REDUCED, TAME, AND EXOTIC FUSION SYSTEMS

KASPER ANDERSEN, BOB OLIVER, AND JOANA VENTURA

Abstract. We define here two new classes of saturated fusion systems, reduced fusion systems and tame fusion systems. These are motivated by our attempts to better understand and search for exotic fusion systems: fusion systems which are not the fusion systems of any finite group. Our main theorems say that every saturated fusion system reduces to a reduced fusion system which is tame only if the original one is realizable, and that every reduced fusion system which is not tame is the reduction of some exotic (nonrealizable) fusion system.

When G is a finite group and S ∈ Syl_{p}(G), the fusion category of G is the category F_{S}(G) whose objects consist of all subgroups of S, and where

Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q) =^{def} {c_{g} ∈ Hom(P, Q) | g ∈ G, gP g^{−1} ≤ Q} .

This provides a means of encoding the p-local structure of G: the conjugacy relations among the subgroups of the Sylow p-subgroup S. An abstract “saturated fusion system” over a finite p-group S is a category whose objects are the subgroups of S, whose morphisms are certain monomorphisms of groups between the subgroups, and which satisfies certain conditions formulated by Puig [Pg2] and stated here in Definition 1.1. In particular, for any finite G as above, F_{S}(G) is a saturated fusion system. A saturated fusion system is called realizable if it is isomorphic to the fusion system of some finite group G, and is called exotic otherwise.

It turns out to be very difficult in general to construct exotic fusion systems, especially over 2-groups. This says something about how close Puig’s definition is to the properties of fusion systems of finite groups.

This paper is centered around the problem of identifying exotic fusion systems. A first step towards doing this was taken in [OV2], where two of the authors developed methods for listing saturated fusion systems over any given 2-group. However, it quickly became clear that in order to have any chance of making a systematic search through all 2-groups (or p-groups) of a given type, one must first find a way to limit the types of fusion systems under consideration, and do so without missing any possible exotic ones.

This leads to the concept of a reduced fusion system. A saturated fusion system is reduced if it contains no nontrivial normal p-subgroups, and also contains no proper normal subsystems of p-power index or of index prime to p. These last concepts will be defined precisely in Definitions 1.2 and 1.21; for now, we just remark that they are analogous to requiring a finite group to have no nontrivial normal p-subgroups and no proper normal subgroups of p-power index or of index prime to p. Thus it is very far from requiring that the fusion system be simple in any sense, but it is adequate for our purposes.

The second concept which plays a central role in our results is that of a tame fusion system. Roughly, a fusion system F is tame if it is realized by a finite group G for which all automorphisms of F are induced by automorphisms of G. The precise (algebraic) definition is given in Definition 2.5. In terms of classifying spaces, F is tame if it is realized by a finite group G such that the natural map from Out(G) to Out(BG_{p}^{∧}) is split surjective, where Out(BG_{p}^{∧}) is the group of homotopy classes of self homotopy equivalences of the space BG_{p}^{∧}.

2000 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D20.

Key words and phrases. Classifying space, p-completion, finite groups, fusion.

B. Oliver is partially supported by UMR 7539 of the CNRS, and by project ANR BLAN08-2 338236, HGRT.

J. Ventura is partially supported by FCT/POCTI/FEDER and grant PDCT/MAT/58497/2004.

For any saturated fusion system F over a finite p-group S, there is a canonical reduction 𝔯𝔢𝔡(F) of F (Definition 2.1). The analogy for a finite group G with maximal normal psubgroup Q would be to set G_{0} = C_{G}(Q)/Q, and then let 𝔯𝔢𝔡(G) ⊴ G_{0} be the smallest normal subgroup such that G_{0}/𝔯𝔢𝔡(G) is p-solvable. Our first main theorem is the following.

Theorem A. For any saturated fusion system F over a finite p-group S, if 𝔯𝔢𝔡(F) is tame, then F is also tame, and in particular F is realizable.

Thus Theorem A says that reduced fusion systems detect all possible exotic fusion systems. If one wants to find all exotic fusion systems over p-groups of order ≤ p^{k} for some p and k, then one first searches for all reduced fusion systems over p-groups of order ≤ p^{k} which are not tame, and then for all other fusion systems which reduce to them.

The proof of Theorem A uses the uniqueness of linking systems associated to the fusion system of a finite group, and through that depends on the classification of finite simple groups. In order to make it clear exactly which part of the result depends on the classification theorem and which part is independent, we introduce another (more technical) concept, that of “strongly tame” fusion systems (Definition 2.9). Using the classification, together with results in [O1] and [O2], we prove that all tame fusion systems are strongly tame (Theorem 2.10). (In fact, the definition of “strongly tame” is such that any tame fusion system which we’re ever likely to be working with can be shown to be strongly tame without using the classification.) Independently of that, and without using the classification theorem, we prove in Theorem 2.20 that F is tame whenever 𝔯𝔢𝔡(F) is strongly tame; and this together with Theorem 2.10 imply Theorem A.

Alternatively, one can also avoid using the classification theorem by restating Theorem A in terms of fusion systems together with associated linking systems.

Albert Ruiz has constructed examples [Rz] which show that the reduction of a tame fusion system need not be tame, and in fact, can be exotic. So there is no equivalence between the tameness of F and tameness of 𝔯𝔢𝔡(F). The next theorem does, however, provide a weaker converse to Theorem A, by saying that for every non-tame reduced fusion system, there is some associated exotic fusion system in the background.

Theorem B. Let F be a reduced fusion system which is not tame. Then there is an exotic fusion system whose reduction is isomorphic to F.

As remarked above, reduced fusion systems can be very far from being simple in any sense. For example, a product of reduced fusion systems is always reduced (Proposition 3.4). The next theorem handles reduced fusion systems which factor as products.

Theorem C. Each reduced fusion system F over a finite p-group S has a unique maximal factorization F = F_{1} × ⋯ × F_{m} as a product of indecomposable fusion systems F_{i} over subgroups S_{i} ⊴ S. If F_{i} is tame for each i, then F is tame.

Here, “unique” means that the indecomposable subsystems are unique as subcategories, not only up to isomorphism. By Theorem C, in order to find minimal reduced fusion systems which are not tame, it suffices to look at those which are indecomposable. In practice, it seems that any reduced indecomposable fusion system which is not simple (which has no proper normal fusion subsystems in the sense of Definition 1.18 or of [Asch, § 6]) has to be

REDUCED, TAME, AND EXOTIC FUSION SYSTEMS 3

over a p-group of very large order. The smallest example of this type we know of is the fusion system of A_{6} ≀ A_{5}, over a group of order 2^{17}.

Using these results and those in [OV2] as starting point, we have started to undertake a systematic computer search for reduced fusion systems over small 2-groups. So far, while details have yet to be rechecked carefully, we seem to have shown that each reduced fusion system over a 2-group of order ≤ 512 is the fusion system of a finite simple group, and is tame. We hope to be able to extend this soon to 2-groups of larger order.

What we really would like to find is an example of a realizable fusion system which is not tame. It seems very likely that such a fusion system exists, but so far, our attempts to find one have been unsuccessful.

The theorems stated above will all be proven in Sections 2 and 3: Theorems A and B as Theorems 2.20 and 2.6, and Theorem C as Proposition 3.6 and Theorem 3.7. They are preceded by a first section containing mostly background definitions and results, and are followed by a fourth section with examples of how to prove certain fusion systems are tame.

All three authors would like to thank Copenhagen University for its hospitality, when letting us meet there for 2-week periods on two separate occasions. We would also like to thank Richard Lyons for his help with automorphisms of certain sporadic groups.

Contents

1. Fusion and linking systems 4 1.1. Background on fusion systems 4 1.2. Background on linking systems 7 1.3. Automorphisms of fusion and linking systems 9 1.4. Normal fusion subsystems 13 1.5. Normal linking subsystems 18

2. Reduced fusion systems and tame fusion systems 22 2.1. Reduced fusion systems and reductions of fusion systems 22 2.2. Tame fusion systems and the proof of Theorem B 24 2.3. Strongly tame fusion systems and linking systems for extensions 27 2.4. Proof of Theorem A 34

3. Decomposing reduced fusion systems as products 45 4. Examples 52

4.1. Dihedral and semidihedral 2-groups 56 4.2. Tameness of some fusion systems studied in [OV2] 58 4.3. Alternating groups 61

References 68

1. Fusion and linking systems

We first collect the basic results about fusion and linking systems and their automorphisms which will be needed in the rest of the paper. Most of this is taken directly from earlier papers, such as [BLO2], [BCGLO1], [BCGLO2] and [O3].

1.1. Background on fusion systems.

We first recall very briefly the definition of a saturated fusion system, in the form given in [BLO2]. In general, for any group G and any pair of subgroups H, K ≤ G, Hom_{G}(H, K) denotes the set of all homomorphisms from H to K induced by conjugation by some element of G. When G is finite and S ∈ Syl_{p}(G), F_{S}(G) (the fusion category of G) is the category whose objects are the subgroups of S, and where for each pair of objects Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q).

A fusion system over a finite p-group S is a category F, where Ob(F) is the set of all subgroups of S, such that for all P, Q ≤ S,

Hom_{S}(P, Q) ⊆ Hom_{F}(P, Q) ⊆ Inj(P, Q);

and each φ ∈ Hom_{F}(P, Q) is the composite of an isomorphism in F followed by an inclusion. Here, Inj(P, Q) denotes the set of injective homomorphisms from P to Q. If F is a fusion system over a finite p-subgroup S, then two subgroups P, Q ≤ S are F-conjugate if they are isomorphic as objects of the category F.

Definition 1.1 ([Pg2], see [BLO2, Definition 1.2]). Let F be a fusion system over a finite p-group S.

• A subgroup P ≤ S is fully centralized in F if |C_{S}(P)| ≥ |C_{S}(P^{∗})| for each P^{∗} ≤ S which is F-conjugate to P.

• A subgroup P ≤ S is fully normalized in F if |N_{S}(P)| ≥ |N_{S}(P^{∗})| for each P^{∗} ≤ S which is F-conjugate to P.

• F is a saturated fusion system if the following two conditions hold:

(I) For each P ≤ S which is fully normalized in F, P is fully centralized in F and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

(II) If P ≤ S and φ ∈ Hom_{F}(P, S) are such that φ(P) is fully centralized in F, and if we set

N_{φ} = {g ∈ N_{S}(P) | φc_{g}φ^{−1} ∈ Aut_{S}(φ(P))},

then there is φ ∈ Hom_{F}(N_{φ}, S) such that φ|_{P} = φ.

If G is a finite group and S ∈ Syl_{p}(G), then the category F_{S}(G) is a saturated fusion system (cf. [BLO2, Proposition 1.3]).

We now list some classes of subgroups of S which play an important role when working with fusion systems over S. Here and elsewhere, for any fusion system F over a finite p-group S, we write for each P ≤ S,

Out_{F}(P) = Aut_{F}(P)/Inn(P) ≤ Out(P) .

Definition 1.2. Fix a prime p, a finite p-group S, and a fusion system F over S. Let P ≤ S be any subgroup.

• P is F-centric if C_{S}(P^{∗}) = Z(P^{∗}) for each P^{∗} which is F-conjugate to P.

• P is F-radical if O_{p}(Out_{F}(P)) = 1; i.e., if Out_{F}(P) contains no nontrivial normal p-subgroups.

