ZJ-THEOREMS FOR FUSION SYSTEMS

Radha Kessar, Markus Linckelmann

Abstract. For p an odd prime, we generalise the Glauberman-Thompson p-nilpotency theorem [5, Ch. 8, Theorem 3.1] to arbitrary fusion systems. We define a notion of Qd(p)- free fusion systems and show that if F is a Qd(p)-free fusion system on some finite p-group P then F is controlled by W (P) for any Glauberman functor W, generalising Glauberman’s ZJ-theorem [3] to arbitrary fusion systems.

1 Introduction

Let p be an odd prime, let G be a finite group, let P be a Sylow-p-subgroup of G and let J(P) be the Thompson subgroup of P (generated by the set of abelian subgroups of P of maximal order). The p-nilpotency theorem of Glauberman and Thompson [5, Ch. 8, Theorem 3.1] states that G is p-nilpotent if and only if N_{G}(Z(J(P))) is p-nilpotent. By a theorem of Frobenius [4, 8.6], G is p-nilpotent if and only if P controls G-fusion in P, or equivalently, if and only if F_{P} (G) = F_{P} (P), where the notation is as described in §2 below. The p-nilpotency theorem has been generalised to p-blocks of finite groups in [7], and the following theorem proves an analogue for arbitrary fusion systems.

Theorem A. Let p be an odd prime and let F be a fusion system on a finite p-group P. We have F = F_{P} (P) if and only if N_{F}(Z(J(P))) = F_{P} (P).

The proof of Theorem A is given in §4. A finite group A is said to be involved in another finite group G if there are subgroups H, K of G such that K ⊴ H and A ≅ H/K. Glauberman’s ZJ-Theorem in [3] asserts that if p is odd and Qd(p) is not involved in G then N_{G}(Z(J(P))) controls strong p-fusion in G, where P is Sylow-p-subgroup of G and where Qd(p) is the semi-direct product (C_{p} ×C_{p})⋊SL_{2}(p) with SL_{2}(p) acting naturally on the elementary abelian group of rank 2. By [4, 14.8] the conclusion holds in fact with ZJ replaced by any Glauberman functor (cf. 1.2 below). In order to extend this to arbitrary fusion systems, we introduce the following notation and terminology. By [1, 4.3], if F is a fusion system on a finite p-group P and Q an F-centric fully normalised

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20C20.

subgroup of P, there is, up to isomorphism, a unique finite group L = L_{Q}^{F} having N_{P} (Q) as Sylow-p-subgroup such that C_{L}(Q) = Z(Q) and N_{F}(Q) = F_{N_{P} (Q)}(L).

Definition 1.1. A fusion system F on a finite p-group P is called Qd(p)-free, if Qd(p) is not involved in any of the groups L_{Q}^{F}, with Q running over the set of F-centric radical fully normalised subgroups of P.

Definition 1.2. (cf. [8, 1.3]) A positive characteristic p-functor is a map sending any finite p-group P to a characteristic subgroup W (P) of P such that W (P) ≠ 1 if P ≠ 1 and such that any isomorphism of finite p-groups P ≅ Q maps W (P) onto W (Q). A Glauberman functor is a positive characteristic p-functor with the following additional property: whenever P is a Sylow-p-subgroup of a finite group L which satisfies C_{L}(O_{p}(L)) = Z(O_{p}(L)) and which does not involve Qd(p), then W (P) is normal in L.

Any of the maps sending a finite p-group P to Z(J(P)) or K_{∞}(P) or K^{∞}(P) are Glauberman functors, where J(P) is the Thompson subgroup of P, and where K_{∞}, K^{∞} are as defined in [4, Section 12].

Theorem B. Let p be an odd prime, let W be a Glauberman functor and let F be a fusion system on a finite p-group P. If F is Qd(p)-free then F = N_{F}(W (P)).

For fusion systems of finite groups this is Glauberman’s ZJ-theorem; for fusion systems of p-blocks of finite groups this has also been noted by G. R. Robinson, generalising [8, 1.4] where it was shown that the conclusion of Theorem B holds under the slightly stronger assumption that SL_{2}(p) is not involved in any of the automorphism groups Aut_{F}(Q), with Q running over the set of F-centric radical subgroups of P. The proof of Theorem B, given in §7, follows the pattern of the proof of [8, 1.4].

Since there exist Glauberman functors mapping P to a subgroup W (P) satisfying C_{P} (W (P)) = Z(W (P)) (for example, K_{∞}, K^{∞} have this property), the above Theorem in conjunction with [1, 4.3] implies that a Qd(p)-free fusion system on a finite p-group P is in fact equal to the fusion system of a finite group L having P as Sylow-p-subgroup and satisfying C_{L}(O_{p}(L)) ⊆ O_{p}(L). In particular, a Qd(p)-free fusion system is the underlying fusion system of a unique p-local finite group in the sense of [2].

2 Background material on fusion systems

Let p be a prime and let P be a finite p-group. Following the terminology of [11], a category on P is a category F with the subgroups of P as objects and with morphism sets Hom_{F}(Q, R) consisting of injective group homomorphisms, for any two subgroups Q, R of P, such that the following hold. Composition of morphisms in F is the usual composition of group homomorphisms, and for any morphism φ : Q → R in F, the induced isomorphism Q ≅ φ(Q), its inverse and the inclusion φ(Q) ⊆ R are all morphisms in F as well. Given a category F on P and a subgroup Q of P, we say that

- Q is fully F-normalised if |N_{P} (Q)| ≥ |N_{P} (φ(Q))| for every morphism φ : Q → P in F;

- Q is fully F-centralised if |C_{P} (Q)| ≥ |C_{P} (φ(Q))| for every morphism φ : Q → P in F;

- Q is F-centric if C_{P} (φ(Q)) = Z(φ(Q)) for every morphism φ : Q → P in F; and - Q is F-radical if O_{p}(Aut_{F}(Q)) = Aut_{Q}(Q).

Following [2, 1.2], if φ : Q → P is a morphism in a category F on P, we denote by N_{φ} the subgroup of N_{P} (Q) consisting of all y ∈ N_{P} (Q) for which there is z ∈ N_{P} (φ(Q)) such that φ(yuy^{−1}) = zφ(u)z^{−1} for all u ∈ Q. If Q, R are subgroups of P we denote by Hom_{P} (Q, R) the set of all group homomorphisms from Q to R induced by conjugation with elements in P. If Q = R we write Aut_{P} (Q) = Hom_{P} (Q, Q); note that Aut_{P} (Q) ≅ N_{P} (Q)/C_{P} (Q).

