Normal Subsystems of Fusion Systems

David A. Craven, University of Oxford

June 2010

Abstract In this article we prove that for any saturated fusion system, that the (unique) smallest weakly normal subsystem of it on a given strongly closed subgroup is actually normal. This has a variety of corollaries, such as the statement that the notion of a simple fusion system is independent of whether one uses weakly normal or normal subsystems. We also develop a theory of weakly normal maps, consider intersections and products of weakly normal subsystems, and the hypercentre of a fusion system.

1 Introduction

The theory of fusion systems is becoming an important topic in algebra, with interactions with group theory, representation theory and topology. This article is concerned with the structure of normal subsystems of fusion systems. There are two notions of a ‘normal’ subsystem in the literature, one stronger than the other. (We will recall their definitions in this article.) We will follow [7] and call the subsystem considered by Aschbacher in [2] a normal subsystem, and the subsystem considered by, among others, Linckelmann in [11] a weakly normal subsystem.

Our first result highlights the exact relationship between normal and weakly normal subsystems.

Theorem A Let F be a saturated fusion system on a finite p-group P. If E is a weakly normal subsystem of F, then O^{p′}(E) is a normal subsystem of F.

Recall that O^{p′}(E) is the smallest (weakly) normal subsystem of E on the same p-group as E (see [12, Section 6.5] or [5, Theorem 5.4] for example, or Section 8 of this article). What Theorem A says is that any weakly normal subsystem can be thought of as a normal subsystem E, together with some p′-automorphisms of E that lie in F. In other words, we have the following corollary.

Corollary B Let F be a saturated fusion system on a finite p-group P. If E is a weakly normal subsystem of F, on a subgroup T, then there exists a normal subsystem E′ of F, also on T, and a p′-subgroup H of Aut_{F}(T) such that E is generated by E′ and H.

Theorem A has several corollaries, such as the following.

Corollary C Let F be a saturated fusion system on a finite p-group P. Then F has no proper, non-trivial normal subsystems if and only if F has no proper, non-trivial weakly normal subsystems. In other words, the notion of a simple fusion system is the same with either normal subsystems or weakly normal subsystems.

We give some more corollaries to Theorem A in Section 9.

Let F be a saturated fusion system on a finite p-group P, and let T be a strongly F-closed subgroup of P. In [2], Aschbacher developed the concept of a normal map, which is a special type of function A(−) on the set of all subgroups U of T, with A(U) a subgroup of Aut_{F}(U). If E is a normal subsystem on T, then A(U) = Aut_{E}(U) for U ⩽ T is a normal map. Conversely, if A(−) is a normal map, then the subsystem generated by A(U) for U ⩽ T is a normal subsystem E′, and A(U) = Aut_{E′}(U) for all U ⩽ T. In the proof of this theorem in [2], the fact that the map was associated to a normal subsystem was pivotal to the proof that the subsystem generated by the map was saturated. Here we decouple the theorem from this requirement, proving a similar result for weakly normal subsystems.

Theorem D Let F be a saturated fusion system on a finite p-group P, and let T be a strongly F-closed subgroup of P. Let A(−) be a function from the set of subgroups U of T, such that A(U) is a subgroup of Aut_{F}(U), satisfying the following conditions.

(i) If ϕ is an F-isomorphism whose domain is U, then A(Uϕ) = A(U)^{ϕ}.

(ii) If U is fully F-normalized, then Aut_{T} (U) ⩽ A(U).

(iii) Aut_{T} (T) is a Sylow p-subgroup of A(T).

(iv) If U is fully F-normalized, then every element of A(U) extends to an element of A(U C_{T} (U)).

(v) If U is fully F-normalized and C_{T} (U) ⩽ U then for any subgroup U ⩽ V ⩽ N_{T} (U), denoting by A(U ⩽ V) the set of automorphisms of V that restrict to automorphisms of U, the restriction map

A(U ⩽ V) → N_{A(U)}(Aut_{V} (U))

is surjective.

If E is the subsystem generated by A(U) for all U ⩽ T, then E is a weakly normal subsystem of F and Aut_{E}(U) = A(U) for all U ⩽ T.

When amalgamated with the axiom needed for weakly normal subsystems to be normal, we get a (slightly) weaker formulation of the axioms for A(−) to be a normal map.

Let F be a saturated fusion system on a finite p-group P. The centre Z (F) of F seems to play an important role in the theory of fusion systems: for example, in [5, Corollary 6.14] it is proved that F is a fusion system of a finite group if and only if F/Z (F) is, and in [2, Lemma 8.10] it is proved that there is a one-to-one correspondence between normal subsystems of F on subgroups containing Z (F) and normal subsystems of F/Z (F). As with finite groups, write Z_{1}(F) = Z (F) and Z_{i}(F) for the preimage in P of Z (F/Z_{i−1}(F)). The series (Z_{i}(F)) eventually stabilizes; write Z_{∞}(F) for this limit, called the hypercentre of F.

Another subsystem that seems important is N_{P} (Q) C_{F}(Q); if Q is a fully F-normalized subgroup of P, then the subsystem N_{P} (Q) C_{F}(Q) is the (saturated) subsystem of F on N_{P} (Q) consisting of all morphisms ϕ : R → S in F such that ϕ extends to ϕ̄ : QR → QS with ϕ̄|_{Q} = c_{g} for some g ∈ N_{P} (Q), where c_{g} denotes conjugation by g. This notion is connected to the hypercentre by the following theorem.

Theorem E Let F be a saturated fusion system on a finite p-group P. If X is a normal subgroup of P, then F = P C_{F}(X) if and only if X ⩽ Z_{∞}(F).

In Section 7 we prove a few results about the hypercentre and central extensions; for example, we prove the following result. (A saturated fusion system F is perfect if there is no surjective morphism F → F_{A}(A) for any non-trivial abelian p-group A.)

Proposition F Let F be a saturated fusion system on a finite p-group P. If F is perfect then Z_{2}(F) = Z (F).

This proposition allows us to define the universal central extension of a perfect fusion system, exactly analogously to perfect finite groups.

We also extend (slightly) Glauberman’s Z^{∗}-theorem, rewriting it in the language of fusion systems at the same time.

Proposition G Let G be a finite group with O_{p′}(G) = 1, and let P be a Sylow p-subgroup of G. Writing F = F_{P} (G), we have that Z_{i}(F) = Z_{i}(G).

We end with three sections applying our results. The first of these deals with intersections and products of subsystems, the second of these deals with corollaries to Theorem A and the final section gives a comparison of weakly normal and normal subsystems, giving two situations in which normal subsystems behave better than weakly normal subsystems.

The notation used in this article is quickly becoming standard in this field, and we refer to [7] for many of the notational issues. We will define weakly normal subsystems and normal subsystems in the next section, as this is still relatively new terminology, but leave the standard definitions to [7] (see also [11] and [6], although note that in the former ‘fusion systems’ there are referred to as ‘saturated fusion systems’ here). Note that in this article maps and homomorphisms are composed from left to right.

2 Definitions and Notation

The definitions of fusion system and saturated fusion system, and fully normalized, fully centralized, centric, radical and essential subgroups, are as in [7], and these are now standard in the literature.

(Notice that our definitions of fully normalized and fully centralized subgroups differ from that of [12].) The definitions of a fully automized subgroup and a receptive subgroup are not common yet, and so we give them now.

Definition 2.1 Let F be a fusion system on a finite p-group P. A subgroup Q of P is fully Fautomized if Aut_{P} (Q) is a Sylow p-subgroup of Aut_{F}(Q). A subgroup R is receptive if, whenever ϕ : S → R is an isomorphism in F, ϕ extends to a map ϕ̄: N_{ϕ} → P, where N_{ϕ} is the (full) preimage of the subgroup Aut_{P} (S) ∩ Aut_{P} (R)^{ϕ^{−1}} under the natural map N_{P} (S) → Aut(S).

If Q is a subgroup of P then by Aut_{P} (Q) we mean the set of automorphisms of Q induced by conjugation by the elements of P (actually N_{P} (Q)). We introduce the notation Aut(Q ⩽ R) for the set of all automorphisms of R that restrict to an automorphism of Q, where Q ⩽ R. We also use the obvious extensions of notation Aut_{F}(Q ⩽ R) and Aut_{P} (Q ⩽ R). A subgroup Q is said to have the surjectivity property if, for any subgroup R with Q C_{P} (Q) ⩽ R ⩽ N_{P} (Q), the map

Aut_{F}(Q ⩽ R) → N_{Aut_{F} (Q)}(Aut_{R}(Q))

obtained by restriction is surjective.

The notion of an F-invariant subsystem was introduced by Puig (see [12], where it is called ‘normal’), and a saturated, F-invariant subsystem is called weakly normal here (and ‘normal’ in [11]). Normal subsystems are defined in [2], with the addition of one more axiom, which involves how the subsystem is embedded with respect to the centralizer of the underlying subgroup.

Definition 2.2 Let F be a saturated fusion system on a finite p-group P, and let T be a strongly F-closed subgroup. A subsystem E ⩽ F, on T, is F-invariant if, whenever Q ⩽ R are subgroups of T, ϕ : Q → R is a morphism in E and ψ : R → T is a morphism in F, the composition ψ^{−1}ϕψ : Qψ → Rψ is a morphism in E. If E is saturated and F-invariant then E is weakly normal in F.

If, in addition, whenever ϕ is an E-automorphism of T, there is an F-automorphism ϕ̄ ∈ Aut_{F}(T C_{P} (T)) such that [ϕ̄, C_{P} (T)] ⩽ Z (T), then E is said to be normal in F.

We also need to fix our notation for quotient subsystems and morphisms, taken from [7].