• P is central in F if P ⊴ S, and every morphism φ ∈ Hom_{F}(Q, R) in F extends to a morphism φ ∈ Hom_{F}(P Q, P R) such that φ|_{P} = Id_{P}.

• P is normal in F (P ⊴ F) if P ⊴ S, and every morphism φ ∈ Hom_{F}(Q, R) in F extends to a morphism φ ∈ Hom_{F}(P Q, P R) such that φ(P) = P.

• P is strongly closed in F if no element of P is F-conjugate to an element of S∖P.

• Z(F) ≤ Z(S) and O_{p}(F) ≤ S denote the largest subgroups of S which are central in F and normal in F, respectively.

It follows directly from the definitions that if P_{1} and P_{2} are both central (normal) in F, then so is P_{1}P_{2}. This is why there always is a largest central subgroup Z(F), and a largest normal subgroup O_{p}(F).

Several forms of Alperin’s fusion theorem have been shown for saturated fusion systems, starting with Puig in [Pg2, § 5]. The following version suffices for what we need in most of this paper. A stronger version will be given in Theorem 4.1.

Theorem 1.3 ([BLO2, Theorem A.10]). For any saturated fusion system F over a finite p-group S, each morphism in F is a composite of restrictions of automorphisms in Aut_{F}(P), for subgroups P which are fully normalized in F, F-centric, and F-radical.

The following elementary result is useful for identifying subgroups which are centric and radical in a fusion system.

Lemma 1.4. Let F be a saturated fusion system over a finite p-group S. If P ≤ S is F-centric and F-radical, then there is no g ∈ N_{S}(P)∖P such that c_{g} ∈ O_{p}(Aut_{F}(P)). Conversely, if P ≤ S is fully normalized in F, and there is no g ∈ N_{S}(P)∖P such that c_{g} ∈ O_{p}(Aut_{F}(P)), then P is F-centric and F-radical.

Proof. Assume P is F-centric and F-radical. Fix g ∈ N_{S}(P) such that c_{g} ∈ O_{p}(Aut_{F}(P)). Then O_{p}(Out_{F}(P)) = 1 since P is F-radical, so c_{g} ∈ Inn(P), and g ∈ P ·C_{S}(P) = P since P is F-centric. This proves the first statement.

Now assume P is fully normalized in F. If P is not F-centric, then C_{S}(P) ≰ P (since P is fully centralized), and hence there is g ∈ N_{S}(P)∖P with c_{g} = 1. If P is not F-radical, then O_{p}(Out_{F}(P)) ≠ 1. This subgroup is contained in each Sylow p-subgroup of Out_{F}(P), and in particular is contained in Out_{S}(P). Thus each nontrivial element of O_{p}(Out_{F}(P)) is induced by conjugation by some element of N_{S}(P)∖P. □

Proposition 1.5. Let F be a saturated fusion system over a finite p-group S. For any normal subgroup Q ⊴ S, Q is normal in F if and only if Q is strongly closed and contained in all subgroups which are centric and radical in F.

Proof. This is shown in [BCGLO1, Proposition 1.6]. Note, however, that wherever “Fradical” appears in the statement and proof of that proposition, it should be replaced by “F-centric and F-radical”. □

Lemma 1.4 shows the importance of being able to identify elements of the subgroup O_{p}(Aut_{F}(P)). The following, very well known property of automorphisms of p-groups is useful in many cases when doing this.

Lemma 1.6. Fix a prime p, a finite p-group P, and a group A ≤ Aut(P) of automorphisms of P. Assume 1 = P_{0} ⊴ P_{1} ⊴ ⋯ ⊴ P_{m} = P is a sequence of normal subgroups such that α(P_{i}) = P_{i} for all α ∈ A and all i. For 1 ≤ i ≤ m, let Ψ_{i}: A ⟶ Aut(P_{i}/P_{i−1}) be the homomorphism which sends α ∈ A to the induced automorphism of P_{i}/P_{i−1}. Then for all α ∈ A, α ∈ O_{p}(A) if and only if Ψ_{i}(α) ∈ O_{p}(Ψ_{i}(A)) for all i = 1, …, m.

Proof. Set Ψ = (Ψ_{1}, …, Ψ_{m}), as a homomorphism from A to ∏_{i=1}^{m} Aut(P_{i}/P_{i−1}). Then Ker(Ψ) is a p-group (cf. [G, Corollary 5.3.3]). If Ψ_{i}(α) ∈ O_{p}(Ψ_{i}(A)) for all i = 1, …, m, then Ψ(α) ∈ O_{p}(Ψ(A)), and so α ∈ O_{p}(A). Conversely, if α ∈ O_{p}(A), then clearly Ψ_{i}(α) ∈ O_{p}(Ψ_{i}(A)) for all i. □

Another elementary lemma which is frequently useful when working with centric and radical subgroups is the following:

Lemma 1.7. Let P and Q be p-subgroups of a finite group G such that P ≤ N_{G}(Q) and Q ≰ P. Then N_{QP} (P) ≩ P, and (Q ∩ N_{G}(P)) ≰ P.

Proof. Since P normalizes Q, QP is also a p-group, and QP ≩ P by assumption. Hence N_{QP} (P) ≩ P (cf. [Sz1, Theorem 2.1.6]). Since N_{QP} (P) = P ·(Q ∩ N_{QP} (P)), we have (Q ∩ N_{QP} (P)) ≰ P. □

We also need to work with certain quotient fusion systems. When F is a saturated fusion system over S and Q ⊴ S is strongly closed in F, we define the quotient fusion system F/Q over S/Q by setting

Hom_{F/Q}(P/Q, R/Q) = Im[Hom_{F}(P, R) ⟶ Hom(P/Q, R/Q)]

for all P, R ≤ S containing Q.

Proposition 1.8. Let F be a saturated fusion system over a finite p-group S, and let Q ⊴ S be a strongly closed subgroup. Then F/Q is a saturated fusion system. For each P ≤ S containing Q, P is fully normalized in F if and only if P/Q is fully normalized in F/Q. If Q is central in F, then P ⊴ F if and only if P/Q ⊴ F/Q.

Proof. By [O1, Lemma 2.6], F/Q is a saturated fusion system, and P is fully normalized if and only if P/Q is. So it remains only to prove the last statement.

By Proposition 1.5, P is normal in F if and only if it is strongly closed in F, and contained in each subgroup which is F-centric and F-radical. For P ≤ S containing Q, clearly P is strongly closed in F if and only if P/Q is strongly closed in F/Q.

We apply the criterion in Lemma 1.4 for detecting subgroups which are centric and radical. Let ρ: Aut_{F}(P) ⟶ Aut_{F/Q}(P/Q) be the homomorphism induced by projection. Then ρ is surjective by definition of F/Q. For α ∈ Aut_{F}(P), we have α|_{Q} = Id_{Q} since Q is central in F, so by Lemma 1.6, α ∈ O_{p}(Aut_{F}(P)) if and only if ρ(α) ∈ O_{p}(Aut_{F/Q}(P/Q)).

If Q ≤ R ≤ S, and R^{∗} is F-conjugate to R and fully normalized in F, then R^{∗}/Q is F/Q-conjugate to R/Q and fully normalized in F/Q. So by Lemma 1.4, R and R^{∗} are centric and radical in F if and only if R/Q and R^{∗}/Q are centric and radical in F/Q. Upon combining this with the above criterion for normality, we see that P ⊴ F if and only if P/Q ⊴ F/Q. □

1.2. Background on linking systems.

We next define abstract linking systems associated to a fusion system F. We use the definition given in [O3], which is more flexible in the choice of objects than the earlier definitions in [BLO2] and [BCGLO1]. This definition also differs slightly from the one given in [BCGLO1, Definition 3.3], in that we include a choice of inclusion morphisms as part of the data in the linking system. All of these definitions are, however, equivalent, aside from having greater freedom in the choice of objects.

For any finite group G and any S ∈ Syl_{p}(G), let T_{S}(G) denote the transporter category of G: the category whose objects are the subgroups of S, and where for all P, Q ≤ S,

Mor_{T_{S}(G)}(P, Q) = N_{G}(P, Q) =^{def} {g ∈ G | gP g^{−1} ≤ Q} .

If H is a set of subgroups of S, then T_{H}(G) ⊆ T_{S}(G) denotes the full subcategory with object set H.

Definition 1.9 ([O3, Definition 3]). Let F be a fusion system over a finite p-group S. A linking system associated to F is a finite category L, together with a pair of functors

T_{Ob(L)}(S) ⟶^{δ} L ⟶^{π} F ,

satisfying the following conditions:

(A) Ob(L) is a set of subgroups of S closed under F-conjugacy and overgroups, and includes all subgroups which are F-centric and F-radical. Each object in L is isomorphic (in L) to one which is fully centralized in F. Also, δ is the identity on objects, and π is the inclusion on objects. For each P, Q ∈ Ob(L) such that P is fully centralized in F, C_{S}(P) acts freely on Mor_{L}(P, Q) via δ_{P} and right composition, and π_{P,Q} induces a bijection

Mor_{L}(P, Q)/C_{S}(P) ⟶^{≅} Hom_{F}(P, Q) .

(B) For each P, Q ∈ Ob(L) and each g ∈ N_{S}(P, Q), π_{P,Q} sends δ_{P,Q}(g) ∈ Mor_{L}(P, Q) to c_{g} ∈ Hom_{F}(P, Q).

(C) For all ψ ∈ Mor_{L}(P, Q) and all g ∈ P, the diagram

\xymatrix{P \ar[r]^{ψ} \ar[d]_{δ_{P} (g)} & Q \ar[d]^{δ_{Q}(π(ψ)(g))} \\ P \ar[r]^{ψ} & Q}

commutes in L.

If L^{∗} is another linking system associated to F with the same set of objects as L, then an isomorphism of linking systems is an isomorphism of categories L ⟶^{≅} L^{∗} which commutes with the structural functors: those coming from T_{Ob(L)}(S) and those going to F.

A p-local finite group is defined to be a triple (S, F, L), where F is a saturated fusion system over a finite p-group S, and where L is a centric linking system associated to F (i.e., one whose objects are the F-centric subgroups of S).

For P ≤ Q in Ob(L), we usually write ι_{P}^{Q} = δ_{P,Q}(1), and regard this as the “inclusion” of P into Q. The definition in [BCGLO1, Definition 3.3] of a (quasicentric) linking system does not include these inclusions, but it is explained there how to choose inclusions in a way so that a functor δ: T_{Ob(L)}(S) ⟶ L can be defined in a unique way with the above properties ([BCGLO1, Lemma 3.7]). Note also that because the above definition includes a choice of inclusions, condition (D)_{q} in [BCGLO1, Definition 3.3] is not needed here.

We have defined linking systems to be as flexible as possible in the choice of objects, but one cannot avoid completely discussing quasicentric subgroups in this context.

Definition 1.10. (a) For any finite group G, a p-subgroup P ≤ G is G-quasicentric if O^{p}(C_{G}(P)) has order prime to p.

(b) For any fusion system F over a finite p-group S, a subgroup P ≤ S is F-quasicentric if for each P^{∗} which is fully centralized in F and F-conjugate to P, C_{F}(P^{∗}) is the fusion system of the p-group C_{S}(P^{∗}). Equivalently, for each Q ≤ P^{∗}·C_{S}(P^{∗}) containing P^{∗}, {α ∈ Aut_{F}(Q) | α|_{P^{∗}} = Id} is a p-group.