A fusion system on P is a category F on P whose morphism sets contain all morphisms induced by conjugation with elements in P, and which has furthermore the following properties. (I-S) Aut_{P} (P) is a Sylow-p-subgroup of Aut_{F}(P); (II-S) for every morphism φ : Q → P such that φ(Q) is fully F-normalised, there is a morphism ψ : N_{φ} → P such that φ = ψ|_{Q}.

This concept is due to Puig [12]; the above definition appears in [9] and is equivalent to the definition of what is called a saturated fusion system in [2, 1.2]; in particular, it is shown in [9] that the axioms (I-S) and (II-S) imply the axioms used in [2, 1.2], (I-BLO) if Q is a fully F-normalised subgroup of P then Q is fully F-centralised and Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q); (II-BLO) for every morphism φ : Q → P such that φ(Q) is fully F-centralised, there is a morphism ψ : N_{φ} → P such that φ = ψ|_{Q}.

If G is a finite group having P as Sylow-p-subgroup we denote by F_{P} (G) the category on P whose morphism sets are the group homomorphisms induced by conjugation with elements in G; that is, Hom_{F_{P} (G)}(Q, R) = Hom_{G}(Q, R) for any two subgroups Q, R of P. It is well-known and easy to verify that F_{P} (G) is a fusion system; we call F_{P} (G) the fusion system of the finite group G on P. In particular, F_{P} (P) is the fusion system on P whose morphisms are exactly those induced by inner automorphisms in P. Note that F_{P} (P) ⊆ F for any fusion system on P.

Let F be a fusion system on P and let Q be a subgroup of P. We denote by C_{F}(Q) the category on C_{P} (Q) such that, for any two subgroups R, R′ of C_{P} (Q), the morphism set in C_{F}(Q) from R to R′ consists of all group homomorphisms φ : R → R′ such that there exists a morphism ψ : RQ → RQ′ in F satisfying ψ|_{R} = φ and ψ|_{Q} = Id_{Q}. Similarly, we denote by N_{F}(Q) the category on N_{P} (Q) such that, for any two subgroups R, R′ of N_{P} (Q), the morphism set in N_{F}(Q) from R to R′ consists of all group homomorphisms φ : R → R′ such that there exists a morphism ψ : RQ → RQ′ in F satisfying ψ|_{R} = φ and ψ(Q) = Q. By a result of Puig, if Q is fully F-centralised then C_{F}(Q) is a fusion system on C_{P} (Q), and if Q is fully F-normalised then N_{F}(Q) is a fusion system on N_{P} (Q). Both statements are in fact particular cases of a more general result; see e.g. [2, Appendix, Prop. A6] for a proof. If F = N_{F}(Q) = N_{F}(R) for normal subgroups Q, R of P then one easily checks that F = N_{F}(QR). Thus the following definition makes sense:

Definition 2.1. Let F be a fusion system on a finite p-group P. We denote by O_{p}(F) the largest normal subgroup of P such that F = N_{F}(O_{p}(F)).

Lemma 2.2. Let F be a fusion system on a finite p-group P. For any subgroup Q of P there is a morphism φ : N_{P} (Q) → P in F such that φ(Q) is fully F-normalised.

Proof. Let ψ : Q → P be a morphism in F such that ψ(Q) is fully F-normalised. Thus Aut_{P} (ψ(Q)) is a Sylow-p-subgroup of Aut_{F}(Q). Therefore ψ ∘ Aut_{P} (Q) ∘ ψ^{−1} is conjugate to a subgroup of Aut_{P} (ψ(Q)); say τ ∘ ψ ∘ Aut_{P} (Q) ∘ ψ^{−1} ∘ τ^{−1} ⊂ Aut_{P} (ψ(Q)) for some τ ∈ Aut_{F}(ψ(Q)). Thus φ = τ ∘ ψ has the property that N_{φ} = N_{P} (Q), hence φ extends to a morphism N_{P} (Q) → P, and φ(Q) = ψ(Q) is fully F-normalised. □

Lemma 2.3. Let F be a fusion system on a finite p-group P. For any fully Fnormalised subgroup Q of P and any morphism φ : N_{P} (Q) → P the subgroup φ(Q) is fully F-normalised.

Proof. If φ : N_{P} (Q) → P is a morphism in F then φ(N_{P} (Q)) ⊆ N_{P} (φ(Q)), hence this inclusion is an equality whenever Q is fully F-normalised. □

Given two subgroups Q, R of P such that Q ≤ R we denote by Aut_{F}(Q ≤ R) the subgroup of Aut_{F}(R) consisting of all φ ∈ Aut_{F}(R) satisfying φ(Q) = Q. Restriction to Q induces a group homomorphism Aut_{F}(Q ≤ R) → Aut_{F}(Q). The following Lemma is a reformulation of a well-known fact following from the extension axiom for fusion systems.

Lemma 2.4. Let F be a fusion system on a finite p-group P and let Q, R be subgroups of P such that QC_{P} (Q) ≤ R ≤ N_{P} (Q). Suppose that Q is fully F-centralised and that Aut_{R}(Q) is normal in Aut_{F}(Q). The group homomorphism Aut_{F}(Q ≤ R) → Aut_{F}(Q) induced by restriction from R to Q is surjective.

Proof. For any y ∈ N_{P} (Q) denote by c_{y} the automorphism of Q given by conjugation with y. Let φ ∈ Aut_{F}(Q). Since Aut_{R}(Q) is normal in Aut_{F}(Q), for any y ∈ R there is z ∈ R such that φ ∘ c_{y} ∘ φ^{−1} = c_{z}. In particular, we have R ⊆ N_{φ}. Thus φ extends to a morphism ψ : R → P. Then, for all u ∈ Q, we have (φ∘c_{y} ∘φ^{−1})(u) = (ψ∘c_{y} ∘ψ^{−1})(u) = ψ(^{y}(ψ^{−1}(u))) = ^{ψ(y)}u, or equivalently, φ ∘ c_{y} ∘ φ^{−1} = c_{ψ(y)}. Since C_{P} (Q) ⊆ R we get ψ(R) = R and hence ψ ∈ Aut_{F}(Q ≤ R). □

Lemma 2.5. Let F be a fusion system on a finite p-group P. Suppose that F = N_{F}(Q) for some normal subgroup Q of P. Then Q is contained in any F-centric radical subgroup R of P.