Definition 2.3 Let F be a saturated fusion system on a finite p-group P, and let T be a strongly Fclosed subgroup of P. Let F/T denote the fusion system on P/T, whose morphisms Hom_{F/T} (Q/T, R/T) (for T ⩽ Q, R ⩽ P) are all morphisms ϕ̄ induced from morphisms ϕ : Q → R in F by taking quotients by T (since ϕ acts as an automorphism of T).

Let F be a fusion system on a finite p-group P, and let Q be a subgroup of P. The subsystem N_{P} (Q) C_{F}(Q) is the subsystem on N_{P} (Q) consisting of all morphisms ϕ : R → S such that ϕ extends to a map ϕ̄ : QR → QS such that ϕ|_{Q} = c_{g} for some g ∈ N_{P} (Q), where c_{g} denotes the conjugation map by g.

Recall that the centre of F, denoted Z (F), is the set of all x ∈ P such that any morphism ϕ : Q → R in F has an extension ϕ̄ : Q⟨x⟩ → R⟨x⟩ such that xϕ̄ = x. We may iterate this construction.

Definition 2.4 Let F be a saturated fusion system on a finite p-group P. Define Z_{1}(F) = Z (F) and Z_{i}(F) inductively by Z_{i}(F) being the preimage in P of Z (F/Z_{i−1}(F)). The largest term of this ascending sequence is the hypercentre of F, and is denoted by Z_{∞}(F).

We introduce a couple more definitions concerning the generation and saturation of fusion systems.

Definition 2.5 Let F be a fusion system on a finite p-group P. Let H be a set of subgroups of P, and let Q be an F-conjugacy class of subgroups of P.

(i) We say that F is H-generated if F is the smallest fusion system on P containing all Fmorphisms between elements of H.

(ii) We say that Q is saturated if it contains a fully automized, receptive subgroup Q of P.

Finally, we have a definition needed for studying hypercentres.

Definition 2.6 Let F be a saturated fusion system on a finite p-group P. We say that F is perfect if there is no morphism F → F_{A}(A) of fusion systems, for any abelian p-group A,.

An equivalent way of defining being perfect is to say that O^{p}(F) = F, where O^{p}(F) is the hyperfocal subsystem (see [5]).

3 Results from the Literature

Having given all of the definitions, we turn to the theorems in the literature that we will need in the next sections. We start with a well-known result on p′-automorphisms of p-groups that we will use so often we will not comment.

Lemma 3.1 ([9, Corollary 5.3.3]) Let P be a finite p-group, and let Q be a normal subgroup of P. The subgroup of Aut(P) of automorphisms that act trivially on both Q and P/Q is a p-group.

The next two propositions and lemma deal with proving saturation in a fusion system.

Proposition 3.2 ([4, Lemma 2.4]) Let F be a fusion system on a finite p-group P, and let H be a union of F-conjugacy classes of subgroups of P. Suppose that F is H-generated, and that every F-conjugacy class in H is saturated. Let Q be an F-conjugacy class of subgroups of P such that all subgroups of P properly containing members of Q lie in H.

(i) Any fully normalized subgroup Q in Q is fully centralized, and for any other subgroup Q̃ ∈ Q, there is an F-morphism ϕ : N_{P} (Q̃) → N_{P} (Q) with Q̃ϕ = Q.

(ii) For any fully normalized subgroup Q in Q, if {Q} is saturated in N_{F}(Q), then Q is saturated in F.

Proposition 3.3 ([4, Lemmas 2.4 and 2.5]) Let F be a fusion system on a finite p-group P, and let H be a union of F-conjugacy classes of subgroups of P. Suppose that F is H-generated, and that every F-conjugacy class in H is saturated. Let Q be an F-conjugacy class of subgroups of P such that all subgroups of P of smaller index than those in Q lie in H. Let H′ denote the set of subgroups of N_{P} (Q) strictly containing Q.

(i) Every N_{F}(Q)-conjugacy class of subgroups of H′ is saturated, and every F-automorphism of Q is the composition of (restrictions of) N_{F}(Q)-morphisms between elements of H′.

(ii) If either Q is not F-centric, or Out_{P} (Q) ∩ O_{p}(Out_{F}(Q)) is non-trivial, then the N_{F}(Q)- conjugacy class {Q} is saturated.

We need a few different equivalent conditions for a fusion system to be saturated.

Theorem 3.4 (Roberts–Shpectorov [13, Theorems 5.2, 5.3]) Let F be a fusion system on a finite p-group P. Then F is saturated if and only if P is fully F-automized and every F-conjugacy class of subgroups of P contains a fully F-normalized, receptive member, and this is equivalent to every F-conjugacy class of subgroups of P containing a fully F-automized, receptive member.

Theorem 3.5 (Cf. Puig [12, Corollary 2.14]) Let F be a fusion system on a finite p-group P, and suppose that P is fully automized. If, for each F-conjugacy class, there is some subgroup Q such that

(i) if R is F-conjugate to Q then there is a map N_{P} (R) → N_{P} (Q) that restricts to a map R → Q, and

(ii) Q has the surjectivity property,

then F is saturated. Conversely, if F is saturated then any fully normalized subgroup Q has the surjectivity property, and if R is F-conjugate to Q then there is a map N_{P} (R) → N_{P} (Q) that restricts to a map R → Q.

The statement here is not quite what is proved in [12] – there it is required that every fully normalized subgroup satisfies (i) and (ii) above – but it is easily seen to be equivalent.

Combining this theorem with results of Broto, Castellana, Grodal, Levi and Oliver from [4], we get the following result.

Theorem 3.6 Let F be a fusion system on a finite p-group P with P fully automized, and let H be the set of all F-centric subgroups of P. If F is generated by {Aut_{F}(U) : U ∈ H}, and in every F-conjugacy class of F-centric subgroups there is a fully normalized subgroup with the surjectivity property, then F is saturated.

Proof: Let H′ denote the union of all saturated conjugacy classes of F, and suppose that Q is a conjugacy class of subgroups of smallest index subject to not being in H′. Let E be the subsystem of F generated by all morphisms in Aut_{F}(R), as R runs over all subgroups of P of larger order than the subgroups in Q. Notice that Hom_{F}(R, S) = Hom_{E}(R, S) if R has larger order than the subgroups in Q, and so such E-conjugacy classes are identical to the F-conjugacy classes, and also saturated. In addition, Q forms a single E-conjugacy class; in particular, a subgroup in Q is F-centric if and only if it is E-centric.

If Q consists of subgroups that are not F-centric, then by Propositions 3.2(ii) and 3.3(ii) Q is saturated in E. Thus we may suppose that Q consists of F-centric subgroups. Let Q be a fully F-normalized subgroup in Q with the surjectivity property, guaranteed by hypothesis. Proposition 3.2(i), applied to Q and E, is exactly the second requirement for Theorem 3.5, and so Q is saturated, as required.

We end with a couple of results on quotients, weakly normal subsystems and normalizers.

Theorem 3.7 (Puig [12, Proposition 6.6]) Let F be a saturated fusion system on a finite pgroup P, and let E be a saturated subsystem of F, on the strongly F-closed subgroup T of P. The following are equivalent:

(i) E is weakly normal in F; and

(ii) every ϕ ∈ Aut_{F}(T) induces an automorphism of E (via ψ ↦ ψ^{ϕ}) and every F-morphism ψ : A → B with A, B ⩽ T may be written as ψ = αβ, where α ∈ Aut_{F}(T) and β ∈ Hom_{E}(Aα, B).

Proposition 3.8 (Kessar–Linckelmann [10, Proposition 3.4]) Let F be a saturated fusion system on a finite p-group P, and let Q be a normal subgroup of P such that F = P C_{F}(Q). If E is a saturated subsystem on a subgroup R with Q ⩽ R ⩽ P, then E = N_{F}(R) if and only if E/Q = N_{F/Q}(R/Q).

Combined with the next lemma, this yields a useful tool for finding subgroups R such that F = N_{F}(R).

Lemma 3.9 Let F be a saturated fusion system on a finite p-group P, and let T be a strongly F-closed subgroup of P. Let Q be a subgroup of P containing T. We have that N_{F}(Q)/T = N_{F/T} (Q/T), and in particular N_{P} (Q)/T = N_{P/T} (Q/T).

Proof: Clearly T g normalizes Q/T if and only if g normalizes Q, and so N_{P} (Q)/T = N_{P/T} (Q/T).

If ϕ : R → S is a morphism in N_{F}(Q), then ϕ extends to a morphism ψ : RQ → SQ such that Qψ = Q. If ϕ̄ denotes the image of ϕ in F/T then ϕ̄ : RT/T → ST/T extends to ψ̄ : RQ/T → SQ/T, with (Q/T)ψ̄ = Q/T. Hence N_{F}(Q)/T ⊆ N_{F/T} (Q/T).

Conversely, let ϕ̄ : A/T → B/T be a morphism in N_{F/T} (Q/T), and let ψ̄ : AQ/T → BQ/T be an extension of ϕ̄ such that (Q/T)ψ̄ = Q/T. Let ψ : QA → QB be a morphism in F whose image in F/T is ψ̄. Since T is strongly F-closed and (Q/T)ψ̄ = Q/T, we must have that Qψ = Q. Similarly, as (A/T)ψ̄ = B/T, we must have that Aψ = B. Hence ϕ = ψ|_{A} is a morphism in N_{F}(Q). It is also clear that the image of ϕ in F/T is ϕ̄, and so N_{F/T} (Q/T) ⊆ N_{F}(Q)/T, completing the proof.

4 Preliminary Results

Lemma 4.1 Let F be a saturated fusion system on a finite p-group P, and let E be a saturated subsystem of F, on the subgroup Q of P. We have that

O_{p}(E) ⩾ O_{p}(F) ∩ Q.

