THE GENERALIZED FITTING SUBSYSTEM OF A FUSION SYSTEM

Michael Aschbacher

California Institute of Technology

We continue our program (begun in [A1]) to translate results from the local theory of finite groups into the setting of saturated fusion systems. See [A1] for motivation and a discussion of saturated fusion systems and their normal subsystems. See [BLO] for basic notation and terminology involving fusion systems, and [FGT] for notation and terminology involving finite groups.

Let p be a prime and F a saturated fusion system on a finite p-group S. In [A1], we defined the notion of a normal subsystem of F. In this paper we begin to establish properties of normal subsystems. In Example 6.4 of [A1], we saw that the intersection E_{1} ∩ E_{2} of normal subsystems E_{i} of F need not be normal in F. However this is not a serious problem, since it develops that E_{1} ∩ E_{2} is not quite the right object to consider. Rather in section 3 we prove:

Theorem 1. Let E_{i} be a normal subsystem of F on a subgroup T_{i} of S, for i = 1, 2. Then there exists a normal subsystem E_{1} ∧ E_{2} of F on T_{1} ∩ T_{2} contained in E_{1} ∩ E_{2}.

The next result probably already appears somewhere in the literature in the special case where F = F_{S}(G) is the system of a finite group G on a Sylow p-subgroup S of G. I have not been able to find a reference to such a theorem, but then I haven’t looked seriously. It would be a bit surprising if the result were not known for finite groups.

Theorem 2. Assume T_{i}, i = 1, 2, are strongly closed in S with respect to F. Then T_{1}T_{2} is strongly closed in S with respect to F.

This work was partially supported by NSF-0504852

If H_{1} and H_{2} are normal subgroups of a group G, then H_{1}H_{2} ⊴ G. The analogue of this result may hold for saturated fusion systems. Here we content ourselves with a proof only in a very special case; this case suffices for our most immediate applications.

Theorem 3. Assume E_{i} ⊴ F on T_{i} for i = 1, 2, and that [T_{1}, T_{2}] = 1. Then there exists a normal subsystem E_{1}E_{2} of F on T_{1}T_{2}. Further if T_{1} ∩ T_{2} ≤ Z(E_{i}) for i = 1, 2, then E_{1}E_{2} is a central product of E_{1} and E_{2}.

In section 2 we discuss the direct product F_{1} × F_{2} of fusion systems F_{1} and F_{2}. Some of the results in this section appear already in section 1 of [BLO]; see in particular 1.5 in [BLO]. A central product F_{1} ×_{Z} F_{2} is a factor system (F_{1} × F_{1})/Z, for some Z ≤ Z(F_{1} × F_{2}) such that Z ∩ F_{i} = 1 for i = 1, 2.

Theorem 3 bears some resemblance to earlier theorems about finite groups due to Gorenstein-Harris in [GH], and Goldschmidt in [Go]. Namely in each of these papers, the authors prove the existence of certain normal subgroups of a group G under the hypothesis that for S ∈ Syl_{2}(G), there are subgroups T_{i} of S for i = 1, 2, such that [T_{1}, T_{2}] = 1 and T_{i} is strongly closed in S with respect to G.

Let E ⊴ F be a system on T. In 6.7 we show the set of subgroups Y of C_{S}(T) such that E ≤ C_{F}(Y) has a largest member C_{S}(E), and C_{S}(E) is strongly closed in S with respect to F. Then in section 6 we define the centralizer in F of E to be a certain fusion system C_{F}(E) on C_{S}(E), and we prove:

Theorem 4. If E ⊴ F then C_{F}(E) ⊴ F, and for X ∈ C_{F}(E)^{fc}, Aut_{C_{F} (E)}(X) = O^{p}(Aut_{C_{F} (T)}(X))Aut_{C_{S}(E)}(X).

In section 7 we find that there is a characteristic subsystem O^{p}(F) of F on the subgroup [S, O^{p}(F)] defined by

[S, O^{p}(F)] = ⟨[U, O^{p}(Aut_{F}(U))] : U ∈ F^{fc}⟩.

In section 8 we show:

Theorem 5. Let E ⊴ F on T, and T ≤ R ≤ S. Then there exists a unique saturated fusion subsystem RE of F on R such that O^{p}(RE) = O^{p}(E). In particular F = SO^{p}(F).

If F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), then by 7.7, O^{p}(F) = F_{S∩O^{p}(G)}(O^{p}(G)).

Define F to be simple if F has no proper nontrivial normal subsystems. Define F to be quasisimple if F = O^{p}(F) and F/Z(F) is simple. Define the components of F to be the subnormal quasisimple subsystems of F. Recall O_{p}(F) is the largest subgroup of S normal in F.

Define E(F) to be the subsystem of F generated by the set Comp(F) of components of F, and set F^{∗}(F) = E(F)O_{p}(F). We call F^{∗}(F) the generalized Fitting subsystem of F. Of course all of these notions are similar to the analogous notions for groups.

In section 9 we prove:

Theorem 6. (1) E(F) is a characteristic subsystem of F.

(2) E(F) is the central product of the components of F.

(3) O_{p}(F) centralizes E(F).

(4) C_{F}(F^{∗}(F)) = Z(F^{∗}(F)).

Finally in section 10 we prove a version of the Gorenstein-Walter theorem on so called L-balance [GW]:

Theorem 7. For each fully normalized subgroup X of S, E(N_{F}(X)) ≤ E(F).

It is worth noting that the proof of L-balance for a group G requires that the components of G/O_{p′}(G) satisfy the Schreier conjecture, or when p = 2, a weak version of the Schreier conjecture due to Glauberman. Our proof of Theorem 7 requires no deep results. The theorem does not quite imply L-balance for groups, since there is not a nice one to one correspondence between quasisimple groups and quasisimple fusion systems. The proof can be translated into the language of groups, but even there at some point one seems to need some result like Theorem A of Goldschmidt in [Go], which is only proved for p = 2 without the classification.

Still, something is going on here, which suggests that in studying fusion systems, one may be lead to new theorems or better proofs of old theorems about finite groups. Indeed it is not improbable that other steps in the Classification of the finite simple groups have easier proofs in the category of saturated 2-fusion systems. If it were then possible to easily translate such results back to the category of finite groups, one would have simplified the corresponding parts of the proof of the Classification.

Section 1. Background

For background involving fusion systems, for the most part we refer the reader to [BLO] or [A1]. However in this section we recall some notation, terminology, and definitions involving fusion systems used frequently in this paper, which may not be quite standard in the field.

Let p be a prime, and F a saturated fusion system on the finite p-group S. Recall that a subsystem E of F on a subgroup T of S is F-invariant if

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

(I2) for each P ≤ Q ≤ T, ϕ ∈ hom_{E}(P, Q), and α ∈ hom_{F}(Q, S), ϕα^{∗} ∈ hom_{E}(P α, T). Here α^{∗} : hom_{F}(P, Q) → hom_{F}(P α, S) is defined by α^{∗} : ϕ ↦ α^{−1}ϕα. Further E is normal in F (written E ⊴ F) if E is F-invariant, E is saturated, and E

satisfies condition (N1) in F:

(N1) Each ϕ ∈ Aut_{E}(T) extends to ϕ̂ ∈ Aut_{F}(T C_{S}(T)) such that [ϕ̂, C_{S}(T)] ≤ Z(T). For T ≤ S strongly closed in S with respect to F, we can define the factor system

F/T as in 8.6 in [A1].

For P ≤ S, set

P^{F} = {P ϕ : ϕ ∈ hom_{F}(P, S)},

and V_{P} = P C_{S}(P).

If F is constrained then the set G(F) of models of F consists of the finite groups G such that S ∈ Syl_{p}(G), F = F_{S}(G), and F^{∗}(G) = O_{p}(G). By a result from [BCGLO], constrained saturated fusion systems possess models. Further from section 2 of [A1], if α : F → F′ is an isomorphism, G′ is a model of F′, and U is a normal p-subgroup of G with C_{G}(U) ≤ U, then the set Ext(α) of isomorphisms from G to G′ described in 2.1 in [A1] is nonempty.

Given T strongly closed in S with respect to F, let F_{T}^{f} be the set of subgroups U of T fully normalized in F, and F_{T}^{fc} the set of U ∈ F_{T}^{f} such that C_{T} (U) ≤ U. In 4.1 of [A1], given U ∈ F_{T}^{f}, we defined a constrained saturated system

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

and given a subsystem E of F on T, we also defined the subsystem

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

Then we obtain the model

G(U) = G_{F,T} (U) ∈ G(D(U)),

and when E ⊴ F, also E(U) ⊴ D(U) by Theorem 2 in [A1], and by Theorem 1 in [A1] we have a unique model

H(U) = H_{F,E}(U) ∈ G(E(U)),

with H(U) ⊴ G(U).

See 5.1 in [A1] for the definition of an F-invariant map on T, and 5.2.3 in [A1] for the definition of a constricted F-invariant map.

A strongly F normalized chain in T is a chain

C = (U_{0} ⊴ ⋯ ⊴ U_{n} = T)

of subgroups of T such that for each 0 ≤ i < n, U_{i} ∈ F^{f}.

Assume E is an F-invariant subsystem on T and U ≤ T. Define C(U) = C_{E}(U) to be the chain (U = U_{0} ≤ ⋯ ≤ U_{n} = T), where for 0 ≤ i < n, U_{i+1} = N_{T} (U_{i}).

(1.1) Assume E is an F-invariant subsystem on T, and let U ≤ T. Then there exists α ∈ hom_{F}(U, T) such that C_{E}(Uα) is a strongly F-normalized chain in T.

Proof. Assume otherwise and choose a counter example with m = |T : U| minimal. If m = 1 then U = T and C(T) = {T } is strongly F-normalized, contrary to the choice of U.

Replacing U by Uβ ∈ F_{T}^{f} for suitable β ∈ hom_{F}(U, T), we may assume U ∈ F_{T}^{f}. Let U_{1} = N_{T} (U). By minimality of m, there exists α ∈ hom_{F}(U_{1}, T) with C(U_{1}α) strongly normalized. As U_{1}α ∈ F_{T}^{f}, we conclude from 1.1.2 in [A1] that we may choose α to extend to γ ∈ hom_{F}(N_{S}(U_{1}), S). As N_{S}(U) ≤ N_{S}(U_{1}),

N_{S}(U)β ≤ N_{S}(U_{1})β ≤ N_{S}(U_{1}β) = N_{S}(U_{1}α),

so as U ∈ F_{T}^{f}, also Uα ∈ F_{T}^{f} and U_{1}α = N_{T} (Uα). Thus C(Uα) is strongly normalized.

Section 2. Direct products

Let p be a prime, and F_{i} be a fusion system on a finite p-group S_{i} for i = 1, 2. Set S = S_{1} × S_{2} and let π_{i} : S → S_{i} be the projection of S on S_{i}. For P_{i}, Q_{i} ≤ S_{i} and ϕ_{i} ∈ hom_{F_{i}}(P_{i}, Q_{i}), define ϕ_{1} × ϕ_{2} ∈ hom(P_{1} × P_{2}, Q_{1} × Q_{2}) by

ϕ_{1} × ϕ_{2} : (x_{1}, x_{2}) ↦ (x_{1}ϕ_{1}, x_{2}ϕ_{2}).

Define the category F = F_{1} ×F_{2} to have objects the subgroups of S, and for P, Q ≤ S, define hom_{F}(P, Q) to consist of the maps ϕ such that ϕ = (ϕ_{1} × ϕ_{2})_{|P}, P ϕ ≤ Q, and ϕ_{i} ∈ hom_{F_{i}}(P π_{i}, Qπ_{i}). Call the map ϕ_{i} the ith projection of ϕ. Observe ϕ_{i} is uniquely determined by the equality ϕπ_{i} = π_{i}ϕ_{i}. Often we write P_{i} for P π_{i}.

Write F̂_{i} for the subcategory of F whose objects are the subgroups of S and with

homF̂_{i}(P, Q) = {ϕ ∈ hom_{F}(P, Q) : ϕ_{3−i} = 1}.

(2.1) F_{1} × F_{2} is a fusion system on S_{1} × S_{2}.

Proof. Let P, Q ≤ S. For ϕ_{i} ∈ hom_{F_{i}}(P_{i}, Q_{i}), ϕ_{i} is a monomorphism of groups, so ϕ_{1} × ϕ_{2} : P_{1} × P_{2} → Q_{1} × Q_{2} is a monomorphism. Thus ϕ = (ϕ_{1} × ϕ_{2})_{|P} : P → Q_{1} × Q_{2} is a monomorphism, so if P ϕ ≤ Q then ϕ : P → Q is a monomorphism. If P ϕ ≤ R ≤ Q then P_{i}ϕ_{i} ≤ R_{i} ≤ Q_{i}, so as F_{i} is a fusion system, ϕ_{i} regarded as a map from P_{i} to R_{i} is in F_{i}, and hence ϕ regarded as a map from P to R is in F. If ϕ is an isomorphism then also ϕ_{i} : P_{i} → Q_{i} is an isomorphism, so as F_{i} is a fusion system, ϕ_{i}^{−1} : Q_{i} → P_{i} is an F_{i}-map, and hence ϕ^{−1} : Q → P is an F-map. Finally for s = (s_{1}, s_{2}) ∈ S, c_{s} = c_{s_{1}} ×c_{s_{2}}, so c_{s} ∈ hom_{F}(S, S), completing the proof.

(2.2) F = ⟨F̂_{1}, F̂_{2}⟩.

Proof. For ϕ ∈ hom_{F}(P, Q), ϕ = (ϕ_{1} × 1)(1 × ϕ_{2}) with ϕ_{1} × 1 ∈ homF̂_{1}(P, S) and 1 × ϕ_{2} ∈ homF̂_{2}(P (ϕ_{1} × 1), Q).

(2.3) Assume E is a fusion system on S and F_{i} is a subsystem of E for i = 1, 2. Assume

(a) For each P, Q ≤ S_{i} and ϕ ∈ hom_{F_{i}}(P, Q), ϕ extends to ϕ̂ ∈ hom_{E}(P S_{3−i}, QS_{3−i}) with ϕ̂ = 1 on S_{3−i}.

(b) E = ⟨ϕ̂ : ϕ ∈ hom_{F_{i}}(P, Q), P, Q ≤ S_{i}, i = 1, 2⟩. That is each morphism in E is a composition of restrictions of such morphisms ϕ̂. Then E = F_{1} × F_{2}.

Proof. Let i = 1 or 2, P, Q ≤ S_{i}, and ϕ ∈ hom_{F_{i}}(P, Q). Then ϕ̂ = ϕ × 1 or 1 × ϕ, so ϕ̂ ∈ hom_{F}(P S_{3−i}, QS_{3−i}), and then E ≤ F by (b). Thus by 2.2, it remains to show F̂_{i} ⊆ E.

Let U, V ≤ S and ψ ∈ homF̂_{1}(U, V). Then ψ = (ψ_{1} × 1)_{|U}, with ψ_{1} ∈ hom_{F_{1}}(U_{1}, V_{1}). But ψ_{1} ×1 ∈ hom_{E}(U_{1}S_{2}, V_{1}S_{2}) by (a), so as E is a fusion system, ψ ∈ hom_{E}(U, S). Then as Uψ ≤ V, also ψ ∈ hom_{E}(U, V), completing the proof.

(2.4) Assume for i = 1, 2 that F_{i} = F_{S_{i}}(G_{i}) for some finite group G_{i} with S_{i} ∈ Syl_{p}(G_{i}). Let G = G_{1} × G_{2}. Then F_{1} × F_{2} = F_{S}(G), so F_{1} × F_{2} is saturated.

Proof. Let E = F_{S}(G); we check that the hypothesis of 2.3 are satisfied by E. For P, Q ≤ S_{i}, hom_{F_{i}}(P, Q) consists of the maps ϕ_{i} = c_{g_{i}|P} : P → Q with g_{i} ∈ N_{G_{i}}(P, Q). Now ϕ̂_{i} = c_{g_{i|P S_{3−i}}} ∈ hom_{E}(P S_{3−i}, QS_{3−i}) is 1 on S_{3−i}, so condition (a) of 2.3 is satisfied. Further for g = g_{1}g_{2} ∈ G with g_{i} ∈ G_{i}, c_{g} = c_{g_{1}}c_{g_{2}} = ϕ̂_{1}ϕ̂_{2}, so condition (b) is also satisfied.

(2.5) For U ≤ S, N_{F}(U) ≤ N_{F}(U_{1}U_{2}) = N_{F_{1}}(U_{1}) × N_{F_{2}}(U_{2}).

Proof. First N_{S}(U) ≤ N_{S_{1}}(U_{1})×N_{S_{2}}(U_{2}). Further for P ≤ N_{S}(U), ϕ ∈ hom_{F}(P, S) is in N_{F}(U) iff ϕ extends to φ ∈ hom_{F}(UP, S) acting on U. In that event, φ is the restriction of φ_{1} × φ_{2} with φ_{i} ∈ hom_{F_{i}}(U_{i}P_{i}, S_{i}) acting on U_{i}, so φ_{1} × φ_{2} ∈ hom_{F}(U_{1}U_{2}P_{1}P_{2}, S) extends ϕ and acts on U_{1}U_{2}. Thus N_{F}(U) ≤ N_{F}(U_{1}U_{2}). Further specializing to the case U = U_{1}U_{2}, the remarks above show N_{F}(U) = N_{F_{1}}(U_{1}) × N_{F_{2}}(U_{2}).

(2.6) Let U ≤ S. Then (1) C_{S}(U) = C_{S_{1}}(U_{1})C_{S_{2}}(U_{2}).

(2) If U ∈ F^{c} then U_{i} ∈ F_{i}^{c}.

(3) If U_{i} ∈ F_{i}^{c} for i = 1, 2, then U_{1} × U_{2} ∈ F^{c}.

(4) Assume F_{i} is saturated for i = 1, 2, and U ∈ F^{fc}. Then there exists ϕ_{i} ∈ hom_{F_{i}}(N_{S_{i}}(U_{i}), S_{i}) with U_{i}ϕ_{i} ∈ F_{i}^{fc} and U(ϕ_{1} × ϕ_{2}) ∈ F^{fc}.

Proof. Part (1) is a straightforward calculation. Suppose U ∈ F^{c}. Then for each ϕ ∈ hom_{F}(U, S), C_{S}(Uϕ) ≤ Uϕ. But (Uϕ)_{i} = U_{i}ϕ_{i}, so by (1), C_{S_{i}}(U_{i}ϕ_{i}) ≤ Uϕ ∩ S_{i} ≤ U_{i}ϕ_{i}, establishing (2).

Assume U_{i} ∈ F_{i}^{c} for i = 1, 2, and set W = U_{1} × U_{2}. Then W ∈ F^{c} iff for each ϕ ∈ hom_{F}(W, S), C_{S}(W ϕ) ≤ W ϕ. But ϕ = ϕ_{1} × ϕ_{2} with ϕ_{i} ∈ hom_{F_{i}}(U_{i}, S_{i}), and C_{S}(W ϕ) = C_{S_{1}}(U_{1}ϕ_{1}) × C_{S}(U_{2}ϕ_{2}) by (1). Now (3) follows.

Finally assume the hypothesis of (4), and using 1.1.2 in [A1], pick ϕ_{i} ∈ hom_{F_{i}}(N_{S_{i}}(U_{i}), S_{i}) with U_{i}ϕ_{i} ∈ F_{i}^{f}. Let ϕ = ϕ_{1} × ϕ_{2}. Then N_{S}(U)ϕ ≤ N_{S_{1}}(U_{1})ϕ_{1} × N_{S_{2}}(U_{2})ϕ_{2} ≤ S, so as U ∈ F^{fc}, also Uϕ ∈ F^{fc}. By (2), U_{i} ∈ F_{i}^{c}, so U_{i}ϕ ∈ F_{i}^{fc}.

(2.7) If F_{1} and F_{2} are saturated then F_{1} × F_{2} is saturated.

Proof. By Theorem A in [BCGLO] it suffices to show:

(a) F = ⟨Aut_{F}(U) : U ∈ F^{c}⟩, and (b) The saturation axioms are satisfied at each member of F^{c}. As F_{i} is saturated, (a) is satisfied in F_{i}. Then by 2.6.3,

F̂_{i} = ⟨AutF̂_{i}(U_{i} × S_{3−i}) : U_{i} ∈ F_{i}^{c}⟩ ≤ ⟨Aut_{F}(U) : U ∈ F^{c}⟩,

so (a) is satisfied by F by 2.2.

Thus it remains to verify (b). Let U ∈ F^{c} and set W = U_{1}U_{2}. As U ∈ F^{c}, U is fully centralized. Suppose U ∈ F^{f} and pick ϕ_{i} and ϕ = ϕ_{1} × ϕ_{2} as in 2.6.4. Then U_{i}ϕ_{i} ∈ F_{i}^{fc}, so as F_{i} is saturated there is G_{i} ∈ G(N_{F_{i}}(U_{i}ϕ_{i})). Let Q_{i} = N_{S_{i}}(U_{i}ϕ_{i}), Q = Q_{1}×Q_{2}, and G = G_{1}×G_{2}. By 2.5, E = N_{F}(W ϕ) = N_{F_{1}}(U_{1}ϕ)×N_{F_{2}}(U_{2}ϕ), so by 2.4, N_{F}(W ϕ) = F_{Q}(G). By 2.5, N_{F}(Uϕ) = N_{E}(Uϕ), so N_{F}(Uϕ) = F_{N_{Q}(Uϕ)}(N_{G}(Uϕ)), and hence N_{F}(Uϕ) is saturated. Thus N_{F}(U) is saturated, so Aut_{S}(U) ∈ Syl_{p}(Aut_{F}(U)), and hence U satisfies axiom I for saturation.

Next suppose ϕ ∈ hom_{F}(U, S). Then

N_{ϕ} = {x ∈ N_{S}(U) : c_{x}ϕ^{∗} ∈ Aut_{S}(Uϕ)} ≤ {(x_{1}, x_{2}) ∈ N_{S_{1}}(U_{1}) × N_{S_{2}}(U_{2}) : c_{x_{i}}ϕ_{i}^{∗} ∈ Aut_{S_{i}}(U_{i}ϕ_{i})} = N_{ϕ_{1}} × N_{ϕ_{2}}.

By 2.6.2, U_{i} ∈ F_{i}^{c}, so ϕ_{i} extends to φ_{i} ∈ hom_{F_{i}}(N_{ϕ_{i}}, S_{i}). Let φ = (φ_{1} × φ_{2})_{|U}. Then φ extends ϕ to N_{ϕ}, verifying axiom II, and completing the proof.

Definition 2.8. Write U = U(F_{1}, F_{2}) for the set of subgroups D of Z(F_{1}) × Z(F_{2}) such that D ∩ Z(F_{i}) = 1 for i = 1, 2. For D ∈ U, define the central product of F_{1} and F_{2} with respect to D to be

F_{1} ×_{D} F_{2} = (F_{1} × F_{2})/D.

(2.9) Let D ∈ U(F_{1}, F_{2}). Then (1) D ≤ Z(F_{1} × F_{2}) = Z(F_{1}) × Z(F_{2}).

(2) Let θ : F → F^{+} = F_{1} ×_{D} F_{2} be the natural map θ : s ↦ s^{+} = sD and ϕθ : x^{+} ↦ (xϕ)^{+}. Then θ is a surjective morphism of fusion systems.

(3) θ : F_{i} → F_{i}^{+} = F_{i}θ is an isomorphism, when F_{i} is regarded as a subsystem of F.

(4) If F_{1} and F_{2} are saturated, then so is F^{+}.

(5) Suppose F_{i} = F_{S_{i}}(G_{i}) for some finite group G_{i} with S_{i} ∈ Syl_{p}(G_{i}), and set G = G_{1} × G_{2}. Then D ≤ Z(G), so G^{+} = G/D is a central product of G_{1} and G_{2}, and F^{+} = F_{S^{+}}(G^{+}).

(6) Assume E is a fusion system on S^{+} and F_{i}^{+} is a saturated subsystem of E on S_{i}^{+} for i = 1, 2. Assume:

(i) For i = 1, 2, each P ∈ F_{i}^{+fc}, and each ϕ ∈ Aut_{F_{i}^{+}}(P), ϕ extends to ϕ̂ ∈ Aut_{F_{i}^{+}}(P S_{3−i}^{+}) with ϕ̂ = 1 on S_{3−i}^{+}.

(ii) E = ⟨ϕ̂: ϕ ∈ Aut_{F_{i}^{+}}(P), P ∈ F_{i}^{+fc}, i = 1, 2⟩. Then E = F_{1} ×_{D} F_{2}. Proof. A straightforward calculation shows Z(F) = Z(F_{1}) × Z(F_{2}), so as D ≤ Z(F_{1}) × Z(F_{2}), (1) holds. Part (2) follows from 8.3 in [A1], and (3) follows from the fact that S_{i} ∩ ker(θ) = S_{i} ∩ D = 1.

If F_{1} and F_{2} are saturated, then so is F by 2.7. Then (4) follows from 8.5 in [A1]. Assume the hypothesis of (5). By 2.4, F = F_{S}(G). Then as D ≤ Z(F), D ≤ Z(G),

so G^{+} is a central product of G_{1} and G_{2} as D ∩ G_{i} = 1. Finally 8.7 in [A1] completes the proof of (5).

Assume the hypothesis of (6). We repeat the proof of 2.3. As in that proof, E ≤ F^{+} by conditions (i) and (ii), and it remains to show F̂_{i}^{+} ⊆ E, since F^{+} = ⟨F̂_{1}^{+}, F̂_{2}^{+}⟩ by 2.2 and (2). As F_{i}^{+} is saturated, F_{i}^{+} = ⟨Aut_{F_{i}^{+}}(P) : P ∈ F_{i}^{+fc}⟩ by A.10 in [BLO]. Thus F̂_{i}^{+} = ⟨ϕ̂ : ϕ ∈ Aut_{F_{i}^{+}}(P) : P ∈ F_{i}^{+fc}⟩, so F̂_{i}^{+} ≤ E by (i) and (ii), completing the proof of (6).

One can extend the notion of the direct product of fusion systems to the notion of a direct product

∏_{i∈I} F_{i}

of a family (F_{i} : i ∈ I) of an arbitrary finite number of fusion systems, either by a recursive construction, or by an obvious generalization of the definitions for families of size 2. In particular from the former point of view, by induction on |I|, the direct product of saturated fusion systems is saturated. Similarly one can extend Definition 2.8 to construct the central product

(∏_{i∈I} F_{i})/Z

of a family of fusion systems with respect to Z ≤ ∏_{i} Z(F_{i}) such that Z ∩ Z(F_{i}) = 1 for each i ∈ I.

(2.10) Let (F_{i} : i ∈ I) and (F̃_{i} : i ∈ I) be families of fusion systems, and (β_{i} : F_{i} → F̃_{i}) a family of morphisms. Then there exists a unique morphism β = ∏_{i} β_{i} : ∏_{i} F_{i} → ∏_{i} F̃_{i} extending β_{i} for each i. Indeed

xβ = (∏_{i} x_{i})β = ∏_{i} x_{i}β_{i}

and

ϕβ = (∏_{i} ϕ_{i})β = ∏_{i} ϕ_{i}β_{i}

for x ∈ S = ∏_{i} S_{i} and ϕ_{i} an F_{i}-map.

Proof. Proceeding by induction on |I| = n, we may take n = 2. It is well known that β : S = S_{1} × S_{2} → S̃ = S̃_{1} × S̃_{2} is the unique group homomorphism extending β_{1} and β_{2}. Let P_{i}, Q_{i} ≤ S_{i} and ϕ_{i} ∈ hom_{F_{i}}(P_{i}, Q_{i}). Then ϕ_{i}β_{i} ∈ homF̃_{i}(P_{i}β_{i}, Q_{i}β_{i}), so ϕ_{1}β_{1} × ϕ_{2}β_{2} ∈ homF̃(P_{1}β_{1} × P_{2}β_{2}, Q_{1}β_{1} × Q_{2}β_{2}). Then if ϕ ∈ hom_{F}(P, Q) with P π_{i} = P_{i} and Qπ_{i} = Q_{i}, and with ϕ = (ϕ_{1} × ϕ_{2})_{|P}, define ϕβ = (ϕ_{1}β_{1} × ϕ_{2}β_{2})_{|P β}. It is straightforward to check that β is a morphism, and using 2.2, β is unique.

(2.11) Let (F_{i} : i ∈ I) be a family of fusion systems, F = ∏_{i} F_{i}, F^{+} = F/Z a central product, and θ : F → F^{+} the natural map. Assume α ∈ Aut(F^{+}) is an isomorphism permuting {F_{i}^{+} = F_{i}θ : i ∈ I}. Then there exists a unique α̂ ∈ Aut(F) such that α̂θ = θα. Moreover F_{i}α̂ = F_{iσ}, where σ ∈ Sym(I) is defined by F_{i}^{+}α = F_{iσ}^{+}.

Proof. By hypothesis F_{i}^{+}α = F_{iσ}^{+} for some σ ∈ Sym(I). Define θ_{i} : F_{i} → F_{i}^{+} to be the restriction of θ to F_{i}, and define α̂_{i} : F_{i} → F_{iσ} by α̂_{i} = θ_{i}αθ_{iσ}^{−1}. Then α̂_{i} is an isomorphism for each i ∈ I, so by 2.10, α̂ = ∏_{i} α̂_{i} is an automorphism of F. By construction α̂_{i}θ = θα_{i} on F_{i}, where α_{i} : F_{i}^{+} → F_{iσ}^{+} is the restriction of α to F_{i}^{+}. Then by 2.10,

θα = θ · ∏_{i} α_{i} = ∏_{i} θα_{i} = ∏_{i} α̂_{i}θ = (∏_{i} α̂_{i}) · θ = α̂θ,

and α̂ is unique subject to this property by 2.10.

