NORMAL SUBSYSTEMS OF FUSION SYSTEMS

Michael Aschbacher

California Institute of Technology

The notion of fusion in groups first arose in the context of finite group theory. The fusion theorems of Alperin [Al] and Goldschmidt [Go], and Glauberman’s Z^{∗}-Theorem [Gl] are highlights of the early work on fusion in finite groups.

Puig defined (in a different language) the category of saturated fusion systems on finite p-groups S, for use in the modular representation theory of finite groups. (While available in preprint for some time, this work has appeared only recently in [P].) Puig’s fusion systems are not in general systems F_{S}(G) determined by a finite group G with Sylow p-subgroup S.

More recently in [O1] and [O2], Oliver proved that for each prime p, and each pair G_{i} of finite groups with Sylow p-groups S_{i}, i = 1, 2, the p-completed classifying spaces of G_{1} and G_{2} are homotopy equivalent if and only if the fusion systems F_{S_{1}}(G_{1}) and F_{S_{2}}(G_{2}) are isomorphic. As part of the proof, Oliver and his collaborators Broto and Levi, introduced in [BLO] the category of p-local finite groups, whose objects consist of a saturated fusion system (possibly not associated to a finite group) together with a so-called centric linking system. Each p-local finite group also has a classifying space and a p-completed classifying space, which can be of great interest.

We seek to investigate saturated fusion systems and p-local finite groups from the algebraic perspective, often using techniques borrowed from the local theory of finite groups. In particular we are interested in examples of “simple” p-local finite groups or fusion systems, which are not associated to finite groups, like the wonderful 2-local groups constructed by Levi and Oliver in [LO], building on earlier work of Solomon and Benson.

We have chosen to concentrate on saturated fusion systems, rather than p-local finite groups. There are two reasons for this choice. First, it it not yet known if each saturated fusion system can be extended to a p-local finite group, so it remains possible that the class of saturated fusion systems is larger than the class of p-local finite groups. But more important, to date there is no accepted notion of a morphism of p-local groups, and hence no category of p-local finite groups. The local theory of finite groups is inextricably tied to the notion of group homomorphism and factor group, so to extend the local theory to a different category, we must at the least be dealing with an actual category.

This work was partially supported by NSF-0203417

As a prerequisite for developing a local theory of fusion systems, and in order to even define the notion of a simple fusion system, one needs a notion of a normal subsystem of a saturated fusion system. Then one must prove that normal subsystems satisfy various properties analogous to those satisfied by normal subgroups of groups.

In this paper, we define the notions of a normal subsystem and factor system of a saturated fusion system F on a finite p-group S, and put into place machinery to prove the necessary theorems about normality. The machinery will be applied in later papers to show that normal subsystems and factor systems behave much like normal subgroups and factor groups of groups. In particular one can define the notion of components of F, and the generalized Fitting subsystem F^{∗}(F) = O_{p}(F)E(F) of F, and establish the balance property of Gorenstein and Walter for fusion systems: For X ≤ S fully normalized in F, E(N_{F}(X)) ≤ E(F). One can also define the notions of composition series and composition factors of F, and prove a Jordon-Holder Theorem for saturated fusion systems.

For basic definitions, notation, and results involving fusion systems, the reader is directed to section 1 and appendix A of [BLO]. Slightly less basic material on fusion systems appears in section 1 of this paper. See [FGT] for background on finite groups. Our convention will be to apply many functions (particularly functions which may be composed, like group homomorphisms) on the right. In the remainder of the introduction, we assume the reader is familiar with basic notation and terminology concerning fusion systems, and we give a brief overview of what goes on in this paper.

Let F be a fusion system on a finite p-group S. If α : F → F̃ is a morphism of fusion systems, then (cf. section 8) the kernel of the associated group homomorphism α : S → S̃ is strongly closed in S with respect to F, but the most natural choice for the members of hom_{F}(P, Q), for P, Q ≤ S, in the kernel leads to nothing interesting. Thus we gain little information about what a “normal subsystem” should look like by inspecting kernels of morphisms.

Assume now that F is saturated, and E is a subsystem of F on T ≤ S. In section 3 we define E to be F-invariant if T is strongly closed in S with respect to F, and one of several equivalent technical conditions on the embedding of E-maps in F is satisfied. The property of F-invariance is natural and has nice inheritance properties, (cf. 3.6) but examples show that F-invariant subsystems need not be saturated. Therefore in section 6, we define E to be normal in F (and write E ⊴ F) if E is F-invariant, E is saturated, and the embedding of E in F satisfies the weak technical condition (N1) of section 6. Again examples show that some such condition is necessary if the normality relation on fusion systems is to behave like the normality relation on groups.

Puig in [P] and Linkelman in [L] also define a notion of “normal subsystem”. In each case these subsystems are the same as our F-invariant subsystems, so their notions are weaker than our notion of normality. It should be noted however that to Linkelman, the term “subsystem” means saturated subsystem, so his notion is closer to ours than that of Puig, differing only in the condition (N1).

Suppose G is a finite group with Sylow p-subgroup S, and F = F_{S}(G). Then by 6.3, for each H ⊴ G, F_{S∩H}(H) ⊴ F. However the converse is false in general: That is there are groups G and normal subsystems of F_{S}(G) which are not of the form F_{S∩H}(H) for any normal subgroup H of G. Still the converse does hold in an important special case, which we now describe.

Let G(F) be the collection of finite groups G such that F = F_{S}(G), S ∈ Syl_{p}(G), and F^{∗}(G) = O_{p}(G). Call G(F) the set of models of F. By the fundamental result Proposition C in [BCGLO], if F is constrained then G(F) ≠ ∅, and indeed all models of F are isomorphic in a strong sense (cf. 2.5). Our first major result says:

Theorem 1. Let F be a saturated constrained fusion system on a finite p-group S, G a model of F, and E ⊴ F. Then there is a unique normal subgroup of G which is a model of E.

Theorem 1 is proved in section 6.

While our definition of normality is natural and easy to state, in practice we want to be able to verify that E ⊴ F from local information: That is by inspecting subsystems of the form N_{F}(P) and N_{E}(P) for suitable fully normalized centric subgroups P of S. In particular write F_{T}^{f} for the set of subgroups U of T which are fully normalized in F, and for U ≤ T set T_{U} = UC_{T} (U) and V_{U} = UC_{S}(U). Then for U ∈ F_{T}^{f}, D(U) = N_{N_{F} (V_{T_{U}})}(U) is a constrained saturated fusion system, E(U) = N_{N_{E}(T_{U})}(U) ≤ D(U), and if E is saturated then E(U) is saturated and constrained. From this discussion, if U ∈ F_{T}^{f} then there is G(U) ∈ G(D(U)). Our second major result is proved in section 7:

Theorem 2. Let F be a saturated fusion system on a finite p-group S, and E a subsystem of F on a subgroup T of S. Then the following are equivalent:

(1) E ⊴ F.

(2) E is F-invariant, E = ⟨Aut_{E}(P) : P ≤ T ⟩, and for each U ∈ F_{T}^{f}, E(U) ⊴ D(U).

(3) E is F-invariant, E = ⟨Aut_{E}(P) : P ≤ T ⟩, and for each U ∈ F_{T}^{f}, E(U) = F_{N_{T} (U)}(H(U)) for some normal subgroup H(U) of G(U) with N_{T} (U) ∈ Syl_{p}(H(U)).

Our third major result is also proved in section 7, and is even more useful:

Theorem 3. Let F be a saturated fusion system on a finite p-group S, and E a subsystem of F on a subgroup T strongly closed in S with respect to F. Then the following are equivalent:

(1) E ⊴ F.

(2) The constriction Aut_{E,c} of Aut_{E} is a normal map on T, and E = 𝔈(Aut_{E,c}).

(3) There exists a normal map A on T such that E = 𝔈(A).

(4) For each U ∈ F_{T}^{fc}, there exists a normal subgroup H(U) of G(U) such that N_{T} (U) ∈ Syl_{p}(H(U)), and

(i) for each P ∈ D(U)^{f} with U ≤ P ≤ T, and for each α ∈ hom_{F}(N_{S}(P), S) with P α ∈ F_{T}^{fc}, Aut_{H(P α)}(Uα) = Aut_{N_{H(U)}(P)}(U)α^{∗}, and

(ii) E = ⟨Aut_{H(U)}(U)φ^{∗} : U ∈ F_{T}^{fc}, φ ∈ Aut_{F}(T)⟩.

The notion of a normal map A on T is defined in 7.6 and the subsystem 𝔈(A) is defined in Definition 5.1. In this formalization, the compatibility conditions in (4i) of Theorem 3 are more attractive, but working with the groups H(U) seems to be more effective.

Theorems 2 and 3 provide a variety of local conditions which can be used to verify normality. The conditions are strong enough to establish the theorems about normal subsystems of saturated fusion systems, necessary to translate results in the local theory of finite groups into the setting of saturated fusion systems.

Finally in section 8, we define the notion of a factor system F/T of F over a subgroup T strongly closed in S with respect to F. We prove a few basic results about factor systems and surjective morphisms of fusion systems, but postpone most of the discussion of these topics to a later paper. See however Remarks 8.7 and 8.11 for an indication of what is true and can be proved.

In Proposition 6.3 of [P], Puig defines the same factor system F/T, and proves there exists a surjective morphism of F onto F/T.

The author would like to thank the referee for numerous suggestions leading to significant improvements in this paper.

Section 1. Background on fusion systems

In this section p is a prime and F is a fusion system on a finite p-group S. We assume the reader is familiar with the most basic notions, definitions, and terminology involving fusion systems, such as can be found in section 1 and appendix A of [BLO].

Given a category C and an isomorphism α : A → B in C, write α^{∗} : Mor(A, A) → Mor(B, B) for the map α^{∗} : β ↦ α^{−1}βα. Similarly if P, Q ≤ S and α ∈ hom_{F}(⟨P, Q⟩, S) then α^{∗} : hom_{F}(P, Q) → hom_{F}(P α, Qα) via α^{∗} : β ↦ α^{−1}βα.

Recall that for P ≤ S and ϕ ∈ hom_{F}(P, S),

N_{ϕ} = {g ∈ N_{S}(P) : c_{g}ϕ^{∗} ∈ Aut_{S}(P ϕ)},

where c_{g} : P → P is conjugation by g.

Write F^{f}, F^{c} for the set of F-fully normalized, F-centric subgroups of S, respectively. Set F^{fc} = F^{f} ∩ F^{c}.

For P ≤ S, let P^{F} = {P ϕ : ϕ ∈ hom_{F}(P, S)} be the orbit of P under F. A subgroup T of S is strongly closed in S with respect to F if for each P ≤ T, P^{F} ⊆ T.

A morphism α : F → F̃ of fusion systems from F to a system F̃ on S̃, is a family (α, α_{P,Q} : P, Q ≤ S) such that α : S → S̃ is a group homomorphism, and α_{P,Q} : hom_{F}(P, Q) → homF̃(P α, Qα) is a function, such that (P ↦ P α, α_{P,Q} : P, Q ≤ S) is a functor from F to F̃, and for all P, Q, ϕα = α(ϕα_{P,Q}). The morphism α is surjective if α : S → S̃ is surjective, and for all P, Q ≤ S, α_{P_{0},Q_{0}} : hom_{F}(P_{0}, Q_{0}) → homF̃(P α, Qα) is surjective, where for X ≤ S, X_{0} is the preimage in S of Xα under α.

Observe that the maps α_{P,Q} are determined by the group homomorphism α via the equalities (xα)(ϕα_{P,Q}) = (xϕ)α for ϕ ∈ hom_{F}(P, Q) and x ∈ P. In particular if α ∈ Aut(F) then α : S → S is an automorphism of groups and α_{P,Q} is the map α^{∗} : ϕ ↦ α^{−1}ϕα defined above. Thus each α ∈ Aut(S) corresponds to at most one automorphism α̇ of F, and α̇ exists iff for all P, Q ≤ S, hom_{F}(P, Q)α^{∗} = hom_{F}(P α, Qα).

A fusion category C on S is a category whose objects are subgroups of S, and the set of C-morphisms from P to Q is a set of injective group homomorphisms from P into Q. A subsystem of F is a fusion system E on a subgroup T of S such that for each P, Q ≤ T, hom_{E}(P, Q) ⊆ hom_{F}(P, Q).

Given a family (F_{i} : i ∈ I) of fusion categories F_{i} on subgroups S_{i} of S, define the fusion system generated by the family to be the fusion system on S obtained by intersecting all fusion systems on S containing each F_{i}. Write

⟨F_{i} : i ∈ I⟩

for this subsystem.

Let F be saturated. A subgroup R of S is normal in F if F = N_{F}(R); that is for each P ≤ S, each ϕ ∈ hom_{F}(P, S) extends to a member of hom_{F}(RP, S) which acts on R. We say F is constrained if there exists a normal centric subgroup of F. Finally the center of F is Z(F) = {z ∈ S : F = C_{F}(z)}.

We close this section by recording a few results from [BLO] that we use fairly frequently:

(1.1) (1) If ϕ ∈ hom_{F}(P, S) with P ϕ fully centralized then ϕ extends to φ ∈ hom_{F}(P C_{S}(P), S).

(2) If ϕ ∈ hom_{F}(P, S) with P ϕ fully normalized then there exists χ ∈ Aut_{F}(P) such that χϕ extends to a member of hom_{F}(N_{S}(P), S). In particular φ = χϕ ∈ hom_{F}(P, S) with P φ = P ϕ and φ extends to a member of hom_{F}(N_{S}(P), S).

Proof. These are special cases of A.2.b in [BLO].

(1.2) (1) If P is fully normalized then N_{F}(P) is saturated.

(2) If P is fully centralized then C_{F}(P) is saturated.

Proof. These are special cases of A.6 in [BLO].

(1.3) Assume P ∈ F^{c}. Then (1) For each ϕ ∈ hom_{F}(P, S), P ϕ ∈ F^{c}.

(2) If P ≤ Q ≤ S then Q ∈ F^{c}.

(3) P is fully centralized.

Proof. As P ∈ F^{c}, C_{S}(P ψ) ≤ P ψ for each ψ ∈ hom(P, S). Let μ ∈ hom_{F}(P ϕ, S). Then ϕμ ∈ hom_{F}(P, S), so C_{S}(P ϕμ) ≤ P ϕμ and hence (1) holds.

Next for η ∈ hom_{F}(Q, S), C_{S}(Qη) ≤ C_{S}(P η) ≤ P η ≤ Qη, so (2) holds.

By (1), C_{S}(P) = Z(P) and C_{S}(P ϕ) = Z(P ϕ), so |C_{S}(P)| = |Z(P)| = |Z(P ϕ)| = |C_{S}(P ϕ)|, so (3) follows.

Section 2. G(F)

In this section p is a prime and F is a saturated fusion system on a finite p-group S. Definition 2.1. Let T be the set of triples (G, R, U) such that G is a finite group, R ∈ Syl_{p}(G), and U is a normal p-subgroup of G such that C_{G}(U) ≤ U. Given τ_{i} = (G_{i}, S_{i}, U_{i}) ∈ T , define a morphism α : τ_{1} → τ_{2} to be isomorphism α : S_{1} → S_{2} of groups such that U_{1}α = U_{2} and G_{1}c_{1}α^{∗} = G_{2}c_{2}, where c_{i} : G_{i} → Aut(U_{i}) is the conjugation map, and α^{∗} : Aut(U_{1}) → Aut(U_{2}) is the isomorphism induced by α : U_{1} → U_{2} as in section 1. Define Ext(α) to be the set of isomorphisms β : G_{1} → G_{2} extending α : S_{1} → S_{2} such that βc_{2} = c_{1}α^{∗}.

Definition 2.2. When F is constrained, write G(F) for the class of finite groups G such that S ∈ Syl_{p}(G), F^{∗}(G) = O_{p}(G), and F = F_{S}(G). Call the members of G(F) the models of F. From Proposition C in [BCGLO], G(F) is nonempty and all models of F are isomorphic.

Example 2.3. Let F_{i}, i = 1, 2, be a pair of saturated constrained fusion systems on p-groups S_{i}, let α : F_{1} → F_{2} be an isomorphism of fusion systems, let G_{i} ∈ G(F_{i}), and let U_{1} be a normal centric subgroup of F_{1}. Set U_{2} = U_{1}α. Then τ_{i} = (G_{i}, S_{i}, U_{i}) ∈ T and α : S_{1} → S_{2} is a morphism α : τ_{1} → τ_{2}. For as U_{1} is normal and centric in F_{1}, its image U_{2} under the isomorphism α is normal and centric in F_{2}. Hence as G_{i} is a model for F_{i}, U_{i} ⊴ G_{i} and C_{G_{i}}(U_{i}) ≤ U_{i}. Further as α : F_{1} → F_{2} is an isomorphism and G_{i} ∈ G(F_{i}) for i = 1, 2, α : S_{1} → S_{2} is an isomorphism and

α^{∗} : G_{1}c_{1} = Aut_{G_{1}}(U_{1}) = Aut_{F_{1}}(U_{1}) → Aut_{F_{2}}(U_{2}) = Aut_{G_{2}}(U_{2}) = G_{2}c_{2}

is an isomorphism.

(2.4) Let τ_{i} = (G_{i}, S_{i}, U_{i}) ∈ T and α : τ_{1} → τ_{2} a morphism. Then (1) Ext(α) ≠ ∅.

(2) Given β ∈ Ext(α), Ext(α) = Z(S_{1})c_{G_{1}} · β, where c_{G_{1}} : G_{1} → Aut(G_{1}) is the conjugation map.

(3) Ext(α) is the set of isomorphisms from G_{1} to G_{2} extending α : S_{1} → S_{2}.

Proof. Let G = Aut_{G_{1}}(U_{1}), V = Z(U_{1}), V_{2} = Z(U_{2}), and π : G → Aut(V) the restriction map ϕ ↦ ϕ_{|V}. Set α_{1} = c_{1} : G_{1} → G, β_{1} = 1_{V} : V → V, α_{2} = c_{2}α^{−∗} : G_{2} → G, and define β_{2} : V_{2} → V to be α_{|V_{2}}^{−1}. As G_{1}c_{1}α^{∗} = G_{2}c_{2}, α^{∗} : G → G_{2}c_{2} is an isomorphism. Thus (G_{i}, β_{i}, α_{i}) is in the set Ext(G, V) of extensions of V by G, defined in section 21 of [3T].