Proof: Let R = O_{p}(F); by [1, (3.7)] (see also [7, Proposition 3.4]), a strongly F-closed subgroup is contained in O_{p}(F) if and only if it possesses a central series each of whose terms is strongly F-closed. Hence R possesses a central series

1 = R_{0} ⩽ R_{1} ⩽ ⋯ ⩽ R_{d} = R

such that each R_{i} is strongly F-closed. We claim that Q_{i} = Q ∩ R_{i} is strongly E-closed; in this case,

1 = Q_{0} ⩽ Q_{1} ⩽ ⋯ ⩽ Q_{d} = Q ∩ R

is a central series for Q ∩ R, each of whose terms is strongly E-closed, yielding that Q ∩ R ⩽ O_{p}(E) (by another application of [1, (3.7)]), as required. It remains to show that Q_{i} is strongly E-closed; however, any morphism in E that originates inside Q_{i} = Q ∩ R_{i} must have image inside Q since E is a subsystem on Q, and must also lie in R_{i} since R_{i} is strongly F-closed, and so Q_{i} is strongly E-closed.

Lemma 4.2 Let F be a saturated fusion system on a finite p-group P. If X is a strongly F-closed subgroup of P then so is X C_{P} (X).

Proof: It is easy to see that C_{P} (X) is strongly N_{F}(X)-closed, and hence X C_{P} (X)/X is strongly N_{F}(X)/X-closed by [7, Theorem 6.1]. However, since N_{F}(X)/X = F/X, by definition of F/X, we see that X C_{P} (X)/X is strongly F/X-closed, so that X C_{P} (X) is strongly F-closed, by [7, Theorem 6.1] again.

Lemma 4.3 Let F be a saturated fusion system on a finite p-group P, and let E be a weakly normal subsystem of F, on the subgroup T of P. If Q is an E-centric subgroup of T, then Q C_{P} (Q)∩T = Q and Q C_{P} (Q) ∩ T C_{P} (T) = Q C_{P} (T).

Proof: Since Q ⩽ T, by the modular law Q C_{P} (Q) ∩ T = Q(C_{P} (Q) ∩ T) = QZ (Q) = Q. Taking the product of both sides of the first equation by C_{P} (T), and by the modular law again,

Q C_{P} (T) = (Q C_{P} (Q) ∩ T) C_{P} (T) = Q C_{P} (Q) ∩ T C_{P} (T),

as needed.

We introduce a piece of notation at this point for a subquotient that will be important in our work to prove Theorem A.

Definition 4.4 Let P be a finite p-group, and let T be a normal subgroup of P. If Q is a subgroup of T such that C_{T} (Q) ⩽ Q, write Y_{Q} for the subquotient Z (Q) C_{P} (T)/Z (Q) of P.

Proposition 4.5 Let F be a saturated fusion system on a finite p-group P, and let E be a weakly normal subsystem of F, on the subgroup T of P. Let Q and R be E-centric subgroups of T, and let ϕ : Q → R be an isomorphism in F. Let ψ be an extension of ϕ to Q C_{P} (T) in F.

(i) im ψ = R C_{P} (T).

(ii) Qψ = R, Z (Q) ψ = Z (R), and (Z (Q) C_{P} (T))ψ = Z (R) C_{P} (T).

(iii) if g ∈ N_{T} (Q), then c_{g} acts trivially on C_{P} (T), and so on Y_{Q}.

Proof: Let Q̄ = Q C_{P} (T) and R̄ = R C_{P} (T). Since Q̄ = Q C_{P} (Q) ∩ T C_{P} (T), and T C_{P} (T) is strongly F-closed, Q̄ψ ⩽ T C_{P} (T). Also, since Q̄ ⩽ Q C_{P} (Q), Q̄ψ ⩽ R C_{P} (R). Hence Q̄ψ ⩽ R C_{P} (R) ∩ T C_{P} (T) = R̄. If |Q̄| = |R̄|, then we have proved (i); however, Q ∩ C_{P} (T) = Z (Q) and R ∩ C_{P} (T) = Z (R), so that indeed they have the same orders.

To prove (ii), clearly Qψ = R, so Z (Q) ψ = Z (R), and certainly CQ̄(Q) = Q̄ ∩ C_{P} (Q) = Z (Q) C_{P} (T), and CQ̄(Q)ψ = CR̄(R).

We move on to (iii). The map c_{g} is an automorphism of Q, and c_{g} acts trivially on C_{P} (T). By (ii), c_{g} acts as an automorphism of Y_{Q}, and this action is trivial since the action on C_{P} (T) is trivial.

We will use the following easy lemma often to move between the various possible interpretations of the condition that [ϕ, C_{P} (T)] is contained in various subgroups of P.

Lemma 4.6 Let F be a saturated fusion system on a finite p-group P, and let E be a weakly normal subsystem of F, on the subgroup T of P. Let Q and R be E-centric subgroups of T, and let ϕ : Q C_{P} (T) → R C_{P} (T) be a morphism in F. The following are equivalent:

(i) [ϕ, C_{P} (T)] ⩽ T;

(ii) [ϕ, C_{P} (T)] ⩽ Z (R); and

(iii) for x ∈ C_{P} (T), we have (Z (Q) x)ϕ = Z (R) x.

Proof: Let x be an element of C_{P} (T). Firstly, notice that [ϕ, x^{−1}] = (xϕ)x^{−1}.

Assume (i). Since x centralizes T it centralizes Q, and so xϕ ∈ C_{P} (R). Also, x centralizes R, so that [ϕ, x^{−1}] ∈ C_{P} (R) ∩ T = Z (R), proving that (i) implies (ii). Clearly (ii) implies (i).

Next, consider the induced action of ϕ on Y_{Q}, which maps it to Y_{R} by Proposition 4.5(ii). As Z (Q) ϕ = Z (R), the induced action is that Z (Q) x is sent to (Z (Q) x)ϕ = Z (R) (xϕ). Hence Z (R) (xϕ) = Z (R) x if and only if (xϕ)x^{−1} ∈ Z (R), proving that (ii) is equivalent to (iii), as needed.

We end with the following trivial lemma, which has been used in various guises in many papers in this field.

Lemma 4.7 Let U and V be subgroups of the finite p-group P, and suppose that U ⊴ V. If ϕ is an automorphism of V that acts trivially on U, and C_{V} (U) ⩽ U, then ϕ is a p-automorphism.

Proof: Since ϕ acts trivially on U, the induced action on Aut(U) is also trivial, and hence ϕ acts trivially on V/Z (U), so also on V/U. The set of all automorphisms in Aut(V) that act trivially on both U and V/U is a p-group, and so in particular ϕ is a p-automorphism.

5 Weakly Normal and Normal Subsystems

In this section, F is a saturated fusion system on a finite p-group P and T is a strongly Fclosed subgroup of P. There is a weakly normal subsystem E on T with O^{p′}(E) = E. Let W denote the subset of Aut_{E}(T) consisting of those automorphisms ϕ that extend to an automorphism ϕ̄ ∈ Aut_{F}(T C_{P} (T)) such that [ϕ̄, C_{P} (T)] ⩽ Z (T).

Let α be an E-automorphism of T. If Q is a subgroup of T, call Q a detecting subgroup for α if there is a morphism β : Q C_{P} (T) → R C_{P} (T) (where R = Qα) in F such that β|_{Q} = α|_{Q} and [β, C_{P} (T)] ⩽ Z (R). Lemma 4.6 gives some equivalent conditions to Q being a detecting subgroup for α in terms of the condition on [β, C_{P} (T)], and we will move between these conditions regularly.

Let E′ denote the subsystem (not necessarily saturated) generated by O^{p′}(Aut_{E}(Q)), as Q runs over all subgroups of T. In [5, Section 5], Broto, Castellana, Grodal, Levi and Oliver prove that for every element α of Aut_{E}(T), there exist E-centric subgroups Q and R of T, with α|_{Q} : Q → R in E′. We will show that such a subgroup Q is a detecting subgroup for α.

Proposition 5.1 If α is an automorphism in Aut_{E}(T) then there is a detecting subgroup for α.

Proof: Notice that any morphism in E′ is a composition of the restriction of p-automorphisms of subgroups of T, with these p-automorphisms lying in E. We begin by examining such automorphisms.

Let g be an element of T, and suppose that g normalizes an E-centric subgroup S of T. Clearly g centralizes C_{P} (T), and so induces an automorphism c_{g} ∈ Aut_{F}(S C_{P} (T)).

If ϕ ∈ Aut_{E}(S) is any other automorphism, then ϕ extends to an F-automorphism of S C_{P} (T). We claim that (c_{g})^{ϕ} acts trivially on Y_{S} = Z (S) C_{P} (T)/Z (S). This is clear since ϕ acts on Y_{S} by Proposition 4.5(ii), and c_{g} acts trivially on Y_{S} by (iii) of the proposition. Hence any p-automorphism of S in E′ extends to an F-automorphism α of S C_{P} (T) acting trivially on Y_{S}. By Lemma 4.6, this is equivalent to [α, C_{P} (T)] ⩽ T.

Let α ∈ Aut_{E}(T), and let Q be an E-centric subgroup such that α|_{Q} is a morphism in E′. The isomorphism α|_{Q} : Q → R is a composition of p-automorphisms θ_{i} of subgroups S_{i} of T in E′. Since, for each θ_{i} there is an extension θ̄_{i} ∈ Aut_{F}(S_{i} C_{P} (T)) with [θ̄_{i}, C_{P} (T)] ⩽ T, taking the composition of these extensions and restricting to Q C_{P} (T) yields an isomorphism ᾱ : Q C_{P} (T) → R C_{P} (T) in F (whose image is such by Proposition 4.5(i)). We evaluate [ᾱ, C_{P} (T)]: since each [θ̄_{i}, C_{P} (T)] lies inside T, and [ᾱ, C_{P} (T)] is contained in the product of conjugates of [θ̄_{i}, C_{P} (T)], we see that [ᾱ, C_{P} (T)] ⩽ T, so that [ᾱ, C_{P} (T)] ⩽ Z (R), proving that Q is a detecting subgroup for α.

