CENTRALIZERS OF SUBSYSTEMS OF FUSION SYSTEMS

J. SEMERARO

Abstract. When (S, F, L) is a p-local finite group and (T, E, L_{0}) is weakly normal in (S, F, L) we show that a definition of C_{S}(E) given by Aschbacher has a simple interpretation from which one can deduce existence and strong closure very easily. We also appeal to a result of Gross to give a new proof that there is a unique fusion system C_{F} (E) on C_{S}(E).

1. Introduction

Let p be prime. By a normal pair ((S, F), (T, E)) of saturated fusion systems we will mean that F is a saturated fusion system on a finite p-group S and E is a normal subsystem of F on T ⩽ S. The centralizer fusion system C_{F}(E) was first considered by Aschbacher in [4, Chapter 6] and is a fusion system of index a power of p in C_{F}(T) on C_{S}(E), the largest subgroup X of C_{S}(T) with the property that E ⊆ C_{F}(X). While this definition is natural and easy-to-state, the proof of uniqueness is rather technical and relies on several of the tools developed in [3] and earlier chapters of [4]. The purpose of this note is twofold. Firstly, we show that C_{S}(E) may be regarded as the centralizer of a linking system of E, where existence, uniqueness and strong closure follow almost trivially. Secondly, we apply Aschbacher’s characterisation of C_{S}(E) as an intersection of local subsystem centralizers in order to give a new proof of the existence of C_{F}(E).

1.1. Main results and structure of the paper. Recall that a p-local finite group is a triple (S, F, L) where S is a finite p-group, F is a saturated fusion system on S and L is a linking system associated to F (see [5, Definition III.4.4]). When ((S, F), (T, E)) is a weakly normal pair of fusion systems, and L_{0} ⊆ L are the associated linking systems, L_{0} is weakly normal in L if a certain pair of conditions hold, mirroring the weak normality conditions for fusion systems [5, Definition III.4.12]. This definition naturally gives rise to the concept of a weakly normal pair ((S, F, L), (T, E, L_{0})) of p-local finite groups. We prove the following result in § 3, which may be regarded as a generalization of [1, Lemma 1.14]:

Theorem A. Let ((S, F, L), (T, E, L_{0})) be a weakly normal pair of p-local finite groups. Then C_{S}(E) fits into an exact sequence

1 ⟶ C_{S}(E) ⟶ Aut_{L}(T) ⟶ Aut_{typ}^{I}(L_{0}) ⟶
Out_{typ}^{L}(L_{0}) ⟶ 1.

In particular, C_{S}(E) is a well-defined strongly F-closed subgroup of S.

The definitions of Aut_{L}(T), Aut_{typ}^{I}(L_{0}) and Out_{typ}^{L}(L_{0}) are all provided or referenced in § 3. A remarkable feature of Theorem A is that, in contrast to the corresponding result of [4, Chapter 6] for fusion systems, the proof is very short and follows quickly from the definitions.

Our next main result, presented in § 4, is a new proof of the existence of the fusion system centralizer C_{F}(E). Recall that that the hyperfocal subgroup, 𝔥𝔶𝔭(F) of a saturated fusion system F has the property that whenever 𝔥𝔶𝔭(F) ⩽ R ⩽ S, there is a unique saturated subsystem F_{R} of F on R contained in F at index a power of p (see [6, Theorem 4.3]). We use this fact to prove the following:

Theorem B. Let ((S, F), (T, E)) be a normal pair of saturated fusion systems. Then

𝔥𝔶𝔭(C_{F}(T)) ⩽ C_{S}(E) ⩽ C_{S}(T).

In particular, there is a unique saturated subsystem C_{F}(E) on C_{S}(E) contained in C_{F}(T) at index a power of p.

For Theorem B, we appeal to the characterisation of C_{S}(E) as an intersection of subsystem centralizers and combine this with an observation of Gross in [10].

In § 5, we discuss some topics for future work, focussing mainly on a conjectured generalisation of Glauberman’s celebrated Z^{∗}-theorem.