In the last part of the final lemma of this section, it is convenient to prove a result about quasisimple systems, a notion not introduced until section 7.

(2.12) Let (F_{i} : 1 ≤ i ≤ p) be a family of fusion systems, F = ∏_{i=1}^{p} F_{i}, F^{+} = F/Z a central product, and α an automorphism of F^{+} of order p such that F_{i}^{+}α = F_{i+1}^{+} for each 1 ≤ i ≤ p, where the indices are read modulo p. Let θ : F → F^{+} be the natural map. Then

(1) Let α̂ ∈ Aut(F) such that α̂θ = θα. Let D = C_{F}(α̂) be the fusion system on C_{S}(α̂) such that for P, Q ≤ C_{S}(α̂), hom_{D}(P, Q) = {ϕ ∈ hom_{F}(P, Q) : ϕα̂ = ϕ}. Then D = F_{1}δ ≅ F_{1}, C_{S}(α̂) is the full diagonal subgroup D of S = S_{1} × ⋯ × S_{p} determined by α̂, δ : S_{1} → S is the map δ : x ↦ (x, xα̂, ⋯, xα̂^{p−1}), and for P, Q ≤ S_{1} and ϕ ∈ hom_{F_{1}}(P, Q), ϕδ = (ϕ, ϕα̂, ⋯, ϕα̂^{p−1})_{|P δ}.

(2) If F_{1} is quasisimple then E^{+} = O^{p}(C_{F^{+}}(α)) = C_{F}(α̂)θ is quasisimple, with E^{+}/Z(E^{+}) ≅ F_{1}/Z(F_{1}), and C_{F^{+}}(α) = E^{+}C_{Z(F^{+})}(α).

Proof. We first prove (1), where appealing to 2.11 and replacing F^{+}, α by F, α̂, we may assume Z = 1 and α = α̂. Visibly F_{1}δ ≤ C_{F}(α) and δ : F_{1} → F_{1}δ is an isomorphism. Further it is well known that D = C_{S}(α). Finally if P, Q ≤ S_{1} and ϕ ∈ hom_{D}(P δ, Qδ) then ϕ = (ϕ_{1} × ⋯ × ϕ_{p})_{|P δ} for some ϕ_{i} ∈ hom_{F_{i}}(P α^{i−1}, Qα^{i−1}). As α centralizes ϕ, ϕ_{i}α = ϕ_{i+1}, so (1) holds.

Now assume F_{1} is quasisimple and remove the assumption that Z = 1. Adopt the notation in section 8 of [A1] in discussing the factor system F^{+}. As α̂θ = θα, α = α̂^{+}, so from section 8 in [A1],

C_{S^{+}}(α) = C_{+}(α̂) = {s^{+} : [s, α̂] ∈ Z}.

As α is free on S and acts on Z(F) ≥ Z, it follows that C_{S^{+}}(α) ≤ C_{S}(α̂)^{+}Z(F^{+}). Similarly C_{F^{+}}(α) ≤ C_{F}(α̂)θZ(F^{+}). By (1), C_{F}(α̂) ≅ F_{1} is quasisimple, so C_{F}(α̂)θ is also quasisimple, and hence C_{F}(α̂)θ = O^{p}(C_{F^{+}}(α)), so (2) holds.

Section 3. E_{1} ∧ E_{2}

In this section F is a saturated fusion system over the finite p-group S, and E_{i}, i = 1, 2, are normal subsystems of F on T_{i}. Set T = T_{1}∩T_{2}. By 3.6.1 in [A1], E_{1}∩E_{2} is F-invariant on T, so in particular T is strongly closed in S with respect to F.

Notation 3.1. Let U ∈ F_{T}^{fc}, and for i = 1, 2, set T_{U,i} = UC_{T_{i}}(U) and T_{U,1,2} = T_{U,1}T_{U,2}. Define D_{1,2}(U) = N_{F}(V), where V = V_{T_{U,1,2}}. Thus D_{1,2}(U) is a fusion system on N_{S}(U).

(3.2) Let U ∈ F_{T}^{fc} and V = V_{T_{U,1,2}}. Then (1) U = T ∩ T_{U,i} = T ∩ T_{U,1,2} = T ∩ V for i = 1, 2.

(2) N_{S}(U) = N_{S}(T_{U,i}) = N_{S}(T_{U,1,2}) = N_{S}(V) for i = 1, 2.

(3) T_{U,i} ∈ F_{T_{i}}^{fc} for i = 1, 2, and V ∈ F^{fc}.

(4) D_{1,2}(U) is saturated and constrained.

(5) Aut_{F}(U) = Aut_{D_{1,2}(U)}(U).

(6) T_{U,i} ≤ V_{T_{U,i}} ≤ V_{U} for i = 1, 2, and T_{U,1,2} ≤ V ≤ V_{U}. Further T_{U,i} = V_{U} ∩ T_{i}.

Proof. First T_{U,i} = UC_{T_{i}}(U) and C_{T} (U) ≤ U as U ∈ F_{T}^{fc}. Thus

T ∩ T_{U,i} = U(T ∩ C_{T_{i}}(U)) = UC_{T} (U) = U.

Similarly U = T ∩ T_{U,1,2} = T ∩ V, establishing (1). Then (1) implies (2), and (3) follows from (2) and the choice of U ∈ F_{T}^{f}. As V ∈ F^{fc} by (3), part (4) follows from 1.2.1 in [A1]. The proof of (5) is the same as that of 4.3.3 in [A1].

By definition, T_{U,i} ≤ T_{U,i}C_{S}(T_{U,i}) = V_{T_{U,i}}, and T_{U,i} = UC_{T_{i}}(U) ≤ UC_{S}(U) = V_{U}. Similarly C_{S}(T_{U,i}) ≤ C_{S}(U) ≤ V_{U}, and T_{U,1,2} ≤ V ≤ V_{U}. As T_{U,i} = UC_{T_{i}}(U), arguing as in (1), we conclude T_{U,i} = T_{i} ∩ V_{U}. That is (6) holds.

Notation 3.3. Let U ∈ F_{T}^{fc}, and for i = 1, 2, form the groups G_{i} = G_{i}(U) = G_{F,T_{i}}(U) and H_{i} = H_{i}(U) = H_{F,E_{i}}(U) of 7.1 and part (3) of Theorem 2 of [A1], with respect to the normal subsystem E_{i} of F. Similarly let D_{i}(U) = D_{F,T_{i}}(U) and E_{i}(U) = E_{F,E_{i}}(U) be the fusion systems of 4.1 in [A1], defined with respect to the normal subsystem E_{i} of F. Finally let D(U) = D_{F,T} (U) be the fusion system of 4.1 in [A1] defined with respect to the strongly closed subgroup T of S, and choose G(U) = G_{F,T} (U) ∈ G(D(U)) as in 7.1 in [A1].

(3.4) Let U ∈ F_{T}^{fc}. Then for i = 1, 2:

(1) H_{i} acts on T_{U,3−i}.

(2) H_{i} acts on V, V_{T_{U,i}}, and T_{U,i}.

(3) E_{i}(U) ≤ D_{1,2}(U).

Proof. Let H = H_{i}, X = T_{U,i}, and Y = T_{U,3−i}. Then working in G = G_{i}, [H, Y] ≤ C_{H}(U)U = K. Further X ∈ Syl_{p}(K) and X ⊴ G as V_{X} ⊴ G and X = T_{i} ∩ V_{X}. Therefore X = O_{p}(K) = F^{∗}(K). Next [X, Y] ≤ X∩Y = U(C_{T_{1}}(U)∩T_{2}) = UC_{T} (U) = U as U ∈ F_{T}^{fc}. Thus C_{Y} (U) centralizes X/U and U, so as F^{∗}(K) = X, [K, C_{Y} (U)] ≤ C_{K}(X/U) ∩ C_{K}(U) ≤ X. Thus K acts on XY. Then as [H, Y] ≤ K, H acts on O_{p}(KY) = XY and hence also on XY ∩ T_{3−i} = Y. This establishes (1).

Next XC_{S}(X) = V_{X} ⊴ G and V ≤ Y V_{X}, so H acts on V_{X} and V = C_{Y V_{X}} (XY), establishing (2). Finally (2) implies (3).

Notation 3.5. Let U ∈ F_{T}^{fc}. By 3.2.4, D = D_{1,2}(U) is saturated and constrained, so by 2.5.1 in [A1], there exists a model G = G_{1,2}(U) for D. Notice X_{i} = N_{T_{i}}(U) is strongly closed in N_{S}(U) with respect to D. By 3.4.3, E_{i}(U) ≤ D, and arguing as in the proof of 4.4 in [A1], E_{i}(U) is D-invariant. As E_{i} is saturated, so is E_{i}(U) by 4.4.2 in [A1]. For h ∈ N_{H_{i}}(X_{i}), h acts on Y = T_{U,3−i} by 3.4.1, and h centralizes C_{S}(X_{i})/Z(X_{i}). Thus ϕ̂ = c_{h|Y V_{X_{i}}} extends ϕ = c_{h|X_{i}} to Y V_{X_{i}} and centralizes C_{S}(X_{i})/Z(X_{i}). In particular ϕ̂ ∈ D, so condition N(D, E_{i}(U)) of [A1] is satisfied. Thus E_{i}(U) ⊴ D, so by Theorem 1 in [A1], E_{i}(U) = F_{X_{i}}(H′_{i}) for a unique normal subgroup H′_{i} of G with X_{i} ∈ Syl_{p}(H′_{i}). By 2.5.2 in [A1], H_{i} ≅ H′_{i} via an isomorphism which is the identity on X_{i}, so we write H_{i} for H′_{i}. By 3.4.2, H_{i} acts on T_{U,j}, V_{T_{U,i}}, and V, so H′_{i} also acts on these groups.

Set H = H(U) = H_{1} ∩ H_{2}. Thus H is a normal subgroup of G, and as T_{i} ∈ Syl_{p}(H_{i}), T = T_{1} ∩ T_{2} is Sylow in H. Next by 3.2.5, Aut_{F}(U) = Aut_{D}(U), so as Aut_{D}(U) = Aut_{G}(U) and H ⊴ G, Aut_{H}(U) ⊴ Aut_{F}(U).

Define A = A_{1,2} on F_{T}^{fc} by A(U) = Aut_{H(U)}(U). Then extend A to a constricted F-invariant map A = A_{1,2} on T as in 5.2.3 in [A1]. Then (as in 5.1 in [A1]) define

E_{1} ∧ E_{2} = 𝔈(A_{1,2}).

Write E for E_{1} ∧ E_{2}.

(3.6) (1) E = ⟨A(Uφ) : U ∈ F_{T}^{f} , φ ∈ Aut_{F}(T)⟩ = ⟨A(Uφ) : U ∈ F_{T}^{fc}, φ ∈ Aut_{F}(T)⟩. (2) E ≤ E_{1} ∩ E_{2}.

Proof. From 3.5, E = 𝔈(A) and A is a constricted F-invariant map. Therefore (1) follows from parts (5) and (6) of 5.5 in [A1].

Let U ∈ F_{T}^{fc} . By construction in 3.5, H_{i}(U) ∈ G(E_{i}(U)), so Aut_{H_{i}(U)}(U) = Aut_{E_{i}(U)}(U) ≤ Aut_{E_{i}}(U). Thus

A(U) = Aut_{H(U)}(U) ≤ Aut_{H_{i}(U)}(U) ≤ Aut_{E_{i}}(U),

as H(U) = H_{1}(U) ∩ H_{2}(U). Then for φ ∈ Aut_{F}(T),

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

as E_{i} is F-invariant. Therefore by (1),

E ≤ ⟨Aut_{E_{i}}(Uφ) : U ∈ F_{T}^{fc}, φ ∈ Aut_{F}(T)⟩ ≤ E_{i},

establishing (2).

(3.7) Let U ∈ F_{T}^{fc}. Then (1) D(U) = N_{D_{1,2}(U)}(V_{U}) ≤ D_{1,2}(U).

(2) G(U) = N_{G_{1,2}(U)}(V_{U}).

(3) H(U) ⊴ G(U).

Proof. By 3.2.6, V ≤ V_{U} and for i = 1, 2, T_{U,i} = V_{U} ∩T_{i}, so T_{U,1,2} = (V_{U} ∩T_{1})(V_{U} ∩T_{2}) ⊴ G(U). Thus (1) holds. Let G = G_{1,2}(U) and H = H(U). From (1), G(U) and N_{G}(V_{U})) are in G(D(U)), so we may choose our member G(U) of G(D(U)) to be N_{G}(V_{U}), and hence (2) holds.

Recall from 3.5 that H ⊴ G. Also [H, C_{S}(U)] ≤ C_{H}(U) = Z(U), so H acts on V_{U}, and hence H ≤ N_{G}(V_{U}) = G(U). Thus (3) holds.

(3.8) Let U ∈ F_{T}^{fc}, V = V_{T_{U,1,2}}, P ∈ D(U)^{f}, with U ≤ P ≤ T, and β ∈ hom_{F}(N_{S}(P), S) with P β ∈ F_{T}^{fc}. Set P_{i} = P T_{U,i} and W_{i} = V V_{T_{U,i}} for i = 1, 2. Then

(1) P = P_{i} ∩ T = P V ∩ T = P W_{i} ∩ T, U = T ∩ W_{i}, and N_{H_{i}(U)}(P_{i}) = N_{H_{i}(U)}(P) for i = 1, 2.

(2) There is an isomorphism β̌ : N_{G_{1,2}(U)}(P) → N_{G_{1,2}(P β)}(V β) extending β : N_{S}(P)∩ N_{S}(U) → N_{S}(P β) ∩ N_{S}(Uβ).

(3) N_{H_{i}(U)}(P_{i})β̌ = N_{H_{i}(P β)}(T_{U,i}β).

(4) N_{H(U)}(P)β̌ = N_{H(P β)}(Uβ).

(5) Aut_{H(P β)}(Uβ) = Aut_{N_{H(U)}(P)}(U)β^{∗}.

Proof. First P_{i} = P C_{T_{i}}(U), so P_{i} ∩ T = P C_{T} (U) = P as C_{T} (U) ≤ U ≤ P. Similarly P = P V ∩ T = P W_{i} ∩ T and U = W_{i} ∩ T. As P = P_{i} ∩ T, N_{H_{i}(U)}(P_{i}) ≤ N_{H_{i}(U)}(P), and the opposite inclusion follows as P_{i} = P T_{U,i} with T_{U,i} ⊴ H_{i}(U). Thus (1) holds.

Part (2) follows as in the proof of parts (8) and (9) of 7.2 in [A1], using 3.2. By (1), P = P_{i} ∩ T so N_{S}(P_{i}) ≤ N_{S}(P). Thus by 7.2 and 7.18 in [A1], applied to

P_{i}, T_{U,i}, E_{i} in the role of P, U, E, it follows that (*) β_{|N_{S}(P_{i})} extends to β̌_{i} : N_{G_{i}(T_{U,i})}(P_{i}) → N_{G_{i}(P_{i}β)}(V_{T_{U,i}}β), and (**) N_{H_{i}(U)}(P_{i})β̌_{i} = N_{H_{i}(P_{i}β)}(T_{U,i}β). Next the maps in N_{N_{F} (V_{T_{U,i}})}(P_{i}V) and N_{N_{F} (V)}(P T V_{T_{U,i}}) are those on P W_{i} acting on

U, so the two systems are equal. Thus (!) their models N_{G_{i}(T_{U,i})}(P_{i}V) and N_{G_{1,2}(U)}(P) are also equal. Similarly (!!) N_{G_{i}(P_{i}β)}(W_{i}β) = N_{G_{1,2}(P β)}(W_{i}β).

As

N_{G_{i}(T_{U,i})}(P_{i}V) = N_{G_{i}(T_{U,i})}(P_{i}) ∩ N_{G_{i}(T_{U,i})}(V)

and

N_{G_{i}(P_{i}β)}(W_{i}β) = N_{G_{i}(P_{i}β)}(V_{T_{U,i}}β) ∩ N_{G_{i}(P_{i}β)}(V β),

we conclude from (*) that N_{G_{i}(T_{U,i})}(P_{i}V)β̌_{i} = N_{G_{i}(P_{i}β)}(W_{i}β). Then it follows from (!) and (!!) that N_{G_{1,2}(U)}(P V_{T_{U,i}})β̌_{i} = N_{G_{1,2}(P β)}(W_{i}β). Then using 2.3 and 2.4.2 in [A1], β̌ = c_{z_{i}} β̌_{i} on N_{G_{1,2}(U)}(P V_{T_{U,i}}), for some z_{i} ∈ Z(N_{S}(P)∩N_{S}(U)) ≤ N_{G_{1,2}(U)}(N_{H_{i}(U)}(P_{i})). Therefore (3) follows from (**) and the observation in 3.5 that H_{i}(U) ≤ N_{G_{1,2}(U)}(V_{T_{U,i}}), so that N_{H_{i}(U)}(P_{i}) ≤ N_{G_{1,2}(U)}(P V_{T_{U,i}}).

Claim K = N_{H(P β)}(Uβ) = N_{H(P β)}(T_{U,1,2}β). For let K_{i} = N_{H_{i}(P β)}(Uβ). Then

[C_{T_{3−i}}(U)β, T_{i} ∩ K_{i}] ≤ C_{T} (Uβ) ≤ Uβ,

so as T_{i} ∩ K_{i} ∈ Syl_{p}(K_{i}) and F^{∗}(K_{i}) = O_{p}(K_{i}), K_{i} acts on T_{U,3−i}β. Thus K = K_{1} ∩ K_{2} acts on T_{U,1,2}β = T_{U,1}βT_{U,2}β, establishing the claim.

By (1),

N_{H_{1}(U)}(P_{1}) ∩ N_{H_{2}(U)}(P_{2}) = N_{H_{1}(U)}(P) ∩ N_{H_{2}(U)}(P) = N_{H(U)}(P),

and similarly

N_{H_{1}(P β)}(T_{U,1}β) ∩ N_{H_{2}(P β)}(T_{U,2}β) = H_{1}(P β) ∩ H_{2}(P β) ∩ N_{G_{1,2}(P β)}(T_{U,1}βT_{U,2}β) = N_{H(P β)}(T_{U,1,2}β) = N_{H(P β)}(Uβ),

by the claim. Now (4) follows from (3) and these two observations. Finally (4) implies (5).

(3.9) Assume F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), and for i = 1, 2, E_{i} = F_{T_{i}}(H_{i}) for some normal subgroup H_{i} of G with T_{i} ∈ Syl_{p}(H_{i}). Set H = H_{1} ∩ H_{2}. Then T ∈ Syl_{p}(H) and E = E_{1} ∧ E_{2} = F_{T} (H). In particular E ⊴ F.

Proof. Let U ∈ F_{T}^{fc}, D = N_{S}(U), Y_{i} = T_{U,i}, Y = Y_{1}Y_{2}, V = V_{Y}, and D = D_{1,2}(U). Since D = N_{F}(V) and F = F_{S}(G), D = F_{D}(N_{G}(V)). Similarly

E_{i}(U) = E_{F,E_{i}}(U) = N_{N_{E_{i}} (Y_{i})}(U) = F_{N_{T_{i}} (U)}(N_{H_{i}}(Y_{i})).

Further from 3.4.2, N_{H_{i}}(Y_{i}) = N_{H_{i}}(V), so E_{i}(U) = F_{N_{T_{i}} (U)}(H_{i}(U)), where H_{i}(U) = N_{H_{i}}(V). From 3.5,

H(U) = H_{1}(U) ∩ H_{2}(U) = N_{H_{1}}(V) ∩ N_{H_{2}}(V) = N_{H}(V).

Thus A(U) = Aut_{N_{H}(V)}(U).

Next [N_{H}(U), Y] ≤ UC_{H}(U), so Y C_{H}(U) ⊴ N_{H}(U). Also Z(U) is Sylow in C_{H}(U), so Y is Sylow in Y C_{H}(U). Hence by a Frattini argument, N_{H}(U) = C_{H}(U)I, where I = N_{H}(Y). Also [C_{S}(Y), I] ≤ C_{I}(Y), so I acts on C_{I}(Y)C_{S}(Y)Y = C_{I}(Y)V. Further C_{T} (Y) ≤ C_{T} (U) = Z(U), so V = C_{S}(Y)Y is Sylow in V C_{I}(Y). Then by another Frattini argument, I = C_{I}(Y)N_{I}(V). Then as U ≤ Y,

Aut_{H}(U) = Aut_{N_{H}(U)}(U) = Aut_{I}(U) = Aut_{N_{I}(V)}(U) ≤ Aut_{N_{H}(V)}(U) = A(U).

Let E′ = F_{T} (H). Then Aut_{E′}(U) = Aut_{H}(U) = A(U), so (in the notation of section 5 in [A1]) Aut_{E′,c} = A. Thus E′ = 𝔈(Aut_{E′,c}) = 𝔈(A) = E, completing the proof.

(3.10) Assume F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G) and F^{∗}(G) = O_{p}(G). Then

(1) For i = 1, 2, E_{i} = F_{T_{i}}(H_{i}) for a unique normal subgroup H_{i} of G with T_{i} ∈ Syl_{p}(H_{i}).

(2) Set H = H_{1} ∩ H_{2}. Then T ∈ Syl_{p}(H) and E = E_{1} ∧ E_{2} = F_{T} (H).

(3) E ⊴ F. Proof. Part (1) is a consequence of Theorem 1 in [A1]. Then (2) and (3) follow from (1) and 3.9.

We are now in a position to prove Theorem 1 by proving E = E_{1} ∧ E_{2} ⊴ F. To do so, we verify the conditions of part (4) of Theorem 3 in [A1] for the groups H(U) we have constructed in 3.5, and then appeal to Theorem 3 in [A1] to conclude that E ⊴ F. Observe that E ≤ E_{1} ∩ E_{2} by 3.6.2.

Let U ∈ F_{T}^{fc}. By 3.7.3, H(U) ⊴ G(U). By 3.6.1, condition (ii) of part (4) of Theorem 3 in [A1] is satisfied. Finally condition (i) of part (4) of Theorem 3 in [A1] is satisfied by 3.8.5. Thus we have established Theorem 1.

Define N (F, T) to be the set of all normal subsystems of F on T. If N (F, T) is nonempty, define the normal subsystem of F generated by T to be:

⟨T^{F}⟩ = ⋀_{E∈N (F,T)} E.

(3.11) If N (F, T) is nonempty then ⟨T^{F}⟩ is the smallest normal subystem of F over T.

Proof. This follows from Theorem 1.

Section 4. The product of strongly closed subgroups

In this section F is a saturated fusion system over the finite p-group S, and T_{i}, i = 1, 2, are nontrivial subgroups of S which are strongly closed in S with respect to F. Set T = T_{1}T_{2}. We prove Theorem 2 in this section. The proof involves a series of reductions. Assume the Theorem is false, and choose Y ≤ T and α ∈ hom_{F}(T, S) such that Y α ≰ T. Subject to this constraint choose Y with n = |T : Y | minimal.

As F is saturated it follows from Alperin’s Fusion Theorem (cf. A.10 in [BLO]) that:

(4.1) F = ⟨Aut_{F}(R) : R ∈ F^{fc}⟩.

In particular by 4.1, we can choose Y and α so that Y ≤ R ∈ F^{fc} and α ∈ Aut_{F}(R). Subject to this constraint, choose R maximal. Let Q = N_{S}(R), Q_{i} = N_{T_{i}}(R), and R_{i} = T_{i} ∩ R. As R ∈ F^{fc}, N_{F}(R) = D(R) is saturated and constrained by 1.2.1 in [A1], so we may choose G ∈ G(D(R)). Observe that as T_{i} is strongly closed in S with respect to F, R_{i} ⊴ G. Further α = c_{g|Y} for some g ∈ G.

(4.2) (1) If ⟨Q, g⟩ ≤ H ≤ G then R = O_{p}(H).

(2) R = O_{p}(G).

Proof. Visibly (1) implies (2). Suppose H is a counter example to (1), and set P = O_{p}(H) and let β ∈ hom_{F}(N_{S}(P), S) with P β ∈ F^{fc}. As R ≤ H ≤ G and R ≤ O_{p}(G), R ≤ P. Then as H is a counter example to (1), R < P. As g ∈ H, γ = c_{g|P} ∈ Aut_{F}(P), and γ extends α.

By 4.1, there exist U_{i} ∈ F^{fc}, 1 ≤ i ≤ m, P_{i} ≤ U_{i}, and β_{i} ∈ A_{F}(U_{i}) such that β = β_{1} ⋯ β_{m}, P_{i+1} = P_{i}β_{i}, and P = P_{1}. Let Y_{1} = Y, Z_{1} = Y α, Y_{i+1} = Y_{i}β_{i}, and Z_{i+1} = Z_{i}β_{i}. As R < P and P_{i} ≤ U_{i} with |P | = |P_{i}|, we have |R| < |U_{i}|. Therefore by maximality of R, Y_{i} ≤ T and Z_{i} ≰ T for each i. Thus Y β = Y_{m+1} ≤ T but (Y β)γβ^{∗} = Y γβ = Z_{m+1} ≰ T. Then as γβ^{∗} ∈ Aut_{F}(P β), we have a contradiction to the maximality of R.

(4.3) (1) For i = 1, 2, T_{i} ≰ R.

(2) For i = 1, 2, Q_{i} ≰ R.

(3) R ≠ Q.

(4) ⟨Y^{N_{G}(Q)}⟩ ≤ T.

Proof. Suppose T_{1} ≤ R. Now for y ∈ Y, y = y_{1}y_{2} with y_{i} ∈ T_{i}. Then y_{1} ∈ T_{1} ≤ R, so y_{2} = y_{1}^{−1}y ∈ R. As R_{i} ⊴ G, y_{i}^{g} ∈ R_{i} for i = 1, 2, so yα = y^{g} ∈ R_{1}R_{2} ≤ T. Thus Y α ≤ T, contrary to the choice of α.

Therefore (1) holds. In particular R < RT_{i}, so Q_{i} = N_{T_{i}}(R) properly contains R_{i}, establishing (2). Then (2) implies (3).

Suppose (4) fails. Then we may take g ∈ N_{G}(Q). Hence R = O_{p}(N_{G}(Q)) by 4.2.1, so R = Q, contrary to (3).

(4.4) (1) Y = T ∩ R.

(2) Y ⊴ Q.

Proof. If Y < T ∩ R, then by minimality of n, T ∩ R ⊴ G, contradicting Y^{g} ≰ T. Thus (1) holds, and as S acts on T, (1) implies (2).

Let H be the set of minimal parabolics of G: The set of all overgroups H of Q in G such that Q is in a unique maximal subgroup of H and Q is not normal in H. By an observation of McBride (cf. B.6.3 in [AS]), G = ⟨H, N_{G}(Q)⟩, so by 4.3.4 we may assume there exists H ∈ H such that Y^{h} ≰ T for some h ∈ H. Without loss h = g, so by 4.2.1:

(4.5) There exist H ∈ H with g ∈ H and R = O_{p}(H).

Let M be the unique maximal overgroup of Q in H and K = O^{p}(H). Set K_{1} = [Q_{1}, K].

(4.6) (1) K = K_{1}.

(2) K centralizes R/R_{1}.

Proof. As Q_{1} ⊴ Q, K_{1} = [K, Q_{1}] is invariant under K and Q, so K_{1} ⊴ KQ = H. Thus K_{1}Q is an overgroup of Q in H, so either K_{1}Q ≤ M or H = K_{1}Q. In the latter case (1) holds. In the former as K_{1}Q_{1} ⊴ H, H = K_{1}N_{H}(P) by a Frattini argument, where P = Q∩K_{1}Q_{1}. Thus as K_{1} ≤ M, N_{H}(P) ≰ M, so H = N_{H}(P) by the uniqueness of M. Therefore P ≤ O_{p}(H), so P ≤ R by 4.5. But now Q_{1} ≤ P ≤ R, contrary to 4.3.2.

Thus (1) is established. Next [R, Q_{1}] ≤ R ∩ T_{1} = R_{1}. Then as R and R_{1} are normal in H, (2) follows from (1).

We are now in a position to obtain a contradiction, and hence establish Theorem 2. By 4.4.1, R_{1} ≤ Y, so by 4.6.2, K acts on Y. Further Q acts on Y by 4.4.2, so Y ⊴ KQ = H, contrary to the choice of H. This completes the proof of Theorem 2.

Section 5. Pairs of commuting strongly closed subgroups

In this section F is a saturated fusion system over the finite p-group S, and T_{i}, i = 1, 2, are subgroups of S strongly closed in S with respect to F, such that [T_{1}, T_{2}] = 1. By Theorem 2, T = T_{1}T_{2} is also strongly closed in S with respect to F.

(5.1) Let T_{1} ∩ T_{2} ≤ U_{i} ≤ T_{i} for i = 1, 2. Then (1) N_{T} (U_{1}U_{2}) = N_{T_{1}}(U_{1})N_{T_{2}}(U_{2}).

(2) C_{T} (U_{1}U_{2}) = C_{T_{1}}(U_{1})C_{T_{2}}(U_{2}).

(3) T_{i} ∩ U_{1}U_{2} = U_{i} for i = 1, 2.

(4) U_{1}U_{2} ∈ F_{T}^{c} iff U_{i} ∈ F_{T_{i}}^{c} for i = 1 and 2.