Now that we have proved that detecting subgroups exist, we choose Q to be a detecting subgroup for an automorphism α. The aim is to show that N_{T} (Q) is also a detecting subgroup for α, for then T is a detecting subgroup for α, and so α lies in the subgroup W defined at the beginning of the section. We will prove this result for p′-automorphisms of T first, and then extend to all E-automorphisms.

Proposition 5.2 Let ϕ : Q → R be an isomorphism in F, where Q and R are E-centric subgroups of T, and let ψ be an extension of ϕ to Q C_{P} (T) in F. Suppose that N_{ϕ} contains N_{T} (Q). If R C_{P} (T) is fully F-normalized then N_{ψ} contains N_{T} (Q), and so ψ extends to θ : N_{T} (Q) C_{P} (T) → N_{T} (R) C_{P} (T) in F.

Proof: Let g be an element of N_{T} (Q), and let h ∈ N_{T} (R) such that c_{g}^{ϕ} = c_{h}. We claim that, as an element of Aut_{F}(R C_{P} (T)), c_{g}^{ψ}c_{h}^{−1} acts trivially on R and Y_{R} = Z (R) C_{P} (T)/Z (R). To see this, obviously c_{g}^{ψ} and c_{h} act the same on R; also, c_{h} acts trivially on Y_{R} by Proposition 4.5(iii), so we need that c_{g}^{ψ} acts trivially on Y_{R}. However, c_{g} acts trivially on Y_{Q} by Proposition 4.5(iii), and ψ maps Y_{Q} to Y_{R} by Proposition 4.5(ii), so that this holds.

Let K denote the set of all elements of Aut_{F}(R C_{P} (T)) that act trivially on R and Y_{R}. It is easy to see that K is a p-group, because a p′-automorphism acting trivially on Z (R) and Z (R) C_{P} (T)/Z (R) acts trivially on Z (R) C_{P} (T), so that (since it acts trivially on R) it acts trivially on R C_{P} (T). In fact, since every F-automorphism of R C_{P} (T) induces an automorphism of R and Y_{R}, K is a normal p-subgroup of Aut_{F}(R C_{P} (T)).

Since R C_{P} (T) is fully F-normalized and K is a normal p-subgroup of Aut_{F}(R C_{P} (T)), K ⩽ Aut_{P} (R C_{P} (T)), so that c_{g}^{ψ}c_{h}^{−1}, which acts trivially on R and Y_{R}, lies in Aut_{P} (R C_{P} (T)). Therefore, c_{g}^{ψ} ∈ Aut_{P} (R C_{P} (T)), and so N_{ψ} contains c_{g}. Thus N_{T} (Q) ⩽ N_{ψ}, so as R C_{P} (T) is fully Fnormalized, ψ extends to χ : N_{ψ} → P in F, which restricts to θ : N_{T} (Q) C_{P} (T) → P. As Qθ = R, N_{T} (Q)θ ⩽ N_{T} (R), so Proposition 4.5(i) implies that the image of θ is contained in N_{T} (R) C_{P} (T), as claimed.

Corollary 5.3 Let ϕ : Q → R be an F-isomorphism, where Q and R are E-centric subgroups of T. Let ψ be an extension of ϕ to Q C_{P} (T) in F. Suppose that Aut_{T} (Q)^{ϕ} = Aut_{T} (R), so that | N_{T} (Q)| = | N_{T} (R)| (this happens for example if ϕ is the restriction of an automorphism of T). There is an extension θ of ψ in F, with θ : N_{T} (Q) C_{P} (T) → N_{T} (R) C_{P} (T) an isomorphism.

Proof: If X is a fully F-normalized subgroup conjugate to R C_{P} (T), then X = S C_{P} (T) for S = T ∩ X, by Proposition 4.5(i). Let χ : N_{T} (R) C_{P} (T) → N_{T} (S) C_{P} (T) be a morphism in F, with Rχ = S. (Such a morphism exists since there is an isomorphism α : R C_{P} (T) → S C_{P} (T) in F such that N_{α} = N_{P} (R C_{P} (T)), which clearly contains N_{T} (R) C_{P} (T); the image of χ is clear by Proposition 4.5(i) again.)

Consider the maps ϕχ : Q → S and ψχ : Q C_{P} (T) → S C_{P} (T). Since N_{ϕχ} contains N_{T} (Q), so does N_{ψχ} by Proposition 5.2, and so there is an extension θ_{1} : N_{T} (Q) C_{P} (T) → N_{T} (S) C_{P} (T) of ψχ in F. We claim that N_{T} (Q)θ_{1} = N_{T} (R)χ. If this is true, then θ = θ_{1}χ^{−1} : N_{T} (Q) C_{P} (T) → N_{T} (R) C_{P} (T) is an isomorphism (since the two groups have the same order) that extends ψ (since on Q C_{P} (T) it acts as ψχχ^{−1} = ψ).

Write N = N_{T} (Q)θ_{1} and M = N_{T} (R)χ. The subgroups N and M are determined by the maps θ_{1}|_{Q} and χ|_{R} respectively, since N and M are the preimages in Aut_{T} (S) of the subgroups Aut_{T} (Q)^{θ_{1}|_{Q}} and Aut_{T} (R)^{χ|_{R}}. However, since Aut_{T} (Q)^{ϕ} = Aut_{T} (R), we see that

Aut_{T} (Q)^{θ_{1}} = Aut_{T} (Q)^{ϕχ} = Aut_{T} (R)^{χ},

so that N = M.

We will use this corollary to prove that N_{T} (Q) is a detecting subgroup whenever Q is, at least for p′-automorphisms.

Proposition 5.4 Let α be a p′-automorphism in Aut_{E}(T). If Q is a detecting subgroup for α, so is N_{T} (Q).

Proof: Since F is saturated, α extends to some automorphism β ∈ Aut_{F}(T C_{P} (T)). Since β acts as an automorphism of C_{P} (T) as well, by raising β to a suitable power, we may assume that β|_{C_{P} (T)} is a p′-automorphism as well.

Let Q be a detecting subgroup for α, so that there exists an isomorphism ϕ : Q C_{P} (T) → R C_{P} (T) in F such that ϕ|_{Q} = α|_{Q} and [ϕ, C_{P} (T)] ⩽ Z (R). By Corollary 5.3, there is a map θ : N_{T} (Q) C_{P} (T) → N_{T} (R) C_{P} (T) in F extending ϕ. Consider the isomorphism γ : N_{T} (Q) C_{P} (T) → N_{T} (R) C_{P} (T) that is the restriction of β to N_{T} (Q) C_{P} (T), and the automorphism χ = θ^{−1}γ of N_{T} (R) C_{P} (T).

Since θ and γ are both extensions of the same morphism on Q, χ|_{R} is the identity. Therefore, since χ induces an automorphism of N_{T} (R), and C_{T} (R) ⩽ R, by Lemma 4.7, χ|_{N_{T} (R)} is a pautomorphism. Since both θ and γ map Y_{Q} to Y_{R}, χ induces an automorphism of Y_{R}; we claim that this is a p′-automorphism. To see this, notice that for x ∈ C_{P} (T), we have (Z (Q) x)θ = Z (R) x and (Z (Q) x)γ = Z (R) (xβ). Therefore (Z (R) x)χ = Z (R) (xβ), so that the order of the automorphism that χ induces on Y_{R} divides the order of β|_{C_{P} (T)}, a p′-automorphism.

Therefore there exists an integer n such that χ^{n}|_{N_{T} (R)} = χ|_{N_{T} (R)} and χ^{n} acts like the identity on Y_{R}. Consider θχ^{n}: this map acts like γ on N_{T} (Q) (since it acts like θχ^{n} = θχ = θθ^{−1}γ = γ), and since χ acts like the identity on Y_{R}, θχ^{n} acts like θ on Y_{Q}; i.e., [θχ^{n}, C_{P} (T)] ⩽ T. Hence θχ^{n} is an extension of α|_{N_{T} (Q)} to N_{T} (Q) C_{P} (T) in F such that [θχ^{n}, C_{P} (T)] ⩽ Z (N_{T} (Q)) (using Lemma 4.6). Thus N_{T} (Q) is a detecting subgroup for α, as required.

From here it is easy to prove the theorem. Recall from the start of this section that W is the set of all ϕ ∈ Aut_{E}(T) that extend to ϕ̄ ∈ Aut_{F}(T C_{P} (T)) with [ϕ̄, C_{P} (T)] ⩽ Z (T). Since Aut_{T} (T) is a Sylow p-subgroup of Aut_{E}(T) and Aut_{T} (T) ⩽ W by Proposition 4.5(iii), and all p′-elements of Aut_{E}(T) lie in W by the previous proposition, we see that W = Aut_{E}(T), as needed.

6 Weakly Normal Maps

In [2], Aschbacher defined invariant maps and normal maps, the former producing (some) invariant subsystems of a fusion system, and the latter producing every normal subsystem of a saturated fusion system. In this section we will define a ‘weakly normal’ map, which will in fact be basically a normal map [2, Remark 7.5] with one axiom removed. In this section we will prove that every weakly normal map determines a weakly normal subsystem, and that every weakly normal subsystem gives rise to a unique weakly normal map, just as with normal subsystems and normal maps.

Because the original proof in [2] uses the fact that the normal map produces a normal subsystem, we need to find another proof.