For any saturated fusion system F, the set of F-quasicentric subgroups is closed under F-conjugacy and overgroups. So a quasicentric linking system as defined in [BCGLO1, § 3] is a linking system in the sense defined here.

Fix a finite group G and S ∈ Syl_{p}(G), and set F = F_{S}(G). Then a subgroup P ≤ S is G-quasicentric if and only if it is F-quasicentric. For any set H of G-quasicentric subgroups of S, define L_{S}^{H}(G) to be the category with object set H, and where for each P, Q ∈ H,

Mor_{L_{S}^{H}(G)}(P, Q) = N_{G}(P, Q)/O^{p}(C_{G}(P)).

Composition is well defined, since for each g ∈ N_{G}(P, Q), g^{−1}Qg ≥ P, so g^{−1}C_{G}(Q)g ≤ C_{G}(P), and thus g^{−1}O^{p}(C_{G}(Q))g ≤ O^{p}(C_{G}(P)). When H is closed under F-conjugacy and overgroups and contains all subgroups of S which are F-centric and F-radical, then L_{S}^{H}(G) is a linking system associated to F. When H is the set of F-centric subgroups of S, we write L_{S}^{c}(G) = L_{S}^{H}(G).

Proposition 1.11. The following hold for any linking system L associated to a saturated fusion system F over a finite p-group S.

(a) For each P, Q ∈ Ob(L), the subgroup E(P) =^{def} Ker[Aut_{L}(P) ⟶ Aut_{F}(P)] acts freely on Mor_{L}(P, Q) via right composition, and π_{P,Q} induces a bijection

Mor_{L}(P, Q)/E(P) ⟶^{≅} Hom_{F}(P, Q) .

(b) For every morphism ψ ∈ Mor_{L}(P, Q), and every P_{0}, Q_{0} ∈ Ob(L) such that P_{0} ≤ P, Q_{0} ≤ Q, and π(ψ)(P_{0}) ≤ Q_{0}, there is a unique morphism ψ|_{P_{0},Q_{0}} ∈ Mor_{L}(P_{0}, Q_{0}) (the “restriction” of ψ) such that ψ ∘ ι_{P_{0}}^{P} = ι_{Q_{0}}^{Q} ∘ ψ|_{P_{0},Q_{0}}.

(b′) For each P, Q ∈ Ob(L) and each ψ ∈ Mor_{L}(P, Q), if we set Q_{0} = π(ψ)(P), then there is a unique ψ_{0} ∈ Iso_{L}(P, Q_{0}) such that ψ = ι_{Q_{0}}^{Q} ∘ ψ_{0}.

(c) The functor δ is injective on all morphism sets.

(d) If P ∈ Ob(L) is fully normalized in F, then δ_{P} (N_{S}(P)) ∈ Syl_{p}(Aut_{L}(P)).

(e) Let P, Q, P , Q ∈ Ob(L) and ψ ∈ Mor_{L}(P, Q) be such that P ⊴ P, Q ≤ Q, and for each g ∈ P there is h ∈ Q such that ι_{Q}^{Q} ∘ ψ ∘ δ_{P} (g) = δ_{Q,Q}(h) ∘ ψ. Then there is a unique

morphism ψ ∈ Mor_{L}(P , Q) such that ψ|_{P,Q} = ψ.

(f) All morphisms in L are monomorphisms and epimorphisms in the categorical sense.

(g) All objects in L are F-quasicentric.

Proof. See [O3, Proposition 4]. Point (b′) is a special case of (b), where ψ_{0} =^{def} ψ|_{P,Q_{0}} is an isomorphism by (a). □

We will also have use for the following “linking system version” of Alperin’s fusion theorem.

Theorem 1.12. For any saturated fusion system F over a finite p-group S and any linking system L associated to F, each morphism in L is a composite of restrictions of automorphisms in Aut_{L}(P), where P is fully normalized in F, F-centric, and F-radical.

Proof. Using Theorem 1.3 together with Proposition 1.11(a), we are reduced to proving the theorem for automorphisms in E(P) = Ker[Aut_{L}(P) ⟶ Aut_{F}(P)] for P ∈ Ob(L). If P is fully centralized, then E(P) = {δ_{P} (g) | g ∈ C_{S}(P)} by axiom (A), and each element δ_{P} (g) is the restriction of δ_{S}(g) ∈ Aut_{L}(S). If P is arbitrary, and Q is fully centralized in F and F-conjugate to P, then there is some ψ ∈ Iso_{L}(P, Q) which satisfies the conclusion of the theorem (choose any φ ∈ Iso_{F}(P, Q), write it as a composite of restrictions of automorphisms, and lift each of those automorphisms to L). Then each element of E(P) has the form ψ^{−1}δ_{Q}(g)ψ for some g ∈ C_{S}(Q), and hence satisfies the conclusion of the theorem. □

1.3. Automorphisms of fusion and linking systems.

Recall that for any linking system L associated to a fusion system F over S, and any pair P ≤ Q of objects in L, the inclusion of P into Q is the morphism ι_{P}^{Q} = δ_{P,Q}(1) ∈ Mor_{L}(P, Q). By Proposition 1.11(b′), each morphism in L splits uniquely as the composite of an isomorphism followed by an inclusion.

As usual, an equivalence of small categories is a functor Φ: C ⟶ D which induces a bijection between the sets of isomorphism classes of objects and bijections between each pair of morphism sets. It is not hard to see that for each such equivalence, there is an “inverse” Ψ: D ⟶ C such that both composites Φ ∘ Ψ and Ψ ∘ Φ are naturally isomorphic to the identities. In particular, the quotient monoid Out(C) of all self equivalences of C modulo natural isomorphisms of functors is a group.

Definition 1.13 ([BLO2, § 8]). Let F be a saturated fusion system over a finite p-group S, and let L be a linking system associated to F.

(a) An automorphism β ∈ Aut(S) is fusion preserving if for each P, Q ≤ S and each φ ∈ Hom_{F}(P, Q), (β|_{Q,β(Q)})φ(β|_{P,β(P)})^{−1} lies in Hom_{F}(β(P), β(Q)). In particular, each such β normalizes Aut_{F}(S). Let Aut(S, F) be the group of all fusion preserving automorphisms of S, and set Out(S, F) = Aut(S, F)/Aut_{F}(S). Note that Out(S, F) is a subquotient of Out(S).

(b) An equivalence of categories α: L ⟶ L is isotypical if α(δ_{P} (P)) = δ_{α(P)}(α(P)) for each P ∈ Ob(L).

(c) Let Out_{typ}(L) be the group of classes of isotypical self equivalences of L modulo natural isomorphisms of functors.

(d) Let Aut_{typ}^{I}(L) be the group of isotypical equivalences of L which send inclusions to inclusions.

Since Out(L) is a group by the above remarks, and is finite since Mor(L) is finite, Out_{typ}(L) is a submonoid of a finite group and hence itself a group. Another proof of this, as well as a proof that Aut_{typ}^{I}(L) is a group, will be given in Lemma 1.14.

One of the main results in [BLO2] (Theorem 8.1) says that for any p-local finite group (S, F, L), Out_{typ}(L) ≅ Out(|L|_{p}^{∧}): the group of homotopy classes of self homotopy equivalences of |L|_{p}^{∧}. This helps to explain the importance of Out_{typ}(L), among other groups of automorphisms of (S, F, L) which we might have chosen.

The next lemma gives an alternative description of Out_{typ}(L), and also of Out(G) — descriptions which will be useful later. For each L associated to F over S, and each γ ∈ Aut_{L}(S), let c_{γ} ∈ Aut_{typ}^{I}(L) be the automorphism which sends P ∈ Ob(L) to γ(P) = π(γ)(P), and sends ψ ∈ Mor_{L}(P, Q) to (γ|_{Q,γ(Q)}) ∘ ψ ∘ (γ|_{P,γ(P)})^{−1}. This is clearly isotypical, since for g ∈ P ∈ Ob(L), c_{γ}(δ_{P} (g)) = δ_{γ(P)}(π(γ)(g)) by axiom (C). For P ≤ Q in Ob(L), c_{γ} sends ι_{P}^{Q} to ι_{γ(P)}^{γ(Q)} by definition of restriction, and thus c_{γ} ∈ Aut_{typ}^{I}(L).

Lemma 1.14. (a) For any saturated fusion system F over a finite p-group S, and any linking system L associated to F, the sequence

1 ⟶ Z(F) ⟶^{δ_{S}} Aut_{L}(S) ⟶^{γ↦c_{γ}} Aut_{typ}^{I}(L) ⟶ Out_{typ}(L) ⟶ 1

is exact. All elements of Aut_{typ}^{I}(L) are automorphisms of L, and hence Aut_{typ}^{I}(L) and Out_{typ}(L) are both groups.

(b) For any finite group G and any S ∈ Syl_{p}(G), the sequence

1 ⟶ Z(G) ⟶^{incl} N_{G}(S) ⟶^{g↦c_{g}} Aut(G, S) ⟶ Out(G) ⟶ 1

is exact, where Aut(G, S) = {α ∈ Aut(G) | α(S) = S}.

Proof. (a) Each equivalence of L (isotypical or not) sends S to itself, since S is the only object which is the target of morphisms from all other objects.

If α ∈ Aut_{typ}^{I}(L), then for each P ∈ Ob(L), α_{P,S} sends ι_{P}^{S} to ι_{α(P)}^{S}, and α_{P} sends δ_{P} (P) to δ_{α(P)}(α(P)). Hence α_{S} sends δ_{S}(P) to δ_{S}(α(P)), and thus determines the action of α on Ob(L). In particular, α permutes the objects of L bijectively, and hence is an automorphism of L. This proves that Aut_{typ}^{I}(L) is a group; and that Out_{typ}(L) is also a group if the above sequence is exact.

We next show that each isotypical equivalence α: L ⟶ L is naturally isomorphic to an isotypical equivalence which sends inclusions to inclusions. For each P ∈ Ob(L), let α(ι_{P}^{S}) = ι_{β(P)}^{S} ∘ ω(P) be the unique decomposition of α(ι_{P}^{S}) as a composite of an isomorphism ω(P) ∈ Iso_{L}(α(P), β(P)) followed by an inclusion (Proposition 1.11(b′)). In particular, ω(S) = Id. Let β be the automorphism of L which on objects sends P to β(P), and which on morphisms sends φ ∈ Mor_{L}(P, Q) to ω(Q) ∘ α(φ) ∘ ω(P)^{−1} in Mor_{L}(β(P), β(Q)). Then β is isotypical by axiom (C) (and since α is isotypical); it sends inclusions to inclusions by construction (and since ω(S) = Id); and ω(−) defines a natural isomorphism from α to β.

This proves that the natural homomorphism from Aut_{typ}^{I}(L) to Out_{typ}(L) is onto. If α ∈ Aut_{typ}^{I}(L) is in the kernel, then it is naturally isomorphic to the identity, via some ω(−) which consists of isomorphisms ω(P) ∈ Iso_{L}(P, α(P)) such that for each ψ ∈ Mor_{L}(P, Q), α(ψ) ∘ ω(P) = ω(Q) ∘ ψ. Since α sends ι_{P}^{S} to ι_{α(P)}^{S}, ω(P) = ω(S)|_{P,α(P)}, and thus α is conjugation by ω(S) ∈ Aut_{L}(S).