Let H_{i} = S_{i}. Then α : (H_{1}, α_{1|H_{1}}, β_{1}) → (H_{2}, α_{2|H_{2}}, β_{2}) is an isomorphism of extensions of V by H = Hα_{1} = Hc_{1} = Hc_{2}α^{−∗} = Hα_{2}, as defined in section 21 of [3T]. Of course |G_{i} : H_{i}| is prime to p, so by 21.7 in [3T], α extends to an isomorphism β : (G_{1}, α_{1}, β_{1}) → (G_{2}, α_{2}, β_{2}) of extensions. That is β : G_{1} → G_{2} is an isomorphism extending α such that α_{1} = βα_{2} and β_{1} = ββ_{2}. As α_{1} = βα_{2}, c_{1} = βc_{2}α^{−∗}, so c_{1}α^{∗} = βc_{2}. Thus β ∈ Ext(α), establishing (1).

Next suppose γ ∈ Ext(α). Then δ = βγ^{−1} ∈ C_{Aut(G_{1})}(S_{1}). Let Ĝ_{1} = G_{1}⟨δ⟩ be the semidirect product of G_{1} by the group ⟨δ⟩ of automorphisms of G_{1}, and let Ĝ_{2} = G_{1}×⟨δ⟩. Let V = V_{1}, Ĝ = Ĝ_{1}/V, and π : Ĝ → Aut(V) the map (ĝV)π = cĝ on V.

As δ centralizes S_{1}, [G_{1}, δ] ≤ C_{G_{1}}(S_{1}) = C_{U_{1}}(S_{1}) = Z(S_{1}) ≤ V_{1}, so Ĝ_{1}/V = G_{1}/V × V ⟨δ⟩/V = Ĝ_{2}/V. Let α_{i} : Ĝ_{i} → Ĝ = Ĝ_{i}/V be the natural map. Let β_{i} : V → V be the identity map. Then (Ĝ_{i}, α_{i}, β_{i}) ∈ Ext(Ĝ, V).

Let Ĥ_{i} = S_{1} × ⟨δ⟩. Then the identity map from (Ĥ_{1}, α_{1|Ĥ_{1}}, β_{1}) to (Ĥ_{2}, α_{2|Ĥ_{2}}, β_{2}) is an isomorphism of extensions. Thus by 21.7 in [3T], 1 : S_{1} × ⟨δ⟩ → S_{1} × ⟨δ⟩ extends to an isomorphism ϵ : Ĝ_{1} → Ĝ_{2}. In particular Ĝ_{1} = G_{1} × ⟨ζ⟩, where ζ = δϵ^{−1} ∈ Z(Ĝ_{1}). Now δ = ζv for some v ∈ V, so as δ and ζ centralize S_{1}, so does v. Thus as an automorphism of G_{1}, δ = c_{δ} = c_{ζv} = c_{v} with v ∈ Z(S_{1}), establishing (2).

Finally let c = c_{G_{1}}, Ex = Ext(α), and E the set of isomorphisms from G_{1} to G_{2} extending α : S_{1} → S_{2}. By definition, Ex ⊆ E. Conversely let β ∈ Ex and suppose γ ∈ E. Then μ = γβ^{−1} ∈ C_{Aut(G_{1})}(S_{1}). We must show γ ∈ Ex, so by (2) it suffices to show μ ∈ Z(S_{1})c. Observe that if 1_{S_{1}} is the identity map on S_{1}, then 1_{S_{1}} : τ_{1} → τ_{1} is a morphism. Hence if μ ∈ Ext(1_{S_{1}}), then μ ∈ Z(S_{1})c by a second application of (2). Thus it suffices to show μc_{1} = c_{1}; that is for each g ∈ G_{1}, c_{gμ} = c_{g} on U_{1}.

Now [G_{1}, μ] ≤ C_{G_{1}}(S_{1}) = Z(S_{1}) ≤ U_{1}, as C_{G_{1}}(U_{1}) ≤ U_{1} ⊴ G_{1}. Thus for g ∈ G_{1},

c_{gμ} = c_{g[g,μ]} = c_{g}c_{[g,μ]} = c_{g}

as [g, μ] ∈ Z(S_{1}), completing the proof.

(2.5) Assume F is a saturated constrained fusion system on S. Then (1) G(F) ≠ ∅.

(2) Let G_{1}, G_{2} ∈ G(F) and U ⊴ F with C_{S}(U) ≤ U ≤ S. Then τ_{i} = (G_{i}, S, U) ∈ T and 1_{F} : τ_{1} → τ_{2} is a morphism, where 1_{F} is the identity map on F. Hence Ext(1_{F}) ≠ ∅.

(3) Let β ∈ Ext(1_{F}). Then Ext(1_{F}) = Z(S)c_{G_{1}} · β, where c_{G_{1}} : G_{1} → Aut(G_{1}) is the conjugation map.

(4) Ext(1_{F}) is the set of isomorphisms from G_{1} to G_{2} which are the identity on S.

Proof. This is essentially Proposition C in [BCGLO]. In particular part (1) follows from that Proposition, and then parts (2)-(4) follow from 2.3 and 2.4 applied to F_{1} = F_{2} = F and 1_{F} in the role of the isomorphism α of 2.3.

Section 3. Invariant subsystems of fusion systems

In this section F is a saturated fusion system over the finite p-group S, and E is a subfusion system of F on a subgroup T of S. Write F_{T}^{f} for the set of nontrivial subgroups P of T such that P is fully normalized in F.

Define E to be F-invariant if:

(I1) T is strongly closed in S with respect to F.

(I2) For each P ≤ Q ≤ T, ϕ ∈ hom_{E}(P, Q), and α ∈ hom_{F}(Q, S), ϕα^{∗} ∈ hom_{E}(P α, T). We say the subsystem E is F-Frattini if for each P ≤ T and γ ∈ hom_{F}(P, S), there

exists φ ∈ Aut_{F}(T) and ϕ ∈ hom_{E}(P φ, S), such that γ = φϕ on P.

(3.1) Assume E is a subsystem of F on a subgroup T of S which is strongly closed in S with respect to F. Suppose P′ ≤ T, P ∈ F^{f}, ϕ ∈ iso_{F}(P′, P), and A ⊴ Aut_{F}(P) such that Aut_{T} (P′)ϕ^{∗} ≤ A. Then

(1) Aut_{S}(P) ∩ A ∈ Syl_{p}(A).

(2) There exists ρ ∈ A such that ϕρ extends to a member of hom_{F}(N_{T} (P′), T).

Proof. As P ∈ F_{T}^{f} and F is saturated, Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)), so as A ⊴ Aut_{F}(P),

(1) holds.

By hypothesis, Aut_{T} (P′)ϕ^{∗} ≤ A. Hence by (1) and Sylow’s Theorem, there exists ρ ∈ A with Aut_{T} (P′)ϕ^{∗}ρ^{∗} ≤ Aut_{S}(P). Thus ϕρ ∈ hom_{F}(P′, P) with N_{T} (P′) ≤ N_{ϕρ}. Then (2) follows as P is fully normalized and F is saturated.

(3.2) Assume E is a subsystem of F on a subgroup T of S, which is strongly closed in S with respect to F. Then (I2) is equivalent to the following condition:

(I2′) For each P ≤ Q ≤ T and each α ∈ hom_{F}(Q, S), α^{∗} : hom_{E}(P, Q) → hom_{E}(P α, Qα) is a bijection.

Proof. The equivalence of (I2) and (I2′) is straightforward.

Let E be a fusion system on a p-group T. We saw in section 1 that each α ∈ Aut(T) corresponds to at most one automorphism α̇ of E, and α̇ exists iff for all P, Q ≤ T, hom_{E}(P, Q)α^{∗} = hom_{E}(P α, Qα). In particular in parts (2) and (3) of the following lemma, the statement Aut_{F}(T) ≤ Aut(E) means that for each α ∈ Aut_{F}(T), α̇ ∈ Aut(E) exists, or equivalently for P, Q ≤ T, hom_{E}(P, Q)α^{∗} = hom_{E}(P α, Qα).

Theorem 3.3. Assume T is a subgroup of S which is strongly closed in S with respect to F and E is a subsystem of F on T. Then the following are equivalent:

(1) E is F-invariant.

(2) Aut_{F}(T) ≤ Aut(E) and E is F-Frattini.

(3) Aut_{F}(T) ≤ Aut(E) and condition (I2A) holds:

(I2A) For each U ∈ F_{T}^{f} there exists a normal subgroup A(U) of Aut_{F}(U), such that for each U′ ≤ T, and each β ∈ iso_{F}(U, U′), Aut_{T} (U′) ≤ A(U)β^{∗} ≤ Aut_{E}(U′).

Proof. We first show that (3) implies (2), so assume (3). Let P ≤ T and H(P) the set of maps γ ∈ hom_{F}(P, S) such that γ = φ_{1}ϕ_{1} ⋯ φ_{r}ϕ_{r} with φ_{i} ∈ Aut_{F}(T) and ϕ_{i} an E-map. We claim hom_{F}(P, S) = H(P). Assume not and pick P and γ ∈ hom_{F}(P, S) − H(P) such that |T : P | = n is minimal. If n = 1 then P = T and γ ∈ Aut_{F}(T) as T is strongly closed, contrary to γ ∉ H(P). Thus n > 1.

By Alperin’s Fusion Theorem (cf. A.10 in [BLO]), there exist subgroups P = P_{1}, …, P_{n+1} = P γ of S, U_{1}, …, U_{n} in F^{frc}, and α_{i} ∈ Aut_{F}(U_{i}), such that P_{i} ≤ U_{i}, P_{i+1} = P_{i}α_{i}, and γ = α_{1} ⋯ α_{n}. Set Q_{i} = T ∩ U_{i}. As T is strongly closed, α_{i} acts on Q_{i}. Now if |P | < |Q_{i}| then α_{i} ∈ H(Q_{i}) by minimality of n. Further if α_{i} ∈ H(Q_{i}) for each i, then γ = α_{1} ⋯ α_{n} ∈ H(P), contrary to the choice of γ. Therefore for some i, |P | = |Q_{i}| and α_{i} ∉ H(Q_{i}), so replacing P, γ by Q_{i}, α_{i}, we may assume γ ∈ Aut_{F}(P).

Let β ∈ hom_{F}(P, S) with P β ∈ F^{f}. We apply 3.1.2 to β and γβ with A = A(P β). By 3.1.2 there exist ρ, μ ∈ A such that βρ and γβμ extend to δ, ϵ ∈ hom_{F}(N_{T} (P), S). By minimality of n, δ, ϵ ∈ H(N_{T} (P)), so restricting to P, βρ, γβμ ∈ H(P). By (I2A), ρ, μ ∈ Aut_{E}(P β), so γ = (γβμ)(μ^{−1}ρ)(βρ)^{−1} ∈ H(P), contrary to the choice of γ. This establishes the claim.

Finally let P ≤ T and γ ∈ hom_{F}(P, S). By the claim, γ = φ_{1}ϕ_{1} ⋯ φ_{r}ϕ_{r} ∈ H(P). Next for Q ≤ T, φ ∈ Aut_{F}(T), and ϕ ∈ hom_{E}(Qφ, T), ϕφ = φ(ϕφ^{∗}) with ϕφ^{∗} ∈ hom_{E}(Q, T) as Aut_{F}(T) ≤ Aut(E) by (3). Using this observation repeatedly,

γ = φ_{1}ϕ_{1} ⋯ φ_{r}ϕ_{r} = φ_{1} ⋯ φ_{r}ϕ = φϕ

for φ = φ_{1} ⋯ φ_{r} ∈ Aut_{F}(T) and some ϕ ∈ hom_{E}(P φ, T). This completes the proof that

(3) implies (2)

Next we show that (2) implies (1), so assume (2). By 3.2, it suffice to verify (I2′), so pick P ≤ Q ≤ T and α as in (I2′). By the Frattini condition, there exists φ ∈ Aut_{F}(T) and β ∈ hom_{E}(P φ, T) with α = φβ on P. By (2), φ ∈ Aut(E), so φ^{∗} : hom_{E}(P, Q) → hom_{E}(P φ, Qφ) is a bijection. As β is an E-map, β^{∗} : hom_{E}(P φ, Qφ) → hom_{E}(P α, Qα) is a bijection. Finally as α = φβ, α^{∗} = φ^{∗}β^{∗}, so (I2′) does hold, showing that (2) implies (1).

Finally we show that (1) implies (3) with A(U) = Aut_{E}(U). Assume (1). Then by (I2′), for α ∈ Aut_{F}(T) and P, P′ ≤ T, α^{∗} : hom_{E}(P, P′) → hom_{E}(P α, P′α) is a bijection, so α ∈ Aut(E). Similarly under the setup of (I2A), (I2′) says that Aut_{E}(U)β^{∗} = Aut_{E}(U′), so indeed (I2A) holds. This completes the proof of the lemma.

(3.4) Assume E is F-invariant on T, and let P ≤ T and ϕ ∈ hom_{F}(P, S). Then (1) P ϕ ≤ T.

(2) Aut_{E}(P) ⊴ Aut_{F}(P).

(3) If P ϕ is fully centralized in F, then |C_{T} (P)| ≤ |C_{T} (P ϕ)|.

(4) If P ϕ ∈ F_{T}^{f} then:

(a) |N_{T} (P)| ≤ |N_{T} (P ϕ)|.

(b) There exists ρ ∈ Aut_{E}(P ϕ) such that ϕρ extends to a member of hom_{F}(N_{T} (P), T).

(5) If P ∈ F_{T}^{f} then P is fully normalized in E.

Proof. Part (1) follows from (I1), and part (2) follows from (I2).

Assume P ϕ is fully normalized in F. Then by 1.1, there exists χ ∈ Aut_{F}(P) such that χϕ extends to a member φ of hom_{F}(N_{S}(P), S). By (I1), N_{T} (P)φ ≤ T, so N_{T} (P)φ ≤ N_{T} (P φ) = N_{T} (P ϕ). Hence (4a) holds, and a similar argument establishes (3). Further (4a) implies (5).

Finally we apply 3.1 to P, P ϕ, ϕ, Aut_{E}(P ϕ) in the roles of P′, P, ϕ and A. Then Aut_{T} (P)ϕ^{∗} ≤ Aut_{E}(P)ϕ^{∗} = A by (I2), so (4b) follows from 3.1.2.

(3.5) Assume E is a F-invariant subsystem on T. Then each orbit of E on subgroups of T contains a member Rφ such that φ ∈ Aut_{F}(T) and R ∈ F_{T}^{f}. For each such R and φ, both R and Rφ are fully normalized and fully centralized in E.

Proof. Let Q ≤ T. Then there exists R ∈ F_{T}^{f} and α ∈ hom_{F}(R, S) with Rα = Q. By 3.3, α = φϕ for some φ ∈ N_{F}(T) and some E-map ϕ. Then Rφ ∈ Q^{E}. By 3.4.1, Rφ ≤ T, and by 3.4.3 and 3.4.5, R is fully normalized and centralized in E. As φ ∈ Aut_{F}(T), N_{T} (Rφ) = N_{T} (R)φ and C_{T} (Rφ) = C_{T} (Rφ), so Rφ is also fully normalized and fully centralized in E by 3.4.3 and 3.4.4. Thus the lemma holds.

(3.6) Assume E is F-invariant and D is a subsystem of F on the subgroup D of S. Then

(1) E ∩ D is a D-invariant subsystem of D on T ∩ D.

(2) If D is F-invariant then E ∩ D is F-invariant on T ∩ D.

Proof. Straightforward.

Section 4. D(U) and E(U)

In this section F is a saturated fusion system over the finite p-group S.

Notation 4.1. Assume E is a subsystem of F on a subgroup T of S such that T is strongly closed in S with respect to F. For X ≤ S, set V_{X} = XC_{S}(X). Let U ∈ F_{T}^{f}, and set T_{U} = UC_{T} (U). Define

D(U) = D_{F,T} (U) = N_{N_{F} (V_{T_{U}})}(U).

Thus D(U) is a fusion system on N_{S}(U). (cf. the proof of 4.2) Define

E(U) = E_{F,E}(U) = N_{N_{E}(T_{U})}(U).

Thus E(U) is a fusion system on N_{T} (U). The system D(U) is the “largest canonical” constrained saturated subsystem of F in which U is normal. It should be viewed as a constrained local subsystem of F, analogous to a constrained local subgroup of a finite group. Similarly if E is saturated, then E(U) plays a similar role in E. As Theorems 2 and 3 suggest, these subsystems are crucial in the machinery developed in section 7 to identify and construct normal subsystems of F.

In the next lemma we adopt the terminology and notation (eg. N_{S}^{K}(P), N_{F}^{K}(P)) from Appendix A in [BLO].

(4.2) Assume E is a subsystem of F on a subgroup T of S such that T is strongly closed in S with respect to F. Let U ∈ F_{T}^{f}. Then

(1) D(U) = N_{F}^{N_{Aut(V_{T_{U}} )}(U)}(V_{T_{U}}).

(2) E(U) = N_{E}^{N_{Aut(T_{U})}(U)}(T_{U}) ≤ D(U).

(3) D(U) is saturated and constrained.

(4) If E is saturated then E(U) is saturated and constrained.

Proof. Let V = V_{T_{U}}, K = N_{Aut(V)}(U), and J = N_{Aut(T_{U})}(U). As T is strongly closed in S, N_{S}(U) acts on T_{U}, and hence also on V. Thus N_{S}(U) = N_{S}(U) ∩ N_{S}(V) = N_{S}^{K}(V). Similarly N_{T} (U) = N_{T} (U) ∩ N_{T} (T_{U}) = N_{T}^{J}(T_{U}). From the definitions in Appendix A of [BLO], N_{F}^{K}(V) = D(U) and N_{E}^{J}(T_{U}) = E(U). In particular (1) is established.

Next we observe that V is fully K-normalized in F: that is |N_{S}^{K}(V)| ≥ |N_{S}^{Kα^{∗}}(V α)| for all α ∈ hom_{F}(V, S). This follows as U ∈ F_{T}^{f}, N_{S}^{K}(V) = N_{S}(U), and N_{S}^{Kα^{∗}}(V α) = N_{S}(Uα). Then D(U) is saturated by (1), the observation, and Proposition A.6 in [BLO]. By construction, V ⊴ D(U), and C_{S}(V) ≤ C_{S}(T_{U}) ≤ V, so V is centric in D(U). Thus D(U) is constrained, completing the proof of (3). The proof of (4) is essentially the same.