Proposition 6.1 Let F be a saturated fusion system on a finite p-group P, and let E be an Finvariant subsystem on a strongly closed subgroup T of P. A subgroup R ⩽ T is E-centric if and only if R is F-conjugate to a fully F-normalized subgroup S of T with C_{T} (S) ⩽ S.

Proof: This follows from [2, Lemma 6.5(2)].

Related to this is the following understanding of the relationship between E-conjugacy classes and F-conjugacy classes of subgroups of a strongly closed subgroup T.

Lemma 6.2 Let F be a saturated fusion system on a finite p-group P, and let E be an F-invariant subsystem on a strongly closed subgroup T of P. If Q is a subgroup of T, then the F-conjugacy class Q containing Q is a disjoint union of E-conjugacy classes Q_{1}, …, Q_{n}, and there are automorphisms ϕ_{i} ∈ Aut_{F}(T) ⩽ Aut(E) of T such that Q_{1}ϕ_{i} = Q_{i}.

Proof: By Theorem 3.7, every F-morphism ϕ between subgroups of T may be written as ϕ = αβ, with α ∈ Aut_{F}(T) and β in E, and Aut_{F}(T) ⩽ Aut(E). This clearly implies the statement.

Theorem 3.6 in fact tells us what we want for the invariant maps of [2] to be weakly normal.

Definition 6.3 Let F be a saturated fusion system on a finite p-group P, and let T be a strongly F-closed subgroup. A weakly normal map on T is a function A(−) on the set of subgroups U of T, with A(U) ⩽ Aut_{F}(U), such that

(i) if ϕ is an F-isomorphism whose domain is U, then A(Uϕ) = A(U)^{ϕ},

(ii) if U is fully F-normalized, then Aut_{T} (U) ⩽ A(U),

(iii) Aut_{T} (T) is a Sylow p-subgroup of A(T),

(iv) if U is fully F-normalized, then every element of A(U) extends to an element of A(U C_{T} (U)), and

(v) if U is fully F-normalized and C_{T} (U) ⩽ U then for any subgroup U ⩽ V ⩽ N_{T} (U), denoting by A(U ⩽ V) the set of automorphisms of V that restrict to automorphisms of U, the restriction map

A(U ⩽ V) → N_{A(U)}(Aut_{V} (U))

is surjective.

The idea is that if E is the subsystem generated by the morphisms in A(U) for all U ⩽ T, then Aut_{E}(U) = A(U).

We will make some short remarks about this definition now. The first two conditions are simply that A(−) is an invariant map in the sense of [2, Section 5]. The third condition is obviously necessary for E = ⟨A(U) : U ⩽ T ⟩ to be saturated, and the fifth condition is simply the surjectivity property for fully F-normalized, E-centric subgroups.

The fourth condition is there to make sure that E is actually generated by the automorphisms of centric subgroups: since every automorphism of a fully F-normalized subgroup U extends to an automorphism of the E-centric subgroup U C_{T} (U), we really do get that E is generated by automorphisms of E-centric subgroups, one of the conditions of Theorem 3.6.

Like with normal maps [2, Section 7], because of Condition (iv), we need only define a weakly normal map on fully F-normalized subgroups U with C_{T} (U) ⩽ U, and then use (i) to extend to all F-conjugates of such subgroups, then (iv) to extend to all fully F-normalized subgroups, and then (i) again to extend to all subgroups of T.

Having discussed the reasons behind the definition of a weakly normal map, we now prove the claimed result.

Theorem 6.4 Let F be a saturated fusion system on a finite p-group P, and let T be a strongly F-closed subgroup of P. If A(−) is a weakly normal map on T, then there exists a weakly normal subsystem E of F on T with Aut_{E}(U) = A(U) for all U ⩽ T.

Proof: Let E denote the subsystem of F generated by A(−). Since A(−) is an invariant map, E is F-invariant by [2, Lemma 5.5], and since Aut_{E}(T) = A(T) for all invariant maps, Aut_{T} (T) is a Sylow p-subgroup of Aut_{E}(T). Let H denote the set of E-centric subgroups of T.

Step 1: E is generated by E-automorphisms of elements of H. Since E is generated by the subgroups A(Q) for Q ⩽ T, it suffices to show that each element of these is a product of (the restriction of) elements of A(U), for U an E-centric subgroup of T. If Q is fully F-normalized then this is true by Condition (iv) of being a weakly normal map. If R is a subgroup of T that is F-conjugate to Q via ϕ, then ϕ = αβ, with α ∈ Aut(E) and β a morphism in E by Theorem 3.7. Since every element of A(Q) is the restriction of an automorphism of some member of H, so is every element of A(Q)^{α} = A(Qα). Finally, if β is an automorphism of Qα (so Qα = R) then we are done; else Qα and R are E-conjugate via β, and hence β is a composition of the restriction of automorphisms of subgroups of T of larger order than that of Q, which by induction are generated by automorphisms of elements of H. Therefore A(R) = A(Q)^{β^{−1}} is generated by E-automorphisms of elements of H, completing the proof.

Let Q be an E-conjugacy class contained in H, and suppose that it consists of subgroups of T of largest order subject to the elements Q of Q not satisfying A(Q) = Aut_{E}(Q). Since Aut_{E}(T) = A(T), Q consists of proper subgroups of T. By Condition (iv), all subgroups in H of larger order than |Q| lie in saturated E-conjugacy classes. Applying an automorphism of E if necessary (and using Lemma 6.2) we may assume that Q contains a fully F-normalized subgroup Q.

Step 2: A(Q) = Aut_{E}(Q). Consider the subsystem N_{E}(Q) of E, and notice that Aut_{E}(Q) = Aut_{N_{E}(Q)}(Q). Since Q is fully F-normalized, by Proposition 3.3(i) every element of Aut_{E}(Q) can be written as the product of the restriction of morphisms in N_{E}(Q) between subgroups of N_{T} (Q) strictly containing Q, and all N_{E}(Q)-conjugacy classes containing such subgroups of N_{T} (Q) are saturated. We first prove that each of these maps ϕ : R → S may be written as the composition of automorphisms of overgroups of Q in N_{E}(Q), proceeding by induction on n = | N_{T} (Q) : R|, the case n = 1 being clear.

Let ψ : S → S̃ be an isomorphism in N_{E}(Q) with S̃ a fully N_{E}(Q)-normalized subgroup of N_{T} (Q). If the result holds for ϕψ and ψ then it holds for ϕ = ϕψ(ψ^{−1}), so we must show that any N_{E}(Q)-morphism whose image is fully N_{E}(T)-normalized may be written as a product of elements of Aut_{N_{E}(Q)}(U_{i}) for various U_{i} ⩽ N_{T} (Q); hence we assume that S is fully N_{E}(Q)-normalized. Since the N_{E}(Q)-conjugacy class containing R is saturated and S is fully normalized, S is fully automized and so there is a map θ : R → S in N_{E}(Q) that extends to a map θ̄ : N_{N_{T} (Q)}(R) → N_{N_{T} (Q)}(S). By induction θ̄, and hence θ, is the restriction of N_{E}(Q)-automorphisms of subgroups of N_{T} (Q) strictly containing Q, and so ϕ can be expressed in such a way if and only if θϕ^{−1} can, which is an automorphism of an overgroup of Q, completing the proof of the claim.

Now let ϕ be an automorphism in Aut_{E}(Q). If ϕ does not lie in A(Q), then it must be the composition of restrictions of N_{E}(Q)-automorphisms ψ_{i} of subgroups R_{i} of N_{T} (Q) strictly containing Q by the previous paragraphs; it suffices to check the case where ϕ is the restriction of a single automorphism of some overgroup R. By choice of Q we have that A(R) = Aut_{E}(R). However, by Condition (v), and the fact that ψ ∈ A(Q ⩽ R), we see that ϕ ∈ A(Q), completing the proof.

Step 3: In any E-conjugacy class of E-centric subgroups of T, there is a fully E-normalized subgroup with the surjectivity property. If Q is a fully F-normalized and E-centric subgroup of T, then Condition (v) of being a weakly normal map implies that Q has the surjectivity property. If α ∈ Aut(E) then Qα also has the surjectivity property, and by Lemma 6.2 there is such a subgroup in every E-conjugacy class of E-centric subgroups.

In particular, we now know that E is saturated by Theorem 3.6.

Step 4: Aut_{E}(Q) = A(Q) for all Q ⩽ T. If Q is fully F-normalized then this follows from Step 1 and Condition (iv) of being a weakly normal map. Since E is F-invariant, if ϕ : Q → R is an F-isomorphism with R a fully F-normalized subgroup of T then Aut_{E}(Q)^{ϕ} = Aut_{E}(R), and since A(Q)^{ϕ} = A(R) as A(−) is an invariant map, we are done.

As an example, we show that the condition that E be generated by automorphisms of centric subgroups is necessary.

Example 6.5 Let P be the group V_{4}, and let F be the fusion system of the alternating group A_{4}. Let E be the subsystem of F, on P, given by all F-maps between subgroups of order 2, but not their extensions to P. Hence Aut_{E}(Q) is trivial for all Q ⩽ P, and so Condition (ii) of Theorem 3.5 is trivially satisfied. It is also clear that E is F-invariant, and that Aut_{P} (P) is a Sylow 2-subgroup of Aut_{E}(P). However, E is obviously not saturated, since for example it does not satisfy the conclusion of Alperin’s fusion theorem.

Hence there are F-invariant subsystems E with Aut_{T} (T) fully E-automized, and with all subgroups having the surjectivity property, that are not saturated.

7 The Hypercentre

In this section we will study the hypercentre and central extensions of fusion systems. We begin with the hypercentral subgroup theorem, proving Theorem E.