Conversely, if γ ∈ Aut_{L}(S), then c_{γ} is naturally isomorphic to Id_{L}, by the natural isomorphism which sends P ∈ Ob(L) to γ|_{P,π(γ)(P)}. This finishes the proof that the above sequence is exact at Aut_{typ}^{I}(L).

It remains to show, for γ ∈ Aut_{L}(S), that c_{γ} = Id_{L} if and only if γ ∈ δ_{S}(Z(F)). If c_{γ} = Id, then since γδ_{S}(g)γ^{−1} = δ_{S}(g) for all g ∈ S, π(γ) = Id_{S} by axiom (C), and γ ∈ δ_{S}(Z(S)) by (A). So assume γ = δ_{S}(a) for some a ∈ Z(S). By Proposition 1.11(e) (and axiom (C) again), γ commutes with a morphism ψ ∈ Mor_{L}(P, Q) if and only if ψ extends to some ψ ∈ Mor_{L}(⟨P, a⟩, ⟨Q, a⟩) such that π(ψ)(a) = a. Thus c_{γ} = Id_{L} if and only if a ∈ Z(F). This finishes the proof that the sequence in (a) is exact.

(b) The natural homomorphism from Aut(G, S) to Out(G) is onto by the Frattini argument (the Sylow p-subgroups of G are permuted transitively by inner automorphisms). The kernel of that map clearly consists of conjugation by elements of N_{G}(S). □

In particular, the group Aut_{typ}^{I}(L) defined here is the same as that defined in [O3], where it was defined explicitly as a group of automorphisms of L rather than of equivalences.

The next lemma describes how elements of Aut_{typ}^{I}(L) induce automorphisms of the associated fusion system. For β ∈ Aut(S, F), let c_{β} ∈ Aut(F) be the automorphism of the category F which sends P ≤ S to β(P), and sends φ ∈ Mor(F) to βφβ^{−1}.

Lemma 1.15 ([O3, Proposition 6]). Let L be a linking system associated to a saturated fusion system F over a finite p-group S, with structure functors T_{Ob(L)}(S) ⟶^{δ} L ⟶^{π} F. Fix α ∈ Aut_{typ}^{I}(L). Let β ∈ Aut(S) be such that α(δ_{S}(g)) = δ_{S}(β(g)) for all g ∈ S. Then β ∈ Aut(S, F), α(P) = β(P) for P ∈ Ob(L), and π ∘ α = c_{β} ∘ π.

Proof. See [O3, Proposition 6]. The relation α(P) = β(P) is not in the statement of the proposition, but it is shown in its proof. It is really part of the statement π ∘α = c_{β} ∘π (since π is the inclusion on objects). □

Lemma 1.15 motivates the following definition. For any saturated fusion system F over a finite p-group S, and any linking system L associated to F, define

μ̃_{L}: Aut_{typ}^{I}(L) ⟶ Aut(S, F)

by setting μ̃_{L}(α) = δ_{S}^{−1} ∘ α_{S} ∘ δ_{S} ∈ Aut(S) for α ∈ Aut_{typ}^{I}(L). By Lemma 1.15, Im(μ̃_{L}) ≤ Aut(S, F). For γ ∈ Aut_{L}(S), μ̃_{L}(c_{γ}) = π(γ) ∈ Aut_{F}(S) by axiom (C) in Definition 1.9. So by Lemma 1.14(a), μ̃_{L} induces a homomorphism

μ_{L}: Out_{typ}(L) ⟶ Out(S, F)

by sending the class of α to that of μ̃_{L}(α). When L = L_{S}^{H}(G) for some finite group G and some set of objects H, we write μ̃_{G}^{H} = μ̃_{L} and μ_{G}^{H} = μ_{L} for short. When L = L_{S}^{c}(G) is the centric linking system, we write μ̃_{G} = μ̃_{L} and μ_{G} = μ_{L}.

Lemma 1.16. For any linking system L associated to a saturated fusion system F, Ker(μ_{L}) is a finite p-group.

Proof. Assume F is a fusion system over the finite p-group S. Since L is a finite category, Aut_{typ}^{I}(L) and Out_{typ}(L) are finite groups. So it suffices to prove that each element of Ker(μ_{L}) has p-power order.

Fix α ∈ Aut_{typ}^{I}(L) such that [α] ∈ Ker(μ_{L}). Thus μ̃_{L}(α) ∈ Aut_{F}(S), and μ̃_{L}(α) = π(γ) for some γ ∈ Aut_{L}(S). So upon replacing α by c_{γ}^{−1} ∘ α, we can assume α ∈ Ker(μ̃_{L}).

Thus α_{S}|_{δ_{S}(S)} = Id. Since α sends inclusions to inclusions, α_{P} |_{δ_{P} (P)} = Id for all P ∈ Ob(L). For each P and each ψ ∈ Aut_{L}(P), ψ and α(ψ) have the same conjugation action on δ_{P} (P), so ψ^{−1}α(ψ) ∈ C_{Aut_{L}(P)}(δ_{P} (P)) ≤ δ_{P} (Z(P)). Hence α(ψ) = ψδ_{P} (g) for some g ∈ Z(P), and α^{k}(ψ) = ψδ_{P} (g^{k}) for all k since α is the identity on δ_{P} (P).

Choose m ≥ 0 such that g^{p^{m}} = 1 for all g ∈ S. Then α^{p^{m}} is the identity on Aut_{L}(P) for each P ∈ Ob(L), and hence (by Theorem 1.12) is the identity on L. □

The kernel of μ_{L} will be studied much more closely in Proposition 4.2.

Since we will need to work with linking systems with different sets of objects associated to the same fusion system, it will be important to know they have the same automorphisms.

Lemma 1.17. Fix a saturated fusion system F over a finite p-group S. Let L_{0} ⊆ L be a pair of linking systems associated to F. Set H_{0} = Ob(L_{0}) and H = Ob(L), and assume H_{0} ⊆ H are both Aut(S, F)-invariant. Then restriction defines an isomorphism

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

Proof. Using Proposition 1.11(a), one sees that L_{0} must be a full subcategory of L. Set P = H∖H_{0}. We can assume, by induction on |H| − |H_{0}|, that all subgroups in P have the same order. Thus all morphisms in L between subgroups in P are isomorphisms.

Since H_{0} is Aut(S, F)-invariant and L_{0} is a full subcategory, there is a well defined restriction homomorphism

Aut_{typ}^{I}(L) ⟶^{Res} Aut_{typ}^{I}(L_{0}).

By assumption, H_{0} contains all subgroups which are F-centric and F-radical. Hence Theorem 1.12 implies that all morphisms in L are composites of restrictions of morphisms in L_{0}. Since each α ∈ Aut_{typ}^{I}(L) sends inclusions to inclusions, it also sends restrictions to restrictions, and hence α|_{L_{0}} = Id_{L_{0}} only if α = Id_{L}. Thus Res is injective. We next show it is surjective, and hence an isomorphism.

Let P_{∗} ⊆ P be a subset consisting of one fully normalized subgroup from each F-conjugacy class in P. For each P ∈ P_{∗}, δ_{P} (N_{S}(P)) ∈ Syl_{p}(Aut_{L}(P)) by Proposition 1.11(d), so there is a unique P̂ ≤ N_{S}(P) such that δ_{P} (P̂) = O_{p}(Aut_{L}(P)). Since P ∉ H_{0}, P is either not F-centric or not F-radical. In either case, P̂ ≩ P by Lemma 1.4. By Proposition 1.11(e), each ψ ∈ Aut_{L}(P) extends to a unique automorphism ψ̂ ∈ Aut_{L}(P̂).

Let ν: P ⟶ P_{∗} be the map which sends P to the unique subgroup ν(P) ∈ P_{∗} which is F-conjugate to P. For each P ∈ P, δ_{ν(P)}(N_{S}(ν(P))) ∈ Syl_{p}(Aut_{L}(ν(P))) by Proposition 1.11(d) (and since ν(P) is fully normalized), and hence there is λ_{P} ∈ Iso_{L}(P, ν(P)) such that

λ_{P} δ_{P} (N_{S}(P))λ_{P}^{−1} ≤ δ_{ν(P)}(N_{S}(ν(P))) .

By Proposition 1.11(e) again, λ_{P} extends to a unique λ̂_{P} ∈ Mor_{L}(N_{S}(P), N_{S}(ν(P))). When P ∈ P_{∗} (so ν(P) = P), we set λ_{P} = Id_{P}, and hence λ̂_{P} = Id_{N_{S}(P)}.

Fix any α_{0} ∈ Aut_{typ}^{I}(L_{0}); we want to extend α_{0} to L. By Lemma 1.15, α_{0} induces some β ∈ Aut(S, F), and α_{0}(P) = β(P) for all P ∈ H_{0}. So define α(P) = β(P) for P ∈ H; this is possible since H is Aut(S, F)-invariant by assumption. By Lemma 1.15 again, for each P, Q ∈ H_{0} and each ψ ∈ Mor_{L_{0}}(P, Q), π(α_{0}(ψ)) = c_{β}(π(ψ)) = β(π(ψ))β^{−1}. In other words,

ψ ∈ Mor_{L_{0}}(P, Q), g ∈ P, π(ψ)(g) = h ∈ Q ⟹ π(α_{0}(ψ))(β(g)) = β(h) .  (1)

We next define α on isomorphisms between subgroups in P. Fix P_{1}, P_{2} ∈ P and ψ ∈ Iso_{L}(P_{1}, P_{2}), and set P_{∗} = ν(P_{1}) = ν(P_{2}). There is a unique ψ_{∗} ∈ Aut_{L}(P_{∗}) such that ψ = λ_{P_{2}}^{−1} ∘ ψ_{∗} ∘ λ_{P_{1}}, and we set

α(ψ) = (α_{0}(λ̂_{P_{2}})|_{α(P_{2}),α(P_{∗})})^{−1} ∘ (α_{0}(ψ̂_{∗})|_{α(P_{∗}),α(P_{∗})}) ∘ (α_{0}(λ̂_{P_{1}})|_{α(P_{1}),α(P_{∗})}) .

Note that λ̂_{P_{1}}, λ̂_{P_{2}}, and ψ̂_{∗} are all in Mor(L_{0}), since all subgroups strictly containing subgroups in P are in H_{0} = Ob(L_{0}) by assumption. Also, the restrictions are well defined (for example, π(α_{0}(λ̂_{P_{i}}))(α(P_{i})) = α(P^{∗})) by (1).