2. Background

2.1. Group Theory. Standard notation and terminology from finite group theory which can be found in any good text on the subject (e.g. [2]) will be adopted. We collect some particularly important notions for future reference. Recall that for a group G, O^{p}(G) (respectively O^{p′}(G)) is the smallest normal subgroup H of G with G/H a p- (respectively p′-) group. Similarly, O_{p}(G) (respectively O_{p′}(G)) is the largest normal p- (respectively p′-) subgroup of G. We say that G has a normal p-complement if O^{p}(G) = O_{p′}(G). A subgroup H of G is G-centric if C_{G}(H) ⩽ H. When O_{p′}(G) = 1, G is p-constrained if there exists a normal G-centric p-subgroup of G.

2.2. Fusion systems. We assume the reader is familiar with the basic definitions and terminology associated with fusion systems which can be found in either of the two recent texts on this subject [5, 8]. In particular, we assume a working knowledge of saturation and Alperin’s fusion theorem, together with definitions and results concerning ‘important’ subsystems such as F-normalizers and F-centralizers of subgroups, and various examples of normal subsystems.

2.3. Linking systems. For an introduction to the theory of abstract linking systems, including a definition and some of its basic consequences, we refer the reader to [5, III.4.1]. An alternative viewpoint (which avoids the categorical definition) is presented in [7, § 2]. A proof that these two viewpoints are equivalent can be found in [7, Appendix A].

3. Centralizers of Linking Systems

In this section, we shall adopt the notation and terminology of [5, § III.4]. Let F be a saturated fusion system on a finite p-group S. For the definition of a linking system associated to F see [5, Definition 4.1]. If ((S, F), (T, E)) is a weakly normal pair of fusion systems and L_{0} and L are linking systems associated to E and F respectively, then [5, Definition 4.12] explains what it means for L_{0} to be weakly normal in L. We begin by describing how L ‘acts’ on L_{0} by conjugation.

Recall the definitions of isotypical, Aut_{typ}^{I}(L) and Out_{typ}(L) given in [5, § III.4.3]. We construct a “conjugation” map

(3.1)  Aut_{L}(T) ⟶^{γ↦c_{γ}} Aut_{typ}^{I}(L_{0})

as follows: For each γ ∈ Aut_{L}(T), let c_{γ} = c_{γ}|_{L_{0}} be the functor from L_{0} to itself defined by setting (P)c_{γ} := P γ := P π(γ) for P ∈ Ob(L_{0}) and (ψ)c_{γ} := (γ|_{P,P γ})^{−1} ∘ ψ ∘ (γ|_{Q,Qγ}) ∈ Hom_{L_{0}}(P γ, Qγ) for each ψ ∈ Hom_{L_{0}}(P, Q).

Note that both of the functors π and δ act on the left (as is conventional), while we let c_{γ} act on the right so that (3.1) defines a group homomorphism. Note also that the above construction makes implicit use of the fact that L_{0} is weakly normal in L. We make the following observation:

Lemma 3.1. Let ((S, F, L), (T, E, L_{0})) be a weakly normal pair of p-local finite groups. Then c_{γ}|_{L_{0}} ∈ Aut_{typ}^{I}(L_{0}) for each γ ∈ Aut_{L}(T).

Proof. By [5, Proposition 4.3 (a)] applied to L, ι_{P}^{Q} ∘ γ|_{Q,Qγ} = γ|_{P,P γ} ∘ ι_{P γ}^{Q} for each P, Q ∈ Ob(L_{0}) so that c_{γ} clearly sends inclusions to inclusions. To see that c_{γ} is isotypical, note that for each P ∈ Ob(L_{0}) and g ∈ P,

δ_{P} (g) ∘ γ|_{P,P γ} = γ|_{P,P γ} ∘ δ_{P γ}(gπ(γ))

by Axiom (C) for L ([5, Definition 4.1]). Thus (δ_{P} (g))c_{γ} = δ_{P γ}(gπ(γ)), as needed. □

Next, we generalise the set Out_{typ}(L_{0}) to define an object dependent on a weakly normal pair of p-local finite groups:

Definition 3.2. Let ((S, F, L), (T, E, L_{0})) be a weakly normal pair of p-local finite groups. Two isotypical equivalences α, β are L-naturally isomorphic if for each R ∈ Ob(L_{0}), there exists η_{R} ∈ Hom_{L}(α(R), β(R)) such that for all φ ∈ Hom_{L_{0}}(R, R′), the diagram

α(R) ⟶^{η_{R}} β(R)
↓^{α(φ)}  ↓^{β(φ)}
α(R′) ⟶^{η_{R′}} β(R′)

commutes. Let Out_{typ}^{L}(L_{0}) denote the set of equivalence classes of Aut_{typ}(L_{0}) under L-natural isomorphism.

Notice that L-natural isomorphism is a coarser equivalence relation than that of natural isomorphism. Thus one has a natural map

Out_{typ}(L_{0}) ⟶ Out_{typ}^{L}(L_{0}).

We will need one more definition before we can state and prove Theorem A.

Definition 3.3. Let ((S, F), (T, E)) be a weakly normal pair of fusion systems. Then

C_{S}(E) := {g ∈ S | E ⊆ C_{F}(⟨g⟩)}.

We now prove Theorem A, which may be regarded as a generalisation of [1, Lemma 1.14(a)].

Theorem 3.4. Let ((S, F, L), (T, E, L_{0})) be a weakly normal pair of p-local finite groups. The sequence

1 ⟶ C_{S}(E) ⟶ Aut_{L}(T) ⟶ Aut_{typ}^{I}(L_{0}) ⟶
Out_{typ}^{L}(L_{0}) ⟶ 1.

is exact. In particular, C_{S}(E) is a strongly F-closed subgroup of S.

Proof. By [5, Lemma III.4.9], Aut_{typ}^{I}(L_{0}) is a group and so C_{S}(E) is a group provided the sequence is exact. First we show that the natural homomorphism from Aut_{typ}^{I}(L_{0}) to Out_{typ}^{L}(L_{0}) is onto. This amounts to showing that each α ∈ Aut_{typ}(L_{0}) is L-naturally isomorphic to an isotypical equivalence which sends inclusions to inclusions. But [5, Lemma III.4.9] implies that any such α is L_{0}-naturally isomorphic to an isotypical equivalence, establishing the claim.

Now suppose that α ∈ Aut_{typ}^{I}(L_{0}) is in the kernel of the natural map from Aut_{typ}^{I}(L_{0}) to Out_{typ}^{L}(L_{0}). Then α is L-naturally isomorphic to the identity functor and so for each P ∈ Ob(L_{0}), there are η_{P} ∈ Iso_{L}(P, α(P)), and for each ψ ∈ Hom_{L_{0}}(P, Q),

η_{P} ∘ α(ψ) = ψ ∘ η_{Q}.

Since α is isotypical, α(ι_{P}^{T}) = ι_{α(P)}^{T} so that η_{P} = η_{T} |_{P,α(P)}, and this shows that α is given by conjugation by η_{T} ∈ Aut_{L}(T). Conversely, if γ ∈ Aut_{L}(T) then c_{γ} is L-naturally isomorphic to the identity functor Id_{L_{0}} via the set of maps {γ|_{P,P π(γ)}}_{P ∈Ob(L_{0})}.