Proof. Let U = U_{1}U_{2}. Then U ∩ T_{i} = U_{i}(U_{3−i} ∩ T_{i}) ≤ U_{i}(T_{1} ∩ T_{2}) = U_{i}, establishing (3). By (3) and as [T_{1}, T_{2}] = 1,

N_{T} (U_{1}U_{2}) = N_{T} (U_{1}) ∩ N_{T} (U_{2}) = T_{2}N_{T_{1}}(U_{1}) ∩ T_{1}N_{T_{2}}(U_{2}) = N_{T_{1}}(U_{1})(T_{2} ∩ T_{1}N_{T_{2}}(U_{2})) = N_{T_{1}}(U_{1})N_{T_{2}}(U_{2})(T_{1} ∩ T_{2}) = N_{T_{1}}(U_{1})N_{T_{2}}(U_{2}),

so (1) holds. A similar argument establishes (2).

Next U ∈ F_{T}^{c} iff for each ϕ ∈ hom_{F}(U, S), C_{T} (Uϕ) ≤ Uϕ iff C_{T_{i}}(U_{i}ϕ) ≤ U_{i}ϕ for i = 1, 2 by (2), iff U_{i} ∈ F_{T_{i}}^{c} for i = 1, 2. Thus (4) holds.

Notation 5.2. Set P = F_{T_{1}}^{c} × F_{T_{2}}^{c}. For i = 1, 2, let P^{i} contains of those (U_{1}, U_{2}) ∈ P such that U_{i} ∈ F_{T_{i}}^{fc} and V_{U_{1}U_{2}} ∈ N_{F}(U_{i})^{fc}.

(5.3) Let (U_{1}, U_{2}) ∈ P and V = V_{U_{1}U_{2}}. Then (1) V = V_{U_{1}} ∩ V_{U_{2}}.

(2) For i = 1, 2, V ∩ T_{i} = U_{i} = V_{U_{i}} ∩ T_{i}.

(3) N_{S}(V) = N_{S}(U_{1}) ∩ N_{S}(U_{2}) = N_{S}(U_{1}U_{2}).

(4) T_{1} ∩ T_{2} ≤ U_{1} ∩ U_{2}.

(5) U_{1}U_{2} ∈ F_{T}^{c}.

Proof. As U_{3−i} ≤ C_{S}(U_{i}),

V_{U_{1}} ∩ V_{U_{2}} = U_{1}C_{S}(U_{1}) ∩ U_{2}C_{S}(U_{2}) = U_{1}(C_{S}(U_{1}) ∩ U_{2}C_{S}(U_{2})) = U_{1}U_{2}(C_{S}(U_{1}) ∩ C_{S}(U_{2})) = U_{2}U_{2}C_{S}(U_{1}U_{2}) = V,

so (1) holds. As U_{i} ∈ F_{T_{i}}^{c}, T_{i} ∩ V_{U_{i}} = U_{i}, so (2) follows from (1). Then (1) and (2) imply (3). As [T_{1}, T_{2}] = 1, T_{1} ∩ T_{2} ≤ Z(T), so (4) follows as U_{i} ∈ F_{T_{i}}^{c}. Then (4) and 5.1.4 imply (5).

(5.4) Let (U_{1}, U_{2}) ∈ P and V = V_{U_{1}U_{2}}. Then (1) The maps

(X_{1}, U_{2})^{N_{F} (U_{2})} ↦ (X_{1}, U_{2})^{F} and (U_{1}, X_{2})^{N_{F} (U_{1})} ↦ (U_{1}, X_{2})^{F}

are bijections of the orbits of N_{F}(U_{2}) on U_{1}^{F} × {U_{2}} and N_{F}(U_{1}) on {U_{1}} × U_{2}^{F} with the orbits of F on U_{1}^{F} × U_{2}^{F}, respectively.

(2) If for i = 1 or 2, U_{i} ∈ F_{T}^{fc}, then U_{1}U_{2} ∈ F_{T}^{f} iff V ∈ F^{f} iff V ∈ N_{F}(U_{i})^{f} iff U_{1}U_{2} ∈ N_{F}(U_{i})^{f}.

(3) If for i = 1 or 2, U_{i} ∈ F_{T}^{fc} then U_{i} = T_{i} ∩ V_{U_{i}} and V_{U_{i}} ∈ F^{fc}.

Proof. Part (1) is a standard result in group theory; the same proof works in the context of fusion systems.

Assume U_{1} ∈ F_{T}^{fc}. By 5.3.3, V ∈ N_{F}(U_{1})^{f} iff U_{1}U_{2} ∈ N_{F}(U_{1})^{f} and V ∈ F^{f} iff U_{1}U_{2} ∈ F_{T}^{f}. If V ∈ F^{f} then |N_{S}(V)| ≥ |N_{S}(W)| for all W ∈ V^{F}, so as N_{S}(V) ≤ N_{S}(U_{1}) by 5.3.3, certainly |N_{S}(V) ∩ N_{S}(U_{1})| ≥ |N_{S}(W) ∩ N_{S}(U_{1})| for all W ∈ V^{N_{F} (U_{1})}. That is V ∈ N_{F}(U_{1})^{f}.

Conversely assume V ∈ N_{F}(U_{1})^{f} and let W ∈ F^{f} ∩ V^{F}. Then |N_{S}(V)| ≤ |N_{S}(W)|. Continue to assume U_{1} ∈ F_{T}^{fc}. By 5.3.2, U_{1} = V ∩ T_{1}, so as U_{1} ∈ F_{T}^{f}, there is α ∈ hom_{F}(N_{S}(W ∩ T_{1}), S) with (W ∩ T_{1})α = U_{1}. By 5.3.3, N_{S}(W) ≤ N_{S}(W ∩ T_{1}) and N_{S}(W ∩ T_{1})α ≤ N_{S}(U_{1}), so N_{S}(W)α ≤ N_{S}(W α), and hence |N_{S}(W)| ≤ |N_{S}(W α)|. Also W α ∈ W^{F} = V^{F}, so by (1), W α ∈ V^{N_{F} (U_{1})}. Thus as V ∈ N_{F}(U_{1})^{f}, |N_{S}(W α)| ≤ |N_{S}(V)|, so |N_{S}(W)| = |N_{S}(V)|, and hence V ∈ F^{f}. Hence (2) holds.

By 5.3.2, U_{i} = T_{i} ∩ V_{U_{i}}, so N_{S}(U_{i}) = N_{S}(V_{U_{i}}). Then (3) follows.

(5.5) Let (U_{1}, U_{2}) ∈ P^{1}, and set V = V_{U_{1}U_{2}}. Then

(1) V and V_{U_{1}} are in F^{fc}, so we can form G(V) = G_{F,S}(V) and G(V_{U_{1}}) = G_{F,S}(V_{U_{1}}), N_{D(V)}(V_{U_{1}}) is a saturated constrained fusion system on X = N_{S}(V)∩N_{S}(V_{U_{1}}), N_{D(V)}(V_{U_{1}}) = N_{D(V_{U_{1}} )}(V), and N_{1} = N_{G(V)}(V_{U_{1}}) and M_{1} = N_{G(V_{U_{1}} )}(V) are in G(N_{D(V)}(V_{U_{1}})), so the set Ext(1) of isomorphisms ˇ1 : N_{1} → M_{1} which extend the identity map on X with ˇ1c_{V} = c_{V} is nonempty.

(2) Let α ∈ hom_{F}(N_{S}(V), S) such that (U_{1}, U_{2})α = (X_{1}, X_{2}) ∈ P^{2}. Then U_{1}U_{2}α = X_{1}X_{2} ∈ F_{T}^{fc}, V α = V_{X_{1}X_{2}} ∈ F^{fc}, and α̇ : D(V) → D(V α) is an isomorphism. Hence there is an isomorphism α̌ : G(V) → G(V α) extending α : N_{S}(V) → N_{S}(V α) such that α̌c_{V α} = c_{V} α^{∗}.

(3) α̇ : N_{D(V)}(V_{U_{2}}) → N_{D(V α)}(V_{X_{2}}) is an isomorphism, N_{2} = N_{G(V)}(V_{U_{2}}) ∈ G(N_{D(V)}(V_{U_{2}}), N_{2}α̌ = M_{2} = N_{G(V α)}(V_{X_{2}}) ∈ G(N_{D(V α)}(V_{X_{2}}), and α̌_{|N_{2}} is in Ext(α).

(4) If E_{i} ⊴ F on T_{i} for i = 1 or 2, then E_{i}(U_{i}) ⊴ D(V), and there exists a unique normal subgroup H_{i}(V) of G(V) with E_{i}(V) = F_{N_{T_{i}} (V)}(H_{i}(V)) and N_{T_{i}}(V) ∈ Syl_{p}(H_{i}(V)).

Proof. By 5.4.3, V_{U_{1}} ∈ F^{fc}, while by 5.4.2, V ∈ F^{f}, so V ∈ F^{fc} as C_{S}(V) ≤ V. Then as V ≤ V_{U_{1}}, we can apply parts (8) and (9) of 7.2 in [A1] to V, V_{U_{1}}, S, 1 in the roles of U, P, T, β to conclude that (1) holds.

Next α exists as in (2) by 5.4.1. By 5.3.5 and 5.4.2, X_{1}X_{2} ∈ F_{T}^{fc} and V_{X_{1}X_{2}} ∈ F^{fc}. Now we apply parts (6) and (7) of 7.2 in [A1] to V, S in the role of U, T to conclude that (2) holds. Then N_{D(V)}(V_{U_{2}})α^{∗} = N_{D(V α)}(V_{U_{2}}α) = N_{D(V α)}(V_{X_{2}}), and N_{2}α̌ = M_{2}, with M_{2} ∈ G(N_{D(V α)}(V_{X_{2}}) via the argument used to establish (1). Thus applying α^{−1}, N_{2} ∈ G(N_{D(V)}(V_{U_{2}}) and (3) holds.

Suppose that E_{1} ⊴ F on T_{1}. Then by Theorem 2 in [A1], there is a normal subgroup H(U_{1}) = H_{F,E_{1}}(U) of G(V_{U_{1}}) such that E_{1}(U_{1}) = F_{Q_{1}}(H(U_{1})) and N_{T_{1}}(U_{1}) = Q_{1} ∈ Syl_{p}(H(U_{1})). Now [H(U_{1}), V_{U_{1}}] ≤ U_{1}, so H(U_{1}) acts on V as U_{1} ≤ V ≤ V_{U_{1}}. Thus E_{1}(U_{1}) ≤ D(V) and setting H_{1}(V) = H(U_{1})ˇ1^{−1}, H_{1}(V) ≤ N_{1}. By 5.3.2, U_{1} = T_{1} ∩ V ⊴ G(V). Then arguing as in the proof of 4.3 in [A1], E_{1}(U_{1}) is D(V)-invariant. From Theorem 2 in [A1], E_{1}(U_{1}) is saturated, and as [H(U_{1}), V_{U_{1}}] ≤ U_{1}, (D(V), E_{1}(U_{1})) satisfies (N1). Therefore E_{1}(U_{1}) ⊴ D(V), so H_{1}(V) ⊴ G(V) by Theorem 1 in [A1].

Suppose on the otherhand that E_{2} ⊴ F on T_{2}. Then by Theorem 2 in [A1], there is H(X_{2}) ⊴ G(V_{X_{2}}) with properties as above. This time H(X_{2}) acts on V α, so E_{1}(X_{2}) ≤ D(V α), and setting H_{2}(V) = H(X_{2})α̌^{−1}, H_{2}(V) ≤ N_{2} and E_{2}(U_{2}) = E_{2}(X_{2})α^{−∗} ≤ D(V α)α^{−∗} = D(V). Then arguing as above, E_{2}(V) ⊴ D(V) and H_{2}(V) ⊴ G(V).

In the remainder of the section we assume:

Hypothesis 5.6. For i = 1, 2, E_{i} ⊴ F on T_{i}, and [T_{1}, T_{2}] = 1. Set T = T_{1}T_{2}.

Notation 5.7. Suppose (Y_{1}, Y_{2}) ∈ P. Then there exists α ∈ hom_{F}(N_{S}(Y_{1}), S) with (U_{1}, U_{2}) = (Y_{1}, Y_{2})α ∈ P^{1}. Set U = U_{1}U_{2} and V = V_{U_{1}U_{2}}. By 5.4.2 and 5.3.5, U ∈ F_{T}^{fc}. Moreover

D(U) = D_{F,T} (V) = N_{F}(V) = D_{F,S}(U) = D(V),

and hence also G(U) = G_{F,T} (U) = G_{F,S}(V) = G(V). By 5.5.4, we can form H_{i}(U) = H_{i}(V) ⊴ G(V). Set H(U) = H_{1}(U)H_{2}(U). Then H(U) ⊴ G(U) and by 5.3.4 and 5.1.1, N_{T} (U) = N_{T_{1}}(U_{1})N_{T_{2}}(U_{2}) ∈ Syl_{p}(H(U)). If in addition Y_{1}Y_{2} ∈ F_{T}^{fc}, then by 2.3 and 2.4 in [A1], there is α̌ ∈ Ext(α), so that α̌ : G(Y_{1}Y_{2}) → G(U) is an isomorphism extending α : N_{S}(Y_{1}Y_{2}) → N_{S}(U). In this case we set H_{i}(Y_{1}Y_{2}) = H_{i}(U)α̌^{−1}, and H(Y_{1}Y_{2}) = H(U)α̌^{−1}. Thus H_{i}(Y_{1}Y_{2}) and H(Y_{1}Y_{2}) are normal in G(Y_{1}Y_{2}).

(5.8) Let (U_{1}, U_{2}) ∈ P with U_{1}U_{2} ∈ F_{T}^{fc}. Then (1) For i = 1, 2, U_{i} ∈ E_{i}^{fc} and N_{T_{i}}(U_{i}) ∈ Syl_{p}(H_{i}(U_{1}U_{2})).

(2) H_{i}(U_{1}U_{2}) acts on Q_{3−i} = N_{T_{3−i}}(U_{3−i}), and indeed on each overgroup of T_{1} ∩ T_{2} in Q_{3−i}, for i = 1, 2.

(3) If T_{1} ∩ T_{2} ≤ Z(E_{1}) ∩ Z(E_{2}) then [H_{1}(U_{1}U_{2}), H_{2}(U_{1}U_{2})] = 1 and T_{1} ∩ T_{2} ≤ Z(H(U_{1}U_{2})).

Proof. Let U = U_{1}U_{2}, and for i = 1, 2, let H_{i} = H_{i}(U) and Q_{i} = N_{T_{i}}(U). Fix i ∈ {1, 2}. By 5.7 there is α ∈ hom_{F}(N_{S}(U_{i}), S) with (U_{1}, U_{2})α ∈ P^{i}. Let W_{i} = U_{i}α and W = W_{1}W_{2}. By 5.3.4 and 5.1, for j = 1, 2, Q_{j} = N_{T_{j}}(U_{j}), N_{T_{j}}(W) = N_{T_{j}}(W_{j}), N_{T} (U) = Q_{1}Q_{2}, and N_{T} (W) = N_{T_{1}}(W_{1})N_{T_{2}}(W_{2}). Then as U ∈ F_{T}^{fc}, N_{T} (U)α = N_{T} (W), so Q_{j}α = N_{T_{j}}(W_{j}). Hence as W_{i} ∈ E_{i}^{fc}, also U_{i} ∈ E_{i}^{fc}, while by 5.7, N_{T_{i}}(W_{i}) is Sylow in H_{i}(W), so Q_{i} = N_{T_{i}}(W_{i})α^{−1} = N_{T_{i}}(W_{i})α̌^{−1} is Sylow in H_{i}(W)α̌^{−1} = H_{i}(U). Thus (1) holds.

By (1), Q_{i} ∈ Syl_{p}(H_{i}), and by 5.7, H_{i} ⊴ G(U). Now Q_{1} centralizes Q_{2}, so [Q_{1}, H_{2}] ≤ C_{H_{2}}(Q_{2}) ≤ U_{2}, and hence H_{2} acts on T_{1} ∩ Q_{1}U_{2} = Q_{1}. Indeed [Q_{1}, H_{2}] ≤ Q_{1} ∩ U_{2} = T_{1} ∩ T_{2}. That is (2) holds.

Assume T_{0} = T_{1} ∩ T_{2} ≤ Z(E_{1}) ∩ Z(E_{2}). Then (cf. 7.11.2) T_{0} is in the center of H_{i}, so T_{0} ≤ Z(H). Further [Q_{1}, H_{2}] ≤ Q_{1} ∩ U_{2} ≤ T_{0} ≤ C_{U_{2}}(H_{2}), so O^{p}(H_{2}) centralizes Q_{1}. Then as Q_{2} centralizes Q_{1}, H_{2} = O^{p}(H_{2})Q_{2} centralizes Q_{1}. Thus [H_{1}, H_{2}] ≤ C_{H_{1}}(Q_{1}) ≤ U_{1} ≤ C_{H_{1}}(H_{2}), so O^{p}(H_{2}) centralizes H_{1}. Then H_{2} = Q_{2}O^{p}(H_{2}) centralizes H_{1}, completing the proof of (3).

Definition 5.9. For each U ∈ F_{T}^{fc}, we define normal subgroups H_{i}(U), i = 1, 2 and H(U) of G(U) = G_{F,T} (U), such that N_{T_{i}}(U) ∈ Syl_{p}(H_{i}(U)) and N_{T} (U) ∈ Syl_{p}(H(U)).

First let U_{i} = T_{i} ∩ U and suppose U_{i} ∈ F_{T_{i}}^{c} for i = 1, 2. Using 5.4, we may choose α ∈ hom_{F}(N_{S}(U_{1}U_{2}), S) such that (U_{1}U_{2})α ∈ F_{T}^{fc}. As N_{S}(U) ≤ N_{S}(U_{1}U_{2}), it follows that Uα ∈ F_{T}^{fc}, so replacing U by Uα, we may assume U_{1}U_{2} ∈ F_{T}^{fc}. In this case define H_{i}(U) = N_{H_{i}(U_{1}U_{2})}(U) and H(U) = N_{H(U_{1}U_{2})}(U). Notice N_{T_{i}}(U) ≤ N_{T_{i}}(U_{1}U_{2}), which is Sylow in H_{i}(U_{1}U_{2}), so N_{T_{i}}(U) is Sylow in H_{i}(U). Similarly N_{T} (U) ∈ Syl_{p}(H(U)).

Next observe C_{S}(U) ≤ V = V_{U_{1}U_{2}} and [H(U), V] ≤ T ∩ V = U_{1}U_{2} ≤ U by 5.1.2, so H(U) acts on V_{U} = UC_{S}(U). Also [U, V] ≤ T ∩ V ≤ U, so V acts on U. Further [U, C_{S}(U_{1}U_{2})] ≤ C_{U}(U_{1}U_{2}) = Z(U_{1}U_{2}) by 5.1.2. Thus Aut_{C_{S}(U_{1}U_{2})}(U) ≤ O_{2}(Aut_{F}(U)), so Aut_{V} (U) ≤ O_{2}(Aut_{F}(U)). Thus for ϕ ∈ Aut_{F}(U), V ≤ N_{ϕ}, so as F is saturated, ϕ ∈ N_{F}(UV). This is Aut_{F}(U) ≤ N_{F}(UV) ≤ N_{F}(V). Therefore V ⊴ G(U). Thus we can take G(U) = N_{G(U_{1}U_{2})}(V_{U}), and we have H(U) and H_{i}(U) normal in G(U) in this case.

In general our definition is recursive, with the recursion based on the parameter n = |C_{T} (U_{1}U_{2}) : Z(U_{1}U_{2})|. Let Û_{i} = C_{T_{i}}(U_{i}) ∩ N_{S}(U) and Û = UÛ_{1}Û_{2}. Then [Û_{i}, U] ≤ Z(U_{i}), so Û_{i} ≤ O_{p}(G(U)) by an argument above. Then Û_{i} = C_{T_{i}}(U_{i}) ∩ O_{p}(G(U)) ⊴ G(U). Also C_{S}(Û) ≤ C_{S}(U), so VÛ ⊴ G(U). Arguing as above, we may assume Û ∈ F_{T}^{fc}. Thus by 7.2.8 in [A1], we can take G(U) = N_{G(Û)}(V_{U}). Proceeding recursively, we define H_{i}(U) = N_{H_{i}(Û)}(U) and H(U) = N_{H(Û)}(U), and observe that H_{i}(U) and H(U) are indeed normal in G(U), with Sylow groups N_{T_{i}}(U) and N_{T} (U).

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

(1) N_{H(U)}(P)β̌ = N_{H(P β)}(Uβ).

(2) Aut_{H(P β)}(Uβ) = Aut_{N_{H(U)}(P)}(U)β^{∗}.

Proof. As usual let U_{i} = T_{i} ∩ U and P_{i} = T_{i} ∩ P.

Suppose first that U_{1}U_{2} ∈ F_{T}^{c}, so that (U_{1}, U_{2}) ∈ P by 5.1.4. As U_{1}U_{2} ≤ P_{1}P_{2}, also P_{1}P_{2} ∈ F_{T}^{c}. Let α_{1} ∈ hom_{F}(N_{S}(U_{1}), S) with X_{1} = U_{1}α_{1} ∈ F_{T_{1}}^{fc}. By 5.4, we may choose α_{1} so that X = (U_{1}U_{2})α_{1} ∈ F_{T}^{fc}. Similarly we may choose β_{1} ∈ hom_{F}(N_{S}(P_{1}), S) so that P_{1}β_{1} ∈ F_{T_{1}}^{fc} and (P_{1}P_{2})β_{1} ∈ F_{T}^{fc}.

Next by construction of H_{1}(U) in 5.9, H_{1}(U) = H_{1}(U_{1}α_{1})α̌_{1}^{−1}, G(U) = G(Uα_{1})α̌_{1}^{−1}, and H_{1}(X) = H_{1}(X_{1}) is the normal subgroup of G(X_{1}) with F_{N_{T_{1}} (X_{1})}(H_{1}(X_{1})) = E_{1}(X_{1}). The analogous statement also holds for P_{1}β_{1}. Thus by 7.18 in [A1] applied to E_{1},

N_{H_{1}(X)}(P_{1}α_{1})α̌_{1}^{−1}β̌_{1} = N_{H_{1}(X_{1})}(P_{1}α_{1})α̌_{1}^{−1}β_{1} = N_{H_{1}(P_{1}β_{1})}(X_{1}α_{1}^{−1}β_{1}) = N_{H_{1}(P_{1}β_{1})}(U_{1}β_{1}).

Then

N_{H_{1}(U_{1}U_{2})}(P_{1})β̌_{1} = N_{H_{1}(X)}(P_{1}α_{1})α̌_{1}^{−1}β̌_{1} = N_{H_{1}(P_{1}β_{1})}(U_{1}β_{1}) = N_{H_{1}((P_{1}P_{2})β_{1})}(U_{1}β_{1}).

Then applying β_{1}^{−1}β to this equality, N_{H_{1}(U_{1}U_{2})}(P_{1})β̌ = N_{H_{1}((P_{1}P_{2})β)}(U_{1}β). By symmetry, N_{H_{2}(U_{1}U_{2})}(P_{2})β̌ = N_{H_{2}((P_{1}P_{2})β)}(U_{2}β), so

N_{H(U_{1}U_{2})}(P_{1}P_{2})β̌ = N_{H_{1}(U_{1}U_{2})}(P_{1})β̌N_{H_{2}(U_{1}U_{2})}(P_{2})β̌ = N_{H_{1}((P_{1}P_{2})β)}(U_{1}β)N_{H_{2}((P_{1}P_{2})β)}(U_{2}β) = N_{H((P_{1}P_{2})β)}((U_{1}U_{2})β),

as H_{i}(U_{1}U_{2}) acts on P_{3−i} and H_{i}((P_{1}P_{2})β) acts on U_{3−i}β by 5.8.2. Therefore

N_{H(U)}(P)β̌ = (N_{H(U_{1}U_{2})}(P_{1}P_{2}) ∩ N(U) ∩ N(P))β̌ = N_{H((P_{1}P_{2})β)}((U_{1}U_{2})β) ∩ N(Uβ) ∩ N(P β) = N_{H(P β)}(Uβ),

so (1) holds in this case. Then (1) implies that (2) also holds in this case.

Now assume we have a counter example with t = |T : U| minimal, subject to this constraint with n = |C_{T} (U_{1}U_{2}) : Z(U_{1}U_{2})| minimal, and subject to these constraint, with m = |P Û : P | minimal, where Û_{i} and Û are defined in 5.9. By the discussion above, n > 1 < t. Set P̂ = P Û. Observe U ⊴ P̂ and N_{S}(U) ≤ N_{S}(Û), so N_{S}(U) ∩ N_{S}(P) acts on P Û = P̂. Let γ ∈ hom_{F}(N_{S}(P̂), S) with P̂γ ∈ F_{T}^{fc}. Then by minimality of n, N_{H(Û)}(P̂)γ̌ = N_{H(P̂γ)}(Ûγ). Thus as above,

N_{H(U)}(P)γ̌ = (N_{H(Û)}(U) ∩ N(P))γ̌ = (N_{H(Û)}(P̂) ∩ N(U) ∩ N(P))γ̌ = N_{H(P̂γ)}(Ûγ) ∩ N(Uγ) ∩ N(P γ) = N_{H(P̂γ)}(Uγ) ∩ N(P γ) = N_{H(P̂γ)}(P γ) ∩ N(Uγ),

which we record as

(*)  N_{H(U)}(P)γ̌ = N_{H(P̂γ)}(P γ) ∩ N(Uγ).

Suppose P ⊴ P̂. Then by minimality of t,

N_{H(P̂γ)}(P γ) = N_{H(P β)}(P̂β)β̌^{−1}γ̌,

and hence it follows from (*) that

(**)  N_{H(U)}(P)γ̌ = N_{H(P β)}(P̂β)β̌^{−1}γ̌ ∩ N(Uγ).

Then applying γ̌^{−1}β̌ to (**), we obtain

(***)  N_{H(U)}(P)β̌ = N_{H(P β)}(P̂β) ∩ N(Uβ).

Thus to establish the lemma in this case, it suffices to show that N_{H(P β)}(Uβ) acts on Ûβ, as then it also acts on P̂β.

Let K = N_{H_{i}(P β)}(Uβ)) and J = ⟨(Û_{i}β)^{K}⟩. As F^{∗}(H_{i}(P β)) = O_{p}(H_{i}(P β)), also F^{∗}(J) = O_{p}(J). But [Û_{i}β, Uβ] ≤ C_{T_{i}}(U_{i}β) ∩ Uβ = Z(U_{i}β), so O^{p}(J) centralizes Uβ. However as Uβ ∈ F_{T}^{c}, Uβ contains a Sylow group of C_{K}(Uβ), so as F^{∗}(J) = O_{p}(J). O^{p}(J) = 1. That is Û_{i}β ⊴ K, so indeed Ûβ ⊴ N_{H(P β)}(Uβ).

Thus we may assume P is not normal in P̂, so P < R = NP̂(P). Observe P Û = RÛ, so |RÛ : R| < |P Û : P | = m. Let δ ∈ hom_{F}(N_{S}(R), S) with Rδ ∈ F_{T}^{fc}. Then by minimality of t, m,

(!)  N_{H(U)}(R)δ̌ = N_{H(Rδ)}(Uδ)

and

(!!)  N_{H(P β)}(Rβ)β̌^{−1}δ̌ = N_{H(Rδ)}(P δ).

Further as P̂ = RÛ,

N_{H(U)}(P) = N_{H(U)}(P̂) ∩ N(P) = N_{H(U)}(R) ∩ N(P),

so by (!),

N_{H(U)}(P)δ̌ = (N_{H(U)}(R) ∩ N(P))δ̌ = N_{H(Rδ)}(Uδ) ∩ N(P δ),

and then applying δ̌^{−1}β̌ to this equality and appealing to (!!) and an argument above,

N_{H(U)}(P)β̌ = N_{H(Rδ)}(P δ)δ̌^{−1}β ∩ N(Uβ) = N_{H(P β)}(Rβ) ∩ N(Uβ) = N_{H(P β)}(Uβ),

completing the proof.

Definition 5.11. Define A to be the constricted F-invariant map on T constructed as in 5.2.3 in [A1] via A(U) = Aut_{H(U)}(U) for U ∈ F_{T}^{fc}. Set

E = E_{1}E_{2} = 𝔈(A).

Theorem 5.12. Assume Hypothesis 5.6. Then E_{1}E_{2} is a normal subsystem of F on T = T_{1}T_{2}. Proof. Let U ∈ F_{T}^{fc}. By construction of the group H(U) in 5.9, H(U) ⊴ G(U) and N_{T} (U) ∈ Syl_{p}(H(U)). Further using 5.5.6 in [A1], condition (ii) of part (4) of Theorem 3 in [A1] is satisfied by A and E as E = 𝔈(A). Also condition (i) of part (4) of Theorem 3 in [A1] is satisfied by 5.10.2. Thus the Theorem follows from Theorem 3 in [A1].

(5.13) Assume T_{1} ∩ T_{2} ≤ Z(E_{i}) for i = 1, 2. Let Q^{j} be the set of pairs (U_{1}, U_{2}) such that U_{j} ∈ F_{T_{j}}^{c}, U_{1}U_{2} ∈ F_{T}^{fc}, and U_{3−j} = T_{3−j}. Then

(1) E = ⟨A(U)φ^{∗} : U = U_{1}U_{2} with (U_{1}, U_{2}) ∈ Q^{j}, φ ∈ Aut_{F}(T), j = 1, 2⟩.

(2) For each P ∈ E_{j}^{fc} and ϕ ∈ Aut_{E_{j}}(P), ϕ extends to ϕ̂ ∈ Aut_{E}(P T_{3−j}) with ϕ̂ = 1 on T_{3−j}.

(3) E = ⟨ϕ̂: ϕ ∈ Aut_{E_{j}}(P) : P ∈ E_{j}^{fc}, j = 1, 2⟩.

(4) E is a central product of E_{1} and E_{2}.