Each morphism in L not in L_{0} factors uniquely as an isomorphism between subgroups in P followed by an inclusion (Proposition 1.11(b′)), and thus these definitions extend to define α as a map from Mor(L) to itself. This clearly preserves composition of isomorphisms between subgroups in P. To prove that α is a functor, it remains to show it preserves composites of inclusions followed by isomorphisms in L_{0}. This means showing, for each P_{1}, P_{2} ∈ P, each P_{i} ≨ Q_{i}, and each ψ ∈ Iso_{L}(P_{1}, P_{2}) which extends to φ ∈ Mor_{L_{0}}(Q_{1}, Q_{2}), that α(ψ) = α_{0}(φ)|_{α(P_{1}),α(P_{2})}. Since N_{Q_{i}}(P_{i}) ≩ P_{i}, we can assume P_{i} ⊴ Q_{i} for i = 1, 2. Set P_{∗} = ν(P_{1}) = ν(P_{2}) again, and set R_{i} = π(λ̂_{P_{i}})(Q_{i}) ≩ P_{∗}. Then P_{∗} ⊴ R_{i} since P_{i} ⊴ Q_{i}. We saw that ψ factors in a unique way ψ = λ_{P_{2}}^{−1} ∘ ψ_{∗} ∘ λ_{P_{1}} for ψ_{∗} ∈ Aut_{L}(P_{∗}). We also have φ = λ_{P_{2}}^{−1} ∘ φ_{∗} ∘ λ_{P_{1}}, where λ_{P_{i}} = λ̂_{P_{i}}|_{Q_{i},R_{i}} and φ_{∗} ∈ Mor_{L_{0}}(R_{1}, R_{2}). Thus α_{0}(λ_{P_{i}}) is a restriction of α_{0}(λ̂_{P_{i}}) (i = 1, 2), and hence an extension of α(λ_{P_{i}}).

It remains to show α(ψ_{∗}) is the restriction of α_{0}(φ_{∗}). By definition, α(ψ_{∗}) is the restriction to α(P_{∗}) of α_{0}(ψ̂_{∗}), where ψ̂_{∗} ∈ Aut_{L}(P̂_{∗}). Set T_{i} = ⟨P̂_{∗}, R_{i}⟩. By Proposition 1.11(e), since ψ_{∗} ∈ Aut_{L}(P_{∗}) extends to ψ̂_{∗} ∈ Aut_{L}(P̂_{∗}) and to φ_{∗} ∈ Mor_{L}(R_{1}, R_{2}), there is φ_{∗} ∈ Mor_{L}(T_{1}, T_{2}) which extends both ψ̂_{∗} and φ_{∗}. Hence α_{0}(φ_{∗}) extends both α_{0}(ψ̂_{∗}) and α_{0}(φ_{∗}) (all of these are in L_{0}), and thus α(ψ_{∗}) is a restriction of each of the latter. This finishes the proof that α is a functor. By construction, α is isotypical, sends inclusions to inclusions, and extends α_{0}; and thus Res is surjective.

We have now shown that restriction defines an isomorphism from Aut_{typ}^{I}(L) to Aut_{typ}^{I}(L_{0}). By Lemma 1.14(a), the outer automorphism groups of L and L_{0} are defined by dividing out by conjugation by elements of Aut_{L}(S). Hence the induced homomorphism

Out_{typ}(L) ⟶^{R} Out_{typ}(L_{0})

is also an isomorphism. □

1.4. Normal fusion subsystems.

Let F be a saturated fusion system over a finite p-group S. By a (saturated) fusion subsystem of F over a subgroup S_{0} ≤ S, we mean a subcategory F_{0} ⊆ F whose objects are the subgroups of S_{0}, and which is itself a (saturated) fusion system over S_{0}.

The following definition of a normal fusion subsystem is equivalent to that of Puig [Pg2, § 6.4], and also to Aschbacher’s definition of an F-invariant subsystem [Asch, § 3] (except that they do not require the subsystem to be saturated). See [Pg2, Proposition 6.6] and [Asch, Theorem 3.3] for proofs of those equivalences.

Definition 1.18. Let F be a saturated fusion system over a finite p-group S, and let F_{0} ⊆ F be a saturated fusion subsystem over S_{0} ≤ S. Then F_{0} is normal in F (F_{0} ⊴ F) if

(i) S_{0} is strongly closed in F;

(ii) for each P, Q ≤ S_{0} and each φ ∈ Hom_{F}(P, Q), there are α ∈ Aut_{F}(S_{0}) and φ_{0} ∈ Hom_{F_{0}}(α(P), Q) such that φ = φ_{0} ∘ α|_{P,α(P)}; and

(iii) for each P, Q ≤ S_{0}, each φ ∈ Hom_{F_{0}}(P, Q), and each β ∈ Aut_{F}(S_{0}), βφβ^{−1} ∈ Hom_{F_{0}}(β(P), β(Q)).

We next list some of the basic properties of normal fusion subsystems, starting with the following technical result.

Lemma 1.19. Fix a saturated fusion system F over a finite p-group S. Let F_{0} ⊆ F be a fusion subsystem (not necessarily saturated) over the subgroup S_{0} ⊴ S, which satisfies conditions (i–iii) in Definition 1.18. Assume P_{0} ≤ S_{0} is F_{0}-centric and fully normalized in F, and Out_{S_{0}}(P_{0}) ∩ O_{p}(Out_{F_{0}}(P_{0})) = 1. Then there is P ≤ S which is F-centric and F-radical and such that P ∩ S_{0} = P_{0}.

Proof. Set

P = {x ∈ N_{S}(P_{0}) | c_{x} ∈ O_{p}(Aut_{F}(P_{0}))} .

If x ∈ P ∩ S_{0}, then c_{x} ∈ O_{p}(Aut_{F}(P_{0})) ∩ Aut_{F_{0}}(P_{0}) = O_{p}(Aut_{F_{0}}(P_{0})) (Aut_{F_{0}}(P_{0}) is normal in Aut_{F}(P_{0}) by 1.18(ii–iii)), so c_{x} ∈ Aut_{S_{0}}(P_{0}) ∩ O_{p}(Aut_{F_{0}}(P_{0})) = Inn(P_{0}), and x ∈ P_{0} since P_{0} is F_{0}-centric. Thus P ∩ S_{0} = P_{0}.

By construction, N_{S}(P) = N_{S}(P_{0}). So if Q if F-conjugate to P and Q_{0} = Q ∩ S_{0}, then |N_{S}(Q)| ≤ |N_{S}(Q_{0})| ≤ |N_{S}(P_{0})| = |N_{S}(P)| since P_{0} is fully normalized in F and F-conjugate to Q_{0}. This proves that P is fully normalized in F.

Now, Aut_{S}(P_{0}) ≥ O_{p}(Aut_{F}(P_{0})) since P_{0} is fully normalized in F. So Aut_{P} (P_{0}) = O_{p}(Aut_{F}(P_{0})), and hence this is normal in Aut_{F}(P_{0}). By the extension axiom, the restriction homomorphism Aut_{F}(P) ⟶ Aut_{F}(P_{0}) is surjective, and thus sends O_{p}(Aut_{F}(P)) into O_{p}(Aut_{F}(P_{0})). So for all x ∈ N_{S}(P) such that c_{x} ∈ O_{p}(Aut_{F}(P)), c_{x} ∈ O_{p}(Aut_{F}(P_{0})), and hence x ∈ P. Since P is fully normalized, it is F-centric and F-radical by Lemma 1.4. □

The following is our main lemma listing properties of normal pairs of fusion systems. Recall that O_{p}(F) is the largest normal p-subgroup of the fusion system F.

Lemma 1.20. Fix a saturated fusion system F over a finite p-group S, and let F_{0} ⊴ F be a normal fusion subsystem over the normal subgroup S_{0} ⊴ S. Then the following hold.

(a) Each F-conjugacy class contains a subgroup P ≤ S such that P and P ∩ S_{0} are both fully normalized in F, and P ∩ S_{0} is fully normalized in F_{0}.

(b) For each P, Q ≤ S_{0} and each φ ∈ Iso_{F}(P, Q), φAut_{F_{0}}(P)φ^{−1} = Aut_{F_{0}}(Q).

(c) The set of F_{0}-centric subgroups of S_{0}, and the set of F_{0}-radical subgroups of S_{0}, are both invariant under F-conjugacy.

(d) If P ≤ S is F-centric and F-radical, then P ∩ S_{0} is F_{0}-centric and F_{0}-radical. Conversely, if P_{0} ≤ S_{0} is F_{0}-centric, F_{0}-radical, and fully normalized in F, then there is P ≤ S which is F-centric and F-radical, and such that P ∩ S_{0} = P_{0}.

(e) O_{p}(F_{0}) is normal in F, and O_{p}(F_{0}) = O_{p}(F) ∩ S_{0}.

Proof. Throughout the proof, whenever P ≤ S, we write P_{0} = P ∩ S_{0} for short.

(a) Fix Q ≤ S. By [BLO2, Proposition A.2(b)], there are subgroups R ≤ S and P_{0} ≤ S_{0} which are fully normalized in F, and morphisms φ ∈ Hom_{F}(N_{S}(Q), N_{S}(R)) and ψ ∈ Hom_{F}(N_{S}(R_{0}), N_{S}(P_{0})) such that φ(Q) = R and ψ(R_{0}) = P_{0}. Set P = ψ(R) (note that P ∩ S_{0} = P_{0} since S_{0} is strongly closed). Since N_{S}(R) ≤ N_{S}(R_{0}), P is also fully normalized in F. Also, P = ψ ∘ φ(Q) is F-conjugate to Q.

By [BLO2, Proposition A.2(b)] again, if P_{0}^{∗} is F-conjugate to P_{0}, then there is a morphism in F from N_{S}(P_{0}^{∗}) to N_{S}(P_{0}) which sends P_{0}^{∗} to P_{0}. In particular, |N_{S_{0}}(P_{0}^{∗})| ≤ |N_{S_{0}}(P_{0})|, and hence P_{0} is also fully normalized in F_{0}.

(b) Fix P, Q ≤ S_{0} and φ ∈ Iso_{F}(P, Q). By condition (ii) in Definition 1.18, there are α ∈ Aut_{F}(S_{0}) and φ_{0} ∈ Iso_{F_{0}}(α(P), Q) such that φ = φ_{0} ∘ α|_{P,α(P)}. Hence

φAut_{F_{0}}(P)φ^{−1} = φ_{0}Aut_{F_{0}}(α(P))φ_{0}^{−1} = Aut_{F_{0}}(Q) ,

where the first equality holds by condition (iii) in Definition 1.18.

(c) Fix P ≤ S_{0}, let P be the F-conjugacy class of P, and let P_{0} be its F_{0}-conjugacy class. If P is F_{0}-centric, then C_{S_{0}}(P^{∗}) = Z(P^{∗}) for all P^{∗} ∈ P_{0}. For all R ∈ P, there is α ∈

Aut_{F}(S_{0}) such that α(R) ∈ P_{0} (condition (ii) in Definition 1.18), and hence C_{S_{0}}(R) = Z(R). Since this holds for all subgroups in P, all of these subgroups are F_{0}-centric.

Now assume P is F_{0}-radical; then O_{p}(Out_{F_{0}}(P^{∗})) = 1 for all P^{∗} ∈ P_{0}. If R ∈ P, and α ∈ Aut_{F}(S_{0}) is such that α(R) ∈ P_{0}, then by condition (iii) in Definition 1.18, conjugation by α sends Out_{F_{0}}(R) ≤ Out_{F}(R) isomorphically to Out_{F_{0}}(α(R)). Since O_{p}(Out_{F_{0}}(α(R))) = 1, O_{p}(Out_{F_{0}}(R)) = 1. So all subgroups in P are F_{0}-radical.

(d) The second statement was shown in Lemma 1.19. It remains to prove the first. Assume P is F-centric and F-radical. We must show that P_{0} is F_{0}-centric and F_{0}-radical.