Next we show that for each γ ∈ Aut_{L}(T), c_{γ} = Id_{L_{0}} if and only if γ ∈ δ_{S}(C_{S}(E)). If c_{γ} = Id_{L_{0}} then since γ^{−1}∘δ_{T} (g)∘γ = δ_{T} (g) for all g ∈ S, we must have π(γ) = Id_{T} by Axiom (C) for L ([5, Definition 4.1]) and the fact that δ_{T} is injective. Hence by Axiom (A2) for L, there is some a ∈ C_{S}(T) such that γ = δ_{T} (a). Now, for each P, Q ∈ Ob(L_{0}) and ψ ∈ Mor_{L_{0}}(P, Q), ψ ∘ δ_{Q}(a) = δ_{P} (a) ∘ ψ so that by [5, Proposition 4.3(b)], there is some \overline{ψ} ∈ Mor_{L}(⟨P, a⟩, ⟨Q, a⟩) such that \overline{ψ}|_{P,Q} = ψ (note that a ∈ C_{S}(T) ⩽ C_{S}(P)). By Axiom (C) for L again and the injectivity of δ, π(\overline{ψ})(a) = a. Hence each morphism in E extends to one which fixes a and a ∈ C_{S}(E). Conversely if γ = δ_{S}(a) for some a ∈ C_{S}(E), then π(γ) = c_{a} is the identity on Ob(L_{0}) and each ψ ∈ Mor(L_{0}) extends to some \overline{ψ} with (a)π(\overline{ψ}) = a. By Axiom (C) for L, \overline{ψ} commutes with δ_{S}(a) and we have c_{γ}(ψ) = ψ. Hence c_{γ} = Id_{L_{0}}, as required.

It remains to prove that C_{S}(E) is strongly F-closed. Let a ∈ C_{S}(E) and choose any subgroup P and morphism φ ∈ Hom_{F}(P, S) with a ∈ P. Let ψ ∈ Hom_{L}(P, S) be such that π(ψ) = φ and note that by Axiom (C) for L, we have ψ^{−1} ∘δ_{P} (a)∘ψ = δ_{S}(aφ). This implies that c_{δ_{S}(aφ)} = c_{ψ}^{−1} ∘ Id_{L_{0}} ∘ c_{ψ} = Id_{L_{0}} (since c_{δ_{P} (a)} = Id_{L_{0}}) and aφ ∈ C_{S}(E) as needed. □

We end this section with an example which shows that Theorem A does not hold for weakly normal pairs of fusion systems:

Example 3.5. Let H := H_{1} × H_{2} × H_{3} with H_{i} ≅ A_{4} for 1 ⩽ i ⩽ 3 be a direct product of three copies of the alternating group on 4 letters. For 1 ⩽ i ⩽ 3 let S_{i} ∈ Syl_{2}(H_{i}) so that S := S_{1} × S_{2} × S_{3} ∈ Syl_{2}(H) and let X_{i} = ⟨x_{i}⟩ be a group of order 3 which acts on S_{i} in such a way that H_{i} = S_{i} ⋊ X_{i} for each i. Set X := ⟨x_{1}x_{2}, x_{1}x_{3}⟩ ≅ C_{3} × C_{3}, G := SX and define

F := F_{S}(G), F_{1} := F_{S_{1}S_{2}}(⟨S_{1}, S_{2}, x_{1}x_{2}⟩) and F_{2} := F_{S_{1}S_{3}}(⟨S_{1}, S_{3}, x_{1}x_{3}⟩).

Clearly F_{i} is normal in F for i = 1, 2 and hence the fusion system E := F_{1}∩F_{2} on S_{1}, whose morphisms consist of those in both F_{1} and F_{2}, is an F-invariant subsystem on S_{1} (intersections of F-invariant subsystems are F-invariant). Furthermore E = F_{S_{1}}(H_{1}) so E is saturated and hence weakly normal. However, E is not normal in F. To see this, let φ ∈ Aut_{E}(S_{1}) be the map induced by conjugation by x_{1} so that φ extends to maps φ_{1} = c_{x_{1}x_{2}} and φ_{2} = c_{x_{1}x_{3}} ∈ Aut_{F}(S). Observe that φ_{i} does not act trivially on C_{S}(S_{1})/Z(S_{1}) = S/S_{1} = S_{2} × S_{3} for i = 1, 2, proving the claim.

Now, for i = 2, 3, E ⊆ C_{F}(S_{i}) = F_{4−i} × F_{S_{i}}(S_{i}), but E is not contained in C_{F}(S_{2} × S_{3}) = F_{S}(S). Hence there is no unique maximal subgroup of S which centralizes E, and hence no weakly normal pair of linking systems associated to (F, E) by Theorem A.

4. Centralizers of Fusion Systems

In order to introduce Aschbacher’s local characterisation of C_{S}(E), we first introduce local subsystems for fusion systems. In this section, ((S, F), (T, E)) is always assumed to be a normal pair of saturated fusion systems.