Proof. Let U ∈ F_{T}^{fc}. We first claim that A(U) = Aut_{A(Ũ)}(U) for some Ũ = Ũ_{1}Ũ_{2} containing U, with Ũ_{i} ∈ E_{i}^{c}. Recall the definition of H(U) in 5.9. If U_{1}U_{2} ∈ F_{T}^{fc} then by definition, H(U) = N_{H(U_{1}U_{2})}(U), so we may take Ũ_{i} to be the projection on T_{i} of the preimage of U in T_{1} × T_{2}, since N_{H(U_{1}U_{2})}(U) acts on Ũ_{i}. In general, the definition of H(U) in 5.9 was recursive on the parameter n defined there. Then proceeding by induction on n, we see that the claim holds.

By the claim,

E = ⟨A(U) : U = U_{1}U_{2}, U_{i} ∈ E_{i}^{c}⟩.

Then by 3.6 and 5.5.3 in [A1],

E = ⟨A(U)φ^{∗} : U = U_{1}U_{2} ∈ F_{T}^{fc}, φ ∈ Aut_{F}(T)⟩.

Further for U = U_{1}U_{2} ∈ F_{T}^{fc}, A(U) = Aut_{H}(U) = Aut_{H_{1}(U)}(U)Aut_{H_{2}(U)}(U), and by 5.8.2, Aut_{H_{j}(U)}(U) = Aut_{Aut_{H_{j} (W)}}(U), where W = U_{j}T_{3−j}. Therefore (1) holds.

Let (U_{1}, U_{2}) ∈ Q^{j}, U = U_{1}U_{2}, and ϕ ∈ Aut_{E_{j}}(U_{j}). Then ϕ = c_{h|U_{j}} for some h ∈ H_{j}(U), and from 5.8.3, ϕ̂ = c_{h|U} extends ϕ to Aut_{E}(U) with ϕ̂ = 1 on T_{3−j}. That is (2) holds when P ∈ F_{T_{j}}^{fc}. In general, given P ∈ E_{j}^{fc}, there is α ∈ hom_{F}(N_{S}(P), S) with P α = U_{j} ∈ F_{T_{j}}^{fc}. Set U = U_{j}T_{3−j} and H = H(U). As E is F-invariant,

Aut_{H}(U)α^{−∗} = Aut_{E}(U)α^{−∗} ≤ Aut_{E}(P T_{3−j}),

and each ϕ ∈ Aut_{E_{j}}(P) extends to the E-map ϕ̂α^{∗}α^{−∗} which is 1 on T_{3−j}. Therefore (2) holds, and for ϕ ∈ Aut_{E_{j}}(P), ϕ̂ is an E-map, so E′ ≤ E, where E′ is the second subsystem in (3).

Next Aut_{H}(U) = ⟨Aut_{H_{1}(U)}(U), Aut_{H_{2}(U)}(U)⟩, with

Aut_{H_{i}(U)}(U) = {ϕ̂: ϕ ∈ Aut_{E_{i}}(U_{i})} ⊆ E′,

and similarly for φ ∈ Aut_{F}(T),

Aut_{H_{i}(U)}(U)φ^{∗} = {ϕ̂: ϕ ∈ Aut_{E_{i}}(U_{i}φ)} ⊆ E′.

Therefore E ⊆ E′, completing the proof of (3).

Part (4) follows from (2), (3), and 2.9.6.

We now establish Theorem 3. In particular we may assume Hypothesis 5.6 holds. By 5.12, E ⊴ F. Further if T_{1} ∩ T_{2} ≤ Z(E_{i}) for i = 1, 2, then by 5.13.4, E is a central product of E_{1} and E_{2}. Thus the Theorem is proved.

Section 6. Centralizers

In this section F is a saturated fusion system over the finite p-group S, and E is a normal subsystem of F over T.

Notation 6.1. For U ∈ F_{T}^{f}, define D(U) = D_{F,T} (U) and E(U) = E_{F,E}(U) as in 4.1 of [A1], and form the group G(U) = G_{F,T} (U) and its normal subgroup H(U) = H_{F,E}(U) of 7.19.3, such that D(U) = F_{N_{S}(U)}(G(U)) with N_{S}(U) ∈ Syl_{p}(G(U)), and E(U) = F_{N_{T} (U)}(H(U)), with N_{T} (U) ∈ Syl_{p}(H(U)). Recall V_{T} = T C_{S}(T). Set

I = ⋂_{U∈F_{T}^{fc}} C_{S}(H(U)) and C_{S}(E) = ⋂_{φ∈Aut_{F} (V_{T})} Iφ.

Denote by X the set of subgroups X of C_{S}(T) such that E ⊆ C_{F}(X).

(6.2) (1) C_{S}(E) ≤ I ≤ C_{S}(T) ≤ V_{T}.

(2) Aut_{F}(V_{T}) acts on C_{S}(E).

(3) X = X′, where

X′ = {X ≤ C_{S}(T) : Aut_{E}(Uφ) ⊆ C_{F}(X) for all U ∈ F_{T}^{fc} and all φ ∈ Aut_{F}(V_{T})}.

(4) Each subgroup of C_{S}(E) is in X.

Proof. Set X = C_{S}(E). By construction, X ≤ I ≤ C_{S}(H(T)) ≤ C_{S}(T), so (1) holds. By (1), Iφ ≤ V_{T} for each φ ∈ Aut_{F}(V_{T}), and of course Aut_{F}(V_{T}) permutes {Iφ : φ ∈ Aut_{F}(V_{T})} via right multiplication. Thus (2) holds.

By Theorem 3 and 5.5.6 in [A1], E = 𝔈(Aut_{E,c}) ⊆ C_{F}(X) for each X ∈ X′, so (3) holds.

By construction, for each U ∈ F_{T}^{fc} and Y ≤ X, Aut_{H(U)}(U) ⊆ C_{F}(Y), and Aut_{H(U)}(U) = Aut_{E}(U). Then by (2), for each φ ∈ Aut_{F}(V_{T}), Aut_{E}(Uφ) ⊆ C_{F}(Y). Thus (3) implies (6.3) Let X ∈ X and U ∈ F_{T}^{fc}. Assume XU ∈ N_{F}(U)^{f} and set V = V_{XU}, D = N_{S}(XU), Q = N_{T} (U), and D(U, X) = N_{N_{F} (V)}(XU). Then

(1) U = T ∩ V.

(2) D(U, X) = N_{N_{F} (U)}^{N_{Aut(V )}(XU)}(V) is a saturated and constrained fusion system on D. Let G = G(U, X) = N_{G(U)}(V) ∈ G(D(U, X)).

(3) E(U) ⊴ D(U, X) and there exists H = H(U, X) ⊴ G with Q ∈ Syl_{p}(H) and E(U) = F_{Q}(H).

(4) F_{D}(HD) = F_{D}(H(U)D), and there exists an isomorphism ι_{X,U} : HD → H(U)D which is the identity on D.

(5) C_{D}(H(U, X)) = C_{D}(H(U)).

(6) There exists α ∈ hom_{F}(N_{S}(Q), S) with Qα and Uα in F_{T}^{fc}, N_{S}(Q)α̌ ≤ N_{S}(Qα), and (XU)α ∈ N_{F}(Uα)^{f}.

(7) There is an isomorphism α̌ : G(U) → G(Uα) extending α : N_{S}(U) → N_{S}(Uα) such that α̌c_{Uα} = c_{U}α^{∗}, where c_{W} : G(W) → Aut(W) is the conjugation map for W ∈ {U, Uα}. Further H(U)α̌ = H(Uα).

(8) N_{G(Qα)}(Uα) = N_{G(U)}(Q)α̌ and N_{H(Qα)}(Uα) = N_{H(U)}(Q)α̌.

Proof. By definition of X, X ≤ C_{S}(T) ≤ C_{S}(U), while C_{T} (U) ≤ U as U ∈ F_{T}^{fc}. Thus

T ∩ V = T ∩ UXC_{S}(UX) = U(T ∩ XC_{S}(UX)) ≤ U(T ∩ C_{S}(U)) = UC_{T} (U) = U,

proving (1).

Argue as in the proof of 4.2 in [A1] to establish (2).

Next [H(U), V_{U}] ≤ H(U) ∩ V_{U} = U by (1), so as V ≤ V_{U}, H(U) acts on XU and V_{U}. Thus E(U) = F_{Q}(H(U)) ≤ D(U, X) = D. Arguing as in in 4.3.1 in [AU], E(U) is D-invariant, and as E is saturated, so is E(U) by 4.3.2 in [A1]. Also [N_{H(U)}(Q), C_{S}(Q)] ≤ C_{H(U)}(Q) = Z(Q), so condition N(D, E(U)) of section 6 in [A1] is satisfied. Therefore E(U) ⊴ D, so by Theorem 1 in [A1], there is a model H for E(U) normal in G. Thus (3) holds.

Let W = UC_{D}(U) and B = N_{D(U)}(XU). As XU ∈ N_{F}(U)^{f}, B is a saturated constrained fusion system on D by 1.2.1 in [A1], and as W ≤ V_{U}, G_{1} = N_{G(U)}(W) is a model for B. As V ≤ W, also B = N_{D(U,X)}(W), so G_{2} = N_{G}(W) is also a model for B. Hence by 2.5 in [A1], the identity map 1_{B} on B gives rise to an isomorphism β : (G_{1}, D, W) → (G_{2}, D, W) in Ext(1_{B}); in particular by 2.5.4 in [A1], β : G_{1} → G_{2} is an isomorphism which is the identity on D. Further as 1_{B} : E(U) → E(U), β maps the unique model H(U) for E(U) normal in G_{1} to the unique module H normal in G_{2}. Therefore (4) holds with ι_{X,U} = β_{|H(U)D}. Then (4) implies (5).

Part (6) follows from parts (5) and (6) of 7.2 in [A1]. The first statement in part (7) follows from 7.2.7 in [A1], and the second statement and part (8) follow from 7.18 in [A1].

(6.4) Let X ≤ C_{S}(T) and ϕ ∈ hom_{F}(X, S). Then (1) If ϕ extends to φ ∈ hom(XT, S) then X ∈ X iff Xϕ ∈ X.

(2) If Xϕ is fully centralized in F then X ∈ X iff Xϕ ∈ X.

(3) If X ∈ X then X^{F} ∩ F^{f} ∩ X ≠ ∅.

(4) If X ∈ X and X′ ∈ X^{F}, then N_{T} (X′) = C_{T} (X′).

Proof. Assume the hypothesis of (1). As T is strongly closed in S, φ acts on T. Let P ≤ T and α ∈ hom_{E}(P, T). If X ∈ X then α extends to α̂ ∈ hom_{F}(P X, S) with α̂ = 1 on X. Then α̂φ^{∗} is an extension of αφ^{∗} with α̂φ^{∗} = 1 on Xϕ, so Xϕ ∈ X. Similarly if Xϕ ∈ X then X ∈ X, so (1) holds. Then (1) implies (2).

Assume X ∈ X. Then there is α ∈ hom_{F}(X, S) with Xα ∈ F^{f} and hence Xα is fully centralized. Thus Xα ∈ X by (2), so (3) holds. Let X′ ∈ X^{F}. By (3) there is Y ∈ X^{F} ∩ F^{f} ∩ X. By 1.1 in [A1] there is β ∈ hom_{F}(N_{S}(X′), S) with X′β = Y. Then N_{T} (X′)β ≤ N_{T} (Y) = C_{T} (Y), so (4) holds.

(6.5) Assume X ∈ X, U ∈ F_{T}^{fc}, and

(*) for each Y ∈ X^{F} with |C_{T} (Y)| > |U|, Y ∈ X. Then X ≤ C_{S}(H(U)).

Proof. Assume otherwise and choose a counter example with |T : U| minimal. Set Q = N_{T} (U) and V = V_{XU}. As X ∈ X, X ≤ C_{S}(Q). Let β ∈ hom_{N_{F} (U)}(N_{S}(UX), S) with UXβ fully normalized in N_{F}(U). Then Q = Qβ ≤ N_{S}(UX)β ≤ N_{S}(UXβ), and as X ≤ C_{S}(Q), also Xβ ≤ C_{S}(Q). If U = T then Xβ ∈ X by 6.4.1, while if U ≠ T then U < Q, so Xβ ∈ X by (*). Further Xβ = X^{b} for some b ∈ G(U), and as H(U) ⊴ G(U), X centralizes H(U) iff X^{b} centralizes H(U). Thus replacing X by X^{b}, we may assume UX is fully normalized in N_{F}(U). Hence we have the hypothesis of 6.3, so we adopt the notation of that lemma, and in particular form the groups G = G(U, X) and H = H(U, X). In particular, to complete the proof it suffices by 6.3.5 to show that H centralizes X.

For h ∈ H, c_{h|U} = ϕ ∈ Aut_{E}(U) extends to ϕ̃ ∈ hom_{F}(UX, T) with ϕ̃ = 1 on X, so there exists g ∈ C_{G}(X) with c_{g} = c_{h} on U. Thus c = hg^{−1} ∈ C_{G}(U). Then [c, H] ≤ C_{H}(U) ≤ U, so Q^{c} = Q. Thus Q^{h} = Q^{cg} = Q^{g}, so as Q centralizes X and g ∈ C_{G}(X), also Q^{h} centralizes X. Thus L = ⟨Q^{H}⟩ ≤ C_{G}(X).

Next H = LB, where B = N_{H}(Q), so B does not centralize X. Assume U ≠ T and pick α as in 6.3.6. As X centralizes Q, Xα centralizes Qα, so Xα ∈ X by (*). As U ≠ T, by minimality of |T : U|, Xα centralizes H(Qα), and hence Xα centralizes B′ = Bι_{X,U} α̌ by parts (4) and (8) of 6.3. But now X centralizes B′α̌^{−1}ι_{X,U}^{−1} = B, contrary to the choice of X and U. Thus U = T = Q and B = H.

Now [H, X] ≤ C_{H}(T) = Z(T), so H acts on W = AZ(T) for each subgroup A of X. Let H^{∗} = H/C_{H}(W). As T centralizes W, H^{∗} is a p′-group. As X is generated by abelian subgroups, we may assume A is abelian, so W is abelian, and it remains to show that A centralizes H. As W is abelian, W = [W, H^{∗}]×C_{W} (H^{∗}). For h ∈ H, an argument above shows h = c_{h}g_{h} for some c_{h} ∈ C_{G}(T) and g_{h} ∈ C_{G}(X). Then g_{h} acts on A, so g_{h} acts on W. Thus J = ⟨g_{h} : h ∈ H⟩ acts on W. Now Aut_{H}([W, H]) ≤ Aut_{J}([W, H]) as h = c_{h}g_{h} and c_{h} centralizes T ≥ [W, H]. But J centralizes A, and hence also its projection on [W, H], so as C_{[W,H]}(H^{∗}) = 1, it follows that A ≤ C_{W} (H), completing the proof of the lemma.

(6.6) For each X ∈ X, X^{F} ⊆ X.

Proof. Assume otherwise and choose Y ∈ X and X ∈ Y^{F} with X ∉ X such that n = |T : C_{T} (X)| is minimal. Set P = C_{T} (X). By 6.4.3, we may assume Y ∈ F^{f}. By 1.1 in [A1], we may pick α ∈ hom_{F}(XP, Y T) with Xα = Y. Set Q = P α. We first observe that by 6.4.1:

(1) n > 1.

Let W = Y Q and pick γ ∈ hom_{F}(W, S) with W γ ∈ F^{f}. By 1.1.2 in [A1], there is χ ∈ Aut_{F}(W γ) such that γχ extends to ψ ∈ hom_{F}(N_{S}(W), N_{S}(W γ)). As T centralizes Y, N_{T} (W) = N_{T} (Q) so

N_{T} (Q)ψ = N_{T} (W)ψ ≤ N_{T} (W γ),

and in particular N_{T} (Q)ψ ≤ T centralizes Y ψ = Y γχ. As |N_{T} (Q)| > |Q| = |P | (since P ≠ T), it follows from minimality of n that

(2) Y γχ = Y ψ ∈ X.

As W γ ∈ F^{f}, it follows from 1.1.2 in [A1] that there exists θ ∈ Aut(W γ) such that αγθ extends to μ ∈ hom_{F}(N_{S}(XP), S). Then as X does not centralize N_{T} (XP), Xμ = Xαγθ = Y γθ does not centralize N_{T} (XP)μ ≤ N_{T} (W γ), which we record as:

(3) Y γθ does not centralize N_{T} (W γ).

Let ρ ∈ hom_{F}(N_{S}(T_{Qγ}), S) such that T_{Qγ}ρ ∈ F^{f}. Then N_{S}(W γ) ≤ N_{S}(Qγ) ≤ N_{S}(T_{Qγ}), so N_{S}(W γ)ρ ≤ N_{S}(W γρ) and hence as W γ is fully normalized, so is W γρ. Thus, replacing γ by γρ, we may assume T_{Qγ} ∈ F^{fc}.

By (2), Y ψ centralizes T_{Qγ}. Form the groups G = G(T_{Qγ}, Y ψ) and H = H(T_{Qγ}, Y ψ) of 6.3. Let K = C_{G}(H) and S_{K} = S ∩ K, so that S_{K} ∈ Syl_{p}(K). By (2) and minimality of n, hypothesis (*) of 6.5 is satisfied for the pair Y ψ, T_{Qγ} in the role of “X, U”. Thus Y ψ ≤ S_{K} by 6.5. As W γ ∈ F^{f}, θ and χ extend to θ̂ and χ̂ on V_{W γ}. These maps act on Qγ and hence on T_{Qγ}, and then also on V = V_{W γT_{Qγ}} = V_{Y ψT_{Qγ}}. Thus θ̂_{|V} = c_{g|V} and χ̂_{|V} = c_{f|V} for some f, g ∈ G. Now Y γχ = Y ψ ≤ S_{K}, so as K ⊴ G, Y γ = Y γχc_{f}^{−1} = (Y ψ)^{f^{−1}} ≤ S_{K}, and then Y γθ = (Y γ)^{g} ≤ S_{K}, contrary to (3). This completes the proof.

(6.7) (1) X is the set of subgroups of C_{S}(E).

(2) C_{S}(E) is strongly closed in S with respect to F.

Proof. We first prove (1). Let X ∈ X. By 6.2.4, each subgroup of C_{S}(E) is in X, so we must show X is contained in C_{S}(E). As Aut_{F}(V_{T}) acts on X, it suffices to show each X ≤ I. Let U ∈ F_{T}^{fc}; it remains to show X centralizes H(U). But by 6.6, hypothesis (*) of 6.5 is satisfied, so X centralizes H(U) by 6.5, completing the proof of (1).

We next prove (2). By (1), we must show that if X ∈ X then X^{F} ⊆ X. But this follows from 6.6.

Notation 6.8. Write T′ for C_{S}(E). By 6.7, T′ is strongly closed in S with respect to F, so we can adopt the notation and appeal to the results involving strongly closed subgroups appearing in earlier sections. Define P = F_{T′}^{c} × F_{T}^{c} and define P^{i}, i = 1, 2, as in 5.2.

Write X^{f}, X^{fc} for F_{T′}^{f}, F_{T′}^{fc}, etc. Let X ∈ X^{fc}, and pick representatives {U_{i} : i ∈ I′} ⊆ N_{F}(X)_{T}^{fc} for the orbits of N_{F}(X) on N_{F}(X)_{T}^{c}. By 5.4, for each i ∈ I′ there exists U ∈ F_{T}^{fc} and X_{i} ∈ X^{F} such that UX_{i} ∈ N_{F}(U)^{f} and there exists α_{i} ∈ iso_{F}(UX_{i}, XU_{i}). Form D(U, X_{i}), G(U, X_{i}), and H(U, X_{i}) as in 6.3. As X ∈ X^{fc} and U_{i} ∈ F_{T}^{c}, D(U, X_{i}) = N_{F}(V_{UX_{i}}). Similarly form D(X, U_{i}) = N_{F}(V_{XU_{i}}). Then as in 5.5.3, α̇_{i} : D(U, X_{i}) → D(X, U_{i}) is an isomorphism, which induces an isomorphism α̌_{i} : G(U, X_{i}) → G(X, U_{i}) = G(D(X, U_{i})). Set H(X, U_{i}) = H(U, X_{i})α̌_{i}. Then N_{T} (U_{i}) ∈ Syl_{p}(H(X, U_{i})) and F_{N_{T} (U_{i})}(H(X, U_{i})) = E(U_{i}).

Define

C(X, U_{i}) = C_{G(X,U_{i})}(H(X, U_{i})) and K(X, U_{i}) = O^{p}(C(X, U_{i}))N_{T′}(X).

Next for X ∈ X^{fc}, define

A(X) = O^{p}(Aut_{C_{F} (T)}(X))Aut_{T′}(X).

As T is strongly closed in C_{S}(X), Aut_{F}(X) = Aut_{Aut_{F} (XT)}(X) and Aut_{C_{F} (T)}(X) ⊴ Aut_{Aut_{F} (XT)}(X). Then as T′ is strongly closed in S, A(X) ⊴ Aut_{F}(X). Hence we can extend A to a constricted F-invariant map on T′ as in 5.2.3 of [A1]. Define

C = C_{F}(E) = 𝔈(A).

(6.9) Let R = S ∩ K(X, T). Then (1) E = ⟨Aut_{E}(U_{i}α) : i ∈ I′, α ∈ Aut_{F}(XT)⟩ = ⟨Aut_{E}(U_{i}^{g}) : i ∈ I′, g ∈ N_{G(X,T)}(R)⟩.

(2) If R = S ∩ K(X, U_{i}) for each i ∈ I′, then R ≤ T′.

Proof. As X ∈ X, E ≤ C_{F}(X). By 3.6.1 in [A1], E is N_{F}(X)-invariant, and as E ⊴ F, E is saturated. Finally [H(X, T), C_{S}(T) ∩ N_{S}(X)] ≤ C_{H(X,T)}(T) = Z(T), so condition N(N_{F}(X), E) of section 6 of [A1] is satisfied. Therefore E ⊴ N_{F}(X). Then by 5.4 in [A1], E^{c} = N_{F}(X)_{T}^{c}. Hence applying 5.5.6 in [A1], and recalling that {U_{i} : i ∈ I′} is a set of representatives for the N_{F}(X) orbits on N_{F}(X)_{T}^{c}, the first equality in (1) holds. Next for α ∈ N_{F}(XT), α extends to V_{XT}, so α = g_{|XT} for some g ∈ G = G(X, T). By a Frattini argument, G = N_{G}(R)K(X, T), so as K(X, T) centralizes T, the second equality in (1) holds.

Assume the hypothesis of (2). From 6.8, there is (X_{i}, U) ∈ P^{2} and α_{i} ∈ isom_{F}(UX_{i}, XU_{i}) inducing an isomorphism α̌_{i} : G(U, X_{i}) → G(X, U_{i}) with H(X, U_{i}) = H(U, X_{i})α̌_{i}. By 6.3.5, C_{N_{S}(UX_{i})}(H(U)) = C_{N_{S}(UX_{i})}(H(U, X_{i})), so as Aut_{E}(U) = Aut_{H(U)}(U), Aut_{E}(U) centralizes R′ = (S ∩ K(X, U_{i}))α̌_{i}^{−1}, and hence Aut_{E}(U_{i}) = Aut_{E}(U)α_{i}^{∗} centralizes R′α̌_{i} = S ∩ K(X, U_{i}) = R. Then for g ∈ N_{G}(R), Aut_{E}(U_{i}^{g}) = Aut_{E}(U_{i})c_{g}^{∗} centralizes Rc_{g} = R^{g} = R, so E centralizes R by (1). That is R ≤ C_{S}(E) = T′.

Recall the notion of a strongly normalized chain from section 1 and the chain C(U) = C_{E}(U) for U ≤ T.

(6.10) Adopt Notation of 6.8 and let i ∈ I such that C(U_{i}) = (U_{i} = U_{i,0} < ⋯ < U_{i,n} = T } is a strongly N_{F}(X)-normalized chain. Then

(1) For each 0 ≤ j < n, there is an isomorphism θ_{j} : N_{G(X,U_{i,j})}(U_{i,j+1}) → N_{G(X,U_{i,j+1})}(U_{i,j}) which is the identity on N_{S}(U_{i,j}), such that N_{H(X,U_{i,j})}(U_{i,j+1})θ_{j} = N_{H(X,U_{i,j+1})}(U_{i,j}).

(2) O^{p}(C(X, U_{i,j}))θ_{j} = O^{p}(C(X, U_{i,j+1})), so K(X, U_{i,j})θ_{j} = K(X, U_{i,j+1}).

(3) ω_{j} = θ_{j} ⋯ θ_{n} maps K(X, U_{i,j}) to K(X, T).

Proof. Part (1) follows from parts (6)-(8) of 6.3. Let U(j) = U_{i,j} and Y_{j} = Y (X, U(j)) for Y ∈ {G, H, C, K}. Set S_{j} = S ∩ G_{j}, and Q = N_{T′}(X).

Recall from 6.8 that for each j, K_{j}Q ≤ C_{j}, so Q ≤ C_{S_{j}}(H_{j}). Then as U_{j+1} ≤ H_{j}, we have K_{j}θ_{j} ≤ C_{G_{j}}(U_{j+1})θ_{j} ≤ G_{j+1}.

Let M_{j} = O^{p}(C_{j}). Then M_{j}θ_{j} ≤ K_{j}θ_{j} ≤ C_{G_{j+1}}(U_{j+1}), so [M_{j}θ, H_{j+1}] ≤ C_{H_{j+1}}(U_{j+1}) = Z(U_{j+1}), and then as M_{j} = O^{p}(M_{j}), M_{j}θ ≤ M_{j+1}. Similarly K_{j+1}θ_{j}^{−1} centralizes U_{j+1} ∈ Syl_{p}(H_{j}), so M_{j+1}θ_{j}^{−1} ≤ M_{j}. Thus (2) is established. Further (2) implies (3).

(6.11) Adopt Notation of 6.8. Then (1) A(X) = Aut_{K(X,T)}(X).

(2) N_{T′}(X) is Sylow in K(X, U_{i}) for each i ∈ I.

Proof. Let V = V_{XT} and form G = G(X, T), H = H(X, T), and K = K(X, T) as in 6.8. Let α ∈ Aut_{C_{F} (T)}(X) be a p′-element. Then α extends to α̂ ∈ Aut_{F}(V) centralizing T, so α̂ = g_{g} for some p′-element g ∈ C_{G}(T). Now [g, H] ≤ C_{H}(T) = Z(T), so g ∈ O^{p}(C_{G}(H)) ≤ K. Therefore A(X) = O^{p}(Aut_{C_{F} (T)}(X))Aut_{T′}(X) ≤ Aut_{K}(X). As K centralizes T, the opposite inclusion is trivial, so (1) holds.

By 1.1 we can choose our representatives U_{i} so that C(U_{i}) is a strongly N_{F}(U)- normalized chain. Adopt the notation of 6.10 and its proof. By 6.10.3, S ∩ K_{j} = S ∩ K_{n} = R_{n} for each j. As this holds for each j ∈ I′, it follows from 6.9.2 that R_{n} ≤ T′. But of course N_{T′}(X) ≤ S ∩ K_{n} ≤ R_{n}, so (2) holds.

(6.12) (1) A is a normal map on T′.

(2) For each P ∈ C_{S}(E)^{c}, O^{p}(A(P)) = O^{p}(Aut_{C_{F} (T)}(P)) and A(P) = Aut_{C_{F} (E)}(P).

(3) For each P ∈ C_{S}(E)^{fc}, A(P) = O^{p}(Aut_{C_{F} (T)}(P))Aut_{C_{S}(E)}(P).

Proof. We first prove (2) and (3). Let P ∈ C^{c} and pick μ ∈ hom_{F}(N_{S}(P), S) with P μ ∈ X^{fc}. Then T μ = T as T is strongly closed in C_{S}(P). Now O^{p}(A(P μ)) = O^{p}(Aut_{C_{F} (T)}(P μ)), so as T μ = T,

O^{p}(A(P)) = O^{p}(A(P μ))μ^{−∗} = O^{p}(Aut_{C_{F} (T)}(P μ))μ^{−∗} = O^{p} ∗ Aut_{C_{F} (T)}(P)),

establishing the first part of (2). Moreover once (1) ie established, 7.10.2 in [A1] gives the second part of (2). Further if P ∈ C^{fc} then N_{T′}(P)μ = NT′(P μ), so the first part of (2) implies (3).

We next prove (1). Let X ∈ X^{fc}; we must verify the three conditions (SA1)-(SA3) of 7.4 in [A1]. By 6.11, A(X) = Aut_{K(X,T)}(X) and N_{T′}(X) ∈ Syl_{p}(K(X, T)), so Aut_{T′}(X) ∈ Syl_{p}(A(X)). Thus (SA1) holds.

Let P ∈ D(X)^{f} with X ≤ P ≤ Q = N_{T′}(X). Suppose α ∈ Aut_{A(P)}(X) is a p′-element. By (2), α extends to α̂ ∈ Aut_{F}(P T) centralizing T. Then α̂_{|X} extends α_{|X}, so α_{|X} ∈ O^{p}(Aut_{C_{F} (T)}(X)). Also for y ∈ P, yα ∈ P and c_{y}^{α} = c_{yα}, so α ∈ N_{Aut(X)}(Aut_{P} (X)). That is Aut_{A(P)}(X) ≤ N_{A(X)}(Aut_{P} (X)).

Conversely let β ∈ N_{A(X)}(Aut_{P} (X)) be a p′-element. Then β ∈ O^{p}(Aut_{C_{F} (T)}(X)), so β extends to β̂ ∈ Aut_{F}(XT) centralizing T. Now β acts on Aut_{P} (X), so β̂ acts on Aut_{P} (XT), and hence P ≤ Nβ̂, so β̂ lifts to a p′-element β̃ ∈ Aut_{F}(P T). Thus β ∈ Aut_{A(P)}(X), so Aut_{A(P)}(X) = N_{A(X)}(Aut_{P} (X)), so that (SA2) holds.