Proof. Let R be a fully F-normalised centric radical subgroup of P. The hypothesis F = N_{F}(Q) implies that Aut_{QR}(R) is normal in Aut_{F}(R). Since R is radical, this forces Aut_{QR}(R) = Aut_{R}(R), hence N_{QR}(R) ⊆ RC_{P} (R). As R is also centric, we get N_{QR}(R) = R, and hence QR = R, or equivalently, Q ⊆ R. □

Besides C_{F}(Q) and N_{F}(Q) we need another particular case of Puig’s result in [2, Appendix, Prop. A6].

Proposition 2.6. Let F be a fusion system on a finite p-group P and let Q be a fully F-normalised subgroup of P. Then there is a fusion system N_{P} (Q)C_{F}(Q) on N_{P} (Q) contained in N_{F}(Q) such that, for any two subgroups R, R′ of N_{P} (Q), the morphism set Hom_{N_{P} (Q)C_{F} (Q)}(R, R′) consists of all group homomorphisms φ : R → R′ such that there exists a homomorphism ψ : RQ → R′Q in N_{F}(Q) and an element y ∈ P satisfying ψ|_{R} = φ and ψ(u) = ^{y}u for all u ∈ Q.

Proof. In the notation of [2, Appendix, Prop. A 6], this is the case where K = Aut_{P} (Q) applied to the fusion system N_{F}(Q) on N_{P} (Q). □

We will frequently use Alperin’s fusion theorem in the following form (see e.g. [2, Appendix, Theorem A 10] for a proof):

Theorem 2.7. (Alperin’s fusion theorem for fusion systems) Let F be a fusion system on a finite p-group P. Any isomorphism in F can be written as a composition of isomorphisms φ : Q ≅ Q′ for which there exists a fully F-normalised centric radical subgroup R of P containing Q, Q′ and an automorphism α ∈ Aut_{F}(R) such that α|_{Q} = φ.

Let F be a fusion system on a finite p-group P such that F = N_{F}(Q) for some normal subgroup Q of P. We define a category F/Q on P/Q as follows: for any two subgroups R, S of P containing Q, a group homomorphism ψ : R/Q → S/Q is a morphism in F/Q if there is a morphism φ : R → S in F satisfying ψ(uQ) = φ(u)Q for all u ∈ R. The following result is due to Puig [12].

Proposition 2.8. Let F be a fusion system on a finite p-group P such that F = N_{F}(Q) for some normal subgroup Q of P. Then the category F/Q is a fusion system on P/Q.

3 On central extensions of fusion systems

The following result is well known.

Proposition 3.1. Let F be a fusion system on a finite p-group P and let Z be a subgroup of Z(P) such that F = C_{F}(Z). Set P̄ = P/Z and F̄ = F/Z. For any subgroup Q of P containing Z the canonical map Q → Q̄ induces a surjective group homomorphism

Aut_{F}(Q) → AutF̄(Q̄)

whose kernel is an abelian p-group, where Q̄ is the image of Q in P̄. In particular, if Q is F-radical then Q̄ is F̄-radical. Moreover, if Q̄ is F̄-centric then Q is F-centric and if Q is F-centric radical, then Q̄ is F-centric radical.

Proof. If φ is an automorphism of Q inducing the identity on Z and on Q̄ then, for all u ∈ Q, we have φ(u) = uζ(u) for some group homomorphism ζ : Q → Z, and hence the group of all such automorphisms is isomorphic to the abelian p-group Hom(Q, Z) with group structure induced by that of Z. Thus if Q is F-radical, the kernel of the map Aut_{F}(Q) → AutF̄(Q̄) is contained in Aut_{Q}(Q). Therefore, in that case, Aut_{F}(Q)/Aut_{Q}(Q) ≅ AutF̄(Q̄)/AutQ̄(Q̄). If Q̄ is F̄-centric, for every subgroup R of P isomorphic to Q in F we have CP̄(R̄) = Z(R̄) ⊆ R̄, and therefore C_{P} (R) ⊆ R, which implies that C_{P} (R) = Z(R) and hence that Q is F-centric. For the last statement we may assume that Q̄ is fully F̄-centralised. The kernel K of the canonical map Aut_{F}(Q) → AutF̄(Q̄) is a p-group. Thus, if Q is F-radical then K ⊆ Aut_{Q}(Q). Let C be the inverse image in P of CP̄(Q̄). That is, the image in P̄ of any element in C centralises Q̄, and hence Aut_{C}(Q) ⊆ K. This implies C ⊆ QC_{P} (Q). Thus, if in addition Q is F-centric we get that C ⊆ Q, and hence CP̄(Q̄) ⊆ Q̄, which shows that Q̄ is F̄-centric. □

Proposition 3.2. Let F, F′ be fusion systems on a finite p-group P and let Z be a subgroup of Z(P) such that F = C_{F}(Z) and F′ = C_{F′}(Z). Suppose that F ⊆ F′. Then F = F′ if and only if F/Z = F′/Z.

Proof. Suppose that F/Z = F′/Z. Let Q be an F′-centric radical subgroup of P. Then, by Proposition 3.1, the kernel K of the canonical map Aut_{F′}(Q) → Aut_{F′/Z}(Q/Z) is contained in Aut_{Q}(Q). Thus K is also the kernel of the canonical map Aut_{F}(Q) → Aut_{F/Z}(Q/Z). Since Aut_{F/Z}(Q/Z) = Aut_{F′/Z}(Q/Z) it follows that Aut_{F}(Q) and Aut_{F′}(Q) have the same order. The assumption F ⊆ F′ implies Aut_{F}(Q) = Aut_{F′}(Q). The equality F = F′ follows then from Theorem 2.7. The converse is trivial. □

Corollary 3.3. Let F be a fusion systems on a finite p-group P, let Z be a subgroup of Z(P) such that F = C_{F}(Z) and let G be a fusion system on P such that G ⊆ F. Set F̄ = F/Z and Ḡ = G/Z. Let Q be a normal subgroup of P containing Z and set Q̄ = Q/Z. We have G = N_{F}(Q) if and only if Ḡ = NF̄(Q̄).

Proof. Suppose that Ḡ = NF̄(Q̄). Then Q̄ is contained in any Ḡ-centric radical subgroup of P̄. It follows from the last statement in Proposition 3.1 that Q is contained in any G-centric radical subgroup of P, and hence G = N_{G}(Q). Since G ⊆ F this implies G ⊆ N_{F}(Q). Clearly N_{F}(Q)/Z = NF̄(Q̄), and thus G = N_{F}(Q) by Proposition 3.2. The converse is trivial. □

Proposition 3.4. Let P be a finite p-group, let Q be a normal subgroup of P and let F, G be fusion systems on P such that F = P C_{F}(Q) and such that G ⊆ F. Let R be a normal subgroup of P containing Q. We have G = N_{F}(R) if and only if G/Q = N_{F/Q}(R/Q).