By (c), this is independent of the choice of P in its F-conjugacy class, and hence by (a), it suffices to prove it when P_{0} is fully normalized in F_{0}. By (b), Aut_{F_{0}}(P_{0}) is normal in Aut_{F}(P_{0}), and hence

O_{p}(Aut_{F_{0}}(P_{0})) ≤ O_{p}(Aut_{F}(P_{0})) .

Let T be the subgroup of all x ∈ N_{S_{0}}(P_{0}) such that c_{x} ∈ O_{p}(Aut_{F_{0}}(P_{0})). If x ∈ T ∩N_{S}(P), then c_{x} ∈ O_{p}(Aut_{F}(P_{0})), and c_{x} induces the identity on P/P_{0} since [x, P] ≤ P ∩[S_{0}, P] ≤ P_{0}. Thus c_{x} ∈ O_{p}(Aut_{F}(P)) by Lemma 1.6, and x ∈ P by Lemma 1.4 since P is centric and radical in F.

Thus T ∩ N_{S}(P) ≤ P_{0}. Also, P normalizes T by construction, so T ≤ P_{0} by Lemma 1.7. Hence P_{0} is centric and radical in F_{0} by Lemma 1.4 again.

(e) Set Q = O_{p}(F_{0}) and R = O_{p}(F) for short. To prove that Q ⊴ F and R_{0} = Q, it suffices to show that Q ⊴ F and R_{0} ⊴ F_{0}. We apply Proposition 1.5, which says that a subgroup is normal in a saturated fusion system if and only if it is strongly closed and contained in all subgroups which are centric and radical. Since an intersection of strongly closed subgroups is strongly closed, R_{0} is strongly closed in F and hence in F_{0}.

If P ≤ S is F-centric and F-radical, then P_{0} is F_{0}-centric and F_{0}-radical by (d), so P ≥ P_{0} ≥ Q. If P_{0} is F_{0}-centric and F_{0}-radical, then the same holds for each subgroup in its F-conjugacy class by (c). So by (d), there is P^{∗} ≤ S which is F-centric and F-radical with P_{0}^{∗} F-conjugate to P_{0}; P^{∗} ≥ R, and hence P_{0}^{∗} and P_{0} both contain R_{0}.

It remains to prove that Q is strongly closed in F. Fix F-conjugate elements g, h ∈ S such that g ∈ Q; we must show h ∈ Q. Since S_{0} is strongly closed in F (since F_{0} ⊴ F), h ∈ S_{0}. Fix φ ∈ Iso_{F}(⟨g⟩, ⟨h⟩) with φ(g) = h. Since F_{0} ⊴ F, there are morphisms χ ∈ Aut_{F}(S_{0}) and φ_{0} ∈ Iso_{F_{0}}(⟨g⟩, ⟨χ^{−1}(h)⟩) such that φ = χ ∘ φ_{0}. Then g′ =^{def} φ_{0}(g) ∈ Q, and h = χ(g′). The invariance condition (iii) in Definition 1.18 implies that χ sends a normal subgroup of F_{0} to another normal subgroup. Thus χ(Q)·Q is also normal in F_{0}, so χ(Q) = Q since Q is the largest subgroup of S_{0} normal in F_{0}, and thus h = χ(g′) ∈ Q. □

We now turn to the specific examples of normal fusion subsystems which we work with in this paper. We first look at those of p-power index and of index prime to p. Two other definitions are first needed. For any saturated fusion system F, the focal subgroup 𝔣𝔬𝔠(F) and the hyperfocal subgroup 𝔥𝔶𝔭(F) are defined by

𝔣𝔬𝔠(F) = ⟨s^{−1}t | s, t ∈ S and F-conjugate⟩ = ⟨s^{−1}α(s) | s ∈ P ≤ S, α ∈ Aut_{F}(P)⟩
𝔥𝔶𝔭(F) = ⟨s^{−1}α(s) | s ∈ P ≤ S, α ∈ O^{p}(Aut_{F}(P))⟩ .

Note that in [BCGLO2], we wrote O_{F}^{p}(S) = 𝔥𝔶𝔭(F).

The following definition also includes many fusion subsystems which are not normal.

Definition 1.21 ([BCGLO2, Definition 3.1]). Let F be a saturated fusion system over a finite p-group S, and let F_{0} ⊆ F be a saturated fusion subsystem over a subgroup S_{0} ≤ S.

(a) F_{0} has p-power index in F if 𝔥𝔶𝔭(F) ≤ S_{0} ≤ S, and Aut_{F_{0}}(P) ≥ O^{p}(Aut_{F}(P)) for all P ≤ S_{0}.

(b) F_{0} has index prime to p in F if S_{0} = S, and Aut_{F_{0}}(P) ≥ O^{p′}(Aut_{F}(P)) for all P ≤ S.

Recall that despite the terminology, these are not analogous to subgroups of a finite group of p-power index or index prime to p. Instead, they are analogous to subgroups which contain a normal subgroup having appropriate index.

The following theorem gives a complete description of all such fusion subsystems.

Theorem 1.22 ([BCGLO2, Theorems 4.3 & 5.4]). The following hold for any saturated fusion system F over a finite p-group S.

(a) For each subgroup S_{0} ≤ S containing the hyperfocal subgroup 𝔥𝔶𝔭(F), there is a unique fusion subsystem F_{0} over S_{0} of p-power index in F. Thus F contains a proper fusion subsystem of p-power index if and only if 𝔥𝔶𝔭(F) ≨ S, or equivalently 𝔣𝔬𝔠(F) ≨ S.

(b) There is a subgroup Γ ⊴ Out_{F}(S) with the following properties. For each subsystem F_{0} ⊆ F of index prime to p, Out_{F_{0}}(S) ≥ Γ. Conversely, for each H ≤ Out_{F}(S) containing Γ, there is a unique subsystem F_{0} ⊆ F of index prime to p with Out_{F_{0}}(S) = H.

Proof. The only part not shown in [BCGLO2] is that 𝔥𝔶𝔭(F) ≨ S implies 𝔣𝔬𝔠(F) ≨ S. By Theorem 1.3,

𝔣𝔬𝔠(F) = ⟨s^{−1}α(s) | s ∈ P ≤ S, P fully normalized in F, α ∈ Aut_{F}(P)⟩ .

Since Aut_{F}(P) = O^{p}(Aut_{F}(P))·Aut_{S}(P) when P is fully normalized, and since s^{−1}α(s) ∈ [S, S] when s ∈ P and α ∈ Aut_{S}(P), we have 𝔣𝔬𝔠(F) = 𝔥𝔶𝔭(F)·[S, S]. Also, 𝔥𝔶𝔭(F) ≨ S implies there is Q ⊴ S such that [S:Q] = p and 𝔥𝔶𝔭(F) ≤ Q. Then [S, S] ≤ Q since S/Q is abelian, and hence 𝔣𝔬𝔠(F) ≤ Q ≨ S. □

One can show, in the situation of Theorem 1.22, that a fusion subsystem of p-power index is normal in F exactly when its underlying p-group is normal in S, and that a fusion subsystem F_{0} ⊆ F of index prime to p is normal in F exactly when Aut_{F_{0}}(S) is normal in Aut_{F}(S). But in fact, we will only be concerned here (in Proposition 1.25(a,b)) with the minimal such fusion subsystems, defined as follows.

Definition 1.23. For any saturated fusion system F over a finite p-group S, O^{p}(F) and O^{p′}(F) denote the unique minimal saturated fusion subsystems of p-power index over 𝔥𝔶𝔭(F), or of index prime to p over S, respectively.

We next recall the definitions of the normalizer fusion systems N_{F}^{K}(Q) (cf. [Pg2, § 2.8] or [BLO2, Definitions A.1, A.3]). For any group G, any subgroup Q ≤ G, and any K ≤ Aut(Q), define

N_{G}^{K}(Q) = {g ∈ N_{G}(Q) | c_{g}|_{Q} ∈ K} .

For example, N_{G}^{Aut(Q)}(Q) = N_{G}(Q) is the usual normalizer, and N_{G}^{{Id}}(Q) = C_{G}(Q) is the centralizer.

Let F be a saturated fusion system over a finite p-group S, and fix Q ≤ S and K ≤ Aut(Q). We say Q is fully K-normalized if for each Q^{∗} which is F-conjugate to Q and each φ ∈ Iso_{F}(Q, Q^{∗}), |N_{S}^{K}(Q)| ≥ |N_{S}^{φKφ^{−1}}(Q^{∗})|. Let N_{F}^{K}(Q) be the fusion system over N_{S}^{K}(Q) defined by setting, for all P, R ≤ N_{S}^{K}(Q),

Hom_{N_{F}^{K}(Q)}(P, R) = {φ ∈ Hom_{F}(P, R) | ∃ φ ∈ Hom_{F}(P Q, RQ),

φ|_{P} = φ, φ(Q) = Q, φ|_{Q} ∈ K} .

As special cases, C_{F}(Q) = N_{F}^{{Id}}(Q) and N_{F}(Q) = N_{F}^{Aut(Q)}(Q). By [Pg2, Proposition 2.15] or [BLO2, Proposition A.6], if Q is fully K-normalized in F, then N_{F}^{K}(Q) is a saturated fusion system. If K ≥ Inn(Q), then Q is normal in N_{F}^{K}(Q) by definition.

This construction is motivated by the following proposition.

Proposition 1.24. Fix a finite group G and S ∈ Syl_{p}(G), and set F = F_{S}(G). For Q ≤ S and K ≤ Aut(Q), Q is fully K-normalized in F if and only if N_{S}^{K}(Q) ∈ Syl_{p}(N_{G}^{K}(Q)). When this is the case, then N_{F}^{K}(Q) = F_{N_{S}^{K}(Q)}(N_{G}^{K}(Q)).

Proof. Fix any T ∈ Syl_{p}(N_{G}^{K}(Q)). Since T normalizes Q, T Q is a p-group, and hence there is a ∈ G such that a(T Q)a^{−1} ≤ S. Set Q^{∗} = aQa^{−1}, φ = c_{a} ∈ Iso_{F}(Q, Q^{∗}), and K^{∗} = φKφ^{−1} ≤ Aut(Q^{∗}). Then aT a^{−1} ∈ Syl_{p}(N_{G}^{K^{∗}}(Q^{∗})) and aT a^{−1} ≤ S, so aT a^{−1} = N_{S}^{K^{∗}}(Q^{∗}). It follows that Q^{∗} is fully K^{∗}-normalized in F; and also that Q is fully K-normalized if and only if |N_{S}^{K}(Q)| = |T |, if and only if N_{S}^{K}(Q) ∈ Syl_{p}(N_{G}^{K}(Q)).

Now assume Q is fully K-normalized. Set N_{F} = N_{F}^{K}(Q), N_{G} = N_{G}^{K}(Q), and N_{S} = N_{S}^{K}(Q) for short, and fix P, R ≤ N_{S}. If φ ∈ Hom_{N_{F}} (P, R), then it extends to some φ = c_{g} ∈ Hom_{F}(P Q, RQ) such that φ(Q) = Q and φ|_{Q} ∈ K, so g ∈ N_{G}, and φ = c_{g} ∈ Hom_{N_{G}}(P, R). This proves that Hom_{N_{F}} (P, R) ⊆ Hom_{N_{G}}(P, R). The opposite inclusion is clear, and thus N_{F} = F_{N_{S}}(N_{G}). □

We now give some examples of normal fusion subsystems: examples which will be important later in the paper. The most obvious example is the inclusion F_{S_{0}}(G_{0}) ⊆ F_{S}(G) when G_{0} ⊴ G are finite groups and S_{0} = S ∩ G_{0}, but this case will be handled later (Proposition 1.28).