Let ϕ ∈ N_{A(Q)}(X) be a p′-element and V = XC_{S}(X). By (SA2), ϕ_{|X} ∈ A(X) acts on Aut_{Q}(X). From 6.11.1, ϕ_{|X} = c_{g|X} for some g ∈ K(X, T). Now g_{|X} acts on Aut_{Q}(X), so as C_{Q}(X) ≤ X, g acts on Q. Let ϕ̂ = c_{g|QC_{S}(Q)}. Then [ϕ̂, C_{S}(Q)] = [g, C_{S}(Q)] ≤ C_{K(X,T)}(Q) = Z(Q), so that (SA3) holds, completing the proof of (1).

We are now in a position to prove Theorem 4. By 6.12.1, A is a normal map on T′, so by Theorem 3 in [A1], C = C_{F}(E) = 𝔈(A) ⊴ F. Further by parts (2) and (3) of 6.12, for P ∈ C^{fc}, Aut_{C}(P) = A(P) = O^{p}(Aut_{C_{F} (T)}(X)Aut_{T′}(P). This completes the proof of Theorem 4.

Section 7. Characteristic and subnormal subsystems

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

Definition 7.1. Define a subsystem E of F to be subnormal if there exists a subnormal series

E = F_{n} ⊴ ⋯ ⊴ F_{0} = F

for E, consisting of subsystems of F such that for each 0 < i ≤ n, F_{i+1} ⊴ F_{i}. Write E ⊴ ⊴ F to indicate that E is subnormal in F.

(7.2) Let E and D be subsystems of F on T and D, respectively. Then (1) If E ⊴ ⊴ F then E is saturated.

(2) If E and D are subnormal in F then there exists a subsystem E ∧ D of F on T ∩ D such that E ∧ D is subnormal in E and D. Further if D ⊴ F, then E ∧ D ⊴ E.

Proof. As normal subsystems of saturated systems are saturated, (1) follows by induction on the length of a subnormal series for E.

Assume the hypothesis of (2). Suppose first that E and D are normal in F. Then I = E ∧ D is normal in F by Theorem 1, and by construction is a system on T ∩ D = I. Adopt the notation of section 3 with E_{1} = E and E_{2} = D. By (1), I is saturated, and by Theorem 1, I is a subsystem of E. As I is F-invariant and I ≤ E, I is E-invariant by 3.6.1 in [A1]. Finally by construction in 3.5, Aut_{E_{i}}(I) = Aut_{H_{i}}(I) for suitable normal subgroups H_{i} of G = G_{1,2}(I), and Aut_{I}(I) = Aut_{H}(I), where H = H_{1} ∩ H_{2}. Then [H, C_{T_{i}}(I)] ≤ C_{H}(I) = Z(I), so (N1) is satisfied for the embedding of I in E_{i}, and hence I ⊴ E_{i}. Thus (2) holds in this case.

Next relax the assumption that E is normal in F, and let E = F_{n} ⊴ ⋯ ⊴ F_{0} = F be a subnormal series for E, with F_{i} a fusion system on T_{i}. Continue to assume however that D ⊴ F. Then F_{1} ∧D is normal in F_{1} and D by the previous case. Then working in F_{1}, by induction on n, there is a normal subsystem E ∧ (F_{1} ∧ D) of E which is subnormal in F ∧D, and hence also subnormal in D. Further this system is on T ∩(T_{1} ∩D) = T ∩D. Thus the lemma also holds in this case.

Finally relax the assumption that D is normal, and let D = D_{m} ⊴ ⋯ ⊴ D_{0} = F be a subnormal series for D, with D_{i} a fusion system on D_{i}. By the previous case, E ∧ D_{1} is a system on T ∩ D_{1} subnormal in E and D_{1}. Then working in D_{1}, by induction on m, (E ∧ D_{1}) ∧ D is a subsystem on T ∩ D_{1} ∩ D = T ∩ D, which is subnormal in D and E ∧ D_{1}, and hence also in E. Thus the proof is at last complete.

(7.3) Let α ∈ Aut(F) and E a subsystem of F on a subgroup T of S.

(1) Eα is a subsystem of F on T α.

(2) If T is strongly closed in S with respect to F, then T α is strongly closed in S with respect to F.

(3) If E is F-invariant, then Eα is F-invariant.

(4) If E ⊴ F then Eα ⊴ F.

(5) If E ⊴ ⊴ F then Eα ⊴ ⊴ F.

Proof. Straightforward.

Define a subsystem E of F to be characteristic in F if E ⊴ F and E is Aut(F)- invariant: that is for each α ∈ Aut(F), Eα = E.

(7.4) If E and D are subsystems of F such that E ⊴ D ⊴ F, D is a system on D ≤ S, and Aut_{F}(D) ≤ Aut(E), then E ⊴ F. In particular if E char D ⊴ F, then E ⊴ F.

Proof. Let E, D be subsystems on T, D, respectively. Then E ⊴ ⊴ F, so E is saturated by 7.2.1.

Let P ≤ T and γ ∈ hom_{F}(P, S). As D ⊴ F, by 3.3 in [A1], there is φ ∈ Aut_{F}(D) and ϕ ∈ hom_{D}(P φ, S) such that γ = φϕ on P. As D ⊴ F, Aut_{F}(D) ≤ Aut(D) (cf. 3.3 in [A1]), and if Aut_{F}(D) ≤ Aut(E) then P φ ≤ T. Moreover if E char D, then Aut_{F}(D) ≤ Aut(E).

Assume Aut_{F}(D) ≤ Aut(E). By the previous paragraph, P φ ≤ T, so P γ = P φϕ ≤ T as ϕ ∈ hom_{D}(P φ, S) and T is strongly closed in D with respect to D. Therefore T is strongly closed in S with respect to F. As E ⊴ D, there exists μ ∈ Aut_{D}(T) and ν ∈ hom_{E}(P φμ, S) with ϕ = μν. Thus γ = φϕ = φμν, with φμ ∈ Aut_{F}(T). That is (in the language of section 3 of [A1]), E is F-Frattini. Specialize to the case P = T. As γ = φϕ with φ ∈ Aut_{F}(D) ≤ Aut(E), and as ϕ ∈ Aut_{D}(T) ≤ Aut(E), we have γ ∈ Aut(E). Therefore Aut_{F}(T) ≤ Aut(E), so E is F-invariant by 3.3 in [A1].

Let V = T C_{D}(T)C_{S}(T C_{D}(T))), D(T) = N_{F}(V), and G ∈ G(D(T)). As D ⊴ F and T ∈ F_{D}^{f}, it follows from Theorem 2 in [A1] that there is H = H(T) ⊴ G with N_{D}(T C_{D}(T)) = F_{D}(H) and D ∈ Syl_{p}(H). Similarly as E ⊴ D, we get K = K(T) ⊴ H with E(T) = N_{E}(T) = F_{T} (K) and T ∈ Syl_{p}(K).

Let X = C_{H}(T) and Y = C_{X}(X/Z(T)). As F^{∗}(X) = O_{p}(X), Y = C_{D∩Y} (T). Let B be a Hall p′-subgroup of K, so that K = BT and [B, X] ≤ Z(T). Then [B, C_{S}(T)] ≤ Y = C_{D∩Y} (T) and [B, C_{D∩Y} (T)] ≤ C_{K}(T) = Z(T), so [B, C_{S}(T)] ≤ Z(T). Therefore (N1) is satisfied by E in F, so E ⊴ F.

Example 7.5. Recall from section 3 that ⟨S^{F}⟩ is the smallest normal subsystem of F on S, and for T ≤ S, N (F, T) is the set of normal subsystems of F on T. Observe ⟨S^{F}⟩ is characteristic in F, using 7.3.4. More generally if:

(i) T is strongly closed in S with respect to F, and

(ii) Aut(F) acts on T, and

(iii) N (F, T) ≠ ∅, then ⟨T^{F}⟩ is characteristic in F.

Definition 7.6. Define

[S, O^{p}(F)] = ⟨S ∩ O^{p}(G(U)) : U ∈ F^{fc}⟩,

where G(U) = G_{F,S}(U) ∈ G(D(U)). We show in the next lemma that [S, O^{p}(F)] is strongly closed in S with respect to F. At that point, for U ∈ F_{[S,O^{p}(F)]}^{fc}, we can define

H^{p}(U) = O^{p}(G_{F,[S,O^{p}(F)]}(U))N_{[S,O^{p}(F)]}(U).

Set A^{p}(U) = Aut_{H^{p}(U)}(U). Let A^{p} be the constricted F-invariant map on [S, O^{p}(F)] determined by A^{p}(U) as in 5.2.3 in [A1], and set

O^{p}(F) = 𝔈(A^{p}).

(7.7) Let T = [S, O^{p}(F)], T̃ = ⟨S ∩ O^{p}(G(V)) : V ∈ F^{frc}⟩, and T̂ = ⟨S ∩ O^{p}(G(V_{U})) : U ∈ F_{T}^{fc}⟩. Then

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

(2) T = T̃ = T̂.

(3) O^{p}(F) is characteristic in F on [S, O^{p}(F)].

(4) O^{p}(O^{p}(F)) = O^{p}(F).

Proof. Assume for the moment that T is strongly closed in S with respect to F. Let V ∈ F^{frc}, U = T ∩ V, D = N_{S}(V), and β ∈ hom_{F}(N_{S}(U), S) with Uβ ∈ F_{T}^{f}. Let G = G(V) and H = O^{p}(G). As V ∈ F^{frc}, V = O_{p}(G). By definition of T, T ∩ H ∈ Syl_{p}(H), so U = O_{p}(HT). Thus [C_{D}(U), H] ≤ C_{H}(U) ≤ Z(U), so C_{D}(U) ≤ V and C_{T ∩D}(U) ≤ U. Then as Dβ ≤ N_{S}(V β) and V ∈ F^{f}, Dβ = N_{S}(V β) and V β ∈ F^{frc}. Hence as C_{Dβ}(Uβ) ≤ V β, C_{S}(Uβ) ∩ N_{S}(V β) = C_{Dβ}(Uβ) ≤ V β, so C_{S}(Uβ) ≤ V β, and hence V_{Uβ} ≤ V β. Then as C_{T ∩D}(U) ≤ U, we conclude that C_{T} (Uβ) ≤ Uβ. Therefore Uβ ∈ F_{T}^{fc} and N_{F}(V)β^{∗} ≤ N_{F}(V_{Uβ}), so by 7.2.8 in [A1], G(V)β̌ = N_{G(V_{Uβ})}(V β), and hence (S ∩ O^{p}(G(V)))β ≤ S ∩ G(V_{Uβ}).

Next we claim that F = D, where D = ⟨N_{F}(V_{W}) : W ∈ F_{T}^{fc}⟩. As F is saturated, F = ⟨Aut_{F}(V) : V ∈ F^{frc}⟩ by A.10 in [BLO]. We argue as in the proof of 5.5.3 in [A1]. Assume the claim is false and pick P ≤ S and α ∈ hom_{F}(P, S) such that α is not a D-map, and m = |S : P | is minimal subject to this constraint. Then m > 1 and we pick V_{i} ∈ F^{frc}, α_{i} ∈ Aut_{F}(V_{i}), and P_{i} as in the proof of 5.5.3 of [A1]. As in that proof, we may take α = α_{1|P} and V_{1} = V. If N_{S}(U) = P then P = V = N_{S}(U), so P = S, contradicting m > 1. Hence P < N_{S}(U), so by minimality of m, β ∈ D, and then as Uβ ∈ F_{T}^{fc} and N_{F}(V)β^{∗} ≤ N_{F}(V_{Uβ}), we have αβ^{∗} ∈ D by definition of D. Then α = (αβ^{∗})β^{−∗} ∈ D. This completes the proof of the claim.

Now we prove (1). Let V ∈ F^{fc}, G = G(V) and H = O^{p}(G)N_{T} (V). As S permutes F^{fc}, T ⊴ S, so as G = O^{p}(G)N_{S}(V), it follows that H ⊴ G. By definition of T, S ∩ O^{p}(G) ≤ T, so Q = N_{T} (V) ∈ Syl_{p}(H). In particular Q is strongly closed in N_{S}(V) with respect to G. Then as F = ⟨Aut_{F}(V) : V ∈ F^{fc}⟩, and as Q is strongly closed in N_{S}(V) with respect to G, (1) holds.

Using the fact that F = ⟨Aut_{F}(V) : V ∈ F^{frc}⟩ and the claim, the same argument shows that T̃ and T̂ are strongly closed. Let W = O_{p}(G(V)). We claim S ∩ O_{p}(G(V)) ≤ T̃. If V = W then V ∈ F^{frc}, so the claim holds. Thus we may assume V < W. Now W ∈ F^{c}, so there is γ ∈ hom_{F}(N_{S}(V), S) with W γ ∈ F^{fc}. By 7.2.8 in [A1], G(V)γ̌ = N_{G(W γ)}(V γ), so (S ∩ O^{p}(G(V)))γ ≤ S ∩ O^{p}(G(W γ)), and by induction on |S : V |, S ∩ O^{p}(G(W γ)) ≤ T̃. Therefore as T̃ is strongly closed, S ∩ O^{p}(G(V)) ≤ T̃, establishing the claim. By the claim, T = T̃. A similar argument using paragraph one shows that T̃ = T̂, completing the proof of (2).

Next suppose V ≤ P ≤ T with P ∈ D(V)^{f}, and β ∈ hom_{F}(N_{S}(P), S) with P β ∈ F^{fc}. Then N_{H}(P) = O^{p}(N_{H}(P))N_{Q}(P), so using 7.2.8 in [A1],

N_{H}(P)β̌ ≤ O^{p}(N_{G(P β)}(V β))(N_{T} (P β) ∩ N_{T} (V β)) ≤ N_{H^{p}(P β)}(V β),

and similarly we obtain the opposite inclusion. Then by Theorem 3 in [A1], E = O^{p}(F) is normal in F.

Visibly for α ∈ Aut(F) and U ∈ F^{fc}, T α = T and A^{p}(U)α = Aut_{H^{p}(Uα)}(Uα) = A^{p}(Uα), so 𝔈(A^{p})α = 𝔈(A^{p}). Thus E is characteristic in F, establishing (3).

Finally let B = O^{p}(F) and U ∈ F_{T}^{fc}. Recall D_{F,S}(U) = N_{N_{F} (V_{U})}(U) and D_{F,T} (U) = N_{N_{F} (V_{T_{U}})}(U), and as U ∈ F_{T}^{fc}, U = T_{U}, so D_{F,S}(U) = D_{F,T} (U) = D_{F,S}(V_{U}). Therefore

G(V_{U}) = G_{F,S}(V_{U}) = G_{F,S}(U) = G_{F,T} (U),

so O^{p}(G(V_{U})) = O^{p}(G_{F,T} (U)).

Next D_{B,T} (U) = N_{B}(U) = E_{F,B}(U) and O^{p}(G_{F,T} (U))N_{T} (U) = H^{p}(U) ∈ G(E_{F,B}(U)), so as G_{B,T} (U) ∈ G(D_{B,T} (U)), G_{B,T} (U) = H^{p}(U). Then as O^{p}(O^{p}(G)) = O^{p}(G) for each finite group G,

O^{p}(G_{B,T} (U)) = O^{p}(O^{p}(G_{F,T} (U)) = O^{p}(G_{F,T} (U)) = O^{p}(G_{F,S}(U)).

Let R = [T, O^{p}(B)]. Then

R = ⟨T ∩O^{p}(G_{B,T} (U)) : U ∈ B^{fc}⟩ = ⟨T ∩O^{p}(G_{F,S}(U)) : U ∈ B^{fc}⟩ = ⟨S∩O^{p}(G_{F,S}(U)) : U ∈ B^{fc}⟩.

Also F_{T}^{fc} ⊆ B^{fc}, so

R ≥ ⟨S ∩ O^{p}(G_{F,S}(U)) : U ∈ F_{T}^{fc}⟩ = ⟨S ∩ O^{p}(G_{F,S}(V_{U})) : U ∈ F_{T}^{fc}⟩ = T̂ = T,

so R = T.

Next

H_{F}^{p}(U) = H^{p}(U) = O^{p}(G_{F,T} (U))N_{T} (U) = O^{p}(G_{B,T} (U))N_{T} (U) = O^{p}(G_{B,R}(U))N_{R}(U) = H_{B}^{p}(U).

Then A_{F}^{p}(U) = Aut_{H_{F}^{p}} (U) = Aut_{H_{B}^{p}}(U) = A_{B}^{p}(U). Also for α a B-map, α is also a F-map and O^{p}(B) ≤ B, so

A_{B}^{p}(Uα) = A_{B}^{p}(U)α^{∗} = A_{F}^{p}(U)α^{∗} = A_{F}^{p}(Uα).

Finally as R = T, B^{f} ⊆ O^{p}(B)^{f}, so using 5.5.4 in [A1],

B = ⟨A_{F}^{p}(P) : P ∈ B^{f}⟩ = ⟨A_{B}^{p}(P) : P ∈ B^{f}⟩ ≤ ⟨A_{B}^{p}(P) : P ∈ O^{p}(B)^{f}⟩ = O^{p}(B),

so B = O^{p}(B), completing the proof of (4).

The first statement in part (2) of the next lemma is essentially Lemma 2.2 in [BC-GLO2], attributed there to Puig.

(7.8) (1) [S, O^{p}(F)] = R_{1} = R_{2}, where

R_{1} = ⟨[U, O^{p}(Aut_{O^{p}(F)}(U))] : U ∈ O^{p}(F)^{fc}⟩ and R_{2} = ⟨[U, O^{p}(Aut_{F}(U))] : U ∈ F^{fc}⟩.

(2) Assume G is a finite group, S ∈ Syl_{p}(G), and F = F_{S}(G). Then [S, O^{p}(F)] = S ∩ O^{p}(G) and O^{p}(F) = F_{S∩O^{p}(G)}(O^{p}(G)).

Proof. Assume either (i) T = [S, O^{p}(F)], E = O^{p}(F), and R = R_{1} or R_{2}, or (ii) E = F = F_{S}(G) for some finite group G and S ∈ Syl_{2}(G), T = S ∩ O^{p}(G), and

R = R_{2}.

Observe R ≤ T. In case (i) the objects of E are contained in T, so R_{1} ≤ T, while R_{2} ≤ T as for U ∈ F^{fc}, [U, O^{p}(Aut_{F}(U))] ≤ S ∩ O^{p}(G_{F,S}) ≤ T. Similarly in case (ii), G_{F,S}(U) = N_{G}(U), and [U, O^{p}(N_{G}(U))] ≤ U ∩ O^{p}(G) ≤ T.

As R ≤ T, to show T = R it remains to show T ≤ R. Suppose not. Then as [T, T] ≤ Φ(T), R[T, T] = Q < T. Let s ∈ T and ϕ ∈ hom_{E}(⟨s⟩, T). By Alperin’s fusion theorem A.10 in [BLO], there exists U_{i} ∈ E^{fc}, 1 ≤ i ≤ n, ϕ_{i} ∈ Aut_{E}(U_{i}), and u_{i} ∈ U_{i}, such that ϕ = ϕ_{1} ⋯ ϕ_{n}, s = u_{1}, and u_{i+1} = u_{i}ϕ_{i}. Then

s^{−1}sϕ = u_{1}^{−1}u_{1}ϕ_{1} · u_{2}^{−1}u_{2}ϕ_{2} ⋯ u_{n}^{−1}u_{n}ϕ_{n},

and u_{i}^{−1}u_{i}ϕ_{i} ∈ Q, so sϕ ∈ sQ. Therefore Q is strongly closed in T with respect to E.

Assume (ii) holds. Then as [T, T] ≤ Q and Q is strongly closed in T with respect to E = F_{S}(G), by a standard transfer argument (cf. 37.4 in [FGT]), Q is Sylow in a normal subgroup L of O^{p}(G) with O^{p}(G)/L ≅ T/Q. Hence as O^{p}(G) = O^{p}(O^{p}(G)), L = O^{p}(G). Then as Q is Sylow in L, Q = T, contrary to an earlier remark. Thus R = T in case (ii).

Next assume (i) holds and let U ∈ F^{fc}. Then S ∩ O^{p}(G(U)) ≤ N_{T} (U) by definition of T, [N_{T} (U), N_{T} (U)] ≤ N_{Q}(U), and N_{Q}(U) is strongly closed in N_{T} (U) with respect to E, and hence with respect to F_{N_{T} (U)}(H^{p}(U)). Hence arguing as in the previous case, S ∩ O^{p}(G(U)) ≤ N_{Q}(U). But then T = ⟨S ∩ O^{p}(G(U)) : U ∈ F^{fc}⟩ ≤ Q, again a contradiction. This contradiction establishes (1).

Now assume the hypothesis of (2) and let H = O^{p}(G). From case (ii), S ∩ O^{p}(G) ≤ R_{2}, while by (1), R_{2} ≤ T = [S, O^{p}(F)]. Further from the definition of T, T ≤ H, so T ∈ Syl_{p}(H). Thus as F = F_{S}(G), U ∈ F_{T}^{fc} iff N_{S}(U) ∈ Syl_{p}(N_{G}(U)) and C_{T} (U) ≤ U. Moreover we may take G(U) = N_{G}(U)/O_{p′}(N_{G}(U)). As T ∈ Syl_{p}(H), O^{p}(N_{G}(U))N_{T} (U) = N_{H}(U), so A^{p}(U) = Aut_{H^{p}(U)}(U) = Aut_{H}(U). Thus

O^{p}(F) = ⟨A^{p}(U^{g}) : U ∈ F_{T}^{fc}, g ∈ N_{G}(T)⟩ = ⟨Aut_{H}(U^{g}) : U ∈ F_{T}^{fc}, g ∈ N_{G}(T)⟩ = F_{T} (H),

completing the proof of (2).

(7.9) Let R ≤ S. Then the following are equivalent:

(1) R ⊴ F.

(2) F_{R}(R) ⊴ F.

Proof. Let E = F_{R}(R). Then E is saturated.

Assume (1) holds. Then for all P ≤ S and ϕ ∈ hom_{F}(P, S), ϕ extends to ϕ̂ ∈ hom_{F}(RP, S) acting on R. In particular if P ≤ R then P ϕ ≤ Rϕ̂ = R, so R is strongly closed in S with respect to F. Also ϕ = ϕ̂_{|P} with ϕ̂ ∈ Aut_{F}(R), so (in the language of section 3 of [A1]) E is F-Frattini. Visibly Aut_{F}(R) ≤ Aut(E), so E is F-invariant by 3.3 in [A1]. As [C_{S}(R), R] ≤ Z(R), condition (N1) is satisfied by E in F, so E is normal in F. That is (1) implies (2).

Now assume (2). Then R is strongly closed in S with respect to F, so by 1.6 in [BCGLO1] it remains to show that R is contained in each U ∈ F^{frc}. Assume U is a counter example; then Q = N_{R}(U) ≰ U. Let G ∈ G(N_{F}(U)), and set W = R ∩ U. As U ∈ F^{frc}, U = F^{∗}(G). As R is strongly closed in S, W ⊴ G. Also [Q, U] ≤ R∩U = W, so Q ≤ H = C_{G}(U/W). Then as U = F^{∗}(G), C_{Q}(W) ≤ W. Let r ∈ Q and ϕ = c_{r|W}. Then for h ∈ H, c_{r^{h}|W} = ϕ^{c_{h}} ∈ Aut_{E}(W), so c_{r^{h}|W} = c_{t|W} for some t ∈ Q. Thus r^{h}t^{−1} ∈ C_{H}(W) ≤ O_{p}(H) ≤ U, so r^{h} ∈ UQ. Thus UQ ⊴ H, so Q ≤ O_{p}(G) = U, contrary to the choice of U.

Recall O_{p}(F) is the largest subgroup of S normal in F. By 7.9, we can also regard O_{p}(F) as the largest normal subsystem of F of the form F_{R}(R) for some R ≤ S.

(7.10) (1) If R ≤ S with R ⊴ F and α ∈ Aut(F), then Rα ⊴ F.

(2) O_{p}(F) is a characteristic subsystem of F.

Proof. Part (1) is straightforward. Then as P = O_{p}(F) is the largest subgroup of S normal in F, (1) implies that P is Aut(F)-invariant. Thus F_{P} (P)α = F_{P α}(P α) = F_{P} (P), so (2) follows from 7.9 and our notational convention that we write O_{p}(F) for F_{P} (P).

Recall that the center of F is

Z(F) = {z ∈ O_{p}(F) : zϕ = z for all ϕ ∈ Aut_{F}(O_{p}(F))} ≤ S.

We also write Z(F) for the fusion system F_{Z}(Z) on the subgroup Z = Z(F) of S.

(7.11) (1) Z(F) is a characteristic subsystem of F.

(2) For each U ∈ F^{fc}, Z(F) ≤ Z(G(U)), where G(U) = G_{F,S}(U) ∈ G(N_{F}(U)).

Proof. Let Z = Z(F) regarded as a subgroup of S, and Z = F_{Z}(Z). For z ∈ Z and s ∈ S, zc_{s} = z by definition of Z, so Z ≤ Z(S). Each F-map ϕ extends to ϕ̂ on R = O_{p}(F), so ϕ̂ centralizes z by definition of Z. In particular Aut_{F}(Z) = 1 and Z ⊴ F, so Z ⊴ F by 7.9. Visibly Z is Aut(F)-invariant, so (1) holds.

Assume the setup of (2). Then Z ≤ Z(S) ≤ U, and then for g ∈ G(U), c_{g|Z} ∈ Aut_{F}(Z) = 1, so Z ≤ Z(G(U)), completing the proof.

(7.12) (1) If U ∈ F^{frc} with G(U) = G_{F,S}(U) p-closed, then U = S.

(2) S = O_{p}(F) iff F = F_{S}(G(S)) iff F^{frc} = {S}.

(3) The following are equivalent:

(a) F = O_{p}(F).

(b) F = F_{S}(S).

(c) For each U ∈ F^{frc}, Aut_{F}(U) is a p-group.

(d) For each U ∈ F^{frc}, G(U) is a p-group.

(e) O^{p}(F) ≤ Z(F).

(f) O^{p}(F) = 1.

Proof. Let U ∈ F^{frc} and set G = G(U). As U ∈ F^{frc}, U = F^{∗}(G). Assume G is p-closed. Then U = F^{∗}(G) = N_{S}(U), and hence U = S. Thus (1) holds.

If S = O_{p}(F) then F is constrained and F = F_{S}(G(S)). If F = F_{S}(G(S)), then S ⊴ F, so N_{S}(U) ⊴ N_{F}(U). Thus G(U) is p-closed, and hence U = S by (1), so F^{frc} = {S}. Finally suppose F^{frc} = {S}. As F is saturated,

F = ⟨Aut_{F}(U) : U ∈ F^{frc}⟩ = ⟨Aut_{F}(S)⟩,

so S ⊴ F, completing the proof of (2).

Next by 7.9, F = O_{p}(F) iff F = F_{S}(S), in which case Aut_{F}(U) = Aut_{S}(U) is a p-group. Further G/Z(U) = Aut_{F}(U), so G is a p-group iff Aut_{F}(U) is a p-group.

Assume G is a p-group for each U ∈ F^{frc}. Then by (1), F^{frc} = {S}, so F = F_{S}(G(S)) by (2). Further O^{p}(G(S)) = 1 and F = F_{S}(S) as G(S) = S is a p-group, so (b) and (f) hold.

Trivially (f) implies (e). Assume (e) holds and let T = [S, O^{p}(F)]. Then S ∩ O^{p}(G) ≤ T ≤ Z(F), so S ∩O^{p}(G) ≤ Z(G). Therefore as F^{∗}(G) = U, O^{p}(G) = 1. Thus (e) implies (d), completing the proof.

(7.13) Assume F = O^{p}(F). Then (1) If F = O_{p}(F), then F = 1.

(2) If E ⊴ F and F = EZ(F) then F = E.

Proof. If F = O_{p}(F), then O^{p}(F) = 1 by 7.12.3, so (1) holds as F = O^{p}(F).

Assume the hypothesis of (2), and let E be defined on T ≤ S and Z = Z(F). Let U ∈ F^{fc} and G = G_{F,S}(U). As F = EZ, U = W Z, where W = U∩T. Then N_{S}(W) = N_{S}(U) and C_{S}(W) = C_{S}(U), so as U ∈ F^{fc}, W ∈ F_{T}^{fc} and U = V_{W}. Thus G = G_{F,T} (W). From 5.9, G = HZ, where H = H_{F,E}(W). Thus as Z ≤ Z(G) by 7.11, O^{p}(G) ≤ H, so O^{p}(G) ∩ S ≤ H ∩ S ≤ T. But by 7.7.2, [S, O^{p}(F)] = ⟨O^{p}(G(U)) ∩ S : U ∈ F^{fc}⟩ ≤ T, and S = [S, O^{p}(F)] as F = O^{p}(F). Thus S = T. Finally as G = HZ, Aut_{F}(U) = Aut_{G}(U) = Aut_{H}(U) = Aut_{E}(U), so F = ⟨Aut_{F}(U) : U ∈ F^{fc}⟩ = ⟨Aut_{E}(U) : U ∈ F^{fc}⟩ = E.

Recall F is quasisimple if F ≠ 1, F = O^{p}(F), and F/Z(F) is simple.

(7.14) Assume F is quasisimple. Then F ≠ O_{p}(F) = Z(F).