Proof. Suppose G/Q = N_{F/Q}(R/Q). In order to show the equality G = N_{F}(R) we proceed by induction over the order of Q. If Q = 1 there is nothing to prove. Suppose Q ≠ 1. Since Q is normal in P, the group Z = Q ∩ Z(P) is non trivial. The assumption F = P C_{F}(Q) implies that F = C_{F}(Z). Set F̄ = F/Z and Ḡ = G/Z. Similarly, set P̄ = P/Z, Q̄ = Q/Z and R̄ = R/Z. Then F̄, Ḡ are fusion systems on P̄ satisfying F̄ = P̄CF̄(Q̄) and Ḡ ⊆ F̄. We have isomorphisms of fusion systems Ḡ/Q̄ ≅ G/Q and N_{F̄/Q̄}(R̄/Q̄) ≅ N_{F/Q}(R/Q) induced by the canonical isomorphism P̄/Q̄ ≅ P/Q. Thus Ḡ/Q̄ = N_{F̄/Q̄}(R̄/Q̄). By induction we get that Ḡ = NF̄(R̄), where R̄ = R/Z. But then Corollary 3.3 implies G = N_{F}(R) as required. The converse is trivial. □

4 Proof of Theorem A

Let p be an odd prime, let P be a finite p-group and let F be a fusion system on P. We have F_{P} (P) ⊆ N_{F}(Z(J(P))) ⊆ F. Thus if F = F_{P} (P) then trivially N_{F}(Z(J(P))) = F_{P} (P). In order to prove the converse, let F be a minimal counter example to Theorem A; that is, the number of morphisms |F| of F is minimal such that N_{F}(Z(J(P))) = F_{P} (P) but F ≠ F_{P} (P). We proceed in a series of steps as in [7].

4.1. Any fusion system G on P which is properly contained in F is equal to F_{P} (P).

Proof. Since N_{G}(Z(J(P))) ⊆ N_{F}(Z(J(P))) = F_{P} (P) we have N_{G}(Z(J(P))) = F_{P} (P) and hence G = F_{P} (P) by the minimality assumption on F. □

4.2. We have O_{p}(F) ≠ 1.

Proof. Since F ≠ F_{P} (P), Alperin’s fusion theorem implies that there is a fully Fnormalised subgroup Q of P such that N_{F}(Q) ≠ F_{R}(R), where R = N_{P} (Q). Amongst all such subgroups choose Q such that R = N_{P} (Q) has maximal possible order. We are going to show that R = P. Assume that R ≠ P. We may choose Q such that Z(J(R)) is also fully F-normalised; indeed, by 2.2 there is a morphism φ : N_{P} (Z(J(R))) → P such that φ(Z(J(R))) is fully F-normalised, and since N_{P} (Q) = R ⊆ N_{P} (R) ⊆ N_{P} (Z(J(R))) it follows from 2.3 that φ(Q) is still fully F-normalised. Having replaced Q by φ(Q), consider the fusion system N_{F}(Z(J(R))) on N_{P} (Z(J(R))). Note that since R is a proper subgroup of P it is also a proper subgroup of N_{P} (R), hence of N_{P} (Z(J(R))). The choice of Q implies that N_{F}(Z(J(R))) = F_{N_{P} (Z(J(R)))}(N_{P} (Z(J(R)))). Then in particular N_{N_{F} (Q)}(Z(J(R))) = F_{R}(R). The minimality assumption on F implies the contradiction N_{F}(Q) = F_{R}(R). Thus R = P, or equivalently, Q ⊴ P. Since N_{F}(Q) ≠ F_{P} (P) we get that N_{F}(Q) = F by 4.1, thus Q ⊆ O_{p}(F). □

Set now Q = O_{p}(F). Note that Q is a proper subgroup of P as the equality Q = P would imply the contradiction F = N_{F}(P) = N_{F}(Z(J(P))) = F_{P} (P), where the second equality uses the fact that Z(J(P)) is characteristic in P. In particular, Aut_{F}(P) = Aut_{N_{F} (Z(J(P)))}(P) = Aut_{P} (P).

4.3. We have P C_{F}(Q) = F_{P} (P).

Proof. Assume that P C_{F}(Q) ≠ F_{P} (P). Then F = P C_{F}(Q) by 4.1. It follows from 3.4 that F/Q ≠ F_{P} (P)/Q = F_{P/Q}(P/Q). Since Q ≠ 1, the minimality assumption on F implies that N_{F/Q}(Z(J(P/Q))) ≠ F_{P/Q}(P/Q). Let R be the inverse image of Z(J(P/Q)) in P. Then R ⊴ P and N_{F}(R) ≠ F_{P} (P), by 3.4. But then N_{F}(R) = F by 4.1, contradicting the fact that R contains Q = O_{p}(F) properly. □

4.4. The subgroup Q of P is F-centric.

Proof. Set R = QC_{P} (Q). In order to show that Q is F-centric it suffices to show that R = Q, since Q = O_{p}(F). Assume that R ≠ Q. Then R ⊴ P but N_{F}(R) ≠ F. Thus N_{F}(R) = F_{P} (P) by 4.1. Since Q is normal in P it is in particular fully F-normalised, and hence the restriction map Aut_{F}(R) → Aut_{F}(Q) is surjective, by the extension axiom (II-S). Since N_{F}(R) = F_{P} (P) we have Aut_{F}(R) = Aut_{P} (R), hence Aut_{F}(Q) = Aut_{P} (Q). Let S be a fully F-normalised centric radical subgroup of P. Then Q ⊆ S by 2.5. Let σ : Aut_{F}(S) → Aut_{F}(Q) = Aut_{P} (Q) be the restriction homomorphism. Then ker(σ) is a subgroup of Aut_{P C_{F} (Q)}(S) = Aut_{P} (S). Thus Aut_{F}(S) = Aut_{P} (S) as claimed. Alperin’s fusion theorem yields the contradiction F = F_{P} (P). Thus R = Q, or equivalently, Q is F-centric. □

We conclude the proof of Theorem A as in [7]. Since F = N_{F}(Q) for some Fcentric normal subgroup Q of P it follows from [1, 4.3] that there is a finite group L having P as Sylow-p-subgroup such that Q ⊴ L, C_{L}(Q) = Z(Q) and F = F_{P} (L). Then N_{F}(Z(J(P))) = F_{P} (N_{L}(Z(J(P))). Since N_{F}(Z(J(P))) = F_{P} (P) it follows from Frobenius’ theorem [4, 8.6] that N_{L}(Z(J(P))) is p-nilpotent. But the nilpotency theorem [5, Ch. 8, Theorem 3.1] of Glauberman and Thompson implies then that L itself is p-nilpotent, or equivalently, F_{P} (L) = F_{P} (P). This however yields the contradiction F = F_{P} (P), and the proof of Theorem A is complete.