Let P ≤ N_{T} (U) and ϕ ∈ hom_{E(U)}(P, N_{T} (U)). To prove (2), we must show ϕ extends to a map in hom_{F}(P V, S) acting on U. As ϕ ∈ hom_{E(U)}(P, N_{T} (U)), ϕ extends to φ ∈ hom_{E}(P T_{U}, N_{T} (U)) acting on U, so replacing P, ϕ by P T_{U},φ, we may assume T_{U} ≤ P.

Let B = N_{F}(U), B = N_{S}(U), Y = C_{S}(T_{U}), and Z = Z(T_{U}). Thus ϕ ∈ hom_{B}(P, B). Now [P, Y] ≤ C_{T} (T_{U}) = Z ≤ P, so Y ≤ N_{B}(P) centralizes the factors in the Aut_{B}(P)- invariant chain 1 ≤ Z ≤ P, and hence Aut_{Y} (P) ≤ O_{p}(Aut_{B}(P)).

Next B is a saturated fusion system on B by 1.2.1. Let β ∈ hom_{B}(N_{B}(P ϕ), B) with P′ = P ϕβ ∈ B^{f}. Thus Aut_{B}(P′) ∈ Syl_{p}(Aut_{B}(P′)) by 1.2.1, so O_{p}(Aut_{B}(P′)) ≤ Aut_{B}(P′). Further

Aut_{Y} (P)(ϕβ)^{∗} ≤ O_{p}(Aut_{B}(P))(ϕβ)^{∗} = O_{p}(Aut_{B}(P′)) ≤ Aut_{B}(P′),

so P Y ≤ N_{ϕβ}. Hence ϕβ extends to ψ ∈ hom_{B}(P Y, B) as B is saturated. Now ψ′ = ψβ^{−∗} ∈ hom_{B}(P Y, B) extends ϕ, and ψ′ ∈ hom_{F}(P V, S) acts on U, completing the proof of (2).

(4.3) Assume E is F-invariant on T, and let U ∈ F_{T}^{f}. Then

(1) D(U) is a saturated constrained fusion system on N_{S}(U), N_{T} (U) is strongly closed in N_{S}(U) with respect to D(U), and E(U) is D(U)-invariant.

(2) If E is saturated then E(U) is saturated and constrained, and N_{E}(U) is saturated.

(3) Aut_{F}(U) = Aut_{D(U)}(U).

(4) N_{F}(U) is a saturated fusion system and N_{E}(U)) is N_{F}(U)-invariant.

Proof. As U is fully normalized in F, 1.2.1 says that N_{F}(U) is saturated.

Suppose P ≤ Q ≤ N_{T} (U), ϕ ∈ hom_{E(U)}(P, Q), and α ∈ hom_{D(U)}(Q, S). As ϕ is an E(U)-map, ϕ extends to φ ∈ hom_{E}(P T_{U}, N_{T} (U)) acting on U, and as α is a D(U)-map, α extends to β ∈ hom_{F}(QV_{T_{U}} , N_{S}(U)). Then φβ^{∗} extends ϕα^{∗} to P αT_{U} acting on U, and as E is F-invariant, φβ^{∗} is a E-map. Thus ϕα^{∗} ∈ hom_{E(U)}(P α, N_{T} (U)), and hence E(U) is D(U)-invariant. Together with 4.2.3, these observations establish (1). Similarly N_{E}(U) is N_{F}(U)-invariant, so (4) holds, and (2) follows using 4.2.4.

Let α ∈ Aut_{F}(U). By 1.1.1, α extends to β ∈ Aut_{F}(V_{U}). Then β acts on T_{U} = T ∩V_{U} and on C_{V_{U}} (T_{U}) = V_{T_{U}}. Thus α ∈ Aut_{D(U)}(U), so (3) holds. Definition 4.4. Given T ≤ S strongly closed in S with respect to F, and P ∈ F_{T}^{f}, define

B(P) = B_{F,T} (P) = ⟨Aut_{T} (P)^{Aut_{F} (P)}⟩,

and define B′(P) = B′_{F,T} (P) to be the preimage in Aut_{F}(P) of O_{p′}(Aut_{F}(P)/B(P)).

(4.5) Assume T is strongly closed in S with respect to F, and let U ∈ F_{T}^{f} and T_{U} ≤ P ≤ N_{T} (U). Then Aut_{Aut_{F} (P)}(U) = N_{Aut_{F} (U)}(Aut_{P} (U)). Proof. Let G = Aut_{F}(P), H = N_{G}(U), and K = Aut_{F}(U). Let ρ : H → K be the restriction map ρ : h ↦ h_{|U}. For x ∈ P, let c_{x} ∈ K be conjugation by x. Then P c = Aut_{P} (U) and c_{x}^{hρ} = c_{xh}, so hρ ∈ J = N_{K}(Aut_{P} (U)). Thus ρ : H → J is a group homomorphism. Further for ϕ ∈ J, P ≤ N_{ϕ}, so as U ∈ F^{f} and F is saturated, ϕ extends to φ ∈ hom_{F}(P, S). As ϕ acts on Aut_{P} (U), C_{S}(U)P φ = C_{S}(U)P, while as T_{U} ≤ P ≤ T, P = T ∩ C_{S}(U)P. Then as T is strongly closed, φ ∈ H. Thus ρ : H → J is surjective, completing the proof.

Section 5. Invariant maps

In this section F is a saturated fusion system over the finite p-group S, and T is a subgroup of S strongly closed in S with respect to F. Definition 5.1. Define an F-invariant map on T to be a function A on the set of subgroups of T such that

(IM1) For each P ≤ T and α ∈ hom_{F}(P, T), A(P)α^{∗} = A(P α) ≤ Aut_{F}(P α).

(IM2) For each P ∈ F_{T}^{f}, Aut_{T} (P) ≤ A(P).

Given an F-invariant map A on T and P ≤ T, we see in 5.3 that A(P) is a normal subgroup of Aut_{F}(P). Set 𝔈(A) = ⟨A(P) : P ≤ T ⟩. Thus 𝔈(A) is a fusion system on T. Example 5.2. Pick a set U of representatives in F_{T}^{f} for the orbits P^{F} of F on the subgroups of T. For U ∈ U, pick a normal subgroup A(U) of Aut_{F}(U) containing Aut_{T} (U). For α ∈ hom_{F}(U, T), define A(Uα) = A(U)α^{∗}. As A(U) ⊴ Aut_{F}(U), the function A is well defined, and by construction A is a F-invariant map on T. From 5.3.1 below, each invariant map can be obtained via this construction.

We have shown that each map A on U such that Aut_{T} (U) ≤ A(U) ⊴ Aut_{F}(U) extends uniquely to a a F-invariant map, which we shall also denote by A. Here are some special cases:

(1) The map B = B_{F,T} : U ↦ B(U) (cf. Definition 4.4) on U defines an invariant map, since by definition Aut_{T} (U) ≤ B(U) ⊴ Aut_{F}(U).

(2) Let E be an F-invariant subsystem of F on T. Then Aut_{E} : U ↦ Aut_{E}(U) defines an invariant map.

(3) Write F_{T}^{fc} for the set of V ∈ F_{T}^{f} such that V = T_{V}, or equivalently C_{T} (V) ≤ V. Set

F_{T}^{c} = ⋃_{V ∈F_{T}^{fc}} V^{F}.

Thus by 3.4.3, F_{T}^{c} consists of those P ≤ T such that C_{T} (P ϕ) ≤ P ϕ for all ϕ ∈ hom_{F}(P, S).

Let V = U∩F_{T}^{fc} and suppose A is a map on V satisfying Aut_{T} (V) ≤ A(V) ⊴ Aut_{F}(V) for each V ∈ V. Then as above, we may extend A to a map on F_{T}^{c} via A(V α) = A(V)α^{∗}. Then for U ∈ U, set A(U) = Aut_{A(T_{U})}(U). We check that A satisfies our conditions: Let K = Aut_{F}(T_{U}), M = N_{K}(U), L = M ∩A(T_{U}), and ρ : α ↦ α_{|U} the restriction map from M into Aut_{F}(U). Thus Lρ = A(U). As U ∈ F_{T}^{f} and F is saturated, ρ is a surjection. For t ∈ N_{T} (U), t acts on T_{U}, so k_{t} = c_{t|T_{U}} ∈ K and k_{t}ρ = c_{t|U}. Thus Aut_{T} (U) ≤ Lρ. Next let α ∈ Aut_{F}(U), β ∈ Lρ, and β̂ ∈ L with β̂ρ = β. As ρ is surjective, there is α̂ ∈ M with α̂ρ = α. Then (β̂α̂^{∗})ρ = βα^{∗}, so Lρ ⊴ Aut_{F}(U).

We say an F-invariant map A is constricted if A(U) = Aut_{A(T_{U})}(U) for each U ∈ F_{T}^{f}. Thus the maps A constructed in this example are constricted.

(4) Here is a special case of (3). Suppose A is an invariant map. Define A_{c} to be the invariant map obtained from the construction in (3) with respect to the restriction of A to V. In particular A(P) = A_{c}(P) for P ∈ F_{T}^{c} and A(U) = Aut_{A(T_{U})}(U) for U ∈ F_{T}^{f}. Call A_{c} the constriction of A. Thus A_{c} is constricted.

For example let E be an F-invariant subsystem of F on T and write Aut_{E,c} for the constriction of the invariant map Aut_{E} defined in (2).

(5.3) Let A be an F-invariant map on T. Then for each U ∈ F_{T}^{f}:

(1) A(U) ⊴ Aut_{F}(U).

(2) B(U) ≤ A(U).

Proof. Part (1) is a consequence of (IM1). Then (2) follows from (1) and (IM2).

(5.4) Let E be an F-invariant subsystem on T. Then E^{c} = F_{T}^{c}.

Proof. If U ∈ F_{T}^{fc} then 3.4.3 says that for each ϕ ∈ hom_{F}(U, S), C_{T} (Uϕ) = Z(Uϕ), so that U and Uϕ are in E^{c}.

On the other hand if P ∈ E^{c} then there exists U ∈ F_{T}^{f} and γ ∈ hom_{F}(P, S) with U = P γ. By 3.3, γ = φϕ for some φ ∈ Aut_{F}(T) and ϕ ∈ hom_{E}(P φ, T). By 3.3, Aut_{F}(T) ≤ Aut(E), so as P ∈ E^{c}, also P φ ∈ E^{c}. Then as ϕ ∈ hom_{E}(P φ, T), U = P γ = P φϕ ∈ E^{c} by 1.3.1. Thus U ∈ F_{T}^{fc}, so P ∈ F_{T}^{c}.

(5.5) Assume A is an F-invariant map on T. Then (1) 𝔈(A) is a F-invariant subsystem of F on T.

(2) A(T) = Aut_{𝔈(A)}(T).

(3) Assume P and Q are families of subgroups of T such that

T ∈ P, P ⊆ ⋃_{Q∈Q} Q^{𝔈(A)}, and 𝔈(A) = ⟨A(P) : P ∈ P⟩.

Then 𝔈(A) = ⟨A(Q) : Q ∈ Q⟩.

(4) 𝔈(A) = ⟨A(U) : U ∈ 𝔈(A)^{f}⟩.

(5) 𝔈(A) = ⟨A(Uφ) : U ∈ F_{T}^{f} , φ ∈ Aut_{F}(T)⟩.

(6) Assume for each U ∈ F_{T}^{f} that A(U) ≤ Aut_{A(T_{U})}(U). Then

𝔈(A) = ⟨A(T_{Uφ}) : U ∈ F_{T}^{f} , φ ∈ Aut_{F}(T)⟩ = ⟨A(V) : V ∈ 𝔈(A)^{fc}⟩ = ⟨A(W_{i}φ) : i ∈ I, φ ∈ Aut_{F}(T)⟩.

where {W_{i} : i ∈ I} is a set of representatives in F_{T}^{fc} for the orbits of F on E^{c}.

Proof. Let A = {A(P) : P ≤ T } and E = 𝔈(A). By (IM1):

(7) For each P ≤ T and α ∈ hom_{F}(P, S), A(P)α^{∗} = A(P α). In particular if β ∈ Aut_{F}(T) and A ⊆ C ≤ F, then for each P ≤ T, Cβ^{∗} contains

A(P)β^{∗} = A(P β), so A = Aβ^{∗} ⊆ Cβ^{∗}. Thus E = Eβ^{∗}, so (8) Aut_{F}(T) ≤ Aut(E). We next observe that:

(9) For each U ∈ F_{T}^{f}, U′ ≤ T, and β ∈ iso_{F}(U, U′), Aut_{T} (U′) ≤ A(U)β^{∗} = A(U′). Namely by (IM1), A(U)β^{∗} = A(Uβ) = A(U′). Also by (IM2), Aut_{T} (U) ≤ A(U).

Further by 1.1.2, we may choose β so that N_{T} (U′)β^{−1} ≤ N_{T} (U), so Aut_{T} (U′) ≤ Aut_{T} (U)β^{∗} ≤ A(U)β^{∗}.

Observe that by (8) and (9) and the equivalence of parts (1) and (3) of 3.3, E is F-invariant. Thus (1) is established.

Let α ∈ Aut_{E}(T). As E = ⟨A(P) : P ≤ T ⟩, there exist U_{i} ≤ T, 1 ≤ i ≤ n, α_{i} ∈ A(U_{i}), and P_{i} ≤ U_{i} such that T = P_{1}, P_{i+1} = P_{i}α_{i} for 1 ≤ i ≤ n, and α = α_{1} ⋯ α_{n} on T. But as T is strongly closed in S with respect to F, T = P_{i} for each i, by induction on i. Thus for each i, T ≤ U_{i} ≤ T, so U_{i} = T, and hence α_{i} ∈ A(T). Thus (2) holds.

Assume the hypothesis of (3), and let D = ⟨A(Q) : Q ∈ Q⟩. Assume (3) fails, and pick P ∈ P and α ∈ hom_{E}(P, T) such that α is not a D-map and m = |T : P | is minimal. As

T^{E} = {T } and T ∈ P ⊆ ⋃_{Q∈Q} Q^{E},

it follows that T ∈ Q. Therefore m > 1. We can pick U_{i}, P_{i}, and α_{i} as in the previous paragraph, except with P = P_{1}. As α = α_{1} ⋯ α_{n}, it suffices to show each α_{i} is a D-map, so we may assume n = 1 and α = α_{1}. Let β ∈ hom_{E}(U_{1}, T) with U_{1}β ∈ Q. If U_{1} ≠ P then by minimality of m, β ∈ D, and as U_{1}β ∈ Q, αβ^{∗} ∈ D by definition of D. Thus α = (αβ^{∗})β^{−∗} ∈ D. Hence we may assume U_{1} = P, and it remains to show we can choose β so that β is a D-map. Apply the argument above, and write β = β_{1} ⋯ β_{k}. As above we are done if P_{i} < U_{i} for each i. But if P_{i} = U_{i} then we replace β by β_{1} ⋯ β_{i−1}β_{i+1} ⋯ β_{k}, and obtain a contradiction by induction on k. This completes the proof of (3).

Part (4) follows from (3) applied with P the set of subgroups of T, and with Q = E^{f}. Similarly (5) follows from (3), (4), and 3.5.

Assume the hypothesis of (6). For U ∈ F_{T}^{f} and φ ∈ Aut_{F}(T), T_{U}φ = (UC_{T} (U)φ = UφC_{T} (Uφ) = T_{Uφ}, so by the hypothesis of (6):

A(Uφ) = A(U)φ^{∗} ≤ Aut_{A(T_{U})}(U)φ^{∗} = Aut_{A(T_{U} φ)}(Uφ),

so that the first equality in (6) follows from (5). Then the second equality follows from (3) applied to P = {T_{Uφ} : U ∈ F_{T}^{f} , φ ∈ Aut_{F}(T)} and Q = E^{fc}, since P ⊆ E^{c} (cf. 6.5.4). Finally the last equality follows from (3) and 3.5.

Define an F-invariant map on T to be proper if for each U ∈ F_{T}^{f}, N_{A(U)}(Aut_{T} (U)) = Aut_{A(N_{T} (U))}(U). (5.6) Assume E is a saturated F-invariant subsystem on T, and set A = Aut_{E}. Then

(1) A is a proper F-invariant map on T.

(2) E = 𝔈(A). Proof. As E is F-invariant, A is an F-invariant map by 5.2.2. As E is saturated, (2) holds by Alperin’s Fusion Theorem (cf. A.10 in [BLO]), and A is proper by 4.5 applied to the saturated system E in the role of F. (5.7) Let A be an F-invariant map on T and U ∈ F_{T}^{f}. Set G = Aut_{F}(U). Then

(1) A(U) = B(U)N_{A(U)}(Aut_{T} (U)).

(2) G = B(U)N_{G}(Aut_{T} (U)).

(3) N_{Aut_{F} (N_{T} (U))}(U) = N_{G}(Aut_{T} (U)).

(4) B(U)Aut_{A(N_{T} (U))}(U) ⊴ G. Proof. Let Q = N_{T} (U), P = Aut_{Q}(U), and L = B(U). By 5.3, L ≤ A(U) ⊴ G. Further as F is saturated and U ∈ F^{f}, R = Aut_{S}(U) ∈ Syl_{p}(G), so R ∩ L ∈ Syl_{p}(L). Then as T is strongly closed in S with respect to F, P is strongly closed in R with respect to G, so by a Frattini argument, G = LN_{G}(P) and A(U) = LN_{A(U)}(P). Thus (1) and (2) hold. Part (3) follows from 4.5. Finally as A(Q) ⊴ Aut_{F}(Q), (3) says that Aut_{A(Q)}(U) ⊴ N_{G}(P), so (4) follows from (2).

(5.8) Assume A_{i}, i = 1, 2, are proper F-invariant maps on T such that A_{1}(T) = A_{2}(T). Then A_{1} = A_{2}.