Proof. As F ≠ 1 and F = O^{p}(F), F ≠ O_{p}(F) by 7.13. Let Z = Z(F). Then F = N_{F}(Z) = C_{F}(Z), and Z ≤ O_{p}(F) ⊴ F. Therefore by 8.10.2 in [A1], O_{p}(F)/Z ⊴ F/Z. But as F is quasisimple, F/Z is simple, so either F = O_{p}(F) or Z = O_{p}(F). We showed the former does not hold, so O_{p}(F) = Z, completing the proof of the lemma.

(7.15) Assume F/Z(F) is simple, and let E ⊴ F. Then either E ≤ Z(F) or F = EZ(F), and in the latter case F = E if Z(F) ≤ E.

Proof. Let Z = Z(F) and E be a system on T ≤ S. As Z(F) = Z, Z ≤ Z(S), so [T, Z] = 1. Thus EZ ⊴ F by Theorem 3, so replacing E by EZ, we may assume Z ≤ E. Adopt the notation of section 8 of [A1]. By 8.10 in [A1], Eθ = E^{+} ⊴ F^{+}, so as F^{+} is simple, E^{+} = F^{+} or 1. In the latter case E ≤ Z(F), and in the former E = F by 8.10 in [A1].

(7.16) Assume F = O^{p}(F). Then F is quasisimple iff each proper normal subsystem of F is contained in Z(F).

Proof. Let Z = Z(F) and F^{+} = F/Z. Assume first that F is quasisimple and let E ⊴ F. By 7.15, either E ≤ Z(F) or F = EZ, and in the latter case F = E by 7.13.2.

Conversely assume each proper normal subsystem of F is contained in Z. Then by 8.10 in [A1], F^{+} is simple, so as F = O^{p}(F), F is quasisimple.

(7.17) Set S^{p} = [S, O^{p}(F)]. Then (1) O^{p}(F) = ⟨O^{p}(Aut_{F}(P))Aut_{S^{p}}(P) : P ∈ O^{p}(F)^{f}⟩.

(2) S^{p} = ⟨[U, O^{p}(Aut_{F}(U))] : U ∈ F^{fc}⟩.

(3) Assume E ≤ F with E saturated. Then O^{p}(E) ≤ O^{p}(F)⟩.

Proof. By 7.6, A = A^{p} is a constricted F-invariant map on S^{p} and O^{p}(F) = 𝔈(A), so by 5.5.4 in [A1], O^{p}(F) = ⟨A^{p}(P) : P ∈ O^{p}(F)^{f}⟩. But for U ∈ F_{S^{p}}^{fc}, A(U) = Aut_{H^{p}(U)}(U) = Aut_{O^{p}(Aut_{F} (U))}(U)Aut_{S^{p}}(U) as H^{p}(U) = O^{p}(G_{F,S^{p}}(U))N_{S^{p}}(U). Then as A is a constricted invariant map, A(P) = Aut_{O^{p}(Aut_{F} (P))}(P)Aut_{S^{p}}(P) for P ∈ O^{p}(F)^{f}, so (1) holds.

Observe that (2) follows from 7.8.1.

Assume the hypothesis of (3) with E a system on R ≤ S. By 7.7.3, O^{p}(E) is saturated, so replacing E by O^{p}(E) and appealing to 7.7.4, we may assume E = O^{p}(E). Let U ∈ E^{fc}.

Then there exists α ∈ hom_{F}(U, S) with Uα ∈ F^{f} and V_{Uα} ∈ F^{fc}. Then Aut_{F}(Uα) = Aut_{Aut_{F} (V_{Uα})}(Uα), so [Uα, O^{p}(Aut_{F}(Uα))] ≤ [V_{Uα}, O^{p}(Aut_{F}(V_{Uα}))] ≤ S^{p} by (2). Then

[U, O^{p}(Aut_{E}(U))]α = [Uα, O^{p}(Aut_{Eα^{∗}}(Uα)] ≤ [Uα, O^{p}(Aut_{F}(Uα)] ≤ S^{p},

so as S^{p} is strongly closed in S with respect to F, we have [U, O^{p}(Aut_{E}(U))] ≤ S^{p}. Then R^{p} ≤ S^{p} by (2).

Similarly for P ∈ E^{f}, there is β ∈ hom_{F}(P, S) with P β ∈ F^{f} and

(O^{p}(Aut_{E}(P)Aut_{R^{p}}(P))β^{∗} = O^{p}(Aut_{Eβ^{∗}}(P β)Aut_{(Rβ)^{p}}(P β) ≤ O^{p}(Aut_{F}(P β))Aut_{S^{p}}(P β) ≤ O^{p}(F),

as (Rβ)^{p} ≤ S^{p} by the previous paragraph. Then as O^{p}(F) is F-invariant, O^{p}(Aut_{E}(P))Aut_{R^{p}}(P) ≤ O^{p}(F), so E = O^{p}(E) ≤ O^{p}(F) by (1). Hence (3) holds.

(7.18) Assume F_{0} ⊴ F is a system on S_{0} ≤ S, and U ∈ F^{frc}. Then U_{0} = U ∩S_{0} ∈ F_{0}^{c}.

Proof. Assume otherwise. Conjugating in F, we may assume U_{0} ∈ F^{f} −F_{0}^{c}, so C_{S_{0}}(U_{0}) ≰ U_{0}. Then W = C_{S_{0}}(U_{0})∩N_{S}(U) ≰ U_{0}. Let G = G_{F,S}(U). Then [U, W] ≤ U∩S_{0} = U_{0}, so [G, W] ≤ C_{G}(U/U_{0}) ∩ C_{G}(U_{0}), and hence W ≤ O_{p}(G) = U. But then W ≤ U ∩ S_{0} = U_{0}, a contradiction.

(7.19) Assume B is a saturated subsystem of F on S with O^{p}(B) = O^{p}(F). Then B = F.

Proof. Let S_{0} = [S, O^{p}(F)], F_{0} = O^{p}(F), and for P ≤ S, set P_{0} = P ∩ S_{0}. If P_{0} ∈ F_{0}^{fc} then from 7.6, G_{F,S_{0}}(P_{0}) = H_{F,F_{0}}(P_{0})N_{S}(P_{0}) ∈ G(N_{F}(V_{P_{0}})), where H_{F,F_{0}}(P_{0}) = O^{p}(G_{F,S_{0}}(P_{0}))N_{S_{0}}(P_{0}) ∈ G(N_{F_{0}}(P_{0})). By symmetry, G_{B,S_{0}}(P_{0}) = H_{B,F_{0}}(P_{0})N_{S}(P_{0}) with H_{B,F_{0}}(P_{0}) ∈ G(N_{F_{0}}(P_{0})). But N_{B}(V_{P_{0}}) ≤ N_{F}(V_{P_{0}}), so we may take G_{B,S_{0}}(P_{0}) ≤ G_{F,S_{0}}(P_{0}), and similarly H_{F,F_{0}}(P_{0}) = H_{B,F_{0}}(P_{0}). Then

G_{B,S_{0}}(P_{0}) = H_{B,F_{0}}(P_{0})N_{S}(P_{0}) = H_{F,F_{0}}(P_{0})N_{S}(P_{0}) = G_{F,S_{0}}(P_{0}).

Then for P ∈ N_{F}(P_{0})^{f},

Aut_{F}(P) = Aut_{N_{G_{F,S_{0}}} (P_{0})}(P) = Aut_{N_{G_{B,S_{0}}} (P_{0})}(P) = Aut_{B}(P).

Next F = ⟨Aut_{F}(P) : P ∈ F^{frc}⟩, so it suffices to show Aut_{F}(P) = Aut_{B}(P) for P ∈ F^{frc}. By 7.18, P_{0} ∈ F_{0}^{c}, so there exists α ∈ hom_{F}(P, S) with U_{0} = P_{0}α ∈ F_{0}^{fc} and U = P α ∈ N_{F}(U_{0})^{f}. Thus by the previous paragraph, Aut_{B}(P)α^{∗} = Aut_{Bα^{∗}}(U) = Aut_{F}(U), so Aut_{B}(P) = Aut_{F}(U)α^{−∗} = Aut_{F}(P), completing the proof.

Section 8. T F_{0}

In this section F is a saturated fusion system over the finite p-group S, F_{0} is a normal subsystem of F on S_{0} ≤ S, and T is a subgroup of S strongly closed in S with respect to F, such that S_{0} ≤ T. Given P ≤ S, we usually write P_{0} for P ∩ S_{0}. For P_{0} ≤ S_{0}, set P_{0}^{T F_{0}} = {P^{t}ϕ : t ∈ T, ϕ ∈ hom_{F_{0}}(P_{0}^{t}, T)} and (T F_{0})_{S_{0}}^{f} = {P_{0} ≤ T : |N_{T} (P_{0})| ≥ |N_{T} (Q_{0})| for all Q_{0} ∈ P_{0}^{T F_{0}}}. Let O, P be the set of orbits of F_{0}, S_{0} on subgroups of S_{0}.

Part of the treatment here (particularly Lemma 8.1) was suggested by the proof of Theorem 4.6 in [BCGLO2].

(8.1) Let P_{0} ≤ S_{0}. Then (1) S is representated on O and P via s : P_{0}^{F_{0}} ↦ P_{0}^{sF_{0}}, s : P_{0}^{S_{0}} ↦ P_{0}^{sS_{0}}.

(2) S acts on F_{0}^{f} and each member of O contains a member of (T F_{0})_{S_{0}}^{f}.

(3) The following are equivalent:

(a) P_{0} ∈ (T F_{0})_{S_{0}}^{f}.

(b) P_{0} ∈ F_{0}^{f} and |N_{T} (P_{0}) : N_{S_{0}}(P_{0})| = |N_{T} (P_{0}^{F_{0}}) : S_{0}|.

(c) P_{0} ∈ F_{0}^{f} and N_{T} (P_{0}^{F_{0}}) acts on P_{0}^{S_{0}}.

(d) P_{0} ∈ F_{0}^{f} and N_{T} (P_{0}^{F_{0}}) = N_{T} (R_{0})S_{0}.

(4) Let Q_{0} ∈ P_{0}^{T F_{0}} ∩ (T F_{0})_{F_{0}}^{f}. Then there exists φ ∈ hom_{F_{0}}(P_{0}, Q_{0}) such that N_{T} (P_{0}) ≤ N_{φ}.

Proof. Part (1) follows as F_{0} is F-invariant. For Q_{0} ∈ F_{0}^{f} and s ∈ S, N_{S_{0}}(Q_{0}^{s}) = N_{S_{0}}(Q_{0})^{s}, so S acts on F_{0}^{f}.

Let R be the set of orbits of S_{0} on F_{0}^{f} contained in P_{0}^{F_{0}}. Then N_{S}(P_{0}^{F_{0}}) acts on R, and by Proposition 1.16 in [BCGL02], (|R|, p) = 1. Thus

(*) N_{S}(P_{0}^{F_{0}}) acts on Q_{0}^{S_{0}} for some Q_{0} ∈ P_{0}^{F_{0}} ∩ F_{0}^{f}.

Next N_{T} (P_{0}^{F_{0}}) acts on P_{0}^{S_{0}} iff N_{T} (P_{0}^{F_{0}}) = N_{T} (P_{0})S_{0} iff m = |N_{T} (P_{0}^{F_{0}}) : S_{0}| = |N_{T} (P_{0})S_{0} : S_{0}| = |N_{T} (P_{0}) : N_{S_{0}}(P_{0})|. Hence parts (b), (c), and (d) of (3) are equivalent. In particular by (*), |N_{T} (Q_{0}) : N_{S_{0}}(Q_{0})| = m. On the other hand N_{T} (P_{0}) ≤ N_{T} (P_{0}^{F_{0}}), so |N_{T} (P_{0}) : N_{S_{0}}(P_{0})| = |N_{T} (P_{0})S_{0} : S_{0}| ≤ m. Thus as Q_{0} ∈ F_{0}^{f} and |N_{T} (Q_{0}) : N_{S_{0}}(Q_{0})| = m, it follows that Q_{0} ∈ (T F_{0})_{S_{0}}^{f} and (a) is equivalent to (b). Thus (3) is established, and as Q_{0} ∈ (T F_{0})_{S_{0}}^{f}, the proof of (2) is also complete.

Assume the hypothesis of (4). For φ ∈ hom_{F_{0}}(P_{0}, S_{0}), let

φS_{0} = {φc_{x} : x ∈ S_{0}}.

Let L = N_{T} (P_{0}). Then as L ≤ N_{T} (P_{0}^{F_{0}}) and N_{T} (P_{0}^{F_{0}}) ≤ N_{T} (Q_{0}^{S_{0}}) by (3), L acts on

X = {φS_{0} : P_{0}φ ∈ Q_{0}^{S_{0}}, φ ∈ hom_{F_{0}}(P_{0}, S_{0})},

via l : φS_{0} ↦ φc_{l}^{∗}S_{0}. Now for X ∈ X, Φ_{X} = {φ ∈ X : P_{0}φ = Q_{0}} is nonempty, and Φ_{X} = φN_{S_{0}}(Q_{0}) for any given φ ∈ Φ_{X}. Further for X, Y ∈ X, φ ∈ Φ_{X}, and ψ ∈ Φ_{Y}, ψ = φα for some α ∈ Aut_{F_{0}}(Q_{0}). Thus |X | = |Aut_{F_{0}}(Q_{0}) : Aut_{S_{0}}(Q_{0})| is prime to p, as Aut_{S_{0}}(Q_{0}) ∈ Syl_{p}(Aut_{F_{0}}(Q_{0})) since Q ∈ F_{0}^{f} and F_{0} is saturated. Therefore L acts on some X ∈ X. Let φ ∈ Φ_{X}. Then for l ∈ L, φc_{l}^{∗} = φc_{x} for some x ∈ S_{0}, so c_{l}φ^{∗} = c_{lx^{−1}}. Then as c_{l}φ^{∗} ∈ Aut_{F}(Q_{0}), c_{lx^{−1}} = c_{l}φ^{∗} ∈ Aut_{T} (Q_{0}); that is L ≤ N_{φ}, completing the proof of (4) and the lemma.

Notation 8.2. Set (T F_{0})_{S_{0}}^{fc} = (T F_{0})_{S_{0}}^{f}∩F_{0}^{c}. For U_{0} ∈ F_{0}^{fc}, write A(U_{0}) for the set of α ∈ hom_{F}(N_{S}(U_{0}), S) such that U_{0}α ∈ F^{f}. For α ∈ A(U_{0}), set G(U_{0}, α) = H(U_{0}α)N_{T} (U_{0})α (where H(U_{0}α) = H_{F,F_{0}}(U_{0}α)), and N (U_{0}, α) = F_{N_{T} (U_{0})α}(G(U_{0}, α))α^{−∗}, regarded as a fusion system on N_{T} (U_{0}).

(8.3) Let U_{0} ∈ F_{0}^{fc}. Then (1) A(U_{0}) ≠ ∅.

(2) For α ∈ A(U_{0}), N_{T} (U_{0})α ∈ Syl_{p}(G(U_{0}, α)).

(3) N (U_{0}, α) = N (U_{0}) is independent of the choice of α ∈ A(U_{0}).

(4) N (U_{0}) is a saturated constrained fusion system on N_{T} (U_{0}), and each α ∈ A(U_{0}) extends to an isomorphism α̌ : G(N (U_{0})) → G(U_{0}, α), where G(N (U_{0})) ∈ G(N (U_{0})).

Proof. Part (1) follows from 1.1.1 in [A1]. Let α ∈ A(U_{0}), H = H(U_{0}α), and R = N_{T} (U_{0})α. Thus G = G(U_{0}, α) = HR. As U_{0} ∈ F_{0}^{f} and N_{S}(U_{0})α ≤ N_{S}(U_{0}α), N_{S_{0}}(U_{0})α = N_{S_{0}}(U_{0}α). Thus R ∈ Syl_{p}(G) since N_{S_{0}}(U_{0}α) ∈ Syl_{p}(H). Thus (2) holds.

Let β ∈ A(U_{0}) and set γ = β^{−1}α regarded as a map from R_{1} = N_{T} (U_{0})β to R. Set δ = γ_{|N_{S_{0}} (U_{0}β)} : N_{S_{0}}(U_{0}β) → N_{S_{0}}(U_{0}α). Then δ^{∗} : N_{F_{0}}(U_{0}β) → N_{F_{0}}(U_{0}α) is an isomorphism of saturated constrained fusion systems, so by 2.3 in [A1], H_{1}c_{1}δ^{∗} = Hc, where H_{1} = H(U_{0}β), c_{1} : G_{1} = H_{1}R_{1} → Aut(U_{0}β) and c : G → Aut(U_{0}α) are the conjugation maps. Then as R_{1}γ = R, G_{1}c_{1}γ^{∗} = Gc. Hence by 2.4 in [A1], there is an isomorphism γ̌ : G_{1} → G extending γ. Then γ̇ : F_{R_{1}}(G_{1}) → F_{R}(G) is an isomorphism, so as γ̇ = β̇^{−1}α̇, β̇γ̇α̇^{−1} : N (U_{0}, β) → N (U_{0}, α) is an isomorphism which is the identity on N_{T} (U_{0}), and hence the identity map. This proves (3).

Next N (U_{0}, α) is a constrained saturated fusion system, so as α̇ : N (U_{0}) → F_{R}(G) is an isomorphism, (4) holds.

Notation 8.4. For U_{0} ∈ (T F_{0})_{S_{0}}^{fc} and U ∈ N (U_{0})^{f} with U ∩ S_{0} = U_{0}, set N (U) = N_{N (U_{0})}(U) and A(U) = Aut_{N (U)}(U). Set

D = ⟨Aut_{N (U)}(U), F_{0} : U_{0} ∈ (T F_{0})_{S_{0}}^{fc}, U ∈ N (U_{0})^{fc}⟩,

regarded as a fusion system over T. Set D_{0}^{f} = {P_{0} ≤ S_{0} : P_{0} ∈ D^{f}} and D_{0}^{fc} = D_{0}^{f} ∩ F_{0}^{c}.

(8.5) (1) F_{0} ⊆ D.

(2) Let P_{0} ≤ S_{0} and α ∈ hom_{D}(P_{0}, T). Then there exists t ∈ T and ϕ ∈ hom_{F_{0}}(P_{0}^{t}, T) such that α = c_{t}ϕ.

(3) D_{0}^{f} = (T F_{0})_{S_{0}}^{f}.

(4) D = ⟨A_{N (U)}(U) : U_{0} ∈ D_{0}^{fc}, U ∈ N (U_{0})^{fc}⟩.

(5) For U_{0} ∈ D_{0}^{fc} and U ∈ N (U_{0})^{f} with U ∩ S_{0} = U_{0}, N (U) is a saturated fusion system, and if U ∈ N (U_{0})^{fc} then N (U) is constrained with N_{G(N (U_{0}))}(U) ∈ G(N (U)).

(6) For U_{0} ∈ D_{0}^{fc}, A(U_{0}) = Aut_{D}(U_{0}). Further N (U_{0}) ≤ D, so for each g ∈ G(N (U_{0})) and D ≤ N_{T} (U_{0}) with D^{g} ≤ T, c_{g|D} ∈ hom_{D}(D, T).

Proof. Part (1) follows from the definition of D. In particular as F_{0} = ⟨Aut_{F_{0}}(P_{0}) : P_{0} ∈ F_{0}^{fc}⟩ ≤ D, we conclude from the definition of D that

(7) D = ⟨Aut_{F_{0}}(V_{0}), Aut_{N (U)}(U) : V_{0} ∈ F_{0}^{fc}, U_{0} ∈ (T F_{0})_{S_{0}}^{fc}, U ∈ N (U_{0})^{fc}⟩.

Assume the setup in (2). Then by (7), there exists V_{i} ∈ F_{0}^{fc}, 1 ≤ i ≤ n, α_{i} ∈ B(V_{i}) with α = α_{1} ⋯ α_{n}, P_{i} ≤ V_{i}, P_{i}α_{i} = P_{i+1}, and P_{0} = P_{1}, where B(V_{i}) = Aut_{F_{0}}(V_{i}) if V_{i} ∉ (T F_{0})_{S_{0}}^{f}, and B(V_{i}) = Aut_{N (V_{i})}(V_{i}) if V_{i} ∈ (T F_{0})_{S_{0}}^{fc}. In the former case α_{i} = φ_{i|P_{i}} for some φ_{i} ∈ Aut_{F_{0}}(V_{i}), while if V_{i} ∈ (T F_{0})_{S_{0}}^{fc} then α_{i} = c_{t_{i}}c_{h_{i}} for some t_{i} ∈ N_{T} (V_{i}) and h_{i} ∈ H(V_{i}, β_{i}). In particular c_{h_{i}} = φ_{i} ∈ Aut_{F_{0}}(V_{i}). Thus in any case, α_{i} = c_{t_{i}}φ_{i}.

Finally given an F_{0}-map ϕ and t ∈ T, ϕc_{t} = c_{t} · ϕc_{t}^{∗} with ϕc_{t}^{∗} an F_{0}-map, so

α = ∏_{i} c_{t_{i}}φ_{i} = ∏_{i} c_{t_{i}} · φ = c_{t}φ,

for a suitable F_{0}-map φ. Thus (2) holds.

By (2), if P_{0} ≤ S_{0} then P_{0}^{D} = P_{0}^{T F_{0}}, so (3) follows.

Let B = ⟨Aut_{N (U)}(U) : U_{0} ∈ D_{0}^{f} U ∈ N (U_{0})^{fc}⟩. Suppose (4) fails and pick P ≤ T and α ∈ hom_{D}(P, T) such that α is not a B-map and m = |T : P | is minimal. As T ∈ N (S_{0})^{fc}, m > 1. We can pick α = α_{1} ⋯ α_{n}, V_{i}, and P_{i} as in the proof of (2). As α = α_{1} ⋯ α_{n}, it suffices to show each α_{i} is a B-map, so we can take n = 1 and α = α_{1}. Then V_{1} ∈ F_{0}^{fc} but V_{1} ∉ D_{0}^{f}. Let β ∈ iso_{F_{0}}(V_{1}, U_{0}) with U_{0} ∈ D_{0}^{f}. If V_{1} ≠ P then by minimality of m, β ∈ B, and as U_{0} ∈ D_{0}^{f}, αβ^{∗} ∈ B, so α = (αβ^{∗})β^{−∗} ∈ B. Thus we may assume V_{1} = P, and it remains to show we can choose β so that β ∈ B. Now apply the argument above and write β = β_{1} ⋯ β_{k}. As above we are done if P_{i} < V_{i} for each i. But if P_{i} = V_{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 (4).

Pick U_{0} and U as in (5). By (3), U_{0} ∈ F_{0}^{fc}. By 8.3.4, N (U_{0}) is a saturated constrained fusion system and G(N (U_{0})) ∈ G(N (U_{0})). Then (5) follows from the definition of N (U) and 1.2.1 in [A1].

Let U_{0} ∈ D_{0}^{fc} and α ∈ Aut_{D}(U_{0}). By (2), α = c_{t}ϕ with t ∈ T and ϕ an F_{0}-map. Then U_{0}^{t} = U_{0}ϕ^{−1} ∈ U_{0}^{F_{0}}, so t ∈ N_{T} (U_{0}^{F_{0}}). Then as U_{0} ∈ (T F_{0})_{S_{0}}^{f} by (3), we conclude from 8.1.3 that t = rs with r ∈ N_{T} (U_{0}) and s ∈ S_{0}. Thus U_{0}c_{s}ϕ = U_{0}c_{t}ϕ = U_{0}α = U_{0}, so φ = c_{s}ϕ ∈ Aut_{F_{0}}(U_{0}) and α = c_{r}φ ∈ A(U), establishing the first statement in (6).

As N (U_{0}) is a saturated fusion system, N (U_{0}) = ⟨Aut_{N (U_{0})}(U) : U ∈ N (U_{0})^{fc}⟩ by Alperin’s Fusion Theorem (cf. A.10 in [BLO1]), so N (U_{0}) ≤ D from 8.4. This completes the proof of (6).

(8.6) Let S_{0} ≤ P ≤ T. Then (1) Aut_{D}(T) = A(T).

(2) hom_{D}(P, T) = hom_{N (S_{0})}(P, T) and P^{D} = P^{T}.

(3) If P ∈ N (S_{0})^{f} then N (P) = N_{D}(P), so A(P) = Aut_{D}(P) and Aut_{T} (P) ∈ Syl_{p}(Aut_{D}(P)).

(4) If α ∈ Aut_{D}(S_{0}) and P ≤ N_{α} then α extends to a member of hom_{D}(P, T).

Proof. The proof of 5.5.2 in [A1] establishes (1). Similarly as S_{0} is strongly closed in T, given α ∈ hom_{D}(P, T), to show that α ∈ hom_{N (S_{0})}(P, T) it suffices (using the argument of 5.5.2 in [A1]) to take α = γ_{P} for some γ ∈ Aut_{N (U)}(U) and some U ∈ N (S_{0})^{fc}. By definition, α is induced in N_{N (S_{0})}(U), so indeed α ∈ hom_{N (S_{0})}(P, T).

Let β ∈ hom_{D}(P, T). By 8.4, N (S_{0}) = F_{T} (G(S_{0})) and G(S_{0}) = HT, where H = H(S_{0}). Thus β = c_{ht|P} for some h ∈ H and t ∈ T, so P^{ht} = P β ≤ T, and hence P^{h} ≤ T. Then [P, h] ≤ T ∩ H = S_{0}, so h acts on P. Then P β = P^{ht} = P^{t}, so P^{D} = P^{T}. Thus (2) holds.

Assume P ∈ N (S_{0})^{f}. Then N (P) = N_{N (S_{0})}(P), so by (2), N (P) = N_{D}(P), and hence Aut_{D}(P) = Aut_{N (P)}(P). By 8.5.5, N (P) is saturated, so Aut_{T} (P) ∈ Syl_{p}(Aut_{N (P)}(P)). By definition, A(P) = Aut_{N (P)}(P). Thus (3) holds.

Assume α ∈ Aut_{D}(S_{0}) and P ≤ N_{α}. By (2), α is induced in N (S_{0}), so as N (S_{0}) is saturated, α extends to hom_{N (S_{0})}(P, T) ⊆ hom_{D}(P, T), establishing (4).

(8.7) Let P_{0} ≤ S_{0} and α ∈ iso_{D}(P_{0}, U_{0}) with U_{0} ∈ D_{0}^{f} ∩ F_{0}^{c} = D_{0}^{fc}. Then α extends to a member of hom_{D}(N_{α}, T).

Proof. Assume false and choose a counter example with m = |S_{0} : P_{0}| minimal. By 8.6.4, m > 1.

By 8.5.2, α = c_{t}α′ for some t ∈ T and α′ ∈ hom_{F_{0}}(P_{0}^{t}, S_{0}). Further N_{α′} = N_{α}^{t}, so replacing P, α by P^{t}, α′, we may assume α ∈ hom_{F_{0}}(P_{0}, T).

Suppose first that P_{0} < N_{α} ∩ S_{0} = Q_{0}. Then as F_{0} is saturated and α is an F_{0}-map, α extends to α̂ ∈ hom_{F_{0}}(Q_{0}, S_{0}). Set R_{0} = Q_{0}α̂ and G = G(N (U_{0})).

Next N_{α}α^{∗} = M ≤ Aut_{T} (U_{0}) acts on R = Aut_{R_{0}}(U_{0}), and R ≤ Aut_{H}(U_{0}), where H = H(U_{0}) ⊴ G = HN_{T} (U_{0}). As U ∈ F_{0}^{c}, Z(U_{0}) is the kernel of the map H → Aut_{H}(U_{0}), so as U_{0} ≤ R_{0}, M ≤ N_{T} (R_{0}).

By 8.1.4, there is β ∈ hom_{F_{0}}(R_{0}, S_{0}) such that R_{0}β ∈ D_{0}^{f} and N_{T} (R_{0}) ≤ N_{β}. Thus α̂β ∈ hom_{F_{0}}(Q_{0}, S_{0}) with N_{α} ≤ N_{α̂β}, so by minimality of m, α̂β extends to γ ∈ hom_{D}(N_{α}, T). Similarly β extends to δ ∈ hom_{D}(N_{T} (R_{0}), T). Now P_{0}γδ^{−1} = U_{0} and Q_{0}γδ^{−1} = R_{0}, so

N_{α}γ ≤ N_{T} (R_{0}β) ∩ N_{T} (U_{0}β) ≤ (N_{T} (R_{0}) ∩ N_{T} (U_{0}))β,

and hence N_{α}γδ^{−1} ≤ N_{T} (R_{0})∩N_{T} (U_{0}). Moreover γδ^{−1} extends α, completing the proof in this case.

So we may assume P ≤ N_{α}, and N_{α} ∩ S_{0} = P_{0}. As m > 1, P_{0} < N_{S_{0}}(P) = Q_{0}. By 8.5.6, A(U_{0}) = Aut_{D}(U_{0}), and Aut_{T} (U_{0}) ∈ Syl_{p}(A(U_{0}) by 8.3.4. Thus there exists χ ∈ Aut_{D}(U_{0}) with Aut_{Q_{0}}(P_{0})(αχ)^{∗} ≤ Aut_{T} (U_{0}). Then applying case one to αχ, αχ extends to γ ∈ hom_{D}(N_{αχ}, T). Further by 8.3.4, χ = c_{g|U_{0}} for some g ∈ G(N (U_{0})). Let δ = γc_{g^{−1}}. Then δ_{|P_{0}} = αχχ^{−1} = α, so as P ≤ N_{α}, we conclude that P δ ≤ T. By 8.5.6, c_{g^{−1}} ∈ hom_{D}(P γ, P δ), and hence δ ∈ hom_{D}(P, T) extends α, completing the proof.

Let

H = {P ≤ T : P ∩ S_{0} ∈ F_{0}^{c}}.

(8.8) Let P ∈ H and set P_{0} = P ∩ S_{0}. Then (1) There exists α ∈ iso_{D}(P_{0}, U_{0}) with U_{0} ∈ D_{0}^{f} and N_{T} (P_{0}) ≤ N_{α}.