5 Local control of characteristic p-functors

Definition 5.1. Let F be a fusion system on a finite p-group P and let W be a positive characteristic p-functor. Let Q be a subgroup of P. Set W_{1}(Q) = Q and P_{1}(Q) = N_{P} (Q). For any positive integer i define inductively W_{i+1}(Q) = W (P_{i}(Q)) and P_{i+1}(Q) = N_{P} (W_{i+1}(Q)). We will say that Q is (F, W)-well-placed if W_{i}(Q) is fully F-normalised for all positive integers i.

Note that for all positive integers i we have W_{i}(Q) ⊆ P_{i}(Q), and if P_{i}(Q) is a proper subgroup of P then in fact P_{i}(Q) is a proper sugroup of P_{i+1}(Q). In particular, P_{i}(Q) = P for i large enough. The following result generalises [4, 5.2], [8, 3.1].

Proposition 5.2. Let F be a fusion system on a finite p-group P, let W be a positive characteristic p-functor and let Q be a subgroup of P. There is a morphism φ : N_{P} (Q) → P such that φ(Q) is (F, W)-well-placed.

Proof. Set W_{i} = W_{i}(Q) and P_{i} = P_{i}(Q) for any positive integer i. Note that P_{i} = N_{P} (W_{i}) for any positive integer i. Let φ_{1} : P_{1} = N_{P} (Q) → P be a morphism in F such that W_{1} = Q is fully F-normalised. Thus, after replacing Q by φ_{1}(Q) we may assume that W_{1} = Q is fully normalised. Assume now that for some positive integer n the subgroups W_{i} are fully F-normalised for 1 ≤ i ≤ n. Let φ_{n+1} : P_{n+1} = N_{P} (W_{n+1}) → P be a morphism in F such that W_{i+1} is fully F-normalised. Since P_{i} = N_{P} (W_{i}) ⊆ P_{n+1} the subgroups φ_{n+1}(W_{i}) are still all fully F-normalised. Note that in particular P_{1} = N_{P} (Q) ⊆ P_{n+1}. Thus we may in fact assume that W_{i} is fully F-normalised for 1 ≤ i ≤ n + 1. The result follows by induction. □

The next result generalises [4, 5.5], [8, 3.2] to arbitrary fusion systems, saying that if a positive characteristic p-functor controls fusion locally, it does so globally.

Proposition 5.3. Let F be a fusion system on a finite p-group P and let W be a positive characteristic p-functor. Assume that for any non-trivial fully F-normalised subgroup Q of P we have N_{F}(Q) = N_{N_{F} (Q)}(W (N_{P} (Q))). Then F = N_{F}(W (P)).

Proof. Suppose the conclusion does not hold. Then there is a fully F-normalised nontrivial subgroup Q of P such that Aut_{N_{F} (W (P))}(Q) is a proper subgroup of Aut_{F}(Q). By the previous Proposition we may assume that Q is (F, W)-well-placed. For any positive integer i, set W_{i} = W_{i}(Q) and P_{i} = P_{i}(Q). Set F_{i} = N_{F}(W_{i}) and G_{i} = N_{F_{i}}(W_{i+1}). Since W_{i} is fully F-normalised, the category F_{i} is a fusion system on P_{i} = N_{P} (W_{i}). Clearly W_{i+1} ⊆ P_{i}, and since W_{i+1} is fully F-normalised, W_{i+1} is also fully F_{i}-normalised, and so G_{i} is a fusion system as well. Note that G_{i} ⊆ F_{i+1}. Clearly F_{1} ⊆ F_{i}. In fact, by the assumptions, we have G_{i} = F_{i}. Thus F_{1} ⊆ F_{i} ⊆ F_{i+1}. If i is large enough then P_{i} = P and hence F_{i} = N_{F}(W (P)). But then also F_{1} ⊆ N_{F}(W (P)). In particular, Aut_{F_{1}}(Q) = Aut_{F}(Q) is contained in Aut_{N_{F} (W (P))}(Q), contradicting our choice of Q. □

6 On Qd(p)-free fusion systems

In this Section we prove that if a fusion system F on a finite p-group P is Qd(p)-free, then so are N_{F}(Q), N_{P} (Q)C_{F}(Q) and N_{F}(Q)/Q for any fully F-normalised subgroup Q of P. In fact, the statements in this section remain true with Qd(p) replaced by any finite group, but we state them as needed in the proof of Theorem B. Given a fusion system F on a finite p-group P and a fully F-normalised centric subgroup Q of P we denote as before by L_{Q}^{F} the p′-reduced p-constrained group from [1, 4.3] for which there is a short exact sequence

1 ⟶ Z(Q) ⟶ L_{Q}^{F} ⟶ Aut_{F}(Q) ⟶ 1

such that N_{P} (Q) is a Sylow-p-subgroup of L_{Q}^{F} and such that N_{F}(Q) = F_{N_{P} (Q)}(L_{Q}^{F}). This short exact sequence is represented by an element in H^{2}(Aut_{F}(Q); Z(Q)). Since Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q) and since Z(Q) is an abelian p-group, the restriction map H^{2}(Aut_{F}(Q); Z(Q)) → H^{2}(Aut_{P} (Q); Z(Q)) is injective. In other words, the group L_{Q}^{F} is, up to isomorphism determined by the group Aut_{F}(Q) and the p-group extension

1 ⟶ Z(Q) ⟶ N_{P} (Q) ⟶ Aut_{P} (Q) ⟶ 1

Proposition 6.1. Let F be a fusion system on a finite p-group P. If Qd(p) is involved in the group L_{Q}^{F} for some fully F-normalised centric subgroup Q of P then Qd(p) is involved in the group L_{R}^{F} for some fully F-normalised centric radical subgroup R of P.