Definition 4.1. Let U be a fully F-normalized T -centric subgroup of T, and let X ⩽ C_{S}(T). Define U_{X} := UXC_{S}(UX), and let D(U, X) := N_{F}^{K}(U_{X}), where

K := {φ ∈ Aut_{F}(U_{X}) | φ|_{UX} ∈ Aut_{F}(UX)}.

The reader will readily check that D(U, 1) is exactly the fusion system D(U) described in [8, Definition 8.37]. The next result, which is a summary of [4, Lemma 6.3], may be regarded as a generalisation of [8, Lemma 8.38], at least in the case where U is T -centric.

Lemma 4.2. Let U be a fully F-normalized T -centric subgroup of T, and let X ⩽ C_{S}(T). If UX is fully N_{F}(U)-normalized then the following hold:

(a) ((N_{S}(UX), D(U, X)), (N_{T} (U), N_{E}(U))) is a normal pair of constrained fusion systems realised by a normal pair (G(U, X), H(U, X)) of finite groups.

(b) H(U) := H(U, 1) is a subgroup of G(U, X) and

C_{S}(H(U)) = C_{S}(H(U, X)).

Proof. (a) is [4, Lemma 6.3.3], while (b) is [4, Lemma 6.3.5]. □

It turns out that one can formulate C_{S}(E) as an intersection of centralizers of the H(U), as the next result describes:

Theorem 4.3. Let U be the set of all fully F-normalized, T -centric subgroups of S. Then

C_{S}(E) = ⋂_{φ∈Aut_{F} (T C_{S}(T))} ( ⋂_{U∈U} C_{S}(H(U))) φ.

Furthermore, C_{S}(E) is a strongly F-closed subgroup of S.

Proof. This is shown in [4, Chapter 6]. □

4.1. Existence of C_{F}(E). A definition of C_{F}(E), due to Aschbacher, has already appeared in Chapter 6 of [4], and the results in this section rely on some of the machinery developed there. Our approach does not however use the theory of normal maps (see [3, § 7]). Instead we rely on a group theoretic result, which is a corollary to the following theorem of Gross in [10]:

Theorem 4.4. Let G be a p-constrained finite group with Sylow p-subgroup S. Assume that O_{p′}(G) = 1 and write

C := C_{Aut(G)}(S) = {φ ∈ Aut(G) | φ|_{S} = Id_{S}}.

Then C has a normal p-complement.

Proof. This follows from (1) and (2) in [10, § 5]: if G is a p-constrained, minimal counterexample to the lemma, (1) and (2) imply that O_{p}(G) = 1, where the minimality of G is applied with the respect to the groups Z/Z(G) and C_{G}(M)/M with M = Z(O_{p}(G)). This supplies the required contradiction. (Note that the assumption p > 2 is not used anywhere in either of these two steps.) □

Corollary 4.5. Let T ⩽ S be finite p-groups and (G, H) be a normal pair of p-constrained finite groups with Sylow p-subgroup (S, T). If O_{p′}(G) = 1 then

O^{p}(C_{G}(T)) ∩ S ⩽ C_{G}(H).

Proof. Let C_{Aut_{G}(H)}(T) be the image of C_{G}(T) under the natural map

Φ : C_{G}(T) ⟶ Aut_{G}(H).

Theorem 4.4 implies that C_{Aut_{G}(H)}(T) has a normal p-complement. In particular, O^{p}(C_{Aut_{G}(H)}(T)) is a p′-group. Since

(O^{p}(C_{G}(T))∩S)C_{G}(H)/C_{G}(H) ⩽ O^{p}(C_{G}(T))C_{G}(H)/C_{G}(H) ⩽ O^{p}(C_{G}(T)/C_{G}(H)),

and this latter group is a p′-group, we must have O^{p}(C_{G}(T)) ∩ S ⩽ C_{G}(H), as needed. □

Recall the definition of the hyperfocal subgroup of a saturated fusion system:

Definition 4.6. For any saturated fusion system F over a finite p-group S, the hyperfocal subgroup 𝔥𝔶𝔭(F) of F is given by:

𝔥𝔶𝔭(F) := ⟨g^{−1} · gα | g ∈ P ⩽ S, α ∈ O^{p}(Aut_{F}(P))⟩

As an immediate consequence of the hyperfocal subgroup theorem ([6, Theorem 4.3]) and [3, Theorem 1] we obtain:

𝔥𝔶𝔭(C_{F}(T)) ⩽ C_{S}(E)

whenever ((S, F), (T, E)) is a pair of constrained fusion systems. In particular, there is a saturated fusion system C_{F}(E) on C_{S}(E) contained in C_{F}(T) by [5, Theorem I.7.4] in this case. Our goal is to prove this in the case where E and F are not constrained. We will achieve this by combining Corollary 4.5 with Theorem 4.3. First, we need a definition:

Definition 4.7. For each X ⩽ C_{S}(T), a chain of subgroups

C(U) := U = U_{0} < U_{1} < ⋯ < U_{n} = T

is strongly (F, X)-normalized if U_{i+1} = N_{T} (U_{i}), U_{i} is fully F-normalized and U_{i}X is fully N_{F}(U_{i})-normalized for each 0 ⩽ i < n.

Observe that strongly (F, 1)-normalized is exactly what Aschbacher calls strongly F-normalized in [4, Chapter 1]. We thus seek a generalisation of [4, 1.1.1]. In other words, under suitable conditions, we would like to assert that each U ⩽ T is Fconjugate to a subgroup W which affords a strongly (F, X)-normalized chain C(W). For this, we need another lemma of Aschbacher which says (among other things) that local subsystems behave well under taking normalizers. Write U for the set of all fully F-normalized T -centric subgroups of T.

Lemma 4.8. Let X ⩽ C_{S}(T) and U ∈ U be such that UX is fully N_{F}(U)- normalized, and define Q := N_{T} (U). There exists α ∈ Hom_{F}(N_{S}(Q), S) with the following properties:

(a) Uα, Qα ∈ U, (XU)α is fully N_{F}(Uα)-normalized and (XQ)α is fully N_{F}(Qα)-normalized;

(b) α extends to a map α̌ : G(U, X) → G(Uα, Xα) with the property that

N_{G(Qα,Xα)}(Uα) = N_{G(U,X)}(Q)α̌ and N_{H(Qα,Xα)}(Uα) = N_{H(U,X)}(Q)α̌;

Proof. This follows from parts (6), (7) and (8) of [4, Lemma 6.6.3], where we observe that the condition X ⊆ C_{S}(E) is never used. □

Following [4, Notation 6.8], for each U ∈ U and X ⩽ C_{S}(T) for which UX is fully N_{F}(U)-normalized, define:

C(U, X) := C_{G(U,X)}(N_{T} (U)) and K(U, X) := O^{p}(C(U, X)).

We next present a slight simplification of [4, 6.10], which shows that we can pass p-local information between elements of U.

Lemma 4.9. Let X ⩽ C_{S}(T) and U ∈ U be such that

C(U) := U = U_{0} < U_{1} < ⋯ < U_{n} = T

is a strongly (F, X)-normalized chain. For each 0 ⩽ j < n, there exists an isomorphism

θ_{j} : N_{G(U_{j},X)}(U_{j+1}) ⟶ N_{G(U_{j+1},X)}(U_{j}),

with

(4.1)  C(U_{j}, X)θ_{j} ⋯ θ_{n} = C(T, X) and K(U_{j}, X)θ_{j} ⋯ θ_{n} = K(T, X).