(2) α extends to α̂ ∈ hom_{D}(N_{T} (P), T).

(3) There exists χ ∈ hom_{N (U_{0})}(N_{T} (P α̂), N_{T} (U_{0})) such that P α̂χ ∈ N (U_{0})^{f}.

(4) If P ∈ D^{f} then P α̂χ ∈ D^{f}.

Proof. Part (1) follows from 8.5.3 and 8.1.4. Then (2) follows from 8.7. As N (U_{0}) is saturated, (3) follows from 1.1.2 in [A?]. Finally if P ∈ D^{f} then as N_{T} (P)α̂χ ≤ N_{T} (P α̂χ), (4) follows.

(8.9) Suppose P ∈ D^{c} − H. Then (1) There exists α ∈ hom_{D}(P, T) such that C_{S_{0}}(P_{0}α) ≰ P_{0}α.

(2) There exists P′ ∈ P^{D} such that Out_{S_{0}}(P′) ∩ O_{p}(Out_{D}(P′)) ≠ 1.

Proof. Assume (1) fails and choose a counter example with m = |S_{0} : P_{0}| minimal. As P ∉ H, P_{0} ∉ F_{0}^{c}, so P_{0} ≠ S_{0} and hence m > 1. Thus P_{0} < N_{S_{0}}(P_{0}) = Q_{0}. Set Q = P Q_{0}. As P ∈ D^{c}, also Q ∈ D^{c}. Thus if Q_{0} ∉ F_{0}^{c} then by minimality of m, there is β ∈ hom_{D}(Q, T) such that C_{S_{0}}(Q_{0}β) ≰ Q_{0}β. But P β ≤ Qβ and C_{S_{0}}(Q_{0}β) ≤ C_{S_{0}}(P_{0}β), contrary to the choice of P.

Therefore Q_{0} ∈ F_{0}^{c}, so by parts (1) and (2) of 8.8, there is γ ∈ hom_{D}(N_{T} (Q), T) with Q_{0}γ = U_{0} ∈ D_{0}^{f}. By 8.1.2 and 8.5.3, there is δ ∈ hom_{F_{0}}(P_{0}γ, S_{0}) with V_{0} = P_{0}γδ ∈ D_{0}^{f}. As F_{0} is saturated, we may choose δ to extend to δ̂ ∈ hom_{F_{0}}(N_{S_{0}}(P_{0}γ), S_{0}). As P_{0} ∉ F_{0}^{c}, C_{S_{0}}(V_{0}) ≰ V_{0}, so X_{0} = C_{S_{0}}(V_{0}) ∩ N_{S_{0}}(U_{0}δ̂) ≰ V_{0}. But (N_{S_{0}}(P_{0}γ) ∩ N_{S_{0}}(U_{0}))δ̂ = N_{S_{0}}(V_{0})∩N_{S_{0}}(U_{0}δ̂) as U_{0} ∈ F_{0}^{f}. Thus Y_{0} = X_{0}δ̂^{−1} ≤ C_{S_{0}}(P_{0}γ)∩N_{S_{0}}(U_{0}), and Y_{0} ≰ P_{0}γ. But now α = γ_{|P} contradicts the choice of P as a counter example. This completes the proof of (1).

Pick α as in (1), and let P′ = P α. Then C_{S_{0}}(P′_{0}) ≰ P′_{0}, so Z_{0} = N_{C_{S_{0}} (P′_{0})}(P′) ≰ P′_{0}, and hence Z_{0} ≰ P′. As P ∈ D^{c}, C_{T} (P′) ≤ P′, so Aut_{Z_{0}}(P′) ≰ Inn(P′). Finally [Z_{0}, P′] ≤ C_{S_{0}}(P′_{0}) ∩ P′ = Z(P′_{0}), so Aut_{Z_{0}}(P′) ≤ O_{p}(Aut_{D}(P′)), establishing (2).

Definition 8.10. For U ∈ H, write A(U) for the set of those α ∈ hom_{D}(N_{T} (U), T) such that U_{0}α ∈ D_{0}^{f} and Uα ∈ N (U_{0})^{f}. Set A(U) = A(Uα)α^{−∗} for α ∈ A(U). Suppose α, β ∈ A(U), and set γ = β^{−1}α : Uβ → Uα. Then by the proof of 8.3.3, γ^{∗} : N (U_{0}β) → N (U_{0}α) is an isomorphism, so

A(Uβ)γ^{∗} = Aut_{N (U_{0}β)}(Uβ)γ^{∗} = Aut_{N (U_{0}α)}(Uα) = A(Uα).

Thus the definition of A(U) is independent of the choice of α ∈ A(U).

Let H^{c} = H ∩ D^{c}. Write H^{fc} for the collection of U ∈ H^{c} such that U_{0} ∈ D_{0}^{fc} and U ∈ N (U_{0})^{fc}.

(8.11) Let P ∈ H. Then (1) A(P) ≠ ∅.

(2) P ∈ H^{c} iff for α ∈ A(P), P α ∈ N (P_{0}α)^{c}.

(3) If U ∈ H^{fc} then Aut_{T} (U) ∈ Syl_{p}(A(U)).

(4) If U ∈ H^{fc} and U ≤ Q with Q ∈ N (U)^{f}, then Aut_{A(Q)}(U) = N_{A(U)}(Aut_{Q}(U)).

Proof. Part (1) follows from parts (1)-(3) of 8.8. Let α ∈ A(P) and set U_{0} = P_{0}α, U = P α, and N = N (U_{0}). Then C_{T} (P)α ≤ N_{T} (P)α ≤ N_{T} (U), so if C_{T} (P) ≰ P then C_{T} (U) ≰ U and hence U ∉ N^{c}. Thus if U ∈ N^{c} then P ∈ H^{c}. Conversely if P ∈ H^{c} then as U = P α, C_{T} (Uϕ) ≤ Uϕ for all ϕ ∈ hom_{N} (U, T), as N ≤ D by 8.5.6, so U ∈ N^{c}. Thus (2) holds.

Suppose U ∈ H^{fc}. By definition in 8.4, A(U) = Aut_{N (U)}(U), and by 8.5.5, N (U) is a saturated fusion system on N_{T} (U), so (3) follows. Indeed from 8.2 and 8.3.4, N = F_{N_{T} (U_{0})}(G) for G ∈ G(N), and there is H = H(U_{0}) ⊴ G with H ∈ G(N_{F_{0}}(U_{0})) and G = HN_{T} (U_{0}). Then N (U) = F_{N_{T} (U)}(N_{H}(U)N_{T} (U)) and hence A(U) = Aut_{H}(U)Aut_{T} (U).

Assume the hypothesis of (4). By parts (1)-(3) of 8.8 applied to S in the role of T, there exists α ∈ hom_{F}(N_{S}(Q), S) such that Q_{0}α ∈ (SF_{0})_{S_{0}}^{f} and Qα is fully normalized in F_{N_{S}(Q_{0}α)}(H(Q_{0}α)N_{S}(Q_{0}α). Hence also Qα ∈ H^{fc}. Let G(X) = G_{F,F_{0}}(X) for X ∈ {U_{0}, Q_{0}α}. By 7.2 in [A1], α extends to an isomorphism α̌ : N_{G(U_{0})}(Q_{0}) → N_{G(Q_{0}α)}(V_{U_{0}α}). Then by 7.18 in [A1], N_{H(U_{0})}(Q_{0})α̌ = N_{H(Q_{0}α)}(U_{0}α). Therefore

N_{H(U_{0})N_{T} (U_{0})}(Q_{0})α̌ = N_{H(Q_{0}α)N_{T} (Q_{0}α)}(U_{0}α),

so N_{G(N (U_{0}))}(Q_{0})α̌ = N_{G(N (Q_{0}α))}(U_{0}α). Then

N_{G(N (U))}(Q)α̌ = (N_{G(N (U_{0}))}(Q_{0}) ∩ N(U) ∩ N(Q))α̌ = N_{G(N (Q_{0}α))}(U_{0}α) ∩ N(Uα) ∩ N(Qα) = N_{G(N (Qα))}(Uα).

Next N_{A(U)}(Aut_{Q}(U)) consists of those ϕ ∈ A(U) induced by g ∈ G = G(N (U)) acting on QC_{G}(U). But C_{G}(U) = C_{T} (U) ≤ U ≤ Q, so these are the members of N_{G}(Q). That is N_{A(U)}(Aut_{Q}(U)) is the image of N_{G(N (U))}(Q) under the conjugation map. Also Aut_{A(Qα)}(Uα) is the image of N_{G(N (Qα))}(Uα) under the conjugation map, so from the last display,

Aut_{A(Q)}(U) = Aut_{A(Qα)}(Uα)α^{−∗} = N_{A(U)}(Aut_{Q}(U))α^{∗}α^{−∗} = N_{A(U)}(Aut_{Q}(U)),

establishing (4).

Theorem 8.12. Let V ∈ H^{c}. Then (1) Each α ∈ hom_{D}(V, T) extends to a member of hom_{D}(N_{α}, T).

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

The proof is almost exactly the same as that of Theorem 7.10 in [A1]. Assume the Theorem is false, and let n be minimal subject to the condition that for some V ∈ H^{c} with n = |T : V |, either A(V) ≠ Aut_{D}(V), or some member α of hom_{D}(V, T) does not extend to a member of hom_{D}(N_{α}, T). Pick V ∈ H^{c} such that n = |T : V |, and pick α ∈ hom_{D}(V, T). Set U = V α. The proof involves a series of reductions.

(8.13) n > 1.

Proof. Suppose n = 1. Then T = V = U = N_{α}, so 8.12.1 holds. Also 8.12.2 follows from 8.6.1. This contradicts the choice of n.

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

Proof. Let U_{1} = V_{1}α_{1}. By minimality of n, α_{1} extends to β ∈ hom_{D}(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_{D}(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}.

Let γ ∈ A(U_{1}), N = N (U_{1}γ), and G ∈ G(N). By minimality of n, Aut_{D}(U_{1}γ) = A(U_{1}γ), and by 8.11.3, Aut_{T} (U_{1}γ) ∈ Syl_{p}(A(U_{1}γ)). Further U ∈ H^{c}, so Uγ ∈ N^{c} by 8.11.2, and hence C_{G}(Uγ) = Z(Uγ), so Aut_{Z(Uγ)}(U_{1}γ) is the kernel of the conjugation map c : N_{A(U_{1}γ)}(Uγ) → Aut_{A(U_{1}γ)}(Uγ). Therefore as Aut_{D}(U_{1}) = A(U_{1}γ)γ^{−∗}, Aut_{Z(U)}(U_{1}) is the kernel of c : N_{Aut_{D}(U_{1})}(U) → Aut_{Aut_{D}(U_{1})}(U). Hence as ψ and c_{t} are in Aut_{D}(U_{1}) with c_{t} = ψ on U, it follows that ψ ∈ c_{t}Aut_{Z(U)}(U_{1}) ⊆ 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_{D}(V_{2}, T) extends α_{1}. Continuing in this manner, α extends to α̂ ∈ hom_{D}(N_{α}, T), completing the proof of the lemma.

(8.15) Assume U ∈ H^{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_{α} such that V < V_{1} and β extends to β_{1} ∈ hom_{D}(V_{1}, T), then β extends to a member of hom_{D}(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 8.11.3. In particular N_{β} = N_{T} (V), so (2) follows from 8.14.

By 8.5.4, there exist P_{i} ∈ H^{fc}, 1 ≤ i ≤ m, V_{i} ≤ P_{i}, and ϕ_{i} ∈ A(P_{i}) such that V = V_{1}, V_{i+1} = V_{i}ϕ_{i}, U = V_{m}ϕ_{m}, and α = ϕ_{1} ⋯ ϕ_{m}.

(8.16) (1) ϕ_{m} extends to a member of hom_{D}(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_{D}(N_{T} (U)).

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

Proof. Let ϕ = ϕ_{m} and N = N_{ϕ}. As P_{m} ∈ H^{fc}, N = N (P_{m}) is a saturated constrained fusion system, and A(P_{m}) = Aut_{G}(P_{m}) for G ∈ G(N) by 8.5.5. In particular ϕ = c_{g|P_{m}} for some g ∈ G.

Suppose first that P_{m} = V_{m}. Then V_{m} = P_{m} = U, and by 8.5.6, ϕ̂ = c_{g|N} ∈ hom_{D}(N, T), and hence 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_{g|W} ∈ hom_{D}(W, T) extends ϕ to W, so the lemma follows in this case from 8.14 and 8.15 applied to V_{m}, ϕ_{m} in place of V, α.

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

Proof. Repeat the proofs of 7.15 and 7.16 in [A1] verbatim.

(8.18) Assume U ∈ H^{fc}. Then (1) Aut_{D}(U) = A(U).

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

Proof. Let α ∈ Aut_{D}(U) and set Q = N_{T} (U). By 8.17 there exists χ ∈ A(U) and γ ∈ N_{Aut_{D}(Q)}(U) such that γ extends αχ. By minimality of n, γ ∈ A(Q), so γ ∈ N_{A(Q)}(U).

By 8.11.4, 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 8.5.5, A(U) = Aut_{G}(U) for G ∈ G(N (U)). Thus α = c_{h|U} for some h ∈ G. 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_{G}(U) = Z(U) ≤ Q. Thus c_{h} : N_{α} = Q ∩ Q^{h^{−1}} → Q extends α to N_{α} and is a D-map by 8.5.6. Thus (2) holds.

We are now in a position to complete the proof of Theorem 8.12.

We first prove 8.12.2. Pick α so that U ∈ H^{fc}. Thus α ∈ A(V), and by 8.18.1, A(U) = Aut_{D}(U), so by 8.10,

A(V) = A(Uα^{−1}) = A(U)α^{−∗} = Aut_{D}(U)α^{−∗} = Aut_{D}(Uα^{−1}) = Aut_{D}(V),

estabishing 8.12.2.

Next we prove 8.12.1 by applying Lemma 2.3 in [BCGLO1] (which we refer to in this paragraph as the Lemma) to D and K = {W ∈ H^{c} : |T : W | ≤ n} in the roles of the system F and the set H of the Lemma. First by 8.18.2, condition (IIB)_{K} of the Lemma holds. Second, by 8.17, condition (IIA)_{K} of the Lemma holds. Therefore by part (b) of the Lemma, condition (II)_{K} of the Lemma holds. Finally as the members of H^{c} are fully centralized in D, condition (II)_{K} is precisely 8.12.1. Thus the proof of Theorem 8.12 is complete.

Definition 8.19 Denote the fusion system D defined in 8.4 by T F_{0}, and call this system the product of T with F_{0}.

Theorem 8.20. Let F_{0} be a normal subsystem on S_{0} of the saturated fusion system F on S. Let S_{0} ≤ T ≤ S with T strongly closed in S with respect to F. Then the product system T F_{0} is saturated.

Proof. We verify the hypothesis of Theorem 2.2 in [BCGLO1] for D = T F_{0} in the role of F and H^{c} in the role of H. By 8.5.4, D is H^{c}-generated, as defined in [BCGLO1]. Further hypothesis (*) of Theorem 2.2 in [BCGLO1] is satisfied by 8.9.2. Visibly H, and hence also H^{c} is closed under conjugation, so it remains to show that D is H^{c}-saturated: That is the axioms for saturation hold for members of H^{c}. Axiom I is satisfied by 8.11.3 and 8.12.2, and axiom II is satisfied by 8.12.1. Thus the proof is complete.

(8.21) (1) F_{0} ⊴ T F_{0}.

(2) T F_{0}/F_{0} ≅ F_{T^{+}}(T^{+}), where T^{+} = T/S_{0}.

(3) The map X ↦ XF_{0} is a bijection between the set of subgroups of T containing S_{0} and the set of saturated subsystems of T F_{0} containing F_{0}.

(4) T F_{0} is the smallest saturated subsystem of F on T containing F_{0}, and the unique saturated subsystem B of F on T such that O^{p}(B) = O^{p}(F_{0}).

(5) For U_{0} ∈ D_{0}^{fc}, N (U_{0}) = D_{T F_{0},S_{0}}(U_{0}) and O^{p}(Aut_{D}(U_{0})) ≤ Aut_{F_{0}}(U_{0}).

(6) (T F_{0})^{frc} ⊆ H^{c}.

Proof. Let U_{0} ∈ D_{0}^{fc} and G ∈ G(N (U_{0})). By 8.3, there is an isomorphism φ ∈ hom_{F}(U_{0}, W) with W ∈ F_{S_{0}}^{fc} extending to an isomorphism φ̌ : G → G(U_{0}, φ) = H(W)N_{T} (U_{0})φ, where H(W) = H_{F,F_{0}}(W). By 8.5.3 and construction in 8.2, U_{0} ∈ F_{0}^{f}, so H(U_{0}φ)φ̌^{−1} = H(U_{0}) = H ⊴ G(U_{0}, φ)φ̌^{−1} = G, and G = H(U_{0})N_{T} (U_{0}). In particular E_{D,F_{0}}(U_{0}) = F_{N_{S_{0}} (U_{0})}(H(U_{0})), so H(U_{0}) = H_{D,F_{0}}(U_{0}).

By 3.6.1 in [A1], F_{0} is D-invariant, and by 8.20, D is saturated. As H(S_{0}) = H_{D,F_{0}}(S_{0}) and [C_{T} (S_{0}), H(S_{0})] ≤ [C_{S}(S_{0}), H(S_{0})] = Z(S_{0}), condition (N1) in the definition of normality is satisfied. Thus (1) is established.

As N (U_{0}) ≤ D_{D,S_{0}}(U_{0}), G ≤ G′ = G_{D,F_{0}}(U_{0}). By 8.5.6, Aut_{D}(U_{0}) = A(U_{0}) = Aut_{N (U_{0})}(U_{0}) = Aut_{G}(U_{0}), so G′ = GC_{G′}(U_{0}) = GC_{T} (U_{0}) = G. Thus N (U_{0}) = F_{N_{T} (U_{0})}(G) = F_{N_{T} (U_{0})}(G′) = D_{D,S_{0}}(U_{0}), establishing the first statement in (5).

As G = H(U_{0})N_{T} (U_{0}),

Aut_{D}(U_{0}) = Aut_{G}(U_{0}) = Aut_{H(U_{0})}(U_{0})Aut_{T} (U_{0}),

so O^{p}(Aut_{D}(U_{0})) ≤ Aut_{H}(U_{0}) ≤ Aut_{F_{0}}(U_{0}), completing the proof of (5).

Part (6) follows from 8.9.2.

By construction in 8.4, D is a subsystem of F on T, and by 8.20, D is saturated. Let U ∈ D^{fc} and set T^{p} = [T, O^{p}(D)]. Suppose first that U ∈ H^{fc}. Then G(U) = N_{G}(U), where G = G(U_{0}), and as we saw above, G = HQ, where H = H(U_{0}) and Q = N_{T} (U_{0}). Hence O^{p}(G(U)) ≤ H, so [U, O^{p}(Aut_{D}(U))] = [U, O^{p}(G(U))] ≤ U ∩ H ≤ S_{0}. Then if T^{p} ≤ S_{0}, we have [U, O^{p}(N_{D}(U))Aut_{T^{p}}(U)] = [U, O^{p}(Aut_{D}(U))][U, T^{p}] ≤ S_{0}. In general, U ≤ V ∈ D^{frc} with G(U) = N_{G(V)}(U) and Aut_{D}(U) = Aut_{Aut_{D}(V)}(U). By (6), V ∈ H^{c}. Then by the first case,

[U, O^{p}(Aut_{D}(U))] ≤ [V, O^{p}(Aut_{D}(V))] ≤ S_{0}.

Thus by 7.8.1,

T^{p} = ⟨[U, O^{p}(Aut_{D}(U))] : U ∈ D^{fc}⟩ ≤ S_{0}.

Let P ∈ O^{p}(D)^{f}. Then P ≤ T^{p} ≤ S_{0}. Hence there exists α ∈ hom_{D}(N_{T} (P), T) with both U = P α and V = UC_{S_{0}}(U) in D_{0}^{f}. Then V ∈ D_{0}^{fc}, so O^{p}(Aut_{D}(V)) ≤ Aut_{F_{0}}(V) by (5). Then as Aut_{D}(U) = Aut_{Aut_{D}(V)}(U), O^{p}(Aut_{D}(U)) ≤ Aut_{F_{0}}(U). Hence

O^{p}(Aut_{D}(P)) = O^{p}(Aut_{D}(U))α^{−∗} ≤ Aut_{F_{0}}(U)α^{−∗} ≤ Aut_{F_{0}}(P).

But by 7.17.1, O^{p}(D) = ⟨O^{p}(Aut_{D}(P))Aut_{T^{p}}(P) : P ∈ O^{p}(D)^{f}⟩, so O^{p}(D) ≤ F_{0}. Then by 7.17.3 and 7.7.4,

O^{p}(D) = O^{p}(O^{p}(D)) ≤ O^{p}(F_{0}) ≤ O^{p}(D),

so O^{p}(D) = O^{p}(F_{0}).

On the other hand suppose B is a saturated subsystem of F on T containing F_{0}, and let U_{0} ∈ D^{fc} and G^{∗} ∈ G(D_{B,T} (U_{0})). As H(U_{0}) ∈ G(D_{F_{0},S_{0}}(U_{0})), G(U_{0}) = H(U_{0})N_{T} (U_{0}) ≤ G^{∗}, so N (U_{0}) ≤ N_{B}(U_{0}). Then for U ∈ N (U_{0})^{fc}, N (U) = N_{N (U_{0})}(U) ≤ N_{B}(U), so A(U) = Aut_{N (U)}(U) ≤ Aut_{B}(U) ≤ B. Therefore D ≤ B by 8.5.4. Thus D is the smallest saturated subsystem of F on T containing F_{0}.

Assume further that O^{p}(B) = O^{p}(F_{0}). Then applying 7.19 to the inclusion D ≤ B, and recalling that O^{p}(D) = O^{p}(F_{0}), we conclude that D = B, completing the proof of (4).

By definition of D/F_{0} in 8.6 in [A1], D/F_{0} = N^{+}, where N = N_{D}(S_{0}) and N^{+} is defined earlier in section 8 of [A1]. But N = F_{T} (G), where G = G(S_{0}), so by 8.8 in [A1], N^{+} = F_{T^{+}}(G^{+}), where G^{+} = G/H and H = H(S_{0}). Then as G = HT, G^{+} = G/H ≅ T/S_{0}, completing the proof of (2).

Let X be the set of normal subgroups of T containing S_{0} and X ∈ X. Then X is the set of overgroups of S_{0} in T strongly closed in T with respect to T, and then as S is strongly closed in T with respect to D, X is the set of overgroups of S_{0} strongly closed in T with respect to D by 8.6.2. Therefore applying 8.20 to the saturated fusion system D in the role of F, XF_{0} is saturated. Further by (4), XF_{0} is the unique saturated subsystem B of T F_{0} on X with O^{p}(B) = O^{p}(F_{0}). Also for any saturated subsystem E of D on X containing F_{0}, O^{p}(F_{0}) ≤ O^{p}(E) ≤ O^{p}(D) = O^{p}(F_{0}) by 7.17 and (4), so XF_{0} is the unique saturated subsystem of D on X containing F_{0}. Then (3) follows by induction of the subnormal length in T of overgroups X of S_{0} in T.

(8.22) F = SO^{p}(F).

Proof. Apply 8.21.4 to O^{p}(F) and S in the roles of F_{0} and T.

Observe that Theorem 5 follows from parts (3) and (4) of 8.21, and from 8.22.

(8.23) Let U ∈ F^{f}. Then (1) N_{F}(U), C_{F}(U), and Inn_{F}(U) are saturated fusion systems.

(2) If E ⊴ F is a system on T and U ≤ T then N_{E}(U) ⊴ N_{F}(U), C_{E}(U) ⊴ C_{F}(U), and Inn_{E}(U) ⊴ Inn_{F}(U).

Proof. Part (1) follows from 1.2 in [A1]. Assume the hypothesis of (2), let U ≤ T, let Q = N_{T} (U), and let (A, B) be (N_{E}(U), N_{F}(U)), (C_{E}(U), C_{F}(U)), or (Inn_{E}(U), Inn_{F}(U)). Then A and B are saturated by (1), while A is B-invariant by 3.6.1 in [A1].

Observe C_{T} (Q) ≤ C_{T} (U) ≤ T_{U} ≤ Q, so Q = T_{Q}. By 6.10.1 in [A1], N(D(U), E(U)) is satisfied. Thus each ϕ ∈ Aut_{E(U)}(Q) extends to ϕ̂ ∈ Aut_{D(U)}(V_{Q}) such that [ϕ̂, C_{S}(Q)] ≤ Z(Q). Let A = Q∩A. Then A = Q, C_{T} (U), T_{U} for A equal to N_{E}(U), C_{E}(U), Inn_{E}(U), respectively. In particular A is ϕ-invariant.

Let ψ ∈ Aut_{A}(A). In the third case where A = T_{U}, ψ certainly acts on T_{U}. In the second ϕ extends to ψ̃ centralizing U as ψ ∈ C_{E}(U). In the first case, A = Q and ψ acts on U as ψ ∈ N_{F}(U), so again ψ acts on T_{U}. Thus in each case an extension ψ̃ of ψ acts on T_{U} and U, so ψ̃ ∈ E(U). In particular by an earlier remark, in the first case, ψ extends to ψ̂ ∈ Aut_{D(U)}(V_{Q}) such that [ψ̂, C_{S}(Q)] ≤ Z(Q). Thus ψ̂ ∈ Aut_{B}(V_{Q}), so (N1) is satisfied by (A, B) in this case, and hence A ⊴ B.

So assume we are in case two or three, where ψ̃ ∈ Aut_{E(U)}(T_{U}). Let H = H(U) be the normal subgroup of G(U) supplied by Theorem 2 in [A1]. Then ψ̃ = c_{h|T_{U}} for some h ∈ H. Now [h, C_{S}(T_{U})] ≤ C_{H}(T_{U}) = Z(T_{U}) = Z(A), so [h, C_{D}(A)] ≤ Z(A) where D = S ∩ B, since C_{D}(A) = C_{S}(T_{U}). Thus c_{h|AC_{D}(A)} is an extension of ψ to AC_{D}(A), so again (A, B) satisfies (N1), and hence A ⊴ B.

(8.24) Let Q ∈ F^{f} with Q ∩ S_{0} = 1. Then (1) N_{F_{0}}(Q) = C_{F_{0}}(Q) = C_{S_{0}}(Q)O^{p}(N_{QF_{0}}(Q)).

(2) N_{F_{0}}(Q) ⊴ N_{F}(Q).

(3) N_{QF_{0}}(Q) = Q × N_{F_{0}}(Q).

Proof. Let B = QF_{0}. Applying 8.21.3 with T = S and X = QS_{0}, we conclude that B is a saturated fusion system on X. Similarly O^{p}(B) = O^{p}(F_{0}) by 8.21.4. Set J = N_{B}(Q) and K = N_{F_{0}}(Q). Thus K ≤ J and applying 8.23.1 to B, we conclude that J is a saturated fusion system on N_{X}(Q) = QN_{S_{0}}(Q). Further by 7.19, O^{p}(J) ≤ O^{p}(B) = O^{p}(F_{0}), so O^{p}(J) ≤ K. By 7.7.3, O^{p}(J) ⊴ J, and by 8.22, J = O^{p}(J)N_{X}(Q) = O^{p}(J)QN_{S_{0}}(Q), so J = QL, where L = N_{S_{0}}(Q)O^{p}(J). By 8.20, L is saturated.

Let P = N_{S}(Q), P_{0} = N_{S_{0}}(Q), R_{0} = S ∩ O^{p}(J), U_{0} ∈ J_{R_{0}}^{f}, G = G_{J ,R_{0}}(U_{0}) and H = H_{J ,O^{p}(J)}(U_{0}). Then S ∩ H = R_{0} ∩ H and [Q, P_{0}] ≤ Q ∩ S_{0} = 1, so Q ≤ G. Then [Q, H] ≤ Q ∩ H ≤ Q ∩ R_{0} = 1, so Q centralizes O^{p}(J). Therefore also Q centralizes P_{0}O^{p}(J) = L. Hence as J = QL, J = Q × L. In particular as K ≤ J, if ϕ ∈ hom_{K}(V_{0}, P_{0}) for some V_{0} ≤ P_{0} then ϕ = 1 × ϕ_{2} for some ϕ_{2} ∈ hom_{L}(V_{0}, P_{0}), so ϕ is an L-map, and hence L = N_{F_{0}}(Q). This establishes (1) and (3).

Let M = N_{F}(Q). Again by 8.23, M is saturated. By 3.6.1 in [A1], L = N_{F_{0}}(Q) is M-invariant, and we showed in paragraph one that L is saturated. Thus to complete the proof of (2), it remains to verify that L satisfies condition (N1) in M. Thus it suffices to show that each p′-element α ∈ Aut_{L}(P_{0}) lifts to α̂ ∈ Aut_{M}(P_{0}C_{P} (P_{0})) with [α̂, C_{P} (P_{0})] ≤ Z(P_{0}). As O^{p}(L) = O^{p}(J) ≤ F_{0}, if this condition holds in SF_{0}, then it also holds in F. Therefore replacing F by SF_{0}, we may assume F = SF_{0}. Therefore arguing as in paragraph one, O^{p}(J) = O^{p}(M) ⊴ M. Specialize to the case where U_{0} = R_{0}, so that H is a model for N_{O^{p}(J)}(R_{0}), and hence we may take H = H_{M,O^{p}(J)}(R_{0}). Set Ḡ = G_{M,R_{0}}(R_{0}). As α is a p′-element, α = g_{|P_{0}} for some p′-element g ∈ NḠ(P_{0}). As O^{p}(J) = O^{p}(M), O^{p}(Ḡ) = O^{p}(H), so g ∈ N_{H}(P_{0}). Finally [N_{H}(P_{0}), C_{P} (P_{0})] ≤ C_{H}(P_{0}) ≤ C_{H}(R_{0}) ≤ R_{0}, so [N_{H}(P_{0}), C_{P} (P_{0})] ≤ C_{R_{0}}(P_{0}) ≤ Z(P_{0}). This completes the proof of (2).