Proof. Let Q be a fully F-normalised centric subgroup of P. We proceed by induction over the order of Q. If Q is F-radical there is nothing to prove. Otherwise, let R be the unique subgroup of N_{P} (Q) containing Q such that Aut_{R}(Q) = O_{p}(Aut_{F}(Q)). Then R is an F-centric subgroup of P which properly contains Q. In particular, Aut_{R}(Q) is normal in Aut_{F}(Q), hence N_{P} (Q) ⊆ N_{P} (R). Let ψ : N_{P} (R) → P be a morphism in F such that ψ(R) is fully F-normalised. Then ψ(Q) is still fully F-normalised. Thus we may assume that both Q and R are fully F-normalised. Let L_{Q≤R} be the inverse image of Aut_{F}(Q ≤ R) in L_{R}^{F}. That is, we have a short exact sequence of groups

1 ⟶ Z(R) ⟶ L_{Q≤R} ⟶ Aut_{F}(Q ≤ R) ⟶ 1

By 2.4 restriction from R to Q induces a surjective group homomorphism Aut_{F}(Q ≤ R) → Aut_{F}(Q). Since Q is F-centric, the kernel of this group homomorphism is Aut_{Z(Q)}(R) ≅ Z(Q)/Z(R) (cf. [11, 1.12]). Thus we have an exact commutative diagram

\xymatrix{1 \ar[dr] &  &  & 1 \ar[d] \\  & Z(Q) \ar[dr] &  & Z(Q)/Z(R) \ar[d] \\ 1 \ar[r] & Z(R) \ar[r] & L_{Q≤R} \ar[r] \ar[dr] & Aut_{F}(Q ≤ R) \ar[d] \ar[r] & 1 \\  &  &  & Aut_{F}(Q) \ar[d] \ar[dr] \\  &  &  & 1 & 1}

Since Q is fully F-normalised, Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q) ≅ N_{P} (Q)/Z(Q), hence its inverse image Aut_{P} (Q ≤ R) ≅ N_{P} (Q ≤ R)/Z(R) is a Sylow-p-subgroup of Aut_{F}(Q ≤ R). Note that since N_{P} (Q) ⊆ N_{P} (R) we have N_{P} (Q) = N_{P} (Q ≤ R), and this is a Sylow-p-subgroup of L_{Q≤R}. Since a group extension by an abelian p-group is determined by its restriction to Sylow-p-subgroups it follows that L_{Q}^{F} ≅ L_{Q≤R} is isomorphic to a subgroup of L_{R}^{F}. Thus, if Qd(p) is involved in L_{Q}^{F} it is involved in L_{R}^{F}, too. The result follows by induction. □

Proposition 6.2. Let F be a fusion system on a finite p-group P, let Q be a fully F-normalised subgroup of P and let G be a fusion system on N_{P} (Q) such that N_{P} (Q)C_{F}(Q) ⊆ G ⊆ N_{F}(Q). Let R be a subgroup of N_{P} (Q) containing Q. Then R is fully G-centralised if and only if R is fully F-centralised. In particular, R is G-centric if and only if R is F-centric.

Proof. If R is fully F-centralised then clearly R is fully G-centralised. Suppose conversely that R is fully G-centralised. Let φ : R → P be a morphism in F such that φ(R) is fully F-centralised. Denote by ψ : φ(Q) → Q the isomorphism which is inverse to φ|_{Q}. We have C_{P} (φ(Q)) ⊆ N_{ψ}, and we also have φ(R) ⊆ N_{ψ}. Indeed, for all r ∈ R and all u ∈ Q we have ψ(φ(r)ψ^{−1}(u)φ(r)^{−1}) = ψ(φ(rur^{−1})) = rur^{−1}. Since Q is fully F-normalised, ψ extends to a morphism τ : φ(R)C_{P} (φ(Q)) → P. Note that τ ∘φ restricts to the identity on Q; in particular, τ ∘ φ : R → P is a morphism in G. Thus S = τ(φ(R)) is a subgroup of N_{P} (Q) containing Q and isomorphic to R in G. Since C_{P} (φ(R)) ⊆ C_{P} (φ(Q)) we get τ(C_{P} (φ(R)) ⊆ C_{P} (S); in particular |C_{P} (φ(R))| ≤ |C_{P} (S)|. As R was chosen fully Gcentralised, it follows that |C_{P} (S)| ≤ |C_{P} (R)|, hence |C_{P} (φ(R))| ≤ |C_{P} (R)|. However, φ(R) is fully F-centralised, and thus so is R. □

Proposition 6.3. Let F be a fusion system on a finite p-group P, let Q be a fully F-normalised subgroup of P and let G be a fusion system on N_{P} (Q) such that N_{P} (Q)C_{F}(Q) ⊆ G ⊆ N_{F}(Q). If F is Qd(p)-free, so is G. In particular, if F is Qd(p)- free, so are N_{F}(Q) and N_{P} (Q)C_{F}(Q).

Proof. Suppose that F is Qd(p)-free. Let R be a fully G-normalised radical centric subgroup of N_{P} (Q). By 2.5 we have Q ⊆ R. By 6.1 it suffices to show that L_{R}^{G} is Qd(p)- free. By 6.2, R is F-centric. By 2.3 there is a morphism φ : N_{P} (R) → P such that φ(R) is fully F-normalised. The plan is to show that L_{R}^{G} is isomorphic to a subgroup of L_{φ(R)}^{F}. Conjugation by φ induces a group isomorphism Aut_{F}(R) ≅ Aut_{F}(φ(R)) sending ρ ∈ Aut_{F}(R) to φ ∘ ρ ∘ φ^{−1}|_{φ(R)}. Restricting this to the subgroup Aut_{G}(R) of Aut_{F}(R) yields an injective group homomorphism

Φ : Aut_{G}(R) → Aut_{F}(φ(R)) .

Consider the canonical group extension

1 ⟶ Z(φ(R)) ⟶ L_{φ(R)}^{F} ⟶^{λ} Aut_{F}(φ(R)) ⟶ 1

Let L be the pullback of λ and Φ; that is,

L = {(y, α) ∈ L_{φ(R)}^{F} × Aut_{G}(R) | λ(y) = Φ(α) }

The canonical projections yield a commutative diagram

1 ⟶ Z(φ(R)) ⟶ L ⟶^{μ} Aut_{G}(R) ⟶ 1
∥  ↓  ↓^{Φ}
1 ⟶ Z(φ(R)) ⟶ L_{φ(R)}^{F} ⟶_{λ} Aut_{F}(φ(R)) ⟶ 1

Since L_{φ(R)}^{F} is Qd(p)-free, so is L. It suffices thus to show that L ≅ L_{R}^{G}. Note that Aut_{P} (φ(R)) is a Sylow-p-subgroup of Aut_{F}(φ(R)) because φ(R) is fully F-normalised, and similarly, Aut_{P} (Q ≤ R) = Aut_{N_{P} (Q)}(R) is a Sylow-p-subgroup of Aut_{G}(R) because R is fully G-normalised. Let S be the inverse image of Aut_{P} (Q ≤ R) in L. Then S is a Sylow-p-subgroup of L. The group L is determined, up to isomorphism, by Aut_{G}(R) and the short exact sequence of p-groups