Proof. Claim for each U ∈ F_{T}^{f} that A_{1}(U) = A_{2}(U). Assume otherwise and choose a counter example U with n = |T : U| minimal. By hypothesis, A_{1}(T) = A_{2}(T), so n > 1. Set Q = N_{T} (U) and let α ∈ hom_{F}(Q, S) with Qα ∈ F^{f}. Then also Uα ∈ F^{f} and A_{1}(Uα) = A_{1}(U)α^{∗} ≠ A_{2}(U)α^{∗} = A_{2}(Uα). Therefore replacing U by Uα, we may assume Q ∈ F^{f}. Thus A_{1}(Q) = A_{2}(Q) by minimality of n. Then as A_{i} is proper,

N_{A_{1}(U)}(P) = Aut_{A_{1}(Q)}(U) = Aut_{A_{2}(Q)}(U) = N_{A_{2}(U)}(P),

so by 5.7.1, A_{1}(U) = B(U)N_{A_{1}(U)}(P) = B(U)N_{A_{2}(U)}(P) = A_{2}(U), contrary to the choice of U. This completes the proof of the claim.

Finally for R ≤ T, there is φ ∈ hom_{F}(R, S) with Rφ ∈ F^{f}. Then by the claim, A_{1}(R)φ^{∗} = A_{1}(Rφ) = A_{2}(Rφ) = A_{2}(R)φ^{∗}, so A_{1}(R) = A_{2}(R). That is A_{1} = A_{2}.

The following lemma was suggested by the referee.

(5.9) Assume E_{1} and E_{2} are saturated F-invariant subsystems on T such that Aut_{E_{1}}(T) = Aut_{E_{2}}(T). Then E_{1} = E_{2}.

Proof. Let A_{i} = Aut_{E_{i}}. By 5.6.1, A_{i} is a proper F-invariant map on T, so by 5.8, A_{1} = A_{2}. Then by 5.6.2, E_{1} = 𝔈(A_{1}) = 𝔈(A_{2}) = E_{2}.

Section 6. Normal subsystems of fusion systems

In this section F is a saturated fusion system over the finite p-group S, and E is a sub-fusion system of F on a subgroup T of S. We continue the notation and terminology from the previous sections. In addition we adopt: Notation 6.1. For P ∈ F_{T}^{f}, define Γ(P) = Γ_{F,E}(P) to consist of those ϕ ∈ Aut_{E}(P) such that ϕ extends to some ϕ̂ ∈ Aut_{F}(V_{P}) such that [ϕ̂, C_{S}(P)] ≤ Z(P). Notation 6.2. Given subsystems D and E of F on subgroups D and T of S, respectively, we denote by N(D, E) the following statement: (N(D, E)) Aut_{D∩E}(T ∩ D) ≤ Γ_{D,D∩E}(T ∩ D).

The subsystem E is normal in F if E is F-invariant and saturated, and the following condition holds:

(N1) N(F, E) is satisfied. That is Aut_{E}(T) ≤ Γ(T), or equivalently each ϕ ∈ Aut_{E}(T) extends to ϕ̂ ∈ Aut_{F}(T C_{S}(T)) such that [ϕ̂, C_{S}(T)] ≤ Z(T).

(6.3) Assume F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G). Let H ⊴ G, T = S ∩ H, and E = F_{T} (H). Then E ⊴ F.

Proof. As H ⊴ G, T = S ∩ H ∈ Syl_{p}(H) and T is strongly closed in S with respect to G. Thus (I1) holds. By a Frattini argument, E is F-Frattini. Let x ∈ N_{G}(T), P ≤ T, and h ∈ N_{H}(P, T). Then h^{x} ∈ N_{H}(P^{x}, T), so c_{x} ∈ Aut(F_{T} (H)), and hence Aut_{F}(T) ≤ Aut(E). Therefore E is F-invariant by 3.3.

As E = F_{T} (H), E is saturated. Thus it remains to verify (N1). Let V = V_{T}, R = C_{S}(T), and K = N_{H}(T). For ϕ ∈ Aut_{E}(T), ϕ = c_{k|T} for some k ∈ K. Then by a Frattini argument, K = C_{H}(T)N_{K}(R), so k = ch for some c ∈ C_{H}(T) and h ∈ N_{K}(R). Now ϕ = c_{k|T} = c_{h|T} and h acts on T R = V with [h, R] ≤ H ∩ R = Z(T). Therefore (N1) holds, completing the proof of the lemma. Example 6.4. Let H = H_{1} × H_{2} × H_{3} be the direct product of three copies H_{i}, 1 ≤ i ≤ 3, of A_{4}. Let X_{i} = ⟨x_{i}⟩ ∈ Syl_{3}(H_{i}), S_{i} = O_{2}(H_{i}), and S = S_{1} × S_{2} × S_{3} ∈ Syl_{2}(H). Let X = ⟨x_{1}x_{2}, x_{1}x_{3}⟩ ≤ H and set G = XS. Then G_{1} = ⟨x_{1}x_{2}, S_{1}, S_{2}⟩ and G_{2} = ⟨x_{1}x_{3}, S_{1}, S_{3}⟩ are normal subgroups of G with Sylow 2-subgroups T_{1} = S_{1}S_{2} and T_{2} = S_{1}S_{3}, respectively. Let F_{i} = F_{T_{i}}(G_{i}), for i = 1, 2, and F = F_{S}(G). As G_{i} ⊴ G, F_{i} ⊴ F by 6.3.

Let E = F_{1} ∩ F_{2}. Then E = F_{S_{1}}(H_{1}) as

ϕ = c_{x_{1}x_{2}|S_{1}} = c_{x_{1}x_{3}|S_{1}} = c_{x_{1}|S_{1}} ∈ Aut_{E}(S_{1}).

In particular E is a saturated fusion system, and by 3.6.1, E is F-invariant. On the other hand E is not normal in F, as (N1) is not satisfied. Namely V_{S_{1}} = S, but ϕ does not extend to ϕ̂ ∈ Aut_{F}(S) with [ϕ̂, S] ≤ S_{1}.

This shows, first, that there exist F-invariant saturated fusion systems which do not satisfy (N1), and, second, that the intersection of normal subsystems is not in general normal. Moreover this also shows, third, that (N1) is a necessary hypothesis if the converse of 6.3 is to hold for constrained fusion systems, since E has no model normal in G.

Put another way, if we are to extend arguments from the local theory of finite groups to the domain of saturated fusion systems, it is crucial that Theorem 1 hold. Thus it is necessary that the definition of “normal subsystem” contain some condition such as (N1). For this reason we feel our notion of normal subsystem is superior to those of Puig [P] and Linkelman [L].

Recall the definitions of B(U) and B′(U) in Definition 4.4.

(6.5) Assume E is F-invariant on T and U ∈ F_{T}^{f}. Set Q = N_{T} (U). Then (1) Γ(U) ⊴ Aut_{F}(U).

(2) If C_{T} (U) ≤ U then B(U) ≤ Γ(U), U ∈ E^{c}, and T_{U} = U.

(3) For each α ∈ hom_{F}(Q, S), C_{T} (Qα) = Z(Qα) and V_{Qα} ∩ T = T_{Qα} = Qα. Thus Q ∈ E^{c}.

(4) T_{U} ∈ E^{c}.

Proof. Let α_{i} ∈ Γ = Γ(U), i = 1, 2. Thus there is an extension α̂_{i} of α_{i} to V = V_{U} with [α̂_{i}, C_{S}(U)] ≤ Z(U). As α̂_{1}α̂_{2} is such an extension of α_{1}α_{2}, Γ is a group. For β ∈ Aut_{F}(U), β extends to β̂ on V by 1.1.1, and α̂β̂ ; 1 is an extension of α_{1}^{β} centralizing C_{S}(U)/Z(U), so (1) holds.

Assume C_{T} (U) ≤ U. As T is strongly closed in S, [N_{T} (U), C_{S}(U)] ≤ C_{T} (U) = Z(U), and as E is a fusion system on T, Aut_{T} (U) ≤ Aut_{E}(U). Thus Aut_{T} (U) ≤ Γ(U), and then the first statement in (2) follows from (1).

Let α ∈ hom_{F}(Q, S) and P = Qα. As C_{T} (U) ≤ U, C_{T} (U) = C_{Q}(U), so as U ∈ F_{T}^{f}, also C_{T} (Uα) = C_{P} (Uα). Thus C_{T} (P) ≤ C_{P} (Uα) ≤ P, so

V_{P} ∩ T = P C_{S}(P) ∩ T = P C_{T} (P) = P = T_{P} ,

establishing (3). Moreover the same argument completes the proof of (2), and establishes (4).

(6.6) Assume F is constrained and T is strongly closed in S with respect to F. Choose G ∈ G(F) and R ≤ S such that R ⊴ G and C_{O_{p}(G)}(R) ≤ R. Set L = ⟨T^{G}⟩ and V = V_{T ∩R}. Then

(1) C_{G}(R) = Z(R), C_{G}(R/(T ∩ R) ∩ C_{G}(T ∩ R) ≤ R, C_{T} (T ∩ R) = Z(T ∩ R), and T ∩ V = T ∩ R.

(2) Assume Aut_{T} (T ∩ R) ∈ Syl_{p}(B(T ∩ R)). Then

(i) T ∈ Syl_{p}(L), F^{∗}(L) = T ∩ O_{p}(G), and L = O^{p′}(L) = ⟨T^{L}⟩.

(ii) For each P ∈ F_{T} (L)^{f}, B(P) = O^{p′}(Aut_{L}(P)).

Proof. Set T_{R} = T ∩ R. By 4.3.1, T is strongly closed in S with respect to G, so as R ⊴ G we conclude:

(a) T_{R} ⊴ G, and

(b) [T, R] ≤ T_{R}.

Then as L = ⟨T^{G}⟩, it follows from (a) and (b) that:

(c) [L, R] ≤ T_{R}.

By hypothesis, C_{O_{p}(G)}(R) ≤ R and F^{∗}(G) = O_{p}(G), so it follows from the Thompson A × B-Lemma (cf. 24.2 in [FGT]) that C_{G}(R) is a p-group. Thus:

(d) C_{G}(R) = C_{O_{p}(G)}(R) ≤ R.

Let c : L → Aut(T_{R}) be the conjugation map. Thus C_{L}(T_{R}) = ker(c). By (c), C_{L}(T_{R}) centralizes R/T_{R} and T_{R}, so as R and T_{R} are normal in G, O^{p}(C_{L}(T_{R})) ≤ O^{p}(C_{G}(R/T_{R}) ∩ C_{G}(T_{R})) ≤ C_{G}(R). Then it follows from (d) that:

(e) C_{L}(T_{R}) ≤ C_{G}(R/T_{R}) ∩ C_{G}(T_{R}) ≤ R.

By (e), C_{T} (T_{R}) = Z(T_{R}), so that T ∩ V = T ∩ T_{R}C_{S}(T_{R}) = T_{R}C_{T} (T_{R}) = T_{R}. Thus (1) is established.

Let A = Aut_{G}(T_{R}) = Aut_{F}(T_{R}). Thus B(T_{R}) = ⟨Aut_{T} (T_{R})^{A}⟩. Assume the hypotheis of (2); that is Aut_{T} (T_{R}) ∈ Syl_{p}(B(T_{R})). Then B(T_{R}) = Aut_{L}(T_{R}) is the image of L in A under c, so by (e):

(f) C_{L}(T_{R})T ∈ Syl_{p}(L).

Claim:

(g) [L, C_{L}(T_{R})] ≤ Z(T_{R}) = T ∩ C_{L}(T_{R}).

For using (e), C_{L}(T_{R}) ≤ R ≤ N_{G}(T), so [T, C_{L}(T_{R})] ≤ T ∩ C_{L}(T_{R}) = C_{T} (T_{R}), while C_{T} (T_{R}) = Z(T_{R}) by an earlier remark. Thus T centralizes C_{L}(T_{R})/Z(T_{R}), so as L = ⟨T^{G}⟩, also L centralizes C_{L}(T_{R})/Z(T_{R}), completing the proof of (g).

Let L^{∗} = L/Z(T_{R}). By (f) and (g), C_{L}(T_{R})^{∗} ≤ Z(L^{∗}) and T^{∗} × C_{L}(T_{R})^{∗} ∈ Syl_{p}(L^{∗}). Now T and C_{L}(T_{R}) are strongly closed in S with respect to G, so by transfer (cf. 37.4 in [FGT]), there is a normal complement L_{0}^{∗} to C_{L}(T_{R})^{∗} in L^{∗} containing T^{∗}. Then for g ∈ G, T^{g} ≤ T^{l}C_{L}(T_{R}) for some l ∈ L, so as T^{l} is strongly closed in T^{l}C_{L}(T_{R}), we have T^{g} = T^{l} ≤ L_{0}. Thus as L = ⟨T^{G}⟩, L_{0} = L. That is T ∈ Syl_{p}(L). As L = ⟨T^{G}⟩, L = O^{p′}(L), so as T ∈ Syl_{p}(L), L = ⟨T^{L}⟩. As F^{∗}(G) = O_{p}(G), F^{∗}(L) = O_{p}(G) ∩ L = O_{p}(G) ∩ T. Thus (2i) is established.

Choose P as in (2ii). Then Aut_{F}(P) = Aut_{G}(P) = N_{G}(P)c′, where c′ : N_{G}(P) → Aut(P) is the conjugation map. As P ∈ F_{T} (L)^{f}, N_{T} (P) ∈ Syl_{p}(N_{L}(P)), so X = O^{p′}(N_{L}(P)) = ⟨N_{T} (P)^{N_{L}(P)}⟩, and Xc′ = B(P). This establishes (2ii).

Theorem 6.7. Assume F is constrained and T is strongly closed in S with respect to F. Choose G ∈ G(F) and R ≤ S such that R ⊴ G and C_{O_{p}(G)}(R) ≤ R. Set L = ⟨T^{G}⟩ and V = V_{T ∩R}. Assume Aut_{T} (T ∩ R) ∈ Syl_{p}(B(T ∩ R)) and Aut_{L}(T) ≤ Σ ≤ Γ(T) ∩ B′(T) with Σ ⊴ Aut_{F}(T), where Γ(T) = Γ_{F,F}(T). Then

(1) There exists a unique normal subgroup H = H_{Σ} of G such that T ∈ Syl_{p}(H) and Aut_{H}(T) = Σ.

(2) H = LO^{p}(K_{Σ}), where K_{Σ} is the preimage in C_{G}(R/(T ∩ R)) ∩ N_{G}(T) of Σ under its action on T via conjugation.

Proof. Observe the hypothesis of 6.6.2 are satisfied, so by 6.6.2.i, T ∈ Syl_{p}(L), and hence G = LN_{G}(T) by a Frattini argument. By hypothesis, Aut_{L}(T) ≤ Σ ≤ Γ(T) ∩ B′(T), so as T ∈ Syl_{p}(L), B(T) = Inn(T) and Σ/B(T) is a p′-group. As F = F_{S}(G), it follows from the definition of Γ(T) that for each p′-element ϕ in Σ, there is a p′-element g = g_{ϕ} ∈ N_{G}(V_{T}) such that c_{g|T} = ϕ and [C_{S}(T), g] ≤ Z(T).

Next as Σ ≤ B′(T) and Inn(T) ∈ Syl_{p}(B′(T)), [ϕ, Aut_{R}(T)] ≤ Inn(T), so [g, R] ≤ V_{T}. As [C_{S}(T), g] ≤ Z(T), [V_{T} , g] ≤ T, while as g is a p′-element, [R, g] = [R, g, g] ≤ [V_{T} , g] ≤ T. Thus [R, g] ≤ T ∩ R = T_{R}. Therefore g ∈ G_{+} = C_{G}(R/T_{R}).

Let K = N_{G_{+}}(T). We have shown that Σ is contained in the image of K under the conjugation map c_{K} : K → Aut(T). Let K_{Σ} and U_{K} be the preimages in K of Σ and a Hall p′-subgroup of Σ under c_{K}, respectively. Set K_{+} = O^{p}(K_{Σ}), U_{+} = O^{p}(U_{K}), and H = LK_{+}. As G_{+} ⊴ G, K and C_{K}(T) are normal subgroups of N_{G}(T). Then as Σ ⊴ Aut_{F}(T), K_{Σ}, and hence also K_{+}, are normal in N_{G}(T). Therefore as G = LN_{G}(T), H = LK_{+} ⊴ G.

As B(T) = Inn(T) and Σ/B(T) is a p′-group, Σ = Inn(T)Aut_{K_{+}}(T). Then as Inn(T) ≤ Aut_{L}(T) ≤ Σ, Σ = Aut_{L}(T)Aut_{K_{+}}(T) = Aut_{H}(T).

As [K, R] ≤ T_{R}, O^{p}(C_{K}(T)) ≤ C_{G}(R) = Z(R). Therefore C_{K}(T) is a p-group. Then as C_{K}(T) ⊴ N_{G}(T), we have C_{K}(T) ≤ C_{S}(T). Thus C_{U^{+}}(T) is a normal p-subgroup of U_{+}, and as [g, C_{S}(T)] ≤ Z(T), also [K_{+}, C_{K_{+}}(T)] ≤ Z(T). As U_{+} = O^{p}(U_{+}) and C_{U^{+}}(T) is a normal p-subgroup of U_{+}, we have C_{U_{+}}(T) = [C_{U_{+}}(T), U_{+}]. Therefore C_{U_{+}}(T) ≤ [C_{K}(T), U_{+}] ≤ Z(T). Then as K_{Σ} = T U_{+}C_{K}(T), T U_{+} is K_{+}-invariant and contains a Hall p′-subgroup of K_{+}, so K_{+} = O^{p}(T U_{+}). In particular T ∩K_{+} ∈ Syl_{p}(K_{+}).

As T ∈ Syl_{p}(L) and T ∩ K_{+} ∈ Syl_{p}(K_{+}), T ∈ Syl_{p}(H). Thus we have shown that H is in the set H of normal subgroups H_{1} of G such that T ∈ Syl_{p}(H_{1}) and Σ = Aut_{H_{1}}(T). Further by construction, H is describe in (2). Thus it remains to assume H_{1} ∈ H, and to show H_{1} = H.

By a Frattini argument, H_{1} = LK_{1}, where K_{1} = N_{H_{1}}(T). Let K_{0} = O^{p}(K_{1}). As T ∈ Syl_{p}(H_{1}), K_{1} ≤ T K_{0}, so H_{1} = LK_{0}, and also [R, K_{0}] ≤ R∩H_{1} = T_{R}. Thus K_{0} ≤ K. As Aut_{H_{1}}(T) = Σ, K_{0} ≤ K_{Σ}, so K_{0} = O^{p}(K_{0}) = O^{p}(H_{1} ∩ K_{+}). Hence as H_{1} ⊴ G, we have K_{0} ⊴ K_{+}. Next as Aut_{K_{1}}(T) = Σ, K_{+} ≤ K_{+}T C_{K}(T). Then as C_{K}(T) is a p-group, and K_{+} = O^{p}(K_{+}), it follows that K_{+} = K_{0}. Thus H = LK_{+} = LK_{0} = H_{1}, completing the proof of the Theorem.