Proof. For each 0 ⩽ j ⩽ n − 1, the isomorphism θ_{j} is the isomorphism α̌ of Lemma 4.8 (b) applied with U = U_{j} and Q = U_{j+1}. It remains to prove that (4.1) holds. Writing

G_{j} := G(U_{j}, X), H_{j} := H(U_{j}, X), C_{j} := C(U_{j}, X) and K_{j} := K(U_{j}, X),

we have

[C_{j}θ_{j}, U_{j+1}] ⩽ [C_{j}θ_{j}, H_{j+1}] ⩽ C_{H_{j+1}}(U_{j+1}) = Z(U_{j+1}),

so that C_{j}θ_{j} ⩽ C_{j+1}. Since, K_{j}θ_{j} ⩽ C_{G_{j}}(U_{j+1})θ_{j} also K_{j}θ_{j} ⩽ K_{j+1}. Conversely C_{j+1}θ_{j}^{−1} and K_{j+1}θ_{j}^{−1} both centralize U_{j+1} which implies that C_{j}θ_{j} = C_{j+1} and K_{j}θ_{j} = K_{j+1}. This completes the proof. □

Corollary 4.10. Let X ⩽ C_{S}(T) and U ∈ U be such that XU is fully N_{F}(U)- normalized. Then there exists α ∈ Hom_{F}(U, S) such that C(Uα) is strongly (F, Xα)- normalized.

Proof. Assuming the lemma is false, let U be a counterexample with n := |T : U| as small as possible and write Q := N_{T} (U). Since T is fully F-normalized and XT is fully N_{F}(T)-normalized by assumption when n = 1, we see that n > 1.

By Lemma 4.8 (a), there is some α ∈ Hom_{F}(N_{S}(Q), S) such that Uα, Qα ∈ U and (XU)α and (XQ)α are fully N_{F}(Uα)- and N_{F}(Qα)-normalized respectively. Furthermore, by (4.1) in Lemma 4.9 above,

Xα ⩽ C_{S}(T)α ⩽ C_{S}(Q)α ⩽ C_{S}(Qα) = C_{S}(T).

By induction C(Qα) is strongly (F, Xα)-normalized, and hence so is C(Uα), contradicting the minimal choice of n. □

We have the following result concerning the hyperfocal subgroup of C_{F}(T):

Proposition 4.11. For each U ∈ U, 𝔥𝔶𝔭(C_{F}(T)) ⩽ C_{N_{S}(U)}(H(U)).

Proof. Let X ⩽ C_{S}(T) and β ∈ O^{p}(Aut_{C_{F} (T)}(X)). It suffices to show that [X, β] ⩽ C_{N_{S}(U)}(H(U)). By Corollary 4.10 there exists α ∈ Hom_{F}(N_{S}(Q), S) such that C(Uα) is strongly (F, Xα)-normalized with Q := N_{T} (U). If [Xα, β^{α}] ⩽ C_{N_{S}(Uα)}(H(Uα)) then

[X, β] = [Xα, β^{α}]α̌^{−1} ⩽ C_{N_{S}(Uα)}(H(Uα))α̌^{−1} = C_{N_{S}(U)}(H(U)).

Thus, on replacing (U, X) by (Uα, Xα) if necessary, we may assume that C(U) is strongly (F, X)-normalized. Suppose we have shown that

(4.2)  O^{p}(Aut_{C_{F} (T)}(X)) = Aut_{K(T,X)}(X).

Then, since X ⩽ N_{S}(U_{j}) for each 0 ⩽ j ⩽ n, Lemma 4.9 implies that Aut_{K(T,X)}(X) = Aut_{K(U,X)}(X) and

[X, β] ⩽ 𝔥𝔶𝔭(C_{D(U,X)}(N_{T} (U))).

Now, the hyperfocal subgroup theorem ([6, Theorem 4.3]) implies that this latter group is equal to O^{p}(C_{G(U,X)}(N_{T} (U)))∩N_{S}(UX), which is contained in C_{S}(H(U, X)) by Corollary 4.5. Since

C_{S}(H(U, X)) = C_{S}(H(U, 1)) = C_{S}(H(U))

by Lemma 4.2 (b), the result follows.