1 ⟶ Z(φ(R)) ⟶ S ⟶ Aut_{P} (Q ≤ R) ⟶ 1

obtained from restricting the first row in the above diagram to Sylow-p-subgroups. In order to show that L ≅ L_{R}^{G} it suffices to show that this short exact sequence is equivalent to

1 ⟶ Z(R) ⟶ N_{P} (Q ≤ R) ⟶ Aut_{P} (Q ≤ R) ⟶ 1

As S is the inverse image in L of Aut_{P} (Q ≤ R), we have

S = {(u, c_{v}) ∈ N_{P} (φ(R)) × Aut_{P} (Q ≤ R) | c_{u} = Φ(c_{v}) } ,

where c_{u}, c_{v} are the automorphisms of φ(R), R, induced by conjugation with u, v, respectively. The equality c_{u} = Φ(c_{v}) is equivalent to c_{u} = φ ∘ c_{v} ∘ φ^{−1}|_{φ(R)} = c_{φ(v)}. Thus S = {(φ(v), c_{v}) | v ∈ N_{P} (Q ≤ R) }. Hence the map ψ : N_{P} (Q ≤ R) → S sending v ∈ N_{P} (Q ≤ R) to (φ(v), c_{v}) is an isomorphism making the diagram

1 ⟶ Z(R) ⟶ N_{P} (Q ≤ R) ⟶ Aut_{P} (Q ≤ R) ⟶ 1
φ|_{Z(R)}↓  ↓^{ψ}  ∥
1 ⟶ Z(φ(R)) ⟶  S  ⟶ Aut_{P} (Q ≤ R) ⟶ 1

commutative. The isomorphism L ≅ L_{R}^{G} follows. □

Proposition 6.4. Let F be a fusion system on a finite p-group P and let Q be a normal subgroup of P such that F = N_{F}(Q). If F is Qd(p)-free then F/Q is Qd(p)-free.

Proof. Let R be a subgroup of P containing Q such that R/Q is F/Q-centric fully normalised. We show first that then R is F-centric fully normalised. Let φ : R → P be a morphism in F and let y ∈ C_{P} (φ(R)). Then yQ ∈ C_{P/Q}(φ(R)/Q) = Z(φ(R)/Q), hence y ∈ φ(R), which implies that R is F-centric. Since N_{P/Q}(R/Q) = N_{P} (R)/Q it follows also that R is fully F-normalised. Consider the group extension

1 ⟶ Z(R) ⟶ L_{R}^{F} ⟶ Aut_{F}(R) ⟶ 1

The fusion system of L = L_{R}^{F} on N_{P} (R) is equal to N_{F}(R), and hence the fusion system of L̄ = L_{R}^{F}/Q is equal to N_{F}(R)/Q = N_{F/Q}(R/Q). Now R/Q is centric in F/Q, hence R/Q is a p-centric subgroup of the finite group L̄. Thus

CL̄(R/Q) = Z(R/Q) × C

where C = O_{p′}(CL̄(R/Q). Let S = C_{R}(R/Q) be the inverse image of Z(R/Q) in R, let K be the inverse image of C in L and let C_{L}(R/Q) be the inverse image of CL̄(R/Q) in L. Then CL̄(R/Q) is the kernel of the composition of canonical maps

L ⟶ Aut_{F}(R) ⟶ Aut_{F/Q}(R/Q)

and we have CL̄(R/Q) = SK and S ∩ K = Q. It follows that the canonical maps induce an exact commutative diagram of the form

1 ⟶ Z(R) ⟶ L ⟶ Aut_{F}(R) ⟶ 1
↓  ↓^{π}  ↓
1 ⟶ Z(R/Q) ⟶ L/K ⟶ Aut_{F/Q}(R/Q) ⟶ 1

The second row, when restricted to Sylow-p-subgroups, yields the exact sequence

1 ⟶ Z(R/Q) ⟶ N_{P/Q}(R/Q) ⟶ Aut_{P/Q}(R/Q) ⟶ 1

and hence L/K ≅ L_{R/Q}^{F/Q}. This group is obviously Qd(p)-free as it is a quotient of the Qd(p)-free group L = L_{R}^{F}. Thus the fusion system F/Q on P/Q is Qd(p)-free. □

Remark 6.5. For future reference we point out that the proofs of 6.1, 6.3 and 6.4 yield the following slightly more precise statements about the groups L_{Q}^{F}, where F is a fusion system on a finite p-group P.

(1) If Q is a fully F-normalised centric subgroup of P and Q ⊆ R ⊆ N_{P} (R) such that Aut_{R}(Q) = O_{p}(Aut_{F}(Q)), then there is a morphism φ : R → P such that both φ(Q), φ(R) are fully F-normalised, and for any such φ the group L_{φ(Q)}^{F} is isomorphic to a subgroup of L_{φ(R)}^{F}.

(2) If Q is a fully F-normalised subgroup of P and G a fusion system on N_{P} (Q) such that N_{P} (Q)C_{F}(Q) ⊆ G ⊆ N_{F}(Q), then for any fully G-normalised centric radical subgroup R of N_{P} (Q) there is a morphism φ : R → P in F such that φ(R) is fully F-normalised centric, and for any such φ the group L_{R}^{G} is isomorphic to a subgroup of L_{φ(R)}^{F}.

(3) If Q is a normal subgroup of P such that F = N_{F}(Q), then for any fully F/Qnormalised centric subgroup R/Q of P/Q, its inverse image R in P is fully F-normalised centric, and the group L_{R}^{F} has a quotient isomorphic to L_{R/Q}^{F/Q}.

7 Proof of Theorem B

Given a fusion system F on a finite p-group P we denote as before by |F| the number of morphisms in F. We argue by induction over |F|. Let F be a counterexample to Theorem B with |F| minimal. That is, F is Qd(p)-free, N_{F}(W (P)) ≠ F, where W is a Glauberman functor, but N_{F}(W (P′)) = F′ for any Qd(p)-free fusion system F′ on some finite p-group P′ such that |F′| < |F|. We show first that

7.1. O_{p}(F) ≠ 1.

Proof. If O_{p}(F) = 1 then for any non trivial fully F-normalised subgroup Q of P we have N_{F}(Q) < F. Using 6.3 we get that N_{F}(Q) is Qd(p)-free. Hence N_{F}(Q) = N_{N_{F} (Q)}(W (N_{P} (Q))) by the induction hypothesis. Then 5.3 implies the contradiction F = N_{F}(W (P)). This proves 7.1. □

We set now Q = O_{p}(F) and R = QC_{P} (Q). We observe next that 7.2. Q < R.