Theorem 7.1 (Hypercentral subgroup theorem) Let F be a saturated fusion system on a finite p-group P, and let Q and R be subgroups of P.

(i) If F = P C_{F}(Q) and F = P C_{F}(R) then F = P C_{F}(QR).

Let X_{F} denote the largest (strongly F-closed) subgroup of P such that F = P C_{F}(X_{F}).

(ii) If Q is a normal subgroup of P contained in X_{F} then Q is strongly F-closed, and X_{F/Q} = X_{F}/Q and O_{p}(F)/Q = O_{p}(F/Q).

(iii) X_{F} = Z_{∞}(F).

Proof: As Q, R ⩽ O_{p}(F), so therefore is QR, and hence F = N_{F}(QR). It remains to show that Aut_{F}(QR) is a p-group, but if ϕ is a p′-automorphism in Aut_{F}(QR) then the restrictions to both Q and R must be trivial, and hence ϕ = 1. Thus F = P C_{F}(QR), proving (i).

Let Q ⊴ P and Q ⩽ X_{F}, and let ϕ : A → B be a morphism with A ⩽ Q. The map ϕ extends to an automorphism of X_{F}, which must be a conjugation map c_{g} for some g ∈ P, and Q^{g} = Q, so that B ⩽ Q. Thus Q is strongly F-closed.

Let R/Q be a subgroup of P/Q such that F/Q = N_{F/Q}(R/Q). As R/Q is normal in P/Q, R ⊴ P and so N_{F}(R) is a saturated subsystem of F. By Lemma 3.9, N_{F}(R)/Q = N_{F/Q}(R/Q) = F/R, and by Proposition 3.8, N_{F}(R)/Q = F/Q implies F = N_{F}(R). Therefore O_{p}(F)/Q = O_{p}(F/Q).

Next, suppose in addition that Aut_{F/Q}(R/Q) is a p-group (i.e., R/Q ⩽ X_{F/Q}). Since Aut_{F}(Q) is a p-group, any p′-automorphism in Aut_{F}(R) must act trivially on both Q and R/Q, and is therefore trivial; thus F = P C_{F}(R), and so X_{F}/Q = X_{F/Q}, proving (ii).

Finally, we will show that X_{F} = 1 if and only if Z (F) = 1. One direction is immediate, so suppose that X_{F} ≠ 1; let Z be the subgroup Z (P) ∩ X_{F}, and note that Z is non-trivial since X_{F} is a non-trivial normal subgroup of P. If Z is strongly F-closed, then since F = P C_{F}(Z) (as Z ⩽ X_{F}) and P acts trivially on Z, we actually have that F = C_{F}(Z), so that Z ⩽ Z (F). If ϕ : A → B is any morphism in F with A ⩽ Z, then ϕ extends to a P -automorphism of X_{F}, which must restrict to a (trivial) automorphism of Z. Hence Z is strongly F-closed, as claimed.

Induction and (ii) of this theorem imply that Z_{∞}(F) ⩽ X_{F}. If we can show that X_{F/Z_{∞}(F)} = 1 then we are done by (ii) again. However, certainly Z (F/Z_{∞}(F)) = 1, and hence X_{F/Z_{∞}(F)} = 1, as needed.

We now want to isolate Z_{i}(F), given Z_{∞}(F); this is very easy to do.

Lemma 7.2 Let F be a saturated fusion system on a finite p-group P. Then

Z_{i}(F) = Z_{∞}(F) ∩ Z_{i}(P).

Proof: Clearly Z_{i}(F) ⩽ Z_{i}(P)∩Z_{∞}(F). Let x be an element in Z_{∞}(F)∩Z_{i}(P), and let ϕ : ⟨x⟩ → P be any morphism in F. We need to show that ϕ acts trivially on ⟨x⟩Z_{i−1}(F)/Z_{i−1}(F), or in other words that [ϕ, x] ∈ Z_{i−1}(F), for then Z_{i−1}(F)x ∈ Z (F/Z_{i−1}(F)); ϕ extends to an automorphism ψ of Z_{∞}(F) (as Z_{∞}(F) ⩽ O_{p}(F)), and by Theorem 7.1, ψ = c_{g} for some g ∈ P. Since [ψ, x] = [g, x] ∈ Z_{i−1}(P) (as x ∈ Z_{i}(P)), by induction [ψ, x] ∈ Z_{i−1}(F), proving that x ∈ Z_{i}(F).

In the case of a perfect fusion system, like with a perfect group, performing a central extension twice still results in a central extension.

Proposition 7.3 Let F be a saturated fusion system on a finite p-group P. If F is perfect then Z_{2}(F) = Z (F).

Proof: Write Z = Z (F), let x be an element of Z_{2}(F), and for g ∈ P let λ_{x} be a map defined by gλ_{x} = [x, g]. We claim that this is a group homomorphism P → Z that induces a morphism of fusion systems F → F_{Z}(Z). Since F is perfect, the homomorphism λ_{x} must have trivial image; i.e., gλ_{x} = 1 for all g ∈ G, so that x ∈ Z (P). Lemma 7.2 will then imply that x ∈ Z (F), as needed.

Firstly, let us prove that λ_{x} is a group homomorphism: since x ∈ Z_{2}(F), in particular x ∈ Z_{2}(P), so that [x, g] ∈ Z (P) for all g ∈ P. Hence

[x, gh] = [x, h][x, g]^{h} = [x, g][x, h],

so that λ_{x} is a group homomorphism.

Let X denote the kernel of λ_{x}, and notice that Z ⩽ X. Let Q and R be subgroups of P with Q ⩽ X, and let ϕ : Q → R be an F-isomorphism. Since x ∈ Z_{2}(F), we may extend ϕ : Q → R to a morphism ψ : Q(Z⟨x⟩) → R(Z⟨x⟩) acting as an automorphism of Z⟨x⟩ that acts trivially on both Z and Z⟨x⟩/Z. Write z = x(xψ^{−1}), so that (zx)ψ = x. Since [x, Q] = 1, we have

[x, Qψ] = [zx, Q]ψ ⩽ ([z, Q]^{x}[x, Q])ψ = 1.

Therefore R ⩽ X, so that X is strongly F-closed. Hence there is a surjective morphism Φ : F → E of fusion systems with kernel X, where E is a fusion system on an abelian p-group. If we can show that any p′-automorphism α of P satisfies αΦ = id then E = F_{P/X}(P/X). Let g be an element in P; we need to show that g^{−1}gα lies in X; i.e., that [x, g^{−1}gα] = 1. Since α is a p′-automorphism, and F = P C_{F}(Z⟨x⟩), we must have that α acts as the identity on Z⟨x⟩, which contains both x and [x, g]. Hence (since [x, g^{−1}] is central in P)

[x, g^{−1}gα] = [x, gα][x, g^{−1}]^{gα} = ([x, g]α)[x, g^{−1}]^{g} = [x, g][x, g^{−1}]^{g} = [x, g^{−1}g] = 1.

Therefore E is the fusion system F_{A}(A) for some abelian p-group A. Since F is perfect, we must have A = 1, so that X = P. Therefore x ∈ Z (P), as needed.

Proposition 7.4 Let G be a finite group with a Sylow p-subgroup P. If O_{p′}(G) = 1 then Z_{i}(G) = Z_{i}(F), where F = F_{P} (G).

Proof: By Glauberman’s Z^{∗}-theorem (for p = 2 it is from [8], and for odd primes it follows from the classification of the finite simple groups), if x is an element of order p in P such that x^{G} ∩ P = {x} then x ∈ Z (G). The condition that x^{G} ∩ P = {x} is the same as x ∈ Z (F). Therefore, Z (G) = 1 if and only if Z (F) = 1, and hence Z_{∞}(G) = Z_{∞}(F).

By Lemma 7.2, Z_{i}(F) = Z_{i}(P) ∩ Z_{∞}(F). We must similarly prove that Z_{i}(G) = Z_{i}(P) ∩ Z_{∞}(G); to see this, again we only have to prove that Z (G) = Z (P) ∩ Z_{∞}(G). Certainly Z (G) ⩽ Z (P) ∩ Z_{∞}(G), so let x be an element of Z (P) ∩ Z_{∞}(G). Any p′-element g of G must centralize Z_{∞}(G), since it acts trivially on Z_{i}(G)/Z_{i−1}(G) for all i. Since P centralizes x as well, C_{G}(x) contains all p′-elements of G and a Sylow p-subgroup of G, so that C_{G}(x) = G, as claimed.

Therefore Z_{i}(G) = Z_{i}(P) ∩ Z_{∞}(G) = Z_{i}(P) ∩ Z_{∞}(F) = Z_{i}(F), as claimed.

8 Intersections and Products of Subsystems

We include an application of the theory of weakly normal maps, proving the existence of a subsystem like the intersection in certain situations.

Theorem 8.1 Let F be a saturated fusion system on a finite p-group P, and let E_{1} and E_{2} be weakly normal subsystems of F, on the strongly F-closed subgroups T and T̄ of P respectively, with T ⩽ T̄. There exists a weakly normal subsystem E, contained in E_{1} ∩ E_{2} and denoted by E_{1} ⋏ E_{2}, such that for any fully F-normalized subgroup Q of T with C_{T} (Q) ⩽ Q, we have

Aut_{E}(Q) = Aut_{E_{1}}(Q) ∩ Aut_{E_{2}}(Q).

Proof: Let A_{1}(−) and A_{2}(−) be the weakly normal maps corresponding to E_{1} and E_{2} respectively, and let A(−) be the map given by A(Q) = A_{1}(Q) ∩ A_{2}(Q) for Q a subgroup of T with C_{T} (Q) ⩽ Q, and then extended to all subgroups using (i) and (iv) of the definition of a weakly normal map, as described earlier in Section 6. We will show that A(−) is again a weakly normal map; clearly the subsystem E generated by A(−) will satisfy the conclusions of the theorem. We prove each of the five conditions in turn.