We are now in a position to prove Theorem 1. Assume the hypothesis of that Theorem. Set R = O_{p}(G). As G is a model for F, C_{G}(R) ≤ R. Next E ⊴ F, so (N1) holds and E is saturated. As (N1) holds, Σ = Aut_{E}(T) ≤ Γ(T), and as E is saturated, Aut_{T} (T_{R}) ∈ Syl_{p}(B(T_{R})) and Σ ≤ B′(P). Thus to show the hypothesis of Theorem 6.7 are satisfied, it remains to show Aut_{L}(T) ≤ Σ. Let α ∈ Aut_{L}(T). We must show α ∈ Σ, and as Inn(T) ≤ Σ, we may assume α is a p′-element. Then α = c_{l|T} for some p′-element l of N_{L}(T). Now Aut_{L}(T_{R}) = B(T_{R}) ≤ Aut_{E}(T_{R}) as E ⊴ F, so β = c_{l|T_{R}} ∈ Aut_{E}(T_{R}). Then T ≤ N_{β}, so as E is saturated, there exists a p′-element γ ∈ Σ extending β. Now γ = c_{g|T} for some p′-element g ∈ N_{G}(T), and g ∈ Xl, where X = N_{G}(T) ∩ C_{G}(T_{R}). By 6.6.1, C_{T} (T_{R}) = Z(T_{R}), so X centralizes T_{R} and T/T_{R}, and hence X/C_{X}(T) is a p-group. Next [L, C_{G}(T_{R})] ≤ C_{L}(T_{R}) = Z(T_{R}), so g ∈ O^{p}(N_{L}(T)X) ≤ O^{p}(N_{L}(T))C_{X}(T). Then gl^{−1} ∈ X ∩ N_{L}(T)C_{X}(T) = (X ∩ L)C_{X}(T) = Z(T_{R})C_{X}(T), so that α ∈ γInn(T) ⊆ Σ.

Thus we have established the hypothesis of 6.7, so we can form the group H = H_{Σ} of 6.7. In particular Σ = Aut_{H}(T). Hence E and F_{T} (H) are both saturated systems on T such that Aut_{F_{T} (H)}(T) = Σ = Aut_{E}(T), so E = F_{T} (H) by 5.9. Therefore H is a model for E normal in G. Moreover by 6.7, H is the unique model. Thus the proof of Theorem 1 is complete. (6.8) Assume E is F-invariant. Then Hypotheses (N1) is satisfied iff N(D(T), E(T)) holds. Proof. First, T_{T} = T, so E(T) = N_{E}(T), and then

Aut_{E}(T) = Aut_{N_{E}(T)}(T) = Aut_{E(T)}(T).

Next, T = T ∩ V_{T}, so D(T) = N_{F}(V_{T}). Thus Aut_{D(T)}(V_{T}) = Aut_{F}(V_{T}), while by 4.2.2, E(T) ≤ D(T). Now the lemma follows. (6.9) Assume E is F-invariant and saturated, and let U ∈ F_{T}^{f}. Then

(1) D(U) is a saturated constrained fusion system on N_{S}(U), N_{T} (U) is strongly closed in N_{S}(U) with respect to D(U), and E(U) is saturated and D(U)-invariant.

(2) N_{F}(U) is saturated and N_{E}(U) is saturated and N_{F}(U)-invariant.

(3) If N(D(U), E(U)) is satisfied then E(U) ⊴ D(U). Proof. First, D(U) is saturated and constrained, N_{T} (U) is strongly closed with respect to D(U), and E(U) is D(U)-invariant by 4.3.1. Similarly by 4.3.4, N_{F}(U) is saturated and N_{E}(U) is N_{F}(U)-invariant. As E is saturated, E(U) and N_{E}(U) are saturated by 4.3.2. Thus (1) and (2) hold. Then (1) implies (3).

(6.10) Assume E ⊴ F. Then (1) For each U ∈ F_{T}^{f}, condition N(D(U), E(U)) is satisfied.

(2) For each U ∈ F_{T}^{f}, D(U) is saturated and constrained, and E(U) ⊴ D(U).

(3) If Q ∈ E^{c} then Aut_{E}(Q) ≤ Γ(Q).

Proof. Let U ∈ F_{T}^{f}. We first prove (1), so we must show N(D(U), E(U)) is satisfied. Assume otherwise and pick a counter example U such that n = |T : U| is minimal. As E ⊴ F, (N1) holds, so N(D(T), E(T)) is satisfied by 6.8. Thus n > 1. Set Q = N_{T} (U) and pick α ∈ hom_{F}(Q, S) such that P = Qα ∈ F_{T}^{f}. By 1.1.2, we may assume N_{S}(Q)α ≤ D = N_{S}(P). Let ϕ ∈ Aut_{E(U)}(Q). Then φ = ϕα^{∗} ∈ Aut_{E}(P), and as P ∈ F_{T}^{f}, φ extends to φ̃ ∈ Aut_{F}(V_{P}) by 1.1.1. Then φ̃ acts on T_{P} = T ∩ V_{P} and hence also on V_{T_{P}} = C_{V_{P}} (T_{P}). Hence φ ∈ D = D(P). By minimality of n, N(D, E(P)) is satisfied. Thus D is saturated and constrained, and E(P) ⊴ D by 6.9. Therefore by 2.5.1 we can choose a model G of D, and by Theorem 1 there is a model H for E(P) normal in G. In particular as φ ∈ Aut_{E}(P), φ = c_{h|P} for some h ∈ H. Next P = T_{P} by 6.5.3, so

[h, C_{S}(P)] = [h, C_{S}(T_{P})] ≤ C_{H}(T_{P}) = Z(T_{P}) = Z(P),

by 6.6.1. Thus φ̂ = c_{h|V_{P}} is an extension of φ to V_{P} with [φ̂, C_{S}(P)] ≤ Z(P). Now ϕ̂ = φ̂α^{−∗} ∈ Aut_{F}(Q) extends φα^{−∗} = ϕ to V_{P} α^{−1} = V_{Q} and [ϕ̂, C_{S}(Q)] ≤ Z(Q). Thus N(D(U), E(U)) is indeed satisfied, and the proof of (1) is complete. Then if Q ∈ E^{c}, we can repeat the proof above, using 6.5.2 in place of 6.5.3, to obtain (3).

Next (1) and 6.9 imply (2).

Section 7. Normal maps

In this section F is a saturated fusion system over the finite p-group S, and T is a subgroup of S strongly closed in F with respect to F. Notation 7.1. Given U ∈ F_{T}^{f}, form the fusion system D(U) = D_{F,T} (U) = N_{N_{F} (V_{T_{U}})}(U) on N_{S}(U) as in 4.1, and pick G(U) = G_{F,T} (U) ∈ G(D(U)), which exists by 4.2.3 and 2.5. Set L(U) = ⟨N_{T} (U)^{G(U)}⟩.

(7.2) Let U ∈ F_{T}^{fc} and set Q = N_{T} (U) and V = V_{U}. Then

(1) D(U) is a saturated constrained fusion system on N_{S}(U), and Q is strongly closed in N_{S}(U) with respect to D(U).

(2) U = T_{U} = V ∩ T, so D(U) = N_{F}(V).

(3) V ⊴ G(U) with C_{S}(V) ≤ C_{S}(U) ≤ V.

(4) Aut_{D(U)}(U) = Aut_{F}(U) = Aut_{G(U)}(U).

(5) There exists α ∈ hom_{F}(N_{S}(Q), S) with Qα ∈ F_{T}^{f} and N_{S}(Q)α ≤ N_{S}(Qα).

(6) Uα and Qα are in F_{T}^{fc} and α̇ : D(U) → D(Uα) is an isomorphism.

(7) The set Ext(α) of isomorphisms α̌ : G(U) → G(Uα) extending α : N_{S}(U) → N_{S}(Uα) such that α̌c_{Uα} = c_{U}α^{∗} is nonempty, where c_{X} : G(X) → Aut(X) is the conjugation map for X ∈ {U, Uα}.

(8) Let P ∈ D(U)^{f} with U ≤ P ≤ T, and let β ∈ hom_{F}(N_{S}(P), S) with P β ∈ F_{T}^{fc}. Then V ≤ N_{S}(P), C_{S}(P β) = C_{V} (P)β, D_{U,P} = N_{D(U)}(P) is a saturated constrained fusion system on X = N_{S}(U) ∩ N_{S}(P), β̇ : D_{U,P} → N_{D(P β)}(V β) = D_{P β,V β} is an isomorphism, N = N_{G(U)}(P) ∈ G(D_{U,P}), and M = N_{G(P β)}(V β) ∈ G(D_{P β,V β}).

(9) The set Ext(β) of isomorphisms β̌ : N → M which extend β : X → Xβ is nonempty.

Proof. Parts (1) and (4) follow from 4.3, applied to the F-invariant system E on T such that hom_{E}(P, T) = hom_{F}(P, T) for P ≤ T. As U ∈ F_{T}^{fc}, C_{Q}(U) ≤ U, so U = T_{U} = T ∩V. Thus U is N_{F}(V)-invariant. Hence by definition, D(U) = N_{F}(V), so V ⊴ G(U) and (2) is established. Also V = UC_{S}(U), so C_{S}(V) ≤ C_{S}(U) ≤ V. Thus (3) holds.

Part (5) is 1.1.2. As U ∈ F_{T}^{fc} and U ≤ Q, Uα and Qα are in F_{T}^{c} by 1.3. Thus Qα ∈ F_{T}^{fc} and as U ∈ F_{T}^{fc} with N_{S}(U)α ≤ N_{S}(Q)α ≤ N_{S}(Qα), also Uα ∈ F_{T}^{fc}. Then for P_{1}, P_{2} ≤ N_{S}(U), hom_{D(U)}(P_{1}, P_{2})α^{∗} = hom_{D(Uα)}(P_{1}α, P_{2}α), so from section 1, α̇ : D(U) → D(Uα) is an isomorphism, establishing (6).

By (1), D(U) is a saturated constrained fusion system, so (7) follows from 2.3 and 2.4. Assume the hypothesis of (8). Then [V, P] ≤ V ∩T = U ≤ P, so V ≤ N_{S}(P). Observe

that E = D_{U,P} is the fusion system on X such that for Y, Y′ ≤ X, hom_{E}(Y, Y′) consist of the maps in hom_{F}(Y, Y′) which extend to a member of hom_{F}(Y P V, Y′P V) acting on P and V. Then

C_{S}(P β) = C_{S}(P β) ∩ C_{S}(Uβ) = C_{S}(P β) ∩ C_{V} (U)β = C_{V β}(P β) = C_{V} (P)β,

so Eβ̇ ≤ N_{D(P β)}(V β), and the opposite inclusion is easy. By (1), D(U) is saturated, and by hypothesis P ∈ D(U)^{f}, so by 1.2.1, E = N_{D(U)}(P) is saturated. Then as V ⊴ E, E is constrained. As D(U) = F_{N_{S}(U)}(G(U)), N ∈ G(E), and similarly M ∈ G(D_{P β,V β}). Thus (8) holds. Then (9) follow from (8), 2.3, and 2.4.

(7.3) Let U ∈ F_{T}^{fc}, P ∈ D(U)^{f} with U ≤ P ≤ T, and β ∈ hom_{F}(N_{S}(P), S) with P_{0} = P β ∈ F_{T}^{fc}. Assume H ⊴ G(U) with N_{T} (U) ∈ Syl_{p}(H), and H_{0} ⊴ G_{0} = G(P_{0}) with N_{T} (P_{0}) ∈ Syl_{p}(H_{0}), such that Aut_{H_{0}}(Uβ) = Aut_{N_{H}(P)}(U)β^{∗}. Adopt the notation of 7.2.8. Then for each β̌ ∈ Ext(β), N_{H}(P)β̌ = N_{H_{0}}(Uβ).

Proof. Let H_{N} = H ∩ N, so that H_{N} = N_{H}(P). Let H_{1} = H_{N} β̌ and U_{0} = Uβ. By hypothesis, Aut_{H_{0}}(Uβ) = Aut_{H_{N}} (U)β^{∗}. Also Aut_{H_{N}} (U)β^{∗} = Aut_{H_{1}}(U_{0}), so Aut_{H_{0}}(U_{0}) = Aut_{H_{1}}(U_{0}). By 7.2.8, C_{S}(P_{0}) ≤ C_{S}(U)β ≤ N_{S}(P_{0}), so as H_{N} centralizes C_{S}(U)/Z(U), H_{1} ≤ K_{0} = C_{G_{0}}(V_{P_{0}}/P_{0}). Further from 5.2.3, C_{N_{T} (P_{0})}(U_{0}) = U_{0}, and hence C_{H_{0}}(U_{0}) = Z(U_{0}). Therefore as Aut_{H_{0}}(U_{0}) = Aut_{H_{1}}(U_{0}), for each p′-element h_{1} ∈ H_{1}, there exists h_{0} ∈ N_{H_{0}}(U_{0}) such that h_{0} = h_{1} on U_{0}. Thus x = h_{1}h_{0}^{−1} ∈ C_{G_{0}}(U_{0}), so [x, P_{0}] ≤ C_{H_{0}}(U_{0}) = Z(U_{0}). Also x ∈ H_{1}H_{0} ≤ K_{0}, so c_{x} is a p-element in Aut(V_{P_{0}}). Hence x is a p-element as C_{G_{0}}(V_{P_{0}}) ≤ O_{p}(G_{0}). Let G_{0}^{∗} = G_{0}/H_{0}. Then h_{1}^{∗} = x^{∗}, so as x^{∗} is a p-element and h_{1}^{∗} a p′-element, h_{1}^{∗} = 1, so h_{1} ∈ H_{0}. Hence O^{p}(H_{1}) ≤ H_{0}, so N_{H}(P)β̌ = H_{1} = ⟨(N_{T} (P)β)^{H_{1}}⟩O^{p}(H_{1}) ≤ H_{0}. Thus as β̌ : N → M is an isomorphism by 7.2.9, H_{1} = H_{N} β̌ ≤ H_{0} ∩ M = H_{M}. But Aut_{H_{N}} (U)β^{∗} = Aut_{H_{0}}(U_{0}), so Aut_{H_{N}} (U)β^{∗} = Aut_{H_{M}} (U_{0}). Then as C_{H}(U) = Z(U) ≤ H_{N} and C_{H_{0}}(U_{0}) = Z(U_{0}) ≤ H_{M}, H_{N} β̌ = H_{M}. Finally N_{H_{0}}(U_{0}) acts on V_{U}β as [N_{H_{0}}(U_{0}), C_{S}(U)β] ≤ [N_{H_{0}}(U_{0}), C_{S}(U_{0})] ≤ C_{H_{0}}(U_{0}) = Z(U_{0}) ≤ V_{U}β, so H_{M} = N_{H_{0}}(U_{0}), completing the proof. Definition 7.4. Define a constricted F-invariant map A on T to be normal if for each U ∈ F_{T}^{fc}:

(SA1) Aut_{S}(U) ∩ A(U) = Aut_{T} (U).

(SA2) For each P ∈ D(U)^{f} with U ≤ P ≤ Q = N_{T} (U), Aut_{A(P)}(U) = N_{A(U)}(Aut_{P} (U)).

(SA3) Each ϕ ∈ N_{A(Q)}(U) extends to ϕ̂ ∈ Aut_{D(U)}(V_{Q}) with [C_{S}(Q), ϕ̂] ≤ Z(Q).

(7.5) Assume A is a normal map on T, and set E = 𝔈(A). Then:

(1) For each P ∈ E^{fc}, Aut_{T} (P) is Sylow in A(P) and B(P).

(2) For each X ∈ E^{c} and each overgroup Y of X in N_{T} (X), Aut_{A(Y)}(X) = N_{A(X)}(Aut_{Y} (X)).

Proof. By 5.5.1, E is F-invariant. Let U ∈ F_{T}^{fc}. By 3.1.1, Aut_{S}(U)∩A(U) ∈ Syl_{p}(A(U)), while by 5.3.2, Aut_{T} (U) ≤ B(U) ≤ A(U). Thus by (SA1), Aut_{T} (U) is Sylow in A(U) and B(U).

Next let P ∈ E^{fc}. By 5.4 and 3.5, there is U ∈ F_{T}^{fc} and β ∈ iso_{F}(P, U). Then A(U) = A(P β) = A(P)β^{∗}. As P and U are fully normalized and centric in E, |Aut_{T} (P)| = |Aut_{T} (U)|, so as Aut_{T} (U) is Sylow in A(U) and B(U), Aut_{T} (P) is Sylow in A(U)β^{−∗} = A(P) and B(U)β^{−∗} = B(P). This establishes (1).

Similarly if X, Y are as in (2) there is U ∈ F_{T}^{fc} and α ∈ iso_{F}(X, U) with P = Y α ∈ D(U)_{T}^{f}. Then by (SA2),

Aut_{A(Y)}(X) = Aut_{A(P α^{−1})}(Uα^{−1}) = Aut_{A(P)}(U)α^{−∗} = N_{A(U)}(Aut_{P} (U))α^{−∗} = N_{A(Uα^{−1})}(Aut_{P α^{−1}}(Uα^{−1}) = N_{A(X)}(Aut_{Y} (X)), so that (2) holds. Remark 7.6. By 7.5, a normal map on T is a function A on the set of subgroups of T such that:

(NM1) For each P ≤ T and α ∈ hom_{F}(P, T), A(P)α^{∗} = A(P α) ≤ Aut_{F}(P α).

(NM2) For each P ≤ T, A(P) = Aut_{A(P C_{T} (P))}(P).

(NM3) For each P ∈ 𝔈(A)^{fc}, Aut_{T} (P) ∈ Syl_{p}(A(P)).

(NM4) For each U ∈ 𝔈(A)^{c} and each overgroup P of U in N_{T} (U), Aut_{A(P)}(U) = N_{A(U)}(Aut_{P} (U)).