It thus remains to prove (4.2). Clearly Aut_{K(T,X)}(X) ⩽ O^{p}(Aut_{C_{F} (T)}(X)) by definition. Conversely, each p′-automorphism α ∈ Aut_{C_{F} (T)}(X) extends to a morphism \overline{α} ∈ Aut_{F}(C_{S}(XT)XT) which fixes T (note that XT is fully N_{F}(T)- normalized), which we may assume is also of p′-order. Thus \overline{α} = c_{g} for some g ∈ K(T, X) and

O^{p}(Aut_{C_{F} (T)}(X)) ⩽ Aut_{K(T,X)}(X),

as required. □

Corollary 4.12. One has 𝔥𝔶𝔭(C_{F}(T)) ⩽ C_{S}(E).

Proof. This follows immediately from Theorem 4.3 and Proposition 4.11, provided 𝔥𝔶𝔭(C_{F}(T)) is left invariant by elements of Aut_{F}(T C_{S}(T)). But such elements clearly normalize O^{p}(Aut_{C_{F} (T)}(P)) for each P ⩽ C_{S}(T), as needed. □

Theorem 4.13. Let T ⩽ S be finite p-groups and ((S, F), (T, E)) be a normal pair of saturated fusion systems. There is a unique subsystem C_{F}(E) on C_{S}(E) contained in C_{F}(T) at index a power of p.

Proof. This follows from Corollary 4.12 and [5, Theorem I.7.4]. □

5. Open questions

The Z^{∗}-theorem of Glauberman ([9, Theorem 1]), which has now been shown to hold for all primes, has the following elementary statement for fusion systems:

Theorem 5.1. Let G be a finite group with S ∈ Syl_{p}(G) If O_{p′}(G) = 1 then Z(F_{S}(G)) = Z(G).

Currently (when p is odd) the proof relies on the classification of finite simple groups, although in recent years there has been some interest in finding a classification-free proof (see [11]). One could hope to generalise Theorem 5.1 as follows:

Conjecture 5.2. Let ((S, F), (T, E)) be a normal pair of fusion systems realised by a normal pair (G, H) of finite groups. If O_{p′}(H) = 1 then C_{S}(E) = C_{G}(H) ∩ S.

So far, all attempts to construct a counterexample to this conjecture have failed. For example, via elementary calculations, it is true when (G, H) = (S_{n}, A_{n}), n ⩾ 6 and p = 2. Pushing this slightly further, we can ask whether the following is true:

Conjecture 5.3. Let ((S, F), (T, E)) be a normal pair of fusion systems realised by a normal pair (G, H) of finite groups. If O_{p′}(H) = 1 then

C_{F}(E) = F_{C_{S}(H)}(C_{G}(H))

To prove Conjecture 5.3, it would be enough to show that F_{C_{S}(H)}(C_{G}(H)) is a fusion system of p-power index in C_{F}(T).

References

[1] Kasper Andersen, Bob Oliver, and Joana Ventura. Reduced, tame and exotic fusion systems. Proc. London Math. Soc., 105 (1):87–152, 2012.

[2] Michael Aschbacher. Finite Group Theory. Cambridge University Press, 1986.

[3] Michael Aschbacher. Normal subsystems of fusion systems. Proc. London Math. Soc., 97:239– 271, 2008.

[4] Michael Aschbacher. The generalized fitting subsystem of a fusion system. Mem. Amer. Math. Soc., 209, 2011.

[5] Michael Aschbacher, Radha Kessar, and Bob Oliver. Fusion Systems in Algebra and Topology. Cambridge University Press, 2011.

[6] C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver. Extensions of p-local finite groups. Trans. Amer. Math. Soc, 359:3791–3858, 2007.

[7] A. Chermak. Fusion systems and localities. Acta Math., 211:47–139, 2013.

[8] David A. Craven. The Theory of Fusion Systems. Cambridge studies in advanced mathematics; 131, 2011.

[9] G. Glauberman. Central elements in core-free groups. J. Algebra., 4:403–420, 1966.

[10] Fletcher Gross. Automorphisms which centralize a sylow p-subgroup. J. Alg., 77:202–233, 1982.

[11] R. Waldecker. Isolated involutions in finite groups, volume 226. Mem. Amer. Math. Soc, 2013.

Heilbronn Institute for Mathematical Research, Department of Mathematics, University of Bristol, U.K.

  E-mail address: js13525@bristol.ac.uk