The first and second properties are clear from the fact that each A_{i}(−) is a weakly normal map. Clearly, since Aut_{T} (T) is a Sylow p-subgroup of A_{1}(T) and contained in A_{2}(T), it is a Sylow p-subgroup of A(T), proving the third condition. The fourth condition follows by the construction of A(−), and so it remains to prove the fifth condition.

Let Q be a fully F-normalized subgroup of T with C_{T} (Q) ⩽ Q, and let R be an overgroup of Q contained in N_{T} (Q). Let ϕ be an automorphism in N_{A(Q)}(Aut_{R}(Q)); since both E_{i} are weakly normal subsystems of F, there are elements ψ_{i} in A_{i}(R) that extend ϕ. (The existence of ψ_{1} is clear, and there is obviously a map ψ_{2} : R → P extending ϕ. To see that ψ_{2} is an automorphism of R, apply the same argument used at the end of the proof of Corollary 5.3.)

Let S be a fully F-normalized subgroup of T that is F-conjugate to R via α : T → S. Notice that Aut_{T} (S)^{α} is a Sylow p-subgroup of A_{1}(R) and is contained in A_{2}(R). Hence A_{2}(R) is a normal subgroup of p′-index of A = A_{1}(R)A_{2}(R), and so all p-subgroups of A are p-subgroups of A_{2}(R).

Let K be the subgroup of A consisting of all automorphisms of R acting trivially on Q; K is a p-subgroup of A by Lemma 4.7. Since all p-subgroups of A are p-subgroups of A_{2}(R), we see that K is contained in A_{2}(R). Notice that ψ_{1}ψ_{2}^{−1} acts trivially on Q so lies in K, and therefore lies in A_{2}(R). Hence ψ_{1} ∈ A_{2}(R), so ψ_{1} ∈ A(R), as needed. Hence Condition (v) of being a weakly normal map is satisfied.

We can slightly relax the conditions of Theorem 8.1 to allow the case where E_{1} is just a saturated subsystem, and not F-invariant. In this case, E is weakly normal in E_{1}, but not necessarily weakly normal in F.

We get two corollaries of Theorem 8.1.

Corollary 8.2 Let F be a saturated fusion system on a finite p-group P, and let E_{1} and E_{2} be weakly normal subsystems of F, on the same strongly F-closed subgroup T of P. Then O^{p′}(E_{1}) = O^{p′}(E_{2}).

This shows that the set of weakly normal subsystems on a given subgroup T, partially ordered by inclusion, has a unique minimal element, which we will denote by R_{F}(T). We also get another corollary, relating these minimal weakly normal subsystems.

Corollary 8.3 Let F be a saturated fusion system on a finite p-group P, and let T and T̄ be strongly F-closed subgroups of P, with T ⩽ T̄. If there are weakly normal subsystems of F on both T and T̄, then R_{F}(T) ⩽ R_{F}(T̄).

In fact, there is also a largest weakly normal subsystem on a given strongly closed subgroup, if the set of such weakly normal subsystems is non-empty. This follows from a result of Puig.

Theorem 8.4 (Puig [12, Proposition 6.10]) Let F be a saturated fusion system on a finite p-group P, and let E be a weakly normal subsystem of F, on a strongly F-closed subgroup T of P. If H is any normal subgroup of Aut_{F}(T) containing Aut_{E}(T), with |H : Aut_{E}(T)| prime to p, then there exists a weakly normal subsystem E′ of F on T, containing E, and such that Aut_{E′}(T) = H. Furthermore, for any Q ⩽ T,

Hom_{E′}(Q, T) = Hom_{E}(Q, T) · H.

In particular, if we take E = R_{F}(T), and H to be the largest such subgroup of Aut_{F}(T), then we get the largest weakly normal subsystem of F on T. We will denote this by R^{F}(T). (Notice that if E′ is any weakly normal subsystem of F on T, then E′ contains E and | Aut_{E′}(T) : Aut_{E}(T)| is prime to p, so every weakly normal subsystem of F on T can be obtained by the method in Theorem 8.4.)

If there are (weakly) normal subsystems of F on T, then we say that T is based in F. (In the next section we note that in [2], Aschbacher proves that there are strongly F-closed subgroups that are not based.) In [3], Aschbacher proves the following theorem.

Theorem 8.5 (Aschbacher) Let F be a saturated fusion system on a finite p-group P, and let T_{1} and T_{2} be strongly F-closed subgroups. If T_{1} and T_{2} are based, so is T_{1} ∩ T_{2}.

An obvious question is to ask whether, in this case, T_{1}T_{2} is also based. In the same article, Aschbacher proves that this is true if T_{1} and T_{2} commute, but the general case is still open. We will use the above theorem to develop a general theory of intersections of weakly normal subsystems.

Let T be a strongly F-closed subgroup of P that is based in F, and let E be a subsystem (not necessarily saturated) of F on some subgroup Q of P, with T ⩽ Q. Suppose that E contains R_{F}(T). It is easy to see via Theorem 8.4 that the set of all weakly normal subsystems of F on T that are contained in E has a unique largest element, analogous to the core of a subgroup of a finite group; we will call such a subsystem the T -core of E. If E_{1} and E_{2} are any two subsystems of F on subgroups Q_{1} and Q_{2} of P, such that T = Q_{1} ∩ Q_{2} is a strongly F-closed subgroup of P based in F, then we can construct the unique maximal weakly normal subsystem of F contained in both E_{i}, simply the T -core of E_{1} ∩E_{2}, as long as R_{F}(T) ⩽ E_{1} ∩E_{2}. In particular, if both E_{i} are weakly normal then such a subsystem exists, and as above we will denote it by E_{1} ⋏ E_{2}. (This notation extends the earlier one, as when both ‘intersections’ are definable they coincide.) This yields a theory of intersections of weakly normal subsystems.

It should be noted that if the E_{i} are both normal in F, E_{1} ⋏E_{2} need not be normal in F. Indeed, the intersection E_{1} ∧ E_{2} constructed by Aschbacher in [3] does not coincide with the subsystem constructed here in general. We will see an example of such a situation in the next section.

Having proved that if T ⩽ T̄ are strongly F-closed subgroups that are based in F then R_{F}(T) ⩽ R_{F}(T̄), we turn our attention to a related question, whether R^{F}(T) ⩽ R^{F}(T̄). For a satisfactory theory of products of weakly normal subsystems to be constructed, this must be true, since else there need not be any weakly normal subsystem on T̄ containing both R^{F}(T) and R_{F}(T̄), an obvious necessity to define a product of those two subsystems.

However, as the following example shows, it is not true in general that R^{F}(T) ⩽ R^{F}(T̄).

Example 8.6 Let Q be elementary abelian of order 9, generated by a and b. Let ϕ be the map fixing b and sending a to ab, and let ψ be the map fixing a and inverting b. The group H = ⟨ψ, ϕ⟩ is isomorphic with S_{3}; let G = Q ⋊ H and write P for the (unique) Sylow 3-subgroup of G. Finally, write R = ⟨b⟩. Notice that R ⊴ G, so that F_{R}(R) is a normal subsystem of F = F_{P} (G). It is easy to see that the subsystem F_{R}(⟨R, ψ⟩) is isomorphic with the fusion system of S_{3} and is equal to R^{F}(R). However, since Aut_{F}(Q) ≅ S_{3}, there can be no weakly normal subsystems of F on Q except for F_{Q}(Q), as there are no normal 3′-subgroups of Out_{F}(Q) = Aut_{F}(Q). Thus R_{F}(Q) = R^{F}(Q) = F_{Q}(Q). Therefore

R^{F}(R) ⩽̸ R^{F}(Q).

(There are three saturated subsystems of F that contain both R^{F}(R) and R_{F}(Q), namely those given by the subgroups ⟨Q, ψϕ^{i}⟩ for i = 0, 1, 2. However none of these is weakly normal in F.)

This means that there can be no ‘reasonable’ theory of products of weakly normal subsystems of fusion systems. It might still be possible to produce a theory of products of normal subsystems of fusion systems.

We end with a couple of examples of situations that show that some obvious candidates for an ‘intersection subsystem’ do not work. To see why we cannot simply take E_{1} ∩ E_{2}, consider the following example.

Example 8.7 Let G be the group A_{4} ×A_{4}, and let P be the Sylow 2-subgroup of G, an elementary abelian group of order 16, generated by a and b in the first factor, and c and d in the second. Let x and y be elements of order 3, with both sending a to b, and x sending c to d and y sending d to c. Let H_{1} be the group isomorphic with A_{4} generated by P and x, and H_{2} be the group generated by P and y.

Let F be the fusion system of G on P, and E_{i} be the subsystem generated by H_{i}. Since H_{i} ⊴ G, the E_{i} are weakly normal in F, but if E = E_{1} ∩ E_{2}, then Aut_{E}(P) is trivial, but there is a non-trivial automorphism on Q = ⟨a, b⟩, which therefore cannot extend to P. Hence E is not saturated.

This example shows why we should define the weakly normal map in Theorem 8.1 only on the fully F-normalized, F-centric subgroups, and then extend it in the unique way, rather than try to define it on all subgroups to begin with, as the subgroup Q above was fully F-normalized, and both Aut_{E_{i}}(Q) were the same, but the ‘correct’ choice for Aut_{E}(Q) was the trivial group.

If the subsystems E_{i} do not lie on the same subgroup then the construction in Theorem 8.1 does not work well; firstly because the E-centric subgroup R need not be E_{i}-centric, but even in this case things can go wrong, even in a fusion system F_{P} (P).