(NM5) For each U ∈ F_{T}^{fc} and each ϕ ∈ N_{A(Q)}(U), ϕ extends to ϕ̂ ∈ hom_{F}(QC_{S}(Q), S) such that [ϕ̂, C_{S}(Q)] ≤ Z(Q), where Q = N_{T} (U).

(7.7) Let A be a constricted F-invariant map on T. Then the following are equivalent: (1) A is normal.

(2) For each U ∈ F_{T}^{fc}:

(i) A(U) = Aut_{H(U)}(U) for a normal subgroup H(U) of G(U) such that Q = N_{T} (U) ∈ Syl_{p}(H(U)).

(ii) For each P ∈ D(U)^{f} with U ≤ P ≤ T, and for each α ∈ hom_{F}(N_{S}(P), S) with P α ∈ F_{T}^{fc}, Aut_{H(P α)}(Uα) = Aut_{N_{H(U)}(P)}(U)α^{∗}.

Proof. Let U ∈ F_{T}^{fc}, Q = N_{T} (U), V = V_{U}, D = D(U), G = G(U), and L = L(U).

Assume now that (1) holds. By 7.5.1, Aut_{Q}(U) = Aut_{T} (U) is Sylow in A(U) and B(U). By parts (1)-(3) of 7.2, the hypothesis of 6.6 are satisfied with D, V, U, Q in the roles of F, R, T ∩ R, T. As Aut_{Q}(U) ∈ Syl_{p}(B(U)), the hypothesis of 6.6.2 is also satisfied. In particular Q ∈ Syl_{p}(L) by 6.6.2.i, and Aut_{L}(U) = B(U) by 6.2.2.ii. By 5.7.1, A(U) = B(U)N_{A(U)}(Aut_{Q}(U)).

Claim the hypothesis of 6.7 are satisfied by Σ = N_{A(Q)}(U). By (SA3), Σ ≤ Γ(Q) = Γ_{D,D}(Q). In particular Σ ≤ Aut_{D}(Q). By 5.3.1, A(Q) ⊴ Aut_{F}(Q), so Σ = N_{A(Q)}(U) ⊴ Aut_{D}(Q). Let B(Q) = B_{D,Q}(Q) and B′(Q) = B_{D,Q}(Q). Thus B(Q) = ⟨Inn(Q)^{Aut_{D}(Q)}⟩ = Inn(Q), and B′(Q)/Inn(Q) = O_{p′}(Aut_{D}(Q)/Inn(Q)). Pick β ∈ hom_{F}(N_{S}(Q), S) as in 7.2.8; thus Qβ ∈ F_{T}^{fc}. Then Aut_{T} (Qβ) ∈ Syl_{p}(A(Qβ)) by 7.5.1. As U ∈ F_{T}^{fc}, Qβ = N_{T} (Qβ) ∩ N_{T} (Uβ), so Inn(Qβ) is Sylow in N_{A(Qβ)}(Uβ). Then as N_{A(Q)}(U)β^{∗} = N_{A(Qβ)}(Uβ), also Inn(Q) is Sylow in N_{A(Q)}(U). Therefore Σ ≤ B′(Q).

It remains to show Aut_{L}(Q) ≤ Σ. As Inn(Q) is Sylow in Aut_{L}(Q) and contained in Σ, it suffices to show each p′-element α in Aut_{L}(Q) is in Σ. Now α = c_{l|Q} for some p′- element l ∈ N_{L}(Q), and by (SA2), c_{l|U} ∈ N_{A(U)}(Aut_{Q}(U)) = Aut_{A(Q)}(U), so c_{l|U} = γ_{|U} for some γ ∈ A(Q). Then γ = c_{g|Q} for some p′-element g ∈ N_{G}(Q). Hence g ∈ Xl, where X = N_{G}(Q) ∩ C_{G}(U). Next [L, C_{G}(U)] ≤ C_{L}(U) ≤ U, so g ∈ O^{p}(N_{L}(Q)X) ≤ O^{p}(N_{L}(Q))C_{G}(Q). Then gl^{−1} ∈ X ∩ LC_{G}(Q) ≤ C_{L}(U)C_{G}(Q) ≤ UC_{G}(Q), so α ∈ γAut_{U}(Q) ⊆ Σ. This completes the proof of the claim.

By the claim and 6.7, there is a normal subgroup H = H(U) of G such that L = O^{p′}(H) and Aut_{H}(Q) = Σ. As Q is Sylow in L = O^{p′}(H), Q is Sylow in H, and by a Frattini argument, H = LN_{H}(Q), so Aut_{H}(U) = Aut_{L}(U)Aut_{N_{H}(Q)}(U).

Next by (SA2) and as Aut_{H}(Q) = Σ = N_{A(Q)}(U),

N_{A(U)}(Aut_{Q}(U)) = Aut_{A(Q)}(U) = Aut_{Σ}(U) = Aut_{N_{H}(Q)}(U),

so that

A(U) = B(U)N_{A(U)}(Aut_{Q}(U)) = Aut_{L}(U)Aut_{N_{H}(Q)}(U) = Aut_{H}(U).

That is (2i) holds.

Let P ∈ D(U)^{f} with U ≤ P ≤ T, and pick α ∈ hom_{F}(N_{S}(P), S) with P α ∈ F_{T}^{fc}. By (SA2), Aut_{A(P)}(U) = N_{A(U)}(Aut_{P} (U)), while by (2i), A(U) = Aut_{H(U)}(U), so

N_{A(U)}(Aut_{P} (U)) = N_{Aut_{H(U)}(U)}(Aut_{P} (U)) = Aut_{N_{H(U)}(P)}(U).

By (2i), Aut_{A(P α)}(Uα) = Aut_{H(P α)}(Uα). Further as A is a normal map,

Aut_{A(P α)}(Uα) = Aut_{A(P)}(U)α^{∗} = N_{A(U)}(Aut_{P} (U))α^{∗} = Aut_{N_{H(U)}(P)}(U)α^{∗},

so (2ii) holds. Thus we have shown that (1) implies (2).

Next assume (2) and now set H = H(U). By (2i), Q ∈ Syl_{p}(H) and A(U) = Aut_{H}(U), so Aut_{Q}(U) is Sylow in A(U). Thus (SA1) holds.

Let Σ = Aut_{H}(Q). As [C_{S}(Q), N_{H}(Q)] ≤ C_{H}(Q) = Z(Q), Σ ≤ Γ(Q). Choose β as in 7.2.8 with Q is the role of P, choose β̌ as in 7.2.9, and let Q_{0} = Qβ, U_{0} = Uβ, and H_{0} = H(Q_{0}). Thus A(Q_{0}) = Aut_{H_{0}}(Q_{0}), so N_{A(Q_{0})}(U_{0}) = N_{Aut_{H_{0}} (Q_{0})}(U_{0}) = Aut_{N_{H_{0}} (U_{0})}(Q_{0}). By (2ii) and 7.3, N_{H}(Q)β̌ = N_{H_{0}}(U_{0}), so

N_{A(Q)}(U) = N_{A(Q_{0})}(U_{0})β^{−∗} = Aut_{N_{H_{0}} (U_{0})}(Q_{0})β^{−∗} = Aut_{N_{H}(Q)β̌}(Q_{0})β^{−∗} = Aut_{N_{H}(Q)}(Q) = Aut_{H}(Q) = Σ.

Thus (SA3) holds.

Choose P and α as in (2ii). By (2ii), Aut_{H(P α)}(Uα) = Aut_{N_{H(U)}(P)}(U)α^{∗}. By (2i), A(U) = Aut_{H(U)}(U) and A(P α) = Aut_{H(P α)}(P α). Thus

Aut_{A(P α)}(Uα) = Aut_{H(P α)}(Uα) = Aut_{N_{H(U)}(P)}(U)α^{∗} = N_{Aut_{H(U)}(U)}(Aut_{P} (U))α^{∗} = N_{A(U)}(Aut_{P} (U))α^{∗}.

But as A is an invariant map,

Aut_{A(P)}(U)α^{∗} = Aut_{A(P)α^{∗}}(P α) = Aut_{A(P α)}(Uα),

so Aut_{A(P)}(U) = N_{A(U)}(Aut_{P} (U)). That is (SA2) holds, and hence (2) implies (1).

(7.8) Let A be a normal map on T, let U ∈ 𝔈(A)^{c}, and P an overgroup of U in T. Then C_{A(P)}(U) = Aut_{Z(U)}(P).

Proof. By 5.5.1, E = 𝔈(A) is F-invariant. By 1.3 and 3.5, there exists β ∈ hom_{F}(P, T) with P β ∈ F_{T}^{fc} and Uβ ∈ E^{c}. If P β, Uβ satisfy the lemma, then

C_{A(P)}(U) = C_{A(P β)}(Uβ)β^{−∗} = Aut_{Z(Uβ)}(P β)β^{−∗} = Aut_{Z(U)}(P),

so the lemma holds. Thus we may assume P ∈ F_{T}^{fc}. Choose H = H(P) as in 7.7.2; then F^{∗}(H) = O_{p}(H), N_{T} (P) ∈ Syl_{p}(H), and Aut_{H}(P) = A(P). As U ∈ E^{c}, C_{T} (U) ≤ U. Thus as F^{∗}(H) = O_{p}(H), N_{T} (P) ∈ Syl_{p}(H), C_{H}(U) = Z(U) by the Thompson (A×B)- Lemma (cf. 24.2 in [FGT]). Therefore Aut_{Z(U)}(P) is the kernel of the conjugation map

c : N_{A(P)}(U) = N_{Aut_{H}(P)}(U) → Aut_{Aut_{H}(P)}(U) = Aut_{A(P)}(U).

That is Aut_{Z(U)}(P) = C_{A(P)}(U).

(7.9) Let A be a normal map on T, E = 𝔈(A), U ∈ F_{T}^{fc}, and Q = N_{T} (U). Then F_{Q}(H(U)) ≤ E(U).

Proof. Let H = H(U) and H = F_{Q}(H). Then H is a saturated fusion system, so H = ⟨Aut_{H}(P) : P ∈ H^{fc}⟩ by Alperin’s Fusion Theorem (cf. A.10 in [BLO]). Indeed as U ⊴ H, Aut_{H}(P) = Aut_{Aut_{H}(P U)}(P), so H = ⟨Aut_{H}(P) : P ∈ H^{fc} and U ≤ P ⟩. Then by 3.5,

H = ⟨Aut_{H}(P φϕ) : U ≤ P ∈ D(U)_{T}^{fc}, φ ∈ Aut_{G}(T), ϕ ∈ hom_{H}(P φ, T)⟩.

If Aut_{H}(P) ≤ E(U) then also Aut_{H}(P φϕ) ≤ E(U) as E(U) is D(U)-invariant. Thus it suffices to show Aut_{H}(P) ≤ E(U) for U ≤ P ∈ D(U)_{T}^{fc}.

We apply parts (8) and (9) of 7.2 to β ∈ hom_{F}(N_{S}(P), S) with P β ∈ F_{T}^{fc}, obtaining the isomorphism β̌ : N = N_{G(U)}(P) to M = N_{G(P β)}(V_{U}β). By 7.7.ii, Aut_{H(P β)}(Uβ) = Aut_{N_{H}(P)}(U)β^{∗}, so by 7.3, N_{H}(P)β̌ = N_{H(P β)}(Uβ) = H_{0}. Next from 7.2.9, β̌c_{P β} = c_{P} β^{∗}, so for h ∈ N_{H}(P),

c_{h}β^{∗} = hc_{P} β^{∗} = hβ̌c_{P β} = c_{hβ̌} ∈ Aut_{H_{0}}(P β) ≤ Aut_{E}(P β),

and hence c_{h} ∈ Aut_{E}(P β)β^{−∗} = Aut_{E}(P). Thus indeed Aut_{H}(P) ⊆ E(U), completing the proof.

Theorem 7.10. Assume A is a normal map on T, and set E = 𝔈(A). Let V ∈ E^{c}. Then

(1) Each α ∈ hom_{E}(V, T) extends to a member of hom_{E}(N_{α}, T).

(2) A(V) = Aut_{E}(V).

Assume the Theorem is false, and let n be minimal subject to the condition that for some V ∈ E^{c} with n = |T : V |, either A(V) ≠ Aut_{E}(V), or some member ϕ of hom_{E}(V, T) does not extend to a member of hom_{E}(N_{ϕ}, T). Pick V ∈ E^{c} such that n = |T : V |, pick α ∈ hom_{E}(V, T), and set U = V α. The proof involves a series of reductions.

(7.11) n > 1.

Proof. Suppose n = 1. Then T = V = U = N_{α}, so 7.10.1 holds. Also 7.10.2 follows from 5.5.2. This contradicts the choice of n.

(7.12) If there exists V_{1} ≤ N_{α} such that V < V_{1} and α extends to α_{1} ∈ hom_{E}(V_{1}, T) then α extends to a member of hom_{E}(N_{α}, T).

Proof. Let U_{1} = V_{1}α_{1}. By minimality of n, α_{1} extends to β ∈ hom_{E}(N_{α_{1}}, T). Let V_{1} ⊴ V_{2} ≤ N_{α}. Claim V_{2} ≤ N_{α_{1}}.

Let x ∈ V_{2} and write c_{x} for conjugation by x on V_{1}. Then ψ = c_{x}α_{1}^{∗} ∈ Aut_{E}(U_{1}) and

ψ_{|U} = c_{x}α_{1}^{∗}res_{U}^{U_{1}} = c_{x}res_{V}^{V_{1}}α^{∗} ∈ Aut_{T} (U)

acts on Aut_{U_{1}}(U) as α_{1}^{∗}res_{U}^{U_{1}} = res_{V}^{V_{1}}α^{∗} on Aut_{V_{2}}(V_{1}). Thus ψ_{|U} = c_{t|U} for some t ∈ N_{T} (U) ∩ N_{T} (U_{1}), since U is centric so the kernel of the map N_{T} (U) → Aut_{T} (U) is Z(U) ≤ U_{1}. By minimality of n, Aut_{E}(U_{1}) = A(U_{1}), and as A is an invariant map, Aut_{T} (U_{1}) ≤ A(U_{1}). Therefore ψ and c_{t} are in A(U_{1}) with c_{t} = ψ on U, so it follows from 7.8 that ψc_{t}^{−1} ∈ C_{A(U_{1})}(U) = Aut_{Z(U)}(U_{1}), and hence ψ ∈ Aut_{T} (U_{1}). This establishes the claim.

As V_{2} ≤ N_{α_{1}} and α_{1} extends to β on N_{α_{1}}, α_{2} = β_{|V_{2}} ∈ hom_{E}(V_{2}, T) extends α_{1}. Continuing in this manner, α extends to α̂ ∈ hom_{E}(N_{α}, T), completing the proof of the lemma.

(7.13) Assume U ∈ E^{fc}. Then:

(1) There exists χ ∈ A(U) such that Aut_{T} (V)α^{∗}χ^{∗} ≤ Aut_{T} (U).

(2) Let β = αχ. If there exists V_{1} ≤ N_{β} such that V < V_{1} and β extends to β_{1} ∈ hom_{E}(V_{1}, T), then β extends to a member of hom_{E}(N_{T} (V), N_{T} (U)).

Proof. Part (1) follows as Aut_{T} (V)α^{∗} ≤ A(U) and Aut_{T} (U) ∈ Syl_{p}(A(U)) by 7.5.1. In particular N_{β} = N_{T} (V), so (2) follows from 7.12.

As E = 𝔈(A), it follows from 5.5.6 that there exist P_{i} ≤ T, 1 ≤ i ≤ m, V_{i} ≤ P_{i}, and ϕ_{i} ∈ A(P_{i}) such that P_{i}φ_{i} ∈ F_{T}^{fc} for some φ_{i} ∈ Aut_{F}(T), V = V_{1}, V_{i+1} = V_{i}ϕ_{i}, U = V_{m}ϕ_{m}, and α = ϕ_{1} ⋯ ϕ_{m}.

(7.14) (1) ϕ_{m} extends to a member of hom_{E}(N_{ϕ_{m}}, T).

(2) Either V_{m} < N_{ϕ_{m}}, or U = V_{m} = P_{m} and there exists ζ ∈ A(U) such that ϕ_{m}ζ extends to a member of Aut_{E}(N_{T} (U)).

(3) If U ∈ E^{fc} then there exists χ_{m} ∈ A(U) such that β_{m} = ϕ_{m}χ_{m} extends to β̂ ∈ hom_{E}(N_{T} (V_{m}), N_{T} (U)).

Proof. Let ϕ = ϕ_{m}. By construction there is φ ∈ Aut_{F}(T) such that P = P_{m}φ ∈ F_{T}^{fc}. By 7.7, A(P) = Aut_{H}(P) for a suitable normal subgroup H of G(P). In particular ϕφ^{∗} = c_{h|P} for some h ∈ H.

Suppose first that P_{m} = V_{m}. Then V_{m} = P_{m} = U, and ϕ̃ = c_{h|N_{ϕφ}} : N_{ϕφ} → Aut_{T} (P) is an E-map by 7.9. Thus ϕ̂ = ϕ̃φ^{−∗} ∈ hom_{E}(N_{ϕ}, T), and ϕ̂ extends ϕ. Thus (1) holds in this case, while (2) and (3) hold with ζ = χ_{m} = 1.

Thus we may assume V_{m} < P_{m}, so that V_{m} < W = N_{P_{m}}(V_{m}). Now (c_{h|W φ})φ^{−∗} ∈ hom_{E}(W, T) extends ϕ to W, so the lemma follows in this case from 7.12 and 7.13 applied to V_{m}, ϕ_{m} in place of V, α.

(7.15) Assume U ∈ E^{fc}. Then for each 1 ≤ k ≤ m there exists χ_{k} ∈ A(U) such that β_{k} = ϕ_{k} ⋯ ϕ_{m}χ_{k} extends to β̂ ∈ hom_{E}(N_{T} (V_{k}), N_{T} (U)).

