ N. Set

𝕂 = {K ⊴ L | KT = N } and 𝕂′ = {K′ ⊴ L | T ≤ K},

and set

O_{L}^{p}(N) = ⋂ 𝕂 and O_{L}^{p′}(N) = ⋂ 𝕂′.

Write O^{p}(L) for O_{L}^{p}(L), and O^{p′}(L) for O_{L}^{p′}(L).

Proposition 7.2. O_{L}^{p}(N) ∈ 𝕂 and O_{L}^{p′}(N) ∈ 𝕂′

Proof. Let K_{1}, K_{2} ∈ 𝕂, set K = K_{1} ∩ K_{2}, and set T_{i} = S ∩ K_{i} (i = 1, 2). Let x ∈ K_{1}. Then x ∈ N = K_{2}T, so we may write x = yt with y ∈ K_{2} and t ∈ T. Then S_{x} = S_{y}, so (y^{−1}, x) ∈ D and y^{−1}x = t. Thus t ∈ S ∩ K_{1}K_{2}, and so t ∈ T_{2}T_{1} by [He1, Theorem A]. This shows that K_{1} ≤ K_{2}(T_{2}T_{1}). As

K_{2}(T_{2}T_{1}) = (K_{2}T_{2})T_{1} = K_{2}T_{1}

by I.2.9, we obtain K_{1} ≤ K_{2}T_{1}. The Dedekind lemma (I.1.10) then yields

K_{1} = K_{1} ∩ K_{2}T_{1} = (K_{1} ∩ K_{2})T_{1} = KT_{1} ≤ KT,

and so K_{1}T ≤ KT. As K_{1}T = N we conclude that KT = N, and thus K ∈ 𝕂. As 𝕂 is finite, iteration of this procedure yields O_{L}^{p}(N) ∈ 𝕂. The proof that O_{L}^{p′}(N) ∈ 𝕂′ is simply the observation that T ≤ ⋂ 𝕂′. □

Lemma 7.3. Let “∗” be either of the symbols “p” or “p′, let (L, Δ, S) be a proper locality, and let (L^{+}, Δ^{+}, S) be an expansion of L. Let N ⊴ L, and for any partial normal subgroup K ⊴ L let K^{+} be the corresponding partial normal subgroup of L^{+} given by Theorem A2. Then O_{L}^{∗}(N)^{+} = O_{L^{+}}^{∗}(N^{+}).

Proof. Write 𝕂^{+} for the set of all K^{+} with K ∈ 𝕂. Then

(⋂ 𝕂)^{+} ≤ ⋂(𝕂^{+})

as ⋂(𝕂^{+}) is a partial normal subgroup of L^{+} containing ⋂ 𝕂. The reverse inclusion is given by Theorem A2, along with the observation that

L ∩ (⋂(𝕂^{+})) = ⋂{L ∩ K^{+}}_{K∈𝕂} = ⋂ 𝕂.

Thus the lemma holds for “p”, and the same argument applies to “p′”. □

Lemma 7.4. Let N and M be partial normal subgroups of L, with N ≤ M. Let “∗” be either of the symbols “p” or “p′. Then O_{L}^{∗}(N) ≤ O_{L}^{∗}(M).

Proof. Since O_{L}^{p′}(M) is a partial normal subgroup of L containg S∩N, we have O_{L}^{p′}(N) ≤ O_{L}^{p′}(M) by definition. Set K = O_{L}^{p}(M), set \overline{L} = L/K, and let ρ : L → \overline{L} be the canonical projection. Then (S ∩M)ρ = Mρ ≥ N ρ, and hence N ρ = (S ∩N)ρ. Subgroup correspondence (I.4.7) then yields N ≤ K(S∩N), and then O_{L}^{p}(N) ≤ K by definition. □

References

[5a] C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, Subgroup families controlling p-local finite groups, Proc. London Math. Soc. 91 (2005), 325-354.

[AKO] Michael Aschbacher, Radha Kessar, and Bob Oliver, Fusion systems in algebra and topology, London Mathematical Society Lecture Note Series, vol. 391, Cambridge University Press, Cambridge, 2011.

[Ch] Andrew Chermak, Finite localities I, arXiv (2016).

[Cr] David Craven, The theory of fusion systems: An algebraic appoach, Cambridge Studies in Advanced Mathematics, vol. . 131, Cambridge University Press, Cambridge, 2011.

[Gold] David Goldschnidt, A conjugation family for finite groups, J.Algebra 16 (1970), 138-142.

[Gor] Daniel Gorenstein, Finite groups, Second Edition, Chelsea, New York, 1980.

[He1] Ellen Henke, Products of partial normal subgroups, (arXiv:1506.01459) (2015).

[He2] , Subcentric linking systems, (arXiv:1506.01458) (2015).

Manhattan Kansas

   E-mail address: chermak@math.ksu.edu