Example 8.8 Let P = D_{8} × C_{2}, with the D_{8} factor generated by an element x of order 4 and y of order 2, and the C_{2} factor being generated by z. Let Q = ⟨x, y⟩, and R = ⟨xz, y⟩; then S = Q ∩ R = ⟨x^{2}, y⟩ is a normal Klein four subgroup of P. If F = F_{P} (P), and E_{1} and E_{2} are the subsystems F_{Q}(Q) and F_{R}(R) respectively, then both E_{1} and E_{2} are weakly normal in F. We see that Aut_{E_{1}}(S) = Aut_{E_{2}}(S). However, the only saturated subsystem of F on S is E = F_{S}(S), for which Aut_{E}(S) is trivial, and so the ‘correct’ subsystem we want inside E_{1} ∩ E_{2} is E itself. Taking the intersection of the Aut_{E_{i}}(S) therefore does not yield a saturated subsystem.

(Notice that in this case, S is both fully F-normalized (indeed, strongly F-closed), and E_{i}-centric for i = 1, 2, so the problem does not lie in not being E_{i}-centric.)

9 Corollaries of Theorem A

Here we collect a few corollaries of Theorem A. We begin with the corollary mentioned in the introduction.

Corollary 9.1 A saturated fusion system has no proper, non-trivial weakly normal subsystems if and only if it has no proper, non-trivial normal subsystems.

Using this corollary, it is easy to show that a fusion system being quasisimple (i.e., a perfect fusion system such that F/Z (F) is simple) is not dependent on which definition of simplicity is used. In [3], Aschbacher proves the existence of the analogue of the generalized Fitting subgroup for fusion systems. This is the central product of O_{p}(F) and all subnormal quasisimple subsystems. It is easy to see that any weakly subnormal (i.e., the transitive closure of being weakly normal) quasisimple subsystem is also subnormal, and therefore we have the following corollary.

Corollary 9.2 The generalized Fitting subsystem of a fusion system F is the central product of O_{p}(F) and all weakly subnormal, quasisimple, subsystems.

In other words, the definition of the generalized Fitting subsystem does not depend on the definition of normality used, just as the definition of a simple fusion system.

A minimal (weakly) normal subsystem of a fusion system F is a (weakly) normal subsystem E of F such that if E′ is a (weakly) normal subsystem of F contained in E, then either E′ = E or E′ = 1. The same methods used for minimal normal subgroups prove that minimal normal subsystems are isomorphic to direct products of isomorphic simple fusion systems.

Corollary 9.3 A subsystem E of a saturated fusion system F is a minimal normal subsystem if and only if it is a minimal weakly normal subsystem.

If E is a normal subsystem of a saturated fusion system F and E is contained in a saturated subsystem F′ of F, then it need not be true that E is normal in F′, even in the case where F′ is itself normal. The following example proves this.

Example 9.4 Let G be the group S_{3}×S_{3}, and let H denote the subgroup of index 2 not containing either direct factor. Let P denote a Sylow p-subgroup of G, and let K denote the first S_{3} factor; write Q = K ∩P. Since K ⊴ G, we have that F_{Q}(K) ⊴ F_{P} (G). Also, as H is a normal subgroup of G, F_{P} (H) ⊴ F_{P} (G). Clearly also, F_{Q}(K) ⩽ F_{P} (H). However, it is not true that F_{Q}(K) ⊴ F_{P} (H). (This example also shows that the Q-core of F_{Q}(K) ∩ F_{P} (H), the subsystem F_{Q}(K) ⋏ F_{P} (H), is not the same as F_{Q}(K) ∧ F_{P} (H), since the former is simply F_{Q}(K), and the latter is F_{Q}(Q).)

It is clear that always E is weakly normal in F′, and this is enough to deduce the following corollary to Theorem A.

Corollary 9.5 Let F be a saturated fusion system on a finite p-group P, and let E be a normal subsystem of F. If F′ is any saturated subsystem of F containing E, and O^{p′}(E) = E, then E ⊴ F′.

In [2, Section 7], Aschbacher gave examples of strongly F-closed subgroups on which there are no normal subsystems of F defined. In view of Theorem A, we therefore have the following corollary.

Corollary 9.6 If F is a saturated fusion system on a finite p-group P, and T is a strongly F-closed subgroup of P, then there need not be a weakly normal subsystem of F defined on T.

10 Comparing Weakly Normal and Normal Subsystems

In this short section we will give an overview of the differences between weakly normal and normal subsystems. We begin with the following easy lemma, needed to start to understand the direct products of fusion systems. (For a definition of the direct product of two fusion systems, see [6, Section 1].)

Lemma 10.1 Let F be a saturated fusion system on the finite p-group P, and suppose that P is the direct product of P_{1} and P_{2}, two strongly F-closed subgroups of P. Write E_{i} for the full subcategory of F on P_{i}. We have that E_{i} is a weakly normal subsystem of F, that F ⩽ E_{1} × E_{2}, and that F = E_{1} × E_{2} if and only if both E_{1} and E_{2} are normal in F. This last condition holds if and only if at least one of the E_{i} is normal in F.

Proof: Let E_{i} be defined as above; obviously it suffices to prove that E_{1} is weakly normal in F to prove that both E_{i} are, so let A(U) = Aut_{F}(U) for U ⩽ T = P_{1}. We prove that A(−) is a weakly normal map: clearly (i) and (ii) are satisfied, and (iii) is satisfied since Aut_{T} (T) = Aut_{P} (T). If U is fully F-normalized and ϕ ∈ Aut_{F}(U), then ϕ extends to an automorphism ϕ̄ of U C_{P} (U) = U C_{T} (U) × P_{2} in F; as T is strongly F-closed, ϕ̄ restricts to an automorphism in Aut_{F}(U C_{T} (U)), proving (iv).

To prove the fifth property, let U be fully F-normalized and U ⩽ V ⩽ N_{T} (U), and let ϕ ∈ N_{Aut_{F} (U)}(Aut_{V} (U)). Clearly ϕ extends to ϕ̄ in Aut_{F}(U × P_{2}) acting trivially on P_{2}, and since V acts trivially on P_{2}, we see that Aut_{V ×P_{2}}(U × P_{2}) is normalized by ϕ̄. Thus we get an extension ψ ∈ Aut_{F}(V × P_{2}) of ϕ̄, which restricts to an F-automorphism of V extending ϕ, completing the proof of (v). Thus the E_{i} are weakly normal in F.

If both E_{i} are normal in F then E_{1} × E_{2} is a normal subsystem of F by [3, Theorem 3], and by the definition of the E_{i}, we see that F = E_{1} × E_{2}. If F = E_{1} × E_{2} on the other hand, it is easy to see that every automorphism of each P_{i} extends to an automorphism of P acting trivially on P_{3−i}, so this equivalence is proved. Finally, suppose that E_{1} is normal in F, and let ϕ_{2} ∈ Aut_{F}(P_{2}). Since F is saturated, it extends to some automorphism ϕ = (ϕ_{1}, ϕ_{2}) on P_{1} × P_{2}, where this notation means that ϕ_{i} ∈ Aut(P_{i}). Since E_{1} ⊴ F, the automorphism (ϕ_{1}, 1) lies in F, and so (1, ϕ_{2}) also lies in F, proving that E_{2} ⊴ F.

Hence if one wants direct products to work, one needs normal subsystems, and not merely weakly normal subsystems. Another deficiency in the definition of weakly normal subsystems is to do with p-power extensions of a fusion system. Let E be a saturated fusion system on a finite p-group T, and suppose that P is a p-group containing a normal subgroup isomorphic to T. (We will identify T with this subgroup.) Suppose that all elements of Aut_{P} (T) induce automorphisms of E. In [5, Section 4.2], it was shown how to construct a saturated fusion system F′ on P containing E, and such that T is strongly F′-closed and F′/T = F_{P/T} (P/T). In other words, this is something like an extension of E by P/T.

In [3, Theorem 5], Aschbacher went further, and showed that if F is any saturated fusion system on P containing E as a normal subsystem then F′ ⩽ F. Since the only requirement for the construction of F′ is that Aut_{P} (T) ⩽ Aut(E) (in other words, something satisfied by weak normality), one might try to relax this condition to E being weakly normal in F.

Proposition 10.2 Let F be a saturated fusion system on a finite p-group P, and let E be a weakly normal subsystem of F, on the subgroup T of P. Let F′ be the saturated fusion system on P containing E such that F′/T = F_{P/T} (P/T). We have that F′ ⩽ F if and only if E is normal in F.

Proof: One direction of this proof is the result above, so assume that F′ ⩽ F. Let ϕ ∈ Aut_{E}(T) be a p′-automorphism; since F′ is saturated, this extends to ϕ̄ ∈ Aut_{F′}(T C_{P} (T)) which we may choose to be a p′-automorphism, and since F′/T = F_{P/T} (P/T), ϕ̄ must act as an inner automorphism on P/T, which must be trivial since ϕ̄ is a p′-automorphism. Hence [ϕ̄, C_{P} (T)] ⩽ Z (T). If ϕ is a p-automorphism of T, then ϕ = c_{g} for some g ∈ T, so that it obviously extends to ϕ̄ = c_{g} ∈ Aut_{F}(T C_{P} (T)) with [ϕ̄, C_{P} (T)] ∈ Z (T). Hence E ⊴ F as required.

This gives two different situations in which the notion of a normal subsystem is better than the notion of a weakly normal subsystem. The chief advantage of weakly normal subsystems is that they are easier to work with in some situations, and one may use Theorem A to pass between normal and weakly normal subsystems with relative ease.

Acknowledgement: I would like to thank Bob Oliver for suggesting an idea that allowed me to complete the proof of Theorem D. I would also like to thank Adam Glesser for listening to several iterations of the proof of Theorem A until the correct version was found.

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