Proof. Assume not and choose a counter example with m − k minimal. Then m − k > 0 by 7.14.3. Let μ = ϕ_{k+1} ⋯ ϕ_{m}. By minimality of m − k, there exists χ ∈ A(U) and β ∈ hom_{E}(N_{T} (V_{k+1}), T) such that β extends μχ. Applying 7.14.1 to ϕ_{1} ⋯ ϕ_{k} in the role of α, ϕ_{k} extends to γ ∈ hom_{E}(N_{ϕ_{k}}, T). By 7.14.2, either V_{k} < N_{ϕ_{k}}, or V_{k} = P_{k} and there is ζ ∈ A(V_{k}) such that ϕ_{k}ζ extends to γ ∈ Aut_{E}(N_{T} (V_{k})). In the latter case ζμ = μξ, where ξ = ζμ^{∗} ∈ A(V_{k})μ^{∗} = A(U), and γβ extends ϕ_{k}μξχ to N_{T} (V_{k}). Here we let β̂ = γβ and χ_{k} = ξχ. In the former case as V_{k} < N_{ϕ_{k}}, it follows from 7.13 applied to V_{k}, ϕ_{k}μχ in place of V, α, that there exists ρ ∈ A(U) such that ϕ_{k}μχρ extends to β̂ ∈ hom_{E}(N_{T} (V_{k}), T). In this case let χ_{k} = χρ; then β̂ extends ϕ_{k}μχ_{k} = ϕ_{k} ⋯ ϕ_{m}χ_{k}, completing the proof.

(7.16) Assume U ∈ E^{fc}. Then for each V ∈ U^{E} and α ∈ hom_{E}(V, U), there exists χ ∈ A(U) such that αχ extends to a member of hom_{E}(N_{T} (V), N_{T} (U)).

Proof. This is immediate from 7.15 applied when k = 1.

(7.17) Assume U ∈ E^{fc}. Then (1) Aut_{E}(U) = A(U).

(2) Each α ∈ A(U) extends to a member of hom_{E}(N_{α}, T).

Proof. Let α ∈ Aut_{E}(U) and set Q = N_{T} (U). By 7.16 there exists χ ∈ A(U) and γ ∈ N_{Aut_{E}(Q)}(U) such that γ extends αχ. By minimality of n, γ ∈ A(Q), so γ ∈ N_{A(Q)}(U). By (SA2), Aut_{A(Q)}(U) = N_{A(U)}(Aut_{Q}(U)), so β = γ_{|U} ∈ A(U) acts on Aut_{Q}(U). Then α = βχ^{−1} ∈ A(U), establishing (1).

Next by 7.7, A(U) = Aut_{H}(U) for a suitable normal subgroup H of G = G(U). Thus α = c_{h|U} for some h ∈ H. Now N_{α} = {g ∈ Q : c_{g}α^{∗} ∈ Aut_{Q}(U)} and c_{g}α^{∗} = c_{g}(c_{h|U})^{∗} = c_{g^{h}}, so

N_{α} = {g ∈ Q : c_{g^{h}} ∈ Aut_{Q}(U)} = {g ∈ Q : g^{h} ∈ Q} = Q^{h^{−1}} ∩ Q

as ker(c) = C_{H}(U) = Z(U) ≤ Q. Thus c_{h} : N_{α} = Q ∩ Q^{h^{−1}} → Q extends α to N_{α}, and is an E-map by 7.9. Thus (2) holds.

We are now in a position to complete the proof of Theorem 7.10.

We first prove 7.10.2. Pick α so that U ∈ E^{fc}. Then by 7.17.1, A(U) = Aut_{E}(U), so as A is F-invariant,

A(V) = A(Uα^{−1}) = A(U)α^{−∗} = Aut_{E}(U)α^{−∗} = Aut_{E}(Uα^{−1}) = Aut_{E}(V).

So 7.10.2 is established.

Next we prove 7.10.1 by applying Lemma 2.3 in [BCGLO] (which we refer to in this paragraph as “the Lemma”) to E and H = {W ∈ E^{c} : |T : W | ≤ n} in the roles of the system F and the set H of the Lemma. First by 7.17.2, condition (IIB)_{H} of the Lemma holds. Second by 7.16, condition (IIA)_{H} of the Lemma holds. Therefore by part (b) of the Lemma, condition (II)_{H} of the Lemma holds. But as the members of E^{c} are fully centralized in E, condition (II)_{H} is precisely 7.10.1. Thus the proof of Theorem 7.10 is complete.

Next we supply a proof of Theorem 3. Note that the hypotheses of that Theorem are the hypotheses of this section.

By Theorem A in [BCGLO], a fusion system E on T is saturated iff the following two conditions hold:

(i) E = ⟨Aut_{E}(U) : U ∈ E^{c}⟩.

(ii) The saturation axioms are satisfied at each member of E^{c}.

Let A = Aut_{E,c}. Assume condition (1) of Theorem 3 holds. Then E is saturated, so (i) and (ii) hold. As E is F-invariant, A is a F-invariant map by 5.2.2. Now for U ∈ E^{c}, Aut_{E}(U) = A(U) from 5.2.3. Thus by (i) and 5.5.6, E = 𝔈(A). By (ii):

(a) For each U ∈ E^{fc}, Aut_{T} (U) ∈ Syl_{p}(Aut_{E}(U)), and

(b) for each U ∈ E^{c} and each φ ∈ hom_{E}(U, T), φ extends to φ̂ ∈ hom_{E}(N_{φ}, T).

Let U ∈ F_{T}^{fc} and set Q = N_{T} (U). Observe that (a) implies (SA1). Let P ∈ D(U)_{T}^{f} with U ≤ P. If φ ∈ Aut_{A(P)}(U) then φ ∈ Aut_{E}(U) = A(U) and φ acts on Aut_{P} (U), so Aut_{A(P)}(U) ≤ N_{A(U)}(Aut_{P} (U)). Conversely if φ ∈ N_{A(U)}(Aut_{P} (U)) then P ≤ N_{φ}, so by (b), φ lifts to φ̂ ∈ hom_{E}(P, T). As φ acts on Aut_{P} (U) and Z(U) = C_{T} (U), φ̂ acts on P. Thus φ̂ ∈ Aut_{E}(P) = A(P), so φ ∈ Aut_{A(P)}(U). Therefore (SA2) holds.

By 6.10.3, N_{A(Q)}(U) = Aut_{E}(Q) ∩ N_{Aut(Q)}(U) ≤ Γ(Q), so (SA3) is satisfied, completing the proof that condition (1) of Theorem 3 implies condition (2) of that Theorem.

Trivially, (2) implies (3). Further (3) and (4) are equivalent by 7.7 and 5.5.6.

Finally assume condition (3) of Theorem 3 holds. As A is an invariant map and E = 𝔈(A), E is F-invariant by 5.5.1.

By 7.10.2, A(V) = Aut_{E}(V) for each V ∈ E^{c}, so as E = 𝔈(A), condition (i) of Theorem A in [BCGLO] is satisfied by 5.5.6. We next verify condition (ii), or equivalently, verify conditions (a) and (b). Condition (a) follows from 7.5.1, while condition (b) follows from 7.10.1. Thus E is saturated. Finally (SA3) at U = T implies (N1). Thus E ⊴ F, so indeed (3) implies (1).

This completes the proof of Theorem 3.

We next prove Theorem 2. Assume the hypothesis of that Theorem. Thus F is a saturated fusion system on S, and under each of the three conditions of the Theorem, E is a F-invariant subsystem of F on a subgroup T of S. In particular the hypothesis of this section are satisfied. Pick U ∈ F_{T}^{f} and let Q = N_{T} (U).

Suppose first that condition (1) of Theorem 2 holds. Then by definition, E is saturated, so E = ⟨Aut_{E}(P) : P ≤ T ⟩ by Alperin’s Fusion Theorem. By 6.10.2, D(U) is saturated and constrained, and E(U) ⊴ D(U). Thus condition (3) of Theorem 2 holds by Theorem 1 applied to the module G(U) of D(U). By 6.3, (3) implies (2).

Finally assume condition (2) of Theorem 2. We wish to show (2) implies (1). For V ≤ T, let A(V) = Aut_{E}(V). By hypothesis, E is F-invariant and E = 𝔈(A). By 5.2.3, A is an invariant map. Further A(U) = Aut_{D(U)}(U) by 4.3.3, so as T_{U} ⊴ D(U), A(U) = Aut_{A(T_{U})}(U). Therefore A is constricted, and we assume U ∈ F_{T}^{fc}. We will verify condition (4) of Theorem 3, and hence conclude from Theorem 3 that (1) holds.

By 7.2.1, D(U) is a saturated constrained fusion system on N_{S}(U), and Q is strongly closed in N_{S}(U) with respect to D(U). Then as E(U) ⊴ D(U), Theorem 1 says that E(U) = F_{Q}(H(U)) for a normal subgroup H(U) of G(U) such that Q ∈ Syl_{p}(H(U)). In particular, A(U) = Aut_{H(U)}(U). Thus by 5.5.6 and earlier remarks,

E = 𝔈(A) = ⟨A(Uφ) : U ∈ F_{T}^{fc}, φ ∈ Aut_{F}(T)⟩

so that condition (4ii) of Theorem 3 holds. Thus it remains only to verify condition (4i).

Let P ∈ D(U)^{f} with U ≤ P ≤ T, and β ∈ hom_{F}(N_{S}(P), S) with P β ∈ F_{T}^{fc}. Set G_{0} = G(P β), H_{0} = H(P β), and N_{H} = N_{H(U)}(P). By 7.2.8, β : D_{U,P} → D_{P β,V_{U} β} is an isomorphism, N = N_{G(U)}(P) ∈ G(D_{U,P}), and M = N_{G(P β)}(V_{U}β) ∈ G(D_{P β,V_{U} β}). As E is F-invariant, Aut_{E(U)}(P)β^{∗} ≤ Aut_{E(P β)}(P β). Then as E(U) = F_{Q}(H(U)) and E(P β) = F_{N_{T} (P β)}(H_{0}), Aut_{N_{H}} (P)β^{∗} ≤ Aut_{H_{0}}(P β). Thus restricting to U, also Aut_{N_{H}} (U)β^{∗} ≤ Aut_{H_{0}}(Uβ).

Next [N_{H_{0}}(Uβ), C_{S}(U)β] ≤ C_{H_{0}}(Uβ) = Z(Uβ) as C_{T} (Uβ) = Z(Uβ). Thus if g ∈ N_{H_{0}}(Uβ) then g acts on V_{U}β, so φ = g_{|Uβ} is a E-map in D_{P β,V_{U} β} = D_{U,P} β^{∗}. Thus φβ^{−∗} ∈ Aut_{N_{H(U)}(P)}(U), and hence Aut_{N_{H(U)}(P)}(U)β^{∗} = Aut_{H(P β)}(Uβ), completing the verification of condition (4i) of Theorem 3.

Recall this also completes the proof of Theorem 2.

Before abandoning this line of inquiry, observe that we have essentially shown:

(7.18) Assume E ⊴ F, U ∈ F_{T}^{fc}, P ∈ D(U)^{f} with U ≤ P ≤ T, and β ∈ hom_{F}(N_{S}(P), S) with P β ∈ F_{T}^{fc}. Then

β : N_{N_{E}(U)}(P) = N_{E(U)}(P) → N_{E(P β)}(Uβ) = N_{N_{E}(P β)}(Uβ)

is an isomorphism, N_{H(U)}(P) ∈ G(N_{E(U)}(P)), N_{H(P β)}(Uβ) ∈ G(N_{E(P β)}(Uβ)), and for each β̌ as in 7.2.9, β̌ : N_{H(U)}(P) → N_{H(P β)}(Uβ) is an isomorphism.

Proof. By 7.3, β̌ : N_{H(U)}(P) → N_{H(P β)}(Uβ) is an isomorphism. As F_{Q}(H(U)) = E(U), F_{N_{Q}(P)}(N_{H(U)}(P)) = N_{E(U)}(P), so N_{H(U)}(P) ∈ G(N_{E(U)}(P)). Similarly N_{H(P β)}(Uβ) ∈ G(N_{E(P β)}(Uβ)), so as β̌ is an isomorphism, so is β : N_{E(U)}(P) → N_{E(P β)}(Uβ).

As U ∈ F_{T}^{fc}, T_{U} = U, so E(U) = N_{N_{E}(T_{U})}(U) = N_{E}(U), and similarly E(P β) = N_{E}(P β). Example 7.19. Here is an example which shows the hypothesis in Theorem 1 that F^{∗}(G) = O_{p}(G) cannot be removed. Assume G is a finite group, S ∈ Syl_{2}(G), and T is a strongly closed abelian subgroup of S contained in Z(S). Then M = N_{G}(T) controls fusion in S. Set E = F_{T} (T). Then E ≤ F and as T is abelian, E consists only of inclusion maps. Therefore E is F-invariant. Let P ≤ T. Then Aut_{E}(P) = 1, so trivially E is saturated and (N1) is satisfied, and hence E ⊴ F.

However there are plenty of examples of this set up in which T is not normal in G. For example take G to be simple and S = T an abelian Sylow group of G. To get examples where T is proper in S, take G simple and S cyclic with |S| > p, and T = Ω_{1}(S). Or take p = 2, G = Sz(2^{n}) or U_{3}(2^{n}), and T = Ω_{1}(S).

On the other hand these examples are a bit deceiving, since F is isomorphic to F_{S}(M), and T is normal in M.

Here is a different sort of example: Take Ĝ to be the split extension of the natural module V for G = Sz(2^{n}) or U_{3}(2^{n}) by G, take T as above, and set T̂ = T V. Then T̂ is strongly closed in Ŝ = SV with respect to Ĝ, but is not Sylow in any normal subgroup of Ĝ. Thus by Theorem 1, there is no normal subsystem of F̂ = FŜ(Ĝ) on T̂. This example shows that even when F̂ is a saturated constrained fusion system, there can be a strongly closed subgroup T̂ of Ŝ in F̂ such that there exists no normal subsystem of F̂ on T̂.

Section 8. Factor systems In this section F is a fusion system over the finite p-group S.

(8.1) Assume α = (α_{0}, α_{P,Q} : P, Q ≤ S) : F → F̃ is a a morphism of fusion systems. Then

(1) T = ker(α_{0}) is strongly closed in S with respect to F.

(2) For P ≤ S define the kernel of α_{P} = α_{P,S} : hom_{F}(P, S) → homF̃(P α_{0}, S̃) to consist of those ϕ such that ϕα_{P} = 1_{P α_{0}} is the identity map on P α_{0}. Then:

(i) ϕ ∈ ker(α_{P}) iff for all x ∈ P, [x, ϕ] = x^{−1}(xϕ) ∈ T.

(ii) If P, Q ≤ T then hom_{F}(P, Q) ≤ ker(α_{P,Q}).

Proof. Write α for α_{0} and regard S^{+} = S/T as a subgroup of S̃, and α as the natural map α : s ↦ sT. For s ∈ S, let s^{+} = sT, and for P ≤ S and ϕ ∈ hom_{F}(P, S), write ϕ^{+} for ϕα_{P}. Observe that

(*)  s^{+}ϕ^{+} = (sα)(ϕα_{P}) = (sϕ)α = (sϕ)^{+}.

Suppose P ≤ T. Then by (*),

(**)  P^{+}ϕ^{+} = P αϕ^{+} = P ϕα = (P ϕ)^{+}.

But as P ≤ T = ker(α), P α = 1, so by (**), P ϕα = P αϕ^{+} = 1ϕ^{+} = 1, and hence P ϕ ≤ ker(α) = T. Thus (1) holds. Also ϕ^{+} is the identity map on P α = 1, so (2ii) holds.

For general P, ϕ^{+} = 1_{P^{+}} iff for each x ∈ P, x^{+} = x^{+}ϕ^{+} = (xϕ)^{+} iff x^{−1}(xϕ) ∈ T, so (2i) holds.

Assume T ≤ S is strongly closed in S with respect to F and set N = N_{F}(T). We define a category F^{+} and a morphism θ : N → F^{+}.

Set S^{+} = S/T and let θ : S → S^{+} be the natural map θ : x ↦ x^{+} = xT. The objects of F^{+} are the subgroups of S^{+}. For P ≤ S and α ∈ hom_{N} (P, S) define α^{+} : P^{+} → S^{+} by x^{+}α^{+} = (xα)^{+}. This is well defined as T is strongly closed in S with respect to F. Now define

hom_{F^{+}}(P^{+}, S^{+}) = {β^{+} : β ∈ hom_{N} (P T, S)},

and define θ_{P} : hom_{N} (P, S) → hom_{F^{+}}(P^{+}, S^{+}) by αθ_{P} = α^{+}. For α ∈ hom_{N} (P, S), α extends to α̂ ∈ hom_{N} (P T, S) and α̂^{+} = α^{+}, so θ_{P} is well defined and θ_{T P} is surjective.

Suppose α ∈ hom_{N} (P, Q) and β ∈ hom_{N} (Q, S). Then for x ∈ P,

(x^{+}α^{+})β^{+} = (xα)^{+}β^{+} = ((xα)β)^{+} = (x(αβ))^{+} = x^{+}(αβ)^{+},

so

(!)  α^{+}β^{+} = (αβ)^{+}

By (!), F^{+} is a category and θ : F → F^{+} is a functor.

Observe x^{+} ∈ ker(α^{+}) iff 1 = x^{+}α^{+} = (xα)^{+} iff xα ∈ T iff x ∈ T iff x^{+} = 1. So the members of hom_{F^{+}}(P^{+}, Q^{+}) are monomorphisms. Similar arguments show:

(8.2) F^{+} is a fusion system on S^{+}.

Moreover from discussion above:

(8.3) θ is a surjective morphism of fusion systems.

For P ≤ S, write C_{+}(P) for C_{S}(P^{+}) = {s ∈ S : [s, P] ≤ T }.

(8.4) Let α : F → F̃ be a surjective morphism of fusion systems, T = ker(α) the kernel of α : S → S̃, and P an overgroup of T in S. Then

(1) N_{S}(P)α = NS̃(P α).

(2) P is fully normalized in F iff P α is fully normalized in F̃.

(3) CS̃(P α) = C_{+}(P)α and P α is fully centralized in F̃ iff |C_{+}(P)| ≥ |C_{+}(P′)| for all P′ ∈ P^{F}.

(4) α : Aut_{F}(P) → AutF̃(P α) has kernel

C_{Aut_{F} (P)}(P^{+}) = {ϕ ∈ Aut_{F}(P) : [ϕ, P] ≤ T }.

(5) For s ∈ N_{S}(P), c_{s}α = c_{sα} ∈ AutF̃(P α), so Aut_{S}(P)α = AutS̃(P α).

(6) Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) iff AutS̃(P α) ∈ Syl_{p}(AutF̃(P α)) and Aut_{C_{+}(P)}(P) ∈ Syl_{p}(C_{Aut_{F} (P)}(P^{+})).