Proposition 1.25. The following hold for any saturated fusion system F over a finite pgroup S.

(a) O^{p}(F) ⊴ F.

(b) O^{p′}(F) ⊴ F.

(c) For each Q ⊴ F and each K ⊴ Aut(Q), N_{F}^{K}(Q) ⊴ F.

Proof. (a,b) The subsystem O^{p′}(F) is a fusion system over S, which is clearly strongly closed. The subsystem O^{p}(F) is a fusion system over the hyperfocal subgroup

𝔥𝔶𝔭(F) = ⟨s^{−1}α(s) | s ∈ P ≤ S, α ∈ O^{p}(Aut_{F}(P))⟩.

We claim 𝔥𝔶𝔭(F) is also strongly closed in F. By Theorem 1.3, it suffices to show, for each P ≤ S fully normalized in F, that P ∩ 𝔥𝔶𝔭(F) is Aut_{F}(P)-invariant. Since Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)), Aut_{F}(P) is generated by Aut_{S}(P) and O^{p}(Aut_{F}(P)). Finally, P ∩ 𝔥𝔶𝔭(F) is Aut_{S}(P)-invariant since 𝔥𝔶𝔭(F) ⊴ S and is O^{p}(Aut_{F}(P))-invariant by definition of 𝔥𝔶𝔭(F), and this proves the claim.

Condition (ii) in Definition 1.18 holds for these two subsystems by Lemma 3.4 and Proposition 3.8(b,c) in [BCGLO2]. Condition (iii) (invariance under conjugation by elements of Aut_{F}(S_{0})) follows from [BCGLO2, Theorems 4.3 & 5.4]: O^{p}(F) is the unique saturated fusion subsystem over 𝔥𝔶𝔭(F) of p-power index in F, and O^{p′}(F) is the unique saturated fusion subsystem over S with its given Aut_{O^{p′}(F)}(S), of index prime to p in F.

(c) Assume g, h ∈ S are F-conjugate. Since Q ⊴ F, there is φ ∈ Hom_{F}(⟨Q, g⟩, ⟨Q, h⟩) such that φ(g) = h and φ(Q) = Q. Set φ_{0} = φ|_{Q} ∈ Aut_{F}(Q). Then c_{h} = φ_{0}c_{g}φ_{0}^{−1} in Aut_{F}(Q). Since K ⊴ Aut(Q), g ∈ N_{S}^{K}(Q) (i.e., c_{g} ∈ K) if and only if h ∈ N_{S}^{K}(Q). This proves that N_{S}^{K}(Q) is strongly closed in F.

Set Aut_{S}^{K}(Q) = K ∩ Aut_{S}(Q) and Aut_{F}^{K}(Q) = K ∩ Aut_{F}(Q). Fix P, R ≤ N_{S}^{K}(Q) and φ ∈ Hom_{F}(P, R). Since Q ⊴ F, there is φ̂ ∈ Hom_{F}(QP, QR) such that φ̂|_{P} = φ and φ̂(Q) = Q. Set φ_{0} = φ̂|_{Q} ∈ Aut_{F}(Q). Since K ⊴ Aut(Q) and Aut_{S}(Q) ∈ Syl_{p}(Aut_{F}(Q)), Aut_{S}^{K}(Q) ∈ Syl_{p}(Aut_{F}^{K}(Q)). Since φ_{0}Aut_{S}^{K}(Q)φ_{0}^{−1} is contained in Aut_{F}^{K}(Q) (again since K is normal), there is χ ∈ Aut_{F}^{K}(Q) such that

(χφ_{0})Aut_{S}^{K}(Q)(χφ_{0})^{−1} = Aut_{S}^{K}(Q) .

By the extension axiom, there is φ ∈ Hom_{F}(N_{S}^{K}(Q)·Q, S) such that φ|_{Q} = χφ_{0}. Furthermore, φ(N_{S}^{K}(Q)) = N_{S}^{K}(Q) since χφ_{0} normalizes Aut_{S}^{K}(Q).

Set P_{1} = φ(P), ψ̂ = φ̂∘(φ|_{P Q,P_{1}Q})^{−1} ∈ Hom_{F}(P_{1}Q, RQ), and ψ = ψ̂|_{P_{1},R}. Then ψ̂|_{Q} = χ^{−1}, so ψ ∈ Hom_{N_{F}^{K}(Q)}(P_{1}, R), and φ = ψ ∘ φ|_{P,P_{1}}. This proves condition (ii) in Definition 1.18. The last condition — the subsystem N_{F}^{K}(Q) is invariant under conjugation by elements of Aut_{F}(N_{S}^{K}(Q)) — is clear. □

We just showed that O^{p′}(F) is normal in F for any F. The following lemma can be thought of as a “converse” to this.

Lemma 1.26. Assume F_{0} ⊴ F is a normal pair of fusion systems over the same finite p-group S. Then F_{0} has index prime to p in F, and thus F_{0} ⊇ O^{p′}(F).

Proof. If P, Q ≤ S are F-conjugate, then by condition (ii) in Definition 1.18, P is F_{0}- conjugate to α(Q) for some α ∈ Aut_{F}(S). Since |N_{S}(Q)| = |N_{S}(α(Q))|, this shows that P is fully normalized in F_{0} if and only if it is fully normalized in F.

If P ≤ S is fully normalized in F_{0} (and hence in F), then Aut_{F_{0}}(P) contains Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)). Also, since Aut_{F_{0}}(P) is normal in Aut_{F}(P) by Lemma 1.20(b), Aut_{F_{0}}(P) contains O^{p′}(Aut_{F}(P)). Since this property depends only on the isomorphism class of P in F_{0}, it holds for all P ≤ S. So F_{0} has index prime to p in F by Definition 1.21(b). □

1.5. Normal linking subsystems.

The following definition of a normal linking subsystem seems to be the most appropriate one for our needs here; it is also the one used in [O3]. In the following definition (and elsewhere), whenever we say that L_{0} ⊆ L is a pair of linking systems associated to F_{0} ⊆ F (or L_{0} is a linking subsystem), it is understood not only that L_{0} is a subcategory of L, but also that the structural functors for L_{0} are the restrictions of the structural functors

T_{Ob(L)}(S) ⟶^{δ} L ⟶^{π} F for L.

Definition 1.27. Fix a pair of saturated fusion systems F_{0} ⊆ F over finite p-groups S_{0} ⊴ S such that F_{0} ⊴ F, and let L_{0} ⊆ L be a pair of associated linking systems. Then L_{0} is normal in L (L_{0} ⊴ L) if

(i) Ob(L) = {P ≤ S | P ∩ S_{0} ∈ Ob(L_{0})};

(ii) for all P, Q ∈ Ob(L_{0}) and ψ ∈ Mor_{L}(P, Q), there are morphisms γ ∈ Aut_{L}(S_{0}) and ψ_{0} ∈ Mor_{L_{0}}(γ(P), Q) such that ψ = ψ_{0} ∘ γ|_{P,γ(P)}; and (iii) for all γ ∈ Aut_{L}(S_{0}) and ψ ∈ Mor(L_{0}), γψγ^{−1} ∈ Mor(L_{0}).

Here, for P, Q ∈ Ob(L_{0}), and ψ ∈ Mor_{L_{0}}(P, Q), we write γ(P) = π(γ)(P), γ(Q) = π(γ)(Q), and

γψγ^{−1} = γ|_{Q,γ(Q)} ∘ ψ ∘ (γ|_{P,γ(P)})^{−1} ∈ Mor_{L}(γ(P), γ(Q))

for short. For any such pair L_{0} ⊴ L, the quotient group L/L_{0} is defined by setting

L/L_{0} = Aut_{L}(S_{0})/Aut_{L_{0}}(S_{0}).

Also, L_{0} is centric in L if for each γ ∈ Aut_{L}(S_{0})∖Aut_{L_{0}}(S_{0}), there is ψ ∈ Mor(L_{0}) such that γψγ^{−1} ≠ ψ.

In the situation of Definition 1.27, we will sometimes say that L_{0} ⊴ L is a normal pair of linking systems associated to F_{0} ⊴ F, or just that (S_{0}, F_{0}, L_{0}) ⊴ (S, F, L) is a normal pair.

One source of normal pairs of linking systems is a normal pair of finite groups; at least, under certain conditions.

Proposition 1.28. Fix a pair G_{0} ⊴ G of finite groups, choose S ∈ Syl_{p}(G), and set S_{0} = S ∩ G_{0} ∈ Syl_{p}(G_{0}). Then F_{S_{0}}(G_{0}) ⊴ F_{S}(G). Assume in addition that H_{0} and H are sets of subgroups of S_{0} and S, respectively, such that L_{S_{0}}^{H_{0}}(G_{0}) and L_{S}^{H}(G) are linking systems associated to F_{S_{0}}(G_{0}) and F_{S}(G), and such that H = {P ≤ S | P ∩ S_{0} ∈ H_{0}}. Then L_{S_{0}}^{H_{0}}(G_{0}) ⊴ L_{S}^{H}(G).

Proof. Fix P, Q ≤ S_{0} and g ∈ N_{G}(P, Q). Then gS_{0}g^{−1} is another Sylow p-subgroup of G_{0}, so there is some h ∈ G_{0} such that (h^{−1}g)S_{0}(h^{−1}g)^{−1} = S_{0}. Set a = h^{−1}g; thus g = ha where a ∈ N_{G}(S_{0}) and h ∈ G_{0}. Thus c_{g} = c_{h} ∘ c_{a} ∈ Hom_{G}(P, Q), where c_{a} ∈ Aut_{G}(S_{0}) and c_{h} ∈ Hom_{G_{0}}(aP a^{−1}, Q). This proves condition (ii) in the definition of a normal fusion system; and condition (ii) in Definition 1.27 follows in a similar way. The other conditions clearly hold. □

When (S_{0}, F_{0}, L_{0}) ⊴ (S, F, L) is a normal pair, then for each γ ∈ Aut_{L}(S_{0}), we let c_{γ} ∈ Aut(L_{0}) denote the automorphism which sends P to γ(P) = π(γ)(P) and sends ψ ∈ Mor_{L_{0}}(P, Q) to (γ|_{Q,γ(Q)}) ∘ ψ ∘ (γ|_{P,γ(P)})^{−1}. The next lemma describes how to tell, in terms only of the fusion system F, whether or not c_{δ(g)} = Id_{L_{0}} for g ∈ S (δ = δ_{S_{0}}).

When L is a linking system associated to F, and A ⊴ F, we say that an automorphism α of L is the identity modulo A if for each P, Q ∈ Ob(L) which contain A and each ψ ∈ Mor_{L}(P, Q), α(P) = P, α(Q) = Q, and α(ψ) = ψ ∘ δ_{P} (a) for some a ∈ A.