Proof. If Q = R = QC_{P} (Q) then Q is F-centric, and we have a short exact sequence of finite groups

1 ⟶ Z(Q) ⟶ L_{Q} ⟶ Aut_{F}(Q) ⟶ 1

where L_{Q} is a finite group having P as Sylow-p-subgroup such that C_{L_{Q}}(O_{p}(L_{Q})) ⊆ O_{p}(L_{Q}) and such that F_{P} (L_{Q}) = F. But L_{Q} is Qd(p)-free by the assumptions. Glauberman’s Theorem implies that F_{P} (L_{Q}) = F_{P} (N_{L_{Q}}(W (P))) and hence F = N_{F}(W (P)), contradicting our choice of F. □

The next step is to prove that

7.3. F = P C_{F}(Q).

Proof. Assume that P C_{F}(Q) < F. Note that then P C_{F}(Q) = N_{P C_{F} (Q)}(W (P)) by induction. We will show that this implies that F = N_{F}(W (P)), contradicting our choice of F. To show this, we will prove by induction over [P : S] that for any Fcentric radical subgroup S of P we have Aut_{F}(S) = Aut_{N_{F} (W (P))}(S). The equality Aut_{F}(P) = Aut_{N_{F} (W (P))}(P) is clear because W (P) is a characteristic subgroup of P. Let S be an F-centric radical subgroup of P. Note that then S contains Q. By Alperin’s fusion theorem, any automorphism of S can be written as product of automorphisms of fully F-normalised centric radical subgroups of P of order at least |S|, and hence we may assume that S is fully F-normalised. Restriction from S to Q induces a group homomorphism

ρ : Aut_{F}(S) ⟶ Aut_{F}(Q) .

Set A = ρ^{−1}(Aut_{S}(Q)). Since Aut_{S}(Q) is normal in Im(ρ) it follows that A is a normal subgroup of Aut_{F}(S), and clearly A ⊆ Aut_{P C_{F} (Q)}(S). Also, since Aut_{P} (S) is a Sylow-p-subgroup of Aut_{F}(S), the intersection Aut_{P} (S) ∩ A is a Sylow-p-subgroup of A. Setting T = N_{P} (S) ∩ SC_{P} (Q) = N_{SR}(S) yields

Aut_{P} (S) ∩ A = Aut_{T} (S) .

The Frattini argument implies that

Aut_{F}(S) = A · N_{Aut_{F} (S)}(Aut_{T} (S)) .

By our initial induction on fusion systems, we get A ⊆ Aut_{P C_{F} (Q)}(S) ⊆ Aut_{N_{F} (W (P))}(S). Thus, in order to prove 7.3 we have to prove that N_{Aut_{F} (S)}(Aut_{T} (S)) ⊆ Aut_{N_{F} (W (P))}(S). We have to consider two cases, depending on whether S contains R or not.

Suppose first that S does not contain R. Then T has bigger order than S. By induction, we get Aut_{F}(T) = Aut_{N_{F} (W (P))}(T). Now every automorphism of S which normalises Aut_{T} (S) extends to T, by the extension axiom, and so N_{Aut_{F} (S)}(Aut_{T} (S)) is contained in the image of the restriction map Aut_{F}(S < T) → Aut_{F}(S). Thus indeed Aut_{F}(S) = Aut_{N_{F} (W (P))}(S) in this case.

Consider second the case where R ⊆ S. Set B = ρ^{−1}(Aut_{Q}(Q)). Then B is normal in Aut_{F}(S) and B ⊆ Aut_{P C_{F} (Q)}(S) ⊆ Aut_{N_{F} (W (P))}(S). As before, since Aut_{P} (S) is a Sylow-p-subgroup of Aut_{F}(S), the intersection Aut_{P} (S)∩B is a Sylow-p-subgroup of B. We claim that Aut_{P} (S) ∩ B = Aut_{R}(S). Indeed, if we denote by c_{y} the automorphism of S given by conjugation with an element y ∈ N_{P} (S), we have Aut_{P} (S) ∩ B = {c_{y} | y ∈ N_{P} (S), there is x ∈ Q such that c_{y}|_{Q} = c_{x}|_{Q}} = {c_{y} | y ∈ R} = Aut_{R}(S). The Frattini argument implies that

Aut_{F}(S) = B · N_{Aut_{F} (S)}(Aut_{R}(S)) .

Since we know already that B is contained in Aut_{N_{F} (W (P))}(S) we need to show that N_{Aut_{F} (S)}(Aut_{R}(S)) ⊆ Aut_{N_{F} (W (P))}(S). To see this we prove first that N_{Aut_{F} (S)}(Aut_{R}(S)) ⊆ Aut_{N_{F} (R))}(S). Indeed, let φ ∈ N_{Aut_{F} (S)}(Aut_{R}(S)) and let x ∈ R. As before, denote by c_{x} the automorphism of S given by conjugation with x. Since φ normalises Aut_{R}(S) we have φ∘c_{x}∘φ = c_{z} for some z ∈ R. But since R ⊆ S we also have φ ∘ c_{x} ∘ φ^{−1} = c_{φ(x)}. Thus z^{−1}φ(x) ∈ C_{P} (S) = Z(S) ⊆ R, and hence φ(R) = R. This shows the inclusion N_{Aut_{F} (S)}(Aut_{R}(S)) ⊆ Aut_{N_{F} (R))}(S). From here we observe that, by induction applied to the fusion system N_{F}(R), we get Aut_{N_{F} (R)}(S) ⊆ Aut_{N_{F} (W (P))}(S) as required.

Thus the assumption P C_{F}(Q) < F yields the contradiction F = N_{F}(W (P)), which concludes the proof of 7.3. □

We use 3.4 to finish the proof of the Theorem. Since F = P C_{F}(Q) is Qd(p)-free, so is the fusion system F/Q on P/Q by 6.4. Thus, by induction, we have F/Q = N_{F/Q}(W (P/Q)). Denoting by V the inverse image of W (P/Q) in P we get from 3.4 that F = N_{F}(V). Hence V ⊆ Q = O_{p}(F). But W (P/Q) ≠ 1, hence V contains Q properly. This contradiction concludes the proof of Theorem B. □

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Radha Kessar Department of Mathematical Sciences University of Aberdeen Meston Building, Aberdeen, AB24 3UE

U.K.

Markus Linckelmann Department of Mathematical Sciences University of Aberdeen Meston Building, Aberdeen, AB24 3UE

U.K.