(7) For β ∈ hom_{F}(P, P β), N_{β}α = N_{βα}.

Proof. Part (1) follows from the homomorphism theorems as T = ker(α) ≤ P.

By (1), |NS̃(P α)| = |N_{S}(P) : T |. Further for ϕ ∈ hom_{F}(P, S), (1) says that N_{S}(P ϕ)α = NS̃((P ϕ)α) = NS̃((P α)(ϕα)), and then that |NS̃((P α)(ϕα))| = |N_{S}(P ϕ) : T |. Then as α is surjective, (2) follows.

The first statement in (3) is a consequence of the definition of the notation C_{+}(P), and implies the second.

Part (4) follows from 8.1.2. For x ∈ P and s ∈ N_{S}(P),

(xα)(c_{s}α) = (xc_{s})α = (x^{s})α = (xα)^{sα} = (xα)c_{sα},

so (5) holds. Part (6) follows from (4) and (5).

Assume the hypothesis of (7). For x ∈ N_{S}(P),

(c_{x}β^{∗})α = (β^{−1}c_{x}β)α = (βα)^{−1}(c_{x}α)(βα) = (βα)^{−1}c_{xα}(βα) = c_{xα}(βα)^{∗},

using (5). This fact together with (1) and another application of (5) shows that

N_{βα} = {xα ∈ NS̃(P α) : c_{xα}(βα)^{∗} ∈ AutS̃((P α)(βα))} = {x ∈ N_{S}(P) : (c_{x}β^{∗}) ∈ Aut_{S}(P β)}α = N_{β}α,

so (7) holds.

(8.5) Assume α : F → F̃ is a surjective morphism of fusion systems, such that F is saturated. Then F̃ is a saturated fusion system.

Proof. Suppose P̃ ≤ S̃ is fully normalized in F̃, and let P be the preimage of P̃ in S under α. Thus P̃ = P α. By 8.4.2, P is fully normalized. Let ϕ̃ ∈ isoF̃(Q̃, P̃). Again as α is surjective there is Q ≤ S and ϕ ∈ iso_{F}(Q, P) with ϕα = ϕ̃. As F is saturated, by 1.1.2 we may assume ϕ extends to φ ∈ hom_{F}(N_{S}(Q), N_{S}(P)). Then φα extends ϕ̃, and by 8.4.1, φα ∈ homF̃(NS̃(Q̃), NS̃(P̃)). Then as C_{+}(X) ≤ N_{S}(X) for X ≤ S, it follows from 8.4.3 that

CS̃(Q̃)(φα) = (C_{+}(Q)α)(φα) = (C_{+}(Q)φ)α ≤ C_{+}(P)α = CS̃(P̃),

so P̃ is fully centralized.

As P ∈ F^{f} and F is saturated, Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)). Then AutS̃(P̃) ∈ Syl_{p}(AutF̃(P̃)) by 8.4.6. Therefore Axiom I for saturation holds. We now verify axiom II: Assume T ≤ P and β̃ ∈ homF̃(P̃, P̃β̃) with P̃β̃ = Q̃ fully centralized. Let Q be the preimage of Q̃ in S under α and β ∈ hom_{F}(P, Q) with βα = β̃. As C_{S}(Q) ≤ C_{+}(Q), Q is fully centralized in F, using 8.4.3. Thus as F is saturated, there exists an extension γ of β with γ ∈ hom_{F}(N_{β}, N_{S}(Q)). Then γα extends β̃, and by 8.4.7,

Nβ̃(γα) = N_{βα}(γα) = (N_{β}α)(γα) = (N_{β}γ)α ≤ N_{S}(Q)α = NS̃(Q̃),

completing the proof. Definition 8.6. Suppose F is saturated and E is a F-invariant subsystem of F on T. Then T is strongly closed in S with respect to F, so we can form the corresponding factor system F^{+}, and write F/T for this factor system. Further we define the factor system of F/E of F over E to be the factor system F/T. Remark 8.7. (1) The definition of the factor system F/E depends only on T, not on E. That this must be so (if we want F/E to be a system on S/T which is a surjective image of F), is a consequence of the following two observations:

(2) It can be shown that there is a surjective morphism Θ : F → F/T of fusion systems extending θ.

(3) It can also be shown that if α : F → F̃ is a surjective morphism of fusion systems with T the kernel of the group homomorphism α : S → S̃, then there exists an isomorphism β : F/T → F̃ such that Θβ = α. Thus the fusion systems F/T, as T ranges over the strongly closed subgroups of S, are precisely the homomorphic images of F.

(8.8) Assume F = F_{S}(G) and E = F_{T} (H) for some normal subgroup H of G with T = S ∩ H. Let M = N_{G}(T) and M^{∗} = M/N_{H}(T). Then

(1) F/E ≅ F_{S^{∗}}(M^{∗}).

(2) F_{S^{∗}}(M^{∗}) ≅ F_{SH/H}(G/H).

Proof. Define α : F_{S^{∗}}(M^{∗}) → F^{+} = F/E by s^{∗}α = s^{+} for s ∈ S; and for P, Q overgroups of T in S, and g ∈ N_{M}(P, Q), define c_{g^{∗}}α = c_{g}^{+}. As S ∩ H = T, the map α : S^{∗} → S^{+} is a well defined isomorphism. Observe that N_{G}(P, Q) = N_{M}(P, Q). Further for h ∈ M, c_{g^{∗}} = c_{h^{∗}} iff [gh^{−1}, P] ≤ T iff c_{gh^{−1}}^{+} = 1 by 8.4.4, iff c_{g}^{+} = c_{h}^{+} using (!). Thus α is a well defined injection on morphisms. As N_{G}(P, Q) = N_{M}(P, Q) and c_{g}^{+} = c_{g^{∗}}α, α is bijective on morphisms. Finally

(s^{∗}α)(c_{g^{∗}}α) = s^{+}c_{g}^{+} = (s^{g})^{+} = ((s^{g})^{∗})α = (s^{∗}c_{g^{∗}})α,

so α · (c_{g^{∗}}α) = c_{g^{∗}} · α, and hence α is an isomorphism of fusion systems, establishing (1).

Let G^{−} = G/H and define β : F_{S^{∗}}(M^{∗}) → F_{S^{−}}(G^{−}) by s^{∗}β = s^{−} and c_{g^{∗}}β = c_{g^{−}}. This time β : S^{∗} → S^{−} is an isomorphism of groups as N_{H}(T) = M ∩ H. Also c_{g^{∗}} = c_{h^{∗}} iff [gh^{−1}, P] ≤ T iff [gh^{−1}, P] ≤ H iff c_{g^{−}} = c_{h^{−}} as T = S ∩ H. So β is bijective on morphisms, and arguing as above, β is then an isomorphism on morphisms. That is (2) holds.

(8.9) Assume α : F → F̃ is a surjective morphism of fusion systems and T = ker(α) is the kernel of α : S → S̃. Then

(1) The map R ↦ Rα is a bijection between the set of overgroups of T in S strongly closed in S with respect to F, and the set of subgroups of S̃ strongly closed with respect to F̃.

(2) Assume F is saturated and let E be a F-invariant subsystem of F on an overgroup R of T in S. Then

(a) Ẽ = Eα is an F̃-invariant subsystem of F̃ on R̃ = Rα.

(b) Assume α : E → Ẽ is surjective and E ⊴ F. Then Ẽ ⊴ F̃.

Proof. Let T ≤ R ≤ S, P ≤ R, and ϕ ∈ hom_{F}(P, S). Then ϕα ∈ homF̃(P α, S̃) and as T is strongly closed in S with respect to F, P ϕ ≤ R iff (P α)(ϕα) = (P ϕ)α ≤ Rα. Thus (1) follows as α is surjective.

Assume the hypothesis of (2). Let P̃ ≤ Q̃ ≤ R̃ = Rα, ϕ̃ ∈ homẼ(P̃, Q̃), and β̃ ∈ homF̃(Q̃, S̃). Let P, Q be the preimages of P̃, Q̃ in S under α, and P_{1}, Q_{1} ≤ R with P_{1}α = P̃, Q_{1}α = Q̃, and ϕ ∈ hom_{E}(P_{1}, Q_{1}) with ϕα = ϕ̃. As α is surjective, there is β ∈ hom_{F}(Q, S) with βα = β̃. As E is F-invariant, ϕβ^{∗} ∈ hom_{E}(P_{1}β, R). Then

ϕ̃(βα)^{∗} = (β^{−1}ϕβ)α = (ϕβ^{∗})α ∈ homẼ(P_{1}βα, R̃) = homẼ((P_{1}α)(βα), R̃) = homẼ(P̃β̃, R̃),

so (2a) holds.

Now assume the hypothesis of (2b). Then E is saturated, so as α : E → Ẽ is a surjection, applying 8.5 to this morphism, Ẽ is saturated. By (a), Ẽ is F̃-invariant. Thus to show that Ẽ ⊴ F̃ and complete the proof of the lemma, it remains to show that Ẽ satisfies (N1) in F̃. Form the groups G = G_{F,R}(R) and H = H_{F,E}(R) of Theorem 2, part (3). Observe T is strongly closed in R, so T ⊴ G. Let V = RC_{S}(R^{+}) = RC_{+}(R). Then Ṽ = V α = VR̃ and AutẼ(Ṽ) = Aut_{H}(V)α. Further [H, C_{+}(R)] ≤ C_{H}(R^{+}) = K. Now C_{+} = C_{+}(R) ∈ Syl_{p}(KC_{+}), so by a Frattini argument, H = KN_{H}(C_{+}). Thus as Kα centralizes R̃, AutẼ(R̃) = Aut_{H}(R̃) = Aut_{N_{H}(C_{+})}(R̃), while [N_{H}(C_{+}), C_{+}]α ≤ C_{H∩C_{+}}(R^{+})α = Z(R̃). Thus for ϕ̃ ∈ AutẼ(R̃), ϕ̃ = c_{h|R̃} for some h ∈ N_{H}(C_{+}), and ϕ̂ = c_{h|R̃CS̃(R̃)} extends ϕ̃ with [ϕ̂, CS̃(R̃)] ≤ Z(R̃). Therefore Ẽ does indeed satisfies (N1) in F̃, completing the proof of the lemma.

(8.10) Let F be saturated, Z = Z(F), and F^{+} = F/Z. Then (1) θ : F → F^{+} is a surjective morphism of fusion systems.

(2) F^{+} is saturated and θ induces a bijection θ : Λ → Λ^{+} of the set Λ of normal subsystems E of F containing Z, with the set Λ^{+} of normal subsystems of F^{+}.

Proof. As Z = Z(F), F = N_{F}(Z), so 8.3 supplies the surjective morphism θ : F → F^{+}, establishing (1). By 8.5, F^{+} is saturated. Let E ∈ Λ. By another application of 8.3, θ : E → Eθ is surjective, so by 8.9.2(b), Eθ ∈ Λ^{+}.

Conversely assume E^{+} ∈ Λ^{+} is a system on T^{+} ≤ S^{+}. Let T be the preimage of T^{+} in S; by 8.9.1, T is strongly closed in S with respect to F.

Let U ∈ F_{T}^{fc}, D = N_{S}(U), Q = N_{T} (U), and G = G(U). Then Z ≤ C_{T} (U) ≤ U, and as Z = Z(F), Z ≤ Z(G). By 8.4.2, U^{+} ∈ F^{+f}. Let E^{+} = T_{U^{+}}^{+}, W^{+} = V_{E^{+}}, and F = C_{+}(U). Now D(U^{+}) = N_{N_{F^{+}} (U^{+})}(W^{+}) = N_{N_{F} (U)}(W)^{+} with D(U^{+}) = F_{D^{+}}(G(U^{+})) and N_{N_{F} (U)}(W) = F_{Q}(N_{G}(W)). Thus we may take G(U^{+}) = G(W)^{+}, where G(W) = N_{G}(W) and G^{+} = G/Z.

By Theorem 2, E^{+}(U^{+}) = F_{Q^{+}}(H(U^{+})) for some H(U^{+}) ⊴ G(U^{+}) with Q^{+} ∈ Syl_{p}(H(U^{+})). Let H = H(U) be the preimage in G of H(U^{+}). As Q^{+} ∈ Syl_{p}(H(U^{+})), Q ∈ Syl_{p}(H). As H^{+} = H(U^{+}) ⊴ G(U^{+}) = G(W)^{+}, H ⊴ G(W).

Let K = C_{G}(U^{+}). Then F ∈ Syl_{p}(K) and F acts on H. Next O^{p}(K) centralizes U, so T ∩ O^{p}(K) ≤ C_{T} (U) = Z(U). Similarly [Q, F ∩ O^{p}(K)] ≤ Q ∩ O^{p}(K) ≤ Z(U), so Q centralizes O^{p}(K). Then [H, F ∩ O^{p}(K)] ≤ H ∩ O^{p}(K) ≤ C_{H}(Q) = Z(Q), so O^{p}(H) centralizes O^{p}(K). Thus K = O^{p}(K)F acts on H.

By a Frattini argument, G = KN_{G}(F). Next E = T ∩ UF ⊴ N_{G}(F), and W^{+} ≤ E^{+}F^{+}, so W^{+} = C_{F^{+}}(E^{+})E^{+} is N_{G}(F)-invariant, and hence W ⊴ N_{G}(F). Thus G = KN_{G}(W), so as H ⊴ N_{G}(W), H ⊴ G.

Next let U ≤ P ≤ T with P ∈ D(U)^{f}. Set P_{1}^{+} = P^{+}C_{E^{+}}(P^{+}) and let P_{1} be the preimage in S of P_{1}^{+}. Let β ∈ hom_{F}(N_{S}(P_{1}), S) with P_{1}β ∈ F^{fc}. Further we can choose β so that P β ∈ F^{fc}. By 8.4.2, P_{1}^{+}β^{+} = (P_{1}β)^{+} ∈ F^{+f}, and by 8.4.1, N_{S^{+}}(P_{1}^{+}) = N_{S}(P_{1})^{+}, so N_{S^{+}}(P_{1}^{+})β^{+} = (N_{S}(P_{1})β)^{+} ≤ S^{+}. Then as E^{+} ⊴ F^{+}, N_{H(U^{+})}(P_{1}^{+})β̌^{+} = N_{H(P_{1}^{+}β^{+}}(Uβ^{+}) by 7.18. But from the discussion above, N_{H}(P)^{+} = N_{H(U^{+})}(P^{+}) = N_{H(U^{+})}(P_{1}^{+})∩N(P^{+}) and N_{H(P β)}(Uβ)^{+} = N_{H(P_{1}^{+}β^{+})}(U^{+}β^{+})∩N(P^{+}β^{+}), so N_{H(U)}(P)β̌ = N_{H(P β)}(Uβ). Therefore by Theorem 3 and 7.7, 𝔈(A) ⊴ F, where A is the constricted invariant map with A(U) = Aut_{H(U)}(U). Set Δ(E^{+}) = 𝔈(A). Then

A(U)^{+} = Aut_{H}(U)^{+} = Aut_{H(U^{+})}(U^{+}) = Aut_{E^{+}}(U^{+}),

so as θ is a morphism,

Δ(E^{+})θ = 𝔈(A)θ = 𝔈(Aut_{E^{+}}) = E^{+}.

Therefore θ ∘ Δ is the identity on Λ^{+}, and θ : Λ → Λ^{+} is a surjection.

Next suppose D ∈ Λ is a system on D ≤ S with Dθ = E^{+}, and set E = Δ(E^{+}). Then as Z ≤ D and Dθ = T^{+}, D = T. Further for ϕ ∈ Aut_{D}(U) a p′-element, there is a p′-element ψ ∈ Aut_{E}(U) with ϕ^{+} = ψ^{+}. Then ϕ and ψ agree on U/Z = U^{+} and Z, so ϕ = ψ. Thus

Aut_{D}(U) = O^{p}(Aut_{D}(U))Aut_{T} (U) = O^{p}(Aut_{E}(U))Aut_{T} (U) = Aut_{E}(U),

so A_{D}(U) = A(U). Hence D = 𝔈(A_{D}) = 𝔈(A) = E, so θ : Λ → Λ^{+} is a bijection with inverse Δ. Remark 8.11. The analogue of 8.10 does not hold for more general factor systems F/V. Namely there can be normal subsystems of F/V whose preimage is not normal in F. For example take p = 2, F to be the system FŜ(Ĝ) with Ĝ the group in Example 7.19, and take V = O_{2}(F) = O_{2}(Ĝ). Then F/V ≅ F_{S}(G) ≅ F_{S}(N_{G}(S)), so F_{T} (T) ⊴ F_{S}(G), where T = Ω_{1}(S) is the strongly closed subgroup of S defined in Example 7.19. However by Theorem 1, the preimage of F_{T} (T) in F is not normal in F, as there is no normal subgroup Ĥ of Ĝ with Sylow group T̂ = T V.

This example also shows that there are saturated fusion systems F and surjective morphisms α : F → F̃ such that for T the kernel of the group homomorphism α : S → S̃, there is no normal subsystem on T.

References

[Al] J. Alperin, Sylow intersection and fusion, J.Algebra 6 (1967), 222-241.

[BCGLO] C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver, Subgroup families controlling p-local finite groups, Proc. London Math. Soc. 91 (2004), 325-354.

[BLO] C. Broto, R. Levi, and Bob Oliver, The homotopy theory of fusion systems, J.Amer.Math.Soc 16 (2003), 779–856.

[FGT] M. Aschbacher, Finite Group Theory, Cambridge University Press, 1986.

[Gl] G. Glauberman, Central elements in core free groups, J.Algebra 4 (1966), 403-420.

[Go] D. Goldschmidt, A conjugation family for finite groups, J.Algebra 16 (1970), 138-142.

[L] M. Linkelman, Simple fusion systems and the Solomon 2-local groups, J.Algebra 296 (2006), 385-401.

[LO] R. Levi and Bob Oliver, Construction of 2-local finite groups of a type studied by Solomon and Benson, Geometry and Topology 6 (2002), 917-990.

[O1] B. Oliver, Equivalences of classifying spaces completed at odd primes, Math. Proc. Camb. Phil. Soc. 137 (2004), 321-347.

[O2] B. Oliver, Equivalences of classifying spaces completed at the prime two, Memoirs AMS.

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

[3T] M. Aschbacher, 3-Transposition Groups, Cambridge University Press, 1997.

Pasadena, California 91125