Section 9. Components

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

(9.1) Assume S/Z(F) is abelian. Then (1) S = O_{p}(F).

(2) F is not quasisimple.

Proof. Let U ∈ F^{frc}, G = G_{F,S}(U), Q = N_{S}(U), and Z = Z(F). Then Z ≤ Z(G) ≤ U by 7.11.12, and as S/Z is abelian, [Q, U] ≤ Z. Hence as U = F^{∗}(G), it follows that Q = U. Thus U = N_{S}(U), so U = S. Therefore S = O_{p}(F) by 7.12.2. Hence (1) holds.

Suppose F is quasisimple. By 7.14, F ≠ O_{p}(F), so F_{S}(S) is a proper normal subsystem of F by 7.9. Thus F_{S}(S) ≤ Z(F) by 7.16, so Aut_{F}(U) is a 2-group for each U ≤ S, and hence F = O_{p}(F) by 7.12.3, contrary to an earlier remark.

(9.2) Assume E_{i} ⊴ F on T_{i} for i = 1, 2, with [T_{1}, T_{2}] = 1 and T_{1} ∩ T_{2} ≤ Z(E_{i}) for i = 1, 2. Then E_{1}E_{2} is a central product of E_{1} and E_{2}, so E_{2} ≤ C_{F}(E_{1}) .

Proof. The first remark follows from Theorem 3, and then the second follows from the first, the definition of the central product in section 2, the definition of C_{F}(E_{1}) in 6.8, 6.7.1, Theorem 4, and Alperin’s Fusion Theorem (cf. A.10 in [BLO]).

A subsystem C of F is a component of F if C is a quasisimple subnormal subsystem of F. Write Comp(F) for the set of components of F.

(9.3) Let C ∈ Comp(F) be a system on T and E ⊴ ⊴ F a system on R. Then either C ∈ Comp(E) or T ∩ R ≤ Z(C).

Proof. By 7.2 there is I = C ∧ E subnormal in C and E on T ∩ R. As I is subnormal in C and C is quasisimple, either I ≤ Z(C) or I = C by 7.16. In the latter case C = I is a quasisimple subnormal subgroup of E, so C ∈ Comp(E). In the former, T ∩ R ≤ Z(C).

(9.4) If F is quasisimple then Comp(F) = {F}.

Proof. If D ∈ Comp(F) then by 9.3 applied to F, D in the roles of C, E, F ∈ Comp(D). In particular F ≤ D ≤ F, so the lemma holds.

(9.5) If F is quasisimple then C_{Aut(F)}(S/Z(F)) = 1.

Proof. Let Z = Z(F) and α ∈ C_{Aut(F)}(S/Z). Then α acts on each U ∈ F^{frc}, and hence on N_{F}(U). Let G = G_{F,S}(U) and Q = N_{S}(U). By 2.3 and 2.4 in [A1], there exists β ∈ Ext(α), an automorphism of G extending α ∈ Aut(Q). Now [Q, α] ≤ Z ≤ Z(G) by 7.11, so β centralizes O^{p}(G). Thus α centralizes S ∩ O^{p}(G), so as F = O^{p}(F), α centralizes S by 7.7.2. Hence α = 1.

(9.6) Let C ∈ Comp(F) be a system on T and E ⊴ ⊴ F a system on R. Then either C ∈ Comp(E) or [T, R] = 1. In particular if C′ ∈ Comp(F) − {C) is a system on T′, then [T, T′] = 1 and T ∩ T′ ≤ Z(C) ∩ Z(C′).

Proof. Note that 9.4 and the first statement in the lemma imply the second statement, so it remains to prove the first statement. Let C = F_{n} ⊴ ⋯ ⊴ F_{0} = F be a subnormal series for C. Assume the lemma fails, and choose a counter example C, E, F with F minimal. By 9.3 and 9.4, n > 0. By minimality of F, for each C^{∗} ∈ Com(F_{1}) − {C} on T^{∗}, [T, T^{∗}] = 1. As the lemma fails, C ∉ Comp(E), so E ≠ F, and T ∩ R ≤ Z = Z(C) by 9.3. Let E^{+} be a proper normal subsystem of F on T^{+} containing E. If C ∈ Comp(E^{+}) then the lemma holds by minimality of F. On the other hand if T centralizes T^{+}, it also centralizes R, so replacing E by E^{+}, we may assume E ⊴ F.

Assume r ∈ R−C_{S}(T). Then c_{r} ∈ Aut(F_{1}), so C′ = Cc_{r} ∈ Comp(F_{1}) by 7.3. If C = C′ then [T, r] ≤ T ∩R ≤ Z, and then 9.5 supplies a contradiction. Thus C′ ≠ C, so [T, T^{r}] = 1 by the previous paragraph. Now [T, r] ≤ T T^{r} ∩ R = R_{0}, and T ∩ R_{0} ≤ T ∩ R = Z. Thus for t ∈ T, [T, [t, r]] ≤ Z. But [t, r] = t^{−1}t^{r} with t^{r} ∈ T^{r} ≤ C_{S}(T), so for all t ∈ T, [T, t] ≤ Z. This is contrary to 9.1.2, so the proof of the lemma is complete.

(9.7) If C ∈ Comp(F) is a system on T, then:

(1) C is the unique component of F on T.

(2) If Z ≤ C_{S}(T) and α ∈ Aut(F) acts on ZT and Z, then Cα = C.

Proof. If C′ is a second component on T, then T is abelian by 9.6 and 9.4. But this contradicts 9.1.2. Hence (1) holds.

Assume the hypothesis of (2). As α ∈ Aut(F), α permutes Comp(F), so Cα is a component on T α. As α acts on T Z and Z, T Z = T αZ. If C ≠ Cα then [T, T α] = 1 by 9.6, so T Z = T αZ centralizes T, and again 9.1.2 supplies a contradiction.

Define E(F) to be the subsystem of F generated by Comp(F).

Theorem 9.8. (1) E(F) is a characteristic subsystem of F.

(2) Let {C_{i} : i ∈ I} be the set of components of F. Then for each J ⊆ I, C_{J} = ∏_{j∈J} C_{j} is a normal subsystem of E(F), which is a central product of the components C_{j}, j ∈ J, and C_{J} ≤ C_{E(F)}(C_{K}) for each K ⊆ I − J.

(3) If E ⊴ F, then E(E) = ∏_{j∈J(E)} E_{j} ⊴ F, where J(E) = {j ∈ I : C_{j} ∈ Comp(E)}.

Proof. Assume F is a minimal counter example and let D be a proper normal subsystem of F. Then by minimality of F,

E(D) = ∏_{j∈J(D)} C_{j}

satisfies the conclusions of the Theorem. In particular E(D) char D, so E(D) ⊴ F by 7.4. Pick D so that J = J(D) is of maximal order m. If J = I then (1) and (2) hold. Further if F ≠ E ⊴ F, then we’ve seen that E(E) is the product of the components C_{k} with k ∈ J(E) and E(E) ⊴ F. Thus (3) holds too. Therefore J ≠ I, so we can pick k_{0} ∈ I − J. The lemma follows from 9.4 if F = C_{k_{0}}, so we may take C_{k_{0}} ∈ Comp(E).

Let Ē = E(E), and E, D, Ē be systems on E, D, Ē ≤ S, respectively, K = J(E) and U = K − J. As E ⊴ F, Aut_{F}(E) ≤ Aut(E) by 3.3 in [A1], so α ∈ Aut_{F}(E) permutes Comp(E). For k ∈ J ∩ K, E_{k} = S ∩ C_{k} ≤ D, so as D is strongly closed in S with respect to F, E_{k}α ≤ D. If C_{k}α ∉ Comp(D), then [E_{k}α, D] = 1 by 9.6, contrary to E_{k}α ≤ D and 9.1.2. Thus C_{k}α ∈ Comp(D), so Aut_{F}(E) permutes {C_{j} : j ∈ J ∩ K}, and hence also Δ = {C_{u} : u ∈ U}. Similarly Aut_{E}(Ē) permutes Δ. Let Û = ∏_{u∈U} C_{u} be the direct product of the members of Δ, and U = ⟨Δ⟩ ≤ Ē. From 2.3, Û is generated by the maps ϕ̂_{u} of 2.3, with ϕ_{u} a C_{u}-map, u ∈ U, so its image U is generated by the images of these maps under the natural morphism θ of 2.11. Therefore as Aut_{F}(E) and Aut_{E}(Ē) permute Δ, it follows that these groups are subgroups of Aut(U). As (2) holds in E, U ⊴ Ē, and then as Aut_{E}(Ē) ≤ Aut(U), it follows from 7.4 that U ⊴ E. Then as Aut_{F}(E) ≤ Aut(U), another application of 7.4 says that U ⊴ F. Further if E(D) and U are systems on T_{J} and T_{U}, respectively, then [T_{J}, T_{U}] = 1 by 9.6. Thus F′ = E(D)U ⊴ F by Theorem 3. Thus as k_{0} ∈ U, F′ = F by maximality of m. In particular F = E(F), so (1) holds.

Next by Theorem 1, V = E(D) ∧ U is a normal subsystem of F on V = T_{U} ∩ T_{J}. As [T_{U}, T_{J}] = 1, V is abelian. Thus V^{frc} = {V }, so O_{p}(V) = V by 7.12.2. Thus V ≤ O_{p}(Y) for Y ∈ {E(D), U}. As (2) holds in Y, each C ∈ Comp(Y) is normal in Y. Let T = S ∩ C and W ∈ Y_{T}^{fc}. Then V is strongly closed in C_{T_{Y}} (W), so V ⊴ G_{Y,T} (W), and then setting H = H_{Y,C}(W), [V, H] ≤ V ∩ H ≤ Z(H) as V ∩ T ≤ Z(C) by 9.3. Thus H = O^{p}(H)N_{T} (W) centralizes V, so C = ⟨Aut_{C}(W) : W ∈ C^{fc}⟩ centralizes V. Thus V ≤ Z(Y). Therefore F is the central product of E(D) and U by 9.2. In particular C_{i} ⊴ F for each i ∈ J ∪ U, so U = {k_{0}} by maximality of m. As k_{0} was arbitrary in I − J, it follows that I = J ∪ U, and hence (2) holds. Finally if E′ ⊴ F, then (3) holds for E′ trivially if E′ = F, and if E′ is proper, we saw that (3) holds earlier.

(9.9) (1) E(F) centralizes O_{p}(F), and E(F)O_{p}(F) is a central product of E(F) and O_{p}(F).

(2) F^{∗}(F) = E(F)O_{p}(F) is a characteristic subsystem of F.

Proof. Let C ∈ Comp(F) on T and R = O_{p}(F). By 9.6, either C ∈ Comp(R) or T centralizes R, and by 7.13.1 the latter holds. Thus R centralizes E, where E(F) is a system on E ≤ S. Next V = R ∩ E ⊴ E(F), so arguing as in the proof of 9.8, V ≤ Z(E(F)). Then (1) follows from 9.2. By (1) and Theorem 3, F^{∗}(F) ⊴ F. Then as E(F) and O_{p}(F) are characteristic, (2) holds.

We call F^{∗}(F) the generalized Fitting subsystem of F.

(9.10) If E = O_{p}(E) ⊴ ⊴ F then E ≤ O_{p}(F).

Proof. This follows by induction on the length of a subnormal series for E.

(9.11) C_{F}(F^{∗}(F)) = Z(F^{∗}(F)).

Proof. Let R = O_{p}(F), E = C_{F}(F^{∗}(F)) be a system on T, and E(F) a system on Q. Then by 9.9.2 and Theorem 4, E ⊴ F.

Next E(E) ≤ E(F) by 9.8, so as T centralizes Q, T ∩ Q is abelian. Thus E(E) = 1 by 9.1. Similarly Z_{0} = O_{p}(E) ≤ R, so as E centralizes F^{∗}(F), Z_{0} ≤ Z = Z(F^{∗}(F)), and of course the opposite inclusion is trivial, so F^{∗}(E) = Z. Further as E centralizes F^{∗}(F), Z = Z(E). Thus replacing F by E, it remains to show that if F^{∗}(F) = Z(F) = Z, then F = Z.

Assume otherwise and choose D to be minimal subject to D ⊴ ⊴ F and D ≰ Z. In particular each proper normal subsystem of D is contained in Z. If O^{p}(D) ≠ D then O^{p}(D) ≤ Z, so D = O_{p}(D) by 7.12.3, contrary to 9.10 and F^{∗}(F) = Z. Thus D = O^{p}(D), D is subnormal in F, and each proper normal subsystem of D is in Z; hence, using 7.16, D ∈ Comp(F), contradicting F^{∗}(F) = Z.

Observe that Theorem 6 follows from 9.8, 9.9, and 9.11.

(9.12) Let E = C_{F}(E(F)) and Q = C_{S}(E(F)). Then (1) F^{∗}(E) = O_{p}(F) = R.

(2) C_{Q}(R) = Z(F^{∗}(F)) = Z(R).

(3) E is constrained, so E = F_{Q}(G), where G ∈ G(E), Q ∈ Syl_{p}(G), and R = F^{∗}(G).

Proof. If C is a component of E then C is also a component of F, and we obtain a contradiction from 9.1 as in the proof of the previous lemma. Thus F^{∗}(E) = O_{p}(E). Hence F^{∗}(E) ≤ R by 9.10. On the other hand R ≤ E, so R ≤ O_{p}(E), and hence (1) holds.

Next C_{Q}(R) ≤ C_{Q}(F^{∗}(E)) ≤ Z(F^{∗}(E)) = Z(R) = Z by 9.11 and (1). Thus (2) holds. By (2), E is constrained, so E = F_{Q}(G) for G ∈ G(E), and Q ∈ Syl_{p}(G) by 2.5 in [A1]. Then as R = O_{p}(E), also R = O_{p}(G) = F^{∗}(G).

(9.13) Let E ⊴ F, J = Comp(F) − Comp(E), and D = ∏_{C∈J} C. Then D ⊴ F, and ED is the central product of E and D.

Proof. By 9.8, D ⊴ E(F). Let E = S ∩ E and D = S ∩ D. As E ⊴ F, Aut_{F}(S ∩ E(F)) acts on J, so D ⊴ F by an argument in the proof of 9.8.

By 7.2.2, V = E ∧ D ⊴ D. By definition of J, V contains no component of D, so F^{∗}(V) = O_{p}(V) ≤ O_{p}(D). By 9.9.1, O_{p}(D) ≤ Z(D), so F^{∗}(V) = Z(V), and hence V = Z(V). Thus V ≤ Z(D).

Next [D, E] ≤ D ∩ E = V ≤ Z(D), so E acts on (S ∩ C)V for each C ∈ Comp(D). Thus by 9.7.2, E acts on C, so [S ∩ C, E] ≤ V ∩ C ≤ Z(C), so E centralizes D by 9.5.

By 9.2, it remains to show that V ≤ Z(E). If not as V ⊴ E there is ϕ ∈ Aut_{E}(V)^{#}. As V ≤ Z(E), ϕ lifts to α ∈ Aut_{E}(E). Then as D centralizes E and E ⊴ F, α lifts to β ∈ Aut_{F}(ED) with [α, D] ≤ Z(E). Then as D is strongly closed, [α, D] ≤ E ∩ D = V. Then as above, α acts on S ∩C with [α, S ∩C] ≤ S ∩C ∩V ≤ Z(C), so by 9.5, [α, S ∩C] = 1. Therefore α centralizes D, and hence also V, for our final contradiction.

(9.14) Let D = E(F) and E be a normal constrained subsystem of F. Then ED is the central product of E and D.

Proof. As E is constrained, F^{∗}(E) = O_{p}(E), so E contains no components of F. Thus the lemma is a special case of 9.13.

Section 10. Balance

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

(10.1) Let U ∈ F^{f} and suppose C is a component of N_{F}(U). Then C ≤ C_{F}(O_{p}(F)).

Proof. Let R = O_{p}(F), D = N_{F}(U), and Q = O_{p}(D). By 1.2.1 in [A1], D is saturated. As C ∈ Comp(D), C centralizes Q by 9.9. Now N_{R}(U) ≤ Q, so C centralizes N_{R}(Q). Let W ∈ C^{fc}. Then each ϕ ∈ Aut_{C}(W) extends to ϕ̃ = 1 on Q, and then as R ⊴ F, ϕ̃ extends to ϕ̂ on W QR. Moreover if ϕ is a p′-element, then we can choose ϕ̂ to be a p′-element. Next as C_{R}(Q) ≤ C_{R}(U) ≤ Q ≤ C_{Q}(ϕ̂), we conclude from the Thompson A × B-Lemma (cf. 24.2 in [FGT]) that the p′-element ϕ̂ centralizes R.

Let P = S ∩ C. By 7.8.1,

P = ⟨[W, ϕ] : W ∈ C^{fc}, ϕ ∈ Aut_{C}(W), ϕ a p′-element⟩.

Then as [W, ϕ] = [W, ϕ̂] and ϕ̂ centralizes R, it follows that P centralizes R, so Aut_{C}(W) = O^{p}(Aut_{C}(W))Aut_{P} (W) ≤ C_{F}(R). Thus

C = ⟨Aut_{C}(W) : W ∈ C^{fc}⟩ ≤ C_{F}(R),

completing the proof.

(10.2) Let U ∈ F^{f}. Then (1) C_{F}(U), and Inn_{F}(U) are normal subsystems of N_{F}(U).

(2) E(N_{F}(U)) = E(C_{F}(U)) = E(Inn_{F}(U)).

Proof. It is straightforward to check that A ∈ {C_{F}(U), Inn_{F}(U)} is invariant under B = N_{F}(U), and that (A, B) satisfies (N1). By 8.23.1, A and B are saturated, so (1) holds.

Next E(B) ≤ A by 9.9.1, so (1) and Theorem 9.8.3 imply (2).

(10.3) Let U ∈ F^{f} and suppose U centralizes E(F). Then E(F) = E(N_{F}(U)).

Proof. Let D = N_{F}(U), E = E(F), and E = S ∩ E. As U centralizes E, E ≤ D. As E ⊴ F, E is saturated, and by 3.6.1 in [A1], E is D-invariant. Each ϕ ∈ Aut_{E}(E) extends to ϕ̂ ∈ Aut_{F}(V_{E}) with [ϕ̂, C_{S}(E)] ≤ Z(E). Further if ϕ is a p′-element, we may pick ϕ̂ to be a p′-element, and

[ϕ̂, C_{S}(E)] ≤ C_{S}(E) ∩ Z(E) = Z(E) ≤ C_{E}(ϕ) = C_{E}(ϕ̂),

so ϕ̂ centralizes C_{S}(E). Then as U ≤ C_{S}(E), ϕ̂ is a D-map. Therefore E ⊴ D, so E ≤ E(D).

Conversely suppose C ∈ Comp(D). Then C centralizes O_{p}(F) by 10.1, and if C ≰ E then C centralizes E, so C ≤ C_{F}(F^{∗}(F)) = Z(F), contradicting C perfect. Hence E = E(D).

Theorem 10.4. Let U ∈ F^{f}. Then E(N_{F}(U)) ≤ E(F).

The proof involves a series of reductions.

Let P = P(F) be the set of pairs (U, C) such that U ∈ F^{f} and C ∈ Comp(N_{F}(U)), but C ≰ E(F).

Assume the Theorem is false and pick F such that P(F) ≠ ∅, and with F minimal subject to this constraint. Let (U, C) ∈ P.

(10.5) U does not centralize E(F).

Proof. This is a consequence of 10.3.

(10.6) If V ≤ S centralizes C, U ≤ N_{S}(V), and V is not normal in F, then for β ∈ hom_{F}(N_{S}(V), S) with V β ∈ F^{f}, Cβ^{∗} ≤ E(N_{F}(V β)), so there exists a component C_{V} of N_{F}(V β) with (V β, C_{V}) ∈ P.

Proof. Let D = N_{F}(V β). Pick α ∈ hom_{D}(Uβ, N_{S}(V β)) with U′ = Uβα ∈ D^{f}. Then C′ = Cβ^{∗}α^{∗} is a component of N_{D}(U′). By hypothesis, D ≠ F. Thus by minimality of F, C′ ≤ E(D). Then as E(D) ⊴ D, Cβ^{∗} ≤ E(D). Thus as C ≰ E(F), also E(D) ≰ E(F), so the lemma follows.

(10.7) Z(F) = 1.

Proof. Assume Z = Z(F) ≠ 1, set F^{+} = F/Z, and let θ : F → F^{+} be the natural map discussed in section 8 of [A1]. By 10.5, U ≰ O_{p}(F), so V = UZ ≰ O_{p}(F). Then appealing to 10.6 and replacing (U, C) by (V β, C_{V}) as in that lemma, we may assume Z ≤ U.

By 8.5 in [A1], F^{+} is saturated, and by 8.4.2 in [A1], U^{+} = Uθ ∈ F^{+f}. As Z ≤ U, N_{F}(U)θ = N_{F^{+}}(U^{+}) by 8.4.1 in [A1]. Then by 8.9.2 in [A1], C^{+} = Cθ is subnormal in N_{F^{+}}(U^{+}), and as C is quasisimple, so is C^{+} using 8.10 in [A1]. Therefore by minimality of F, C^{+} ≤ E(F^{+}).

Let D^{+} be a component of F^{+} and D its preimage in F. By 7.15, D = O^{p}(D)Z. Further using 8.10 in [A1], O^{p}(D) is quasisimple and subnormal in F, so O^{p}(D) ∈ Comp(F). Thus E(F)θ = E(F^{+}) and C = O^{p}(C) ≤ E(F), contrary to the choice of C.

(10.8) O_{p}(F) = 1.

Proof. Assume R = O_{p}(F) ≠ 1, and pick V of order p in Z(S)∩R. Then V ∈ F^{f} centralizes C and U, so if V is not normal in F, then by 10.6, there exists C_{V} ∈ Comp(N_{F}(V)) with (V, C_{V}) ∈ P. But this is contrary to 10.5.

Thus V ⊴ F. Let E = C_{F}(V). By 10.7, E is proper in F. As V ≤ Z(S), S = E ∩ S. By 10.2, E ⊴ F. By 8.23.2, N_{E}(U) ⊴ N_{F}(U), so by 9.8, C ∈ Comp(N_{E}(U)). Then by minimality of F, C ≤ E(E), contrary to 10.2.2.

(10.9) There exists (U, C) ∈ P such that either U ≤ S ∩ E(F), or U ∩ E(F) = 1.

Proof. If V = U ∩ E(F) ≠ 1 then by 10.6 and 10.8, we may replace (U, C) by (V β, C_{V}) ∈ P, and V β ≤ E(F).

In the remainder of the proof let E = E(F) and set E = S ∩ E. By 10.9, we may assume that either U ≤ E or U ∩ E = 1. Let T = S ∩ C, E_{U} = N_{E}(U), Q = N_{S}(U), Q_{T} = N_{Q}(T), F_{U} = N_{F}(U), and E_{U} = N_{E}(U).

(10.10) E_{U} ⊴ F_{U}.

Proof. If U ≤ E, this follows from 8.23.2, while if U ∩ E = 1 it follows from 8.24.2.

(10.11) (1) C centralizes E_{U}.

(2) There exists β ∈ hom_{F}(N_{S}(E_{U}), S) such that E_{U}β ∈ F^{f}, and there is (E_{U}β, C_{E}) ∈ P.

Proof. As C ≰ E, (1) follows from 9.13 applied to F_{U}, E_{U} in the roles of F and E. Let β ∈ hom_{F}(N_{S}(E_{U}), S) with E_{U}β ∈ F^{f}. Now (2) follows from 10.6.

Appealing to 10.11.2 and replacing (U, C) by (E_{U}, C_{E}), we may assume U ≤ E and U = N_{E}(U), so U = E. Let Z be of order p in Z(S)∩E. Then Z ∈ F^{f}. As Z centralizes C and E acts on Z, 10.6 says that C ≤ E(F_{Z}), where F_{Z} = C_{F}(Z), and (Z, C_{Z}) ∈ P for some C_{Z} ∈ Comp(F_{Z}). Replacing (E, C) by (Z, C_{Z}), we may assume C ∈ Comp(F_{Z}). Set E_{Z} = C_{E}(Z). By 8.23.3, E_{Z} ⊴ F_{Z}.

Let Comp(F_{Z}) = {C_{i} : i ∈ I}, J = {j ∈ I : C_{j} ≰ E_{Z}}, and D = ∏_{j∈J} C_{j}. By 9.13, D ⊴ F_{Z} and E_{Z}D is the central product of E_{Z} and D. By the choice of C, C = C_{j_{0}} for some j_{0} ∈ J.

Let X ∈ F_{E}^{fc} and F_{X} = N_{F}(X). Then Z ≤ X, so C_{F}(X) = C_{F_{Z}} (X). Then D = C_{D}(X) ⊴ C_{F_{Z}} (X) = C_{F}(X)). Claim D ⊴ C_{F}(X). First D is C_{F}(X)-invariant by 3.6.1 in [A1], and D is saturated. Set D = D ∩ S. We must show each ϕ ∈ Aut_{D}(D) lifts ϕ̃ ∈ Aut_{C_{F} (X)}(DC_{S}(DX)) with [ϕ̃, C_{S}(DX)] ≤ Z(D), and as Inn(D) ∈ Syl_{p}(Aut_{D}(D), we may take ϕ to be a p′-element. But ϕ lifts to ϕ̂ ∈ Aut_{F_{Z}} (DC_{S}(DZ)) with [ϕ̂, C_{S}(DZ)] ≤ Z(D), and we may take ϕ̂ to be a p′-element. Then [E, ϕ̂] ≤ E ∩ Z(D) ≤ Z(D) as E is strongly closed in S, and ϕ̂_{|Z(D)} = ϕ_{|Z(D)} = 1, so ϕ̂ centralizes E as ϕ̂ is a p′-element. Thus we may take ϕ̃ = ϕ̂_{|DC_{S}(XD)}, to complete the proof of the claim. By the claim, C ∈ Comp(F_{X}).

Next E_{X} = N_{E}(X) ⊴ F_{X} by 8.23. Further F^{∗}(E_{X}) = O_{p}(E_{X}) as X ∈ F_{E}^{fc}. Thus C ≤ E(F_{X}) ≤ C_{F_{X}} (E_{X}) by 9.14. As this holds for each X ∈ F_{E}^{fc}, it follows from the definition of C_{F}(E(F)), that T ≤ C_{F}(E(F)). This is impossible by 9.12.1 and 10.8.

This completes the proof of Theorem 10.4.

Observe that Theorem 7 is just a restatement of Theorem 10.4, so we have established Theorem 7.

(10.12) Assume X = ⟨x⟩ is of order p in S and X ∈ F^{f}. Let {O_{i} : i ∈ I) be the orbits of X on Comp(F), F_{i} = ∏_{B∈O_{i}} B, and S_{i} = S ∩ F_{i}. Then

(1) N_{E(F)}(X) ⊴ N_{F}(X).

(2) F_{i} ⊴ XE(F) and N_{F_{i}}(X) ⊴ N_{E(F)}(X).

(3) Suppose C ∈ Comp(N_{F}(X)). Then either:

(a) C is a full diagonal subsystem of F_{i} for some orbit O_{i} of length p. In particular C/Z(C) ≅ B/Z(B) for B ∈ O_{i}.

(b) C ∈ Comp(N_{XF_{i}}(X)) for some orbit O_{i} of length 1.

Proof. As X induces a group of automorphisms of F, X permutes Comp(F). As |X| = p the orbits of X on Comp(F) are of length 1 or p. By 9.8, F_{i} ⊴ E(F).

Let E = S ∩ E(F) and E_{i} = S ∩ F_{i}. As X is of order p, either X ≤ E or X ∩ E = 1. In the former case, (1) follows from 8.23.2, and in the latter, (1) follows from 8.24.2. Similarly if we can establish the first statement in (2), then the second statement follows from the same argument.

By 8.21, XE(F) is a saturated fusion system, and by 10.4 and 9.8, components of N_{F}(X) are components of N_{XE(F)}(X). Thus to prove the remainder of the lemma, we may replace F by XE(F), and assume F = XE(F).

By 9.8, E(F) is the central product of its components. Further x induces an automorphism of E(F) of order 1 or p permutating the components, so by 2.11, x lifts to an automorphism x̂ of the direct product Ê(F) of the components of F. Let Z = Z(Ê(F)).

Next as X permutes O_{i}, X acts on E_{i} and induces a group of automorphism of F_{i}. Thus as F_{i} ⊴ E(F) and F = XE(F), we conclude from 7.4 that F_{i} ⊴ F, completing the proof of (2).

Finally assume the hypothesis of (3), and let T = S ∩ C. Then T ≤ E. Let T̂ be the preimage of T in Ê(F) and T̂_{i} the projection of T̂ on Ê_{i}. By 9.1, T/Z(C) is not abelian, so for some i, T̂_{i}/(Z ∩ T̂_{i}) is nonabelian. Thus as x̂ centralizes T̂, [T̂, CÊ_{i}(x̂)] ≰ Z, so T does not centralize C_{E_{i}}(x). We conclude from 9.13 applied to N_{F}(X) and N_{F_{i}}(X) in the roles of F and E, that C ≤ F_{i}. In particular if O_{i} is of order 1 then (3b) holds, while if O_{i} is of order p then (3a) holds by 2.12.

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