Lemma 1.29. Let (S_{0}, F_{0}, L_{0}) ⊴ (S, F, L) be a normal pair such that all objects in L are F-centric. Fix A ⊴ F_{0}. Then for g ∈ S, c_{δ(g)} ∈ Aut(L_{0}) is the identity modulo A if and only if [g, S_{0}] ≤ A, and for each P, Q ≤ S_{0} and φ ∈ Mor_{F_{0}}(P, Q), φ extends to some φ ∈ Mor_{F}(⟨P A, g⟩, ⟨QA, g⟩) such that φ(g) ∈ gA.

Proof. Fix g ∈ S. Set γ = δ_{S_{0}}(g) and B = ⟨g, A⟩ for short.

Assume c_{γ} ∈ Aut(L_{0}) is the identity modulo A. Then [g, S_{0}] ≤ A since [γ, δ_{S_{0}}(s)] ∈ δ_{S_{0}}(A) for s ∈ S_{0} (and δ_{S_{0}} is injective by Proposition 1.11(c)). Since F_{0} is generated by morphisms between objects of L_{0} which contain A (by Theorem 1.3 and Proposition 1.5, and since A ⊴ F_{0}), it suffices to prove the extension property for such morphisms. Fix P, Q ∈ Ob(L_{0}) such that A ≤ P and A ≤ Q, fix φ ∈ Hom_{F_{0}}(P, Q), and choose a lifting of φ to ψ ∈ Mor_{L_{0}}(P, Q). By assumption, δ_{Q}(g) ∘ ψ ∘ δ_{P} (g)^{−1} = ψ ∘ δ_{P} (a) for some a ∈ A. So by Proposition 1.11(e), there is a unique morphism ψ ∈ Mor_{L}(P B, QB) such that ψ|_{P,Q} = ψ, and δ_{QB}(g) ∘ ψ ∘ δ_{P B}(ag)^{−1} = ψ by the uniqueness of the extension. Set φ = π(ψ); then φ ∈ Hom_{F}(P B, QB), φ|_{P} = φ, and φ(ag) = g (so φ(g) ∈ gA) by axiom (C).

Now assume [g, S_{0}] ≤ A, and g has the above extension property: each φ ∈ Hom_{F_{0}}(P, Q) extends to φ ∈ Hom_{F}(P B, QB) such that φ(g) ∈ gA. We claim c_{γ} ∈ Aut(L_{0}) is the identity modulo A. Since [g, S_{0}] ≤ A, gP g^{−1} = P for all P ∈ Ob(L_{0}) which contain A. Fix ψ ∈ Mor_{L_{0}}(P, Q), where A ≤ P, Q. By assumption, π(ψ) extends to some φ ∈ Hom_{F}(P B, QB) such that φ(g) ∈ gA, and this lifts to ψ̂ ∈ Mor_{L}(P B, QB). Since P is F-centric, ψ̂|_{P,Q} = ψ ∘ δ_{P} (x) for some x ∈ Z(P). Upon replacing ψ̂ by ψ̂ ∘ δ_{P B}(x)^{−1} and φ by φ ∘ c_{x}^{−1}, we can assume ψ̂|_{P,Q} = ψ. By axiom (C), the conjugation action of δ_{S}(g) fixes ψ̂ modulo δ_{P B}(A), and hence c_{γ} ∈ Aut(L_{0}) sends ψ into ψ ∘ δ_{P} (A). □

The next lemma describes another way to construct normal pairs of linking systems.

Lemma 1.30. Fix a normal pair of fusion systems F_{0} ⊴ F over p-groups S_{0} ⊴ S. Let H_{0} be a set of subgroups of S_{0} such that

• H_{0} is closed under F-conjugacy and overgroups, and contains all subgroups of S_{0} which are F_{0}-centric and F_{0}-radical; and

• H =^{def} {P ≤ S | P ∩ S_{0} ∈ H_{0}} is contained in the set of F-centric subgroups.

Assume F has an associated centric linking system L^{c}. Let L ⊆ L^{c} be the full subcategory with object set H. Let L_{0} ⊆ L be the subcategory with object set H_{0}, where for P, Q ∈ H_{0},

Mor_{L_{0}}(P, Q) = {ψ ∈ Mor_{L}(P, Q) | π(ψ) ∈ Hom_{F_{0}}(P, Q)} .  (2)

Then L_{0} ⊴ L is a normal pair of linking systems associated to F_{0} ⊴ F. For any such pair L_{0} ⊴ L with Ob(L_{0}) = H_{0} and Ob(L) = H, L_{0} is centric in L.

Proof. Since Ob(L) is closed under F-conjugacy and under overgroups, and contains all subgroups which are F-centric and F-radical by Lemma 1.20(d) and the assumptions on H_{0}, L is a linking system associated to F. Since all objects in L_{0} are F-centric, they are also F_{0}-centric, and hence fully centralized in F_{0}. Axiom (A) for L_{0} thus follows from axiom (A) for L, together with the assumptions on H_{0} = Ob(L_{0}). Axioms (B) and (C) for L_{0} follow immediately from those for L, and L_{0} is thus a linking system associated to F_{0}.

Condition (i) in Definition 1.27 holds by assumption, while conditions (ii) and (iii) follow from (2) and since F_{0} is normal in F. Thus L_{0} ⊴ L.

Fix any such pair L_{0} ⊴ L associated to F_{0} ⊴ F. Assume γ ∈ Aut_{L}(S_{0}) is such that γψγ^{−1} = ψ for each ψ ∈ Mor(L_{0}). Since γ(δ_{S_{0}}(g))γ^{−1} = δ_{S_{0}}(π(γ)(g)) for g ∈ S_{0} by axiom (C) for the linking system L, π(γ) = Id_{S_{0}}. Since S_{0} ∈ H_{0} is F-centric, this means that γ = δ_{S_{0}}(z) for some z ∈ Z(S_{0}), and in particular, that γ ∈ Aut_{L_{0}}(S_{0}). So L_{0} is centric in L. □

We now list the examples of normal pairs of linking systems which motivated Definition 1.27, and which we need to refer to later.

Proposition 1.31. Let F be a saturated fusion system over the finite p-group S, let F_{0} ⊴ F be a normal fusion subsystem over S_{0} ⊴ S, and let H_{0} be a set of subgroups of S_{0}. Assume that F has an associated centric linking system L^{c}, and that one of the following three conditions holds.

(a) F_{0} = O^{p}(F), S_{0} = 𝔥𝔶𝔭(F), and H_{0} is the set of F_{0}-centric subgroups of S_{0}.

(b) F_{0} = O^{p′}(F), S_{0} = S, and H_{0} is the set of F_{0}-centric subgroups of S_{0}.

(c) For some normal p-subgroup Q ⊴ F and some normal subgroup K ⊴ Aut(Q) containing Inn(Q), F_{0} = N_{F}^{K}(Q), S_{0} = N_{S}^{K}(Q), and H_{0} is the set of all F_{0}-centric subgroups of S_{0} which contain Q.

Set H = {P ≤ S | P ∩ S_{0} ∈ H_{0}}. Then there is a normal pair of linking systems L_{0} ⊴ L associated to F_{0} ⊴ F with Ob(L_{0}) = H_{0} and Ob(L) = H. For any such normal pair L_{0} ⊴ L, L_{0} is centric in L in cases (b) and (c), and in case (a) if Z(F) = 1. Furthermore, in cases (a) and (b), and in case (c) if Q = O_{p}(F) and K = Inn(Q), H_{0} is Aut(S_{0}, F_{0})-invariant, H is Aut(S, F)-invariant, and L_{0} ⊴ L can be chosen such that L_{0} is Aut_{typ}^{I}(L)-invariant.

Proof. In all cases, F_{0} ⊴ F by Proposition 1.25. Also, H_{0} is Aut(S_{0}, F_{0})-invariant and H is Aut(S, F)-invariant: this is clear in cases (a) and (b), and holds in case (c) when Q = O_{p}(F) (since Q = O_{p}(F_{0}) by Lemma 1.20(e)).

(a) Set S_{0} = 𝔥𝔶𝔭(F), F_{0} = O^{p}(F), and H_{0} = Ob(F_{0}^{c}). By [BCGLO2, Theorem 4.3(a)], a subgroup of S_{0} is F_{0}-quasicentric if and only if it is F-quasicentric. In particular, every F_{0}-centric subgroup of S_{0} is F-quasicentric and hence all subgroups in H are F-quasicentric. By Lemma 1.20(d), H contains all subgroups which are F-centric and F-radical. By Lemma 1.20(c), H_{0} is closed under F-conjugacy, so H is closed under F-conjugacy (and it is clearly closed under overgroups). Hence if L^{q} ⊇ L^{c} is the quasicentric linking system which contains L^{c} constructed in [BCGLO1, Proposition 3.4], then the full subcategory L ⊆ L^{q} with object set H is also a linking system associated to F.

By [BCGLO2, Proposition 2.4], there is a unique map λ: Mor(L^{q}) ⟶ S/S_{0} which sends composites to products and inclusions in L^{q} to the identity, and such that λ(δ_{S}(g)) = [g] for all g ∈ S. By [BCGLO2, Theorem 3.9], there is a p-local finite group (S_{0}, F′_{0}, L_{0}) where for P, Q ∈ Ob(L_{0}),

Mor_{L_{0}}(P, Q) = {ψ ∈ Mor_{L^{q}}(P, Q) | λ(ψ) = 1} .  (3)

Furthermore, F′_{0} is constructed using [BCGLO2, Proposition 3.8] (cf. the proof of [BCGLO2, Theorem 3.9]), and hence (by part (b) of that proposition) it has p-power index in F. Thus F′_{0} = F_{0} by Theorem 1.22(a).

Now, Ob(L_{0}) = H_{0} since L_{0} is a centric linking system. Condition (i) in Definition 1.27 holds for L_{0} ⊆ L by definition of H, condition (iii) (γL_{0}γ^{−1} = L_{0} for γ ∈ Aut_{L}(S_{0})) holds by construction, and condition (ii) holds since λ|_{δ_{S_{0}}(S)} is surjective. So L_{0} ⊴ L.

We next check that L_{0} is Aut_{typ}^{I}(L)-invariant. Fix α ∈ Aut_{typ}^{I}(L) and set β = μ̃_{L}(α) ∈ Aut(S, F). Then β(S_{0}) = S_{0} since S_{0} = 𝔥𝔶𝔭(F), and β|_{S_{0}} ∈ Aut(S_{0}, F_{0}) by the uniqueness of F_{0} (Theorem 1.22(a) again). Since α(P) = β(P) for P ∈ Ob(L_{0}) (Lemma 1.15), α sends Ob(L_{0}) = H_{0} to itself. By Lemma 1.17, α = α|_{L} for some α ∈ Aut_{typ}^{I}(L^{q}), λ∘α = β ∘λ (where β ∈ Aut(S/S_{0}) is induced by β) by the uniqueness of λ, and hence α(Mor(L_{0})) = Mor(L_{0}) by (3).

Now let L_{0} ⊴ L be any normal pair of linking systems associated to F_{0} ⊴ F with these objects. Assume Z(F) = 1; we must show L_{0} is centric in L. Assume γ ∈ Aut_{L}(S_{0}) is such that γψγ^{−1} = ψ for each ψ ∈ Mor(L_{0}). Since γ(δ_{S_{0}}(g))γ^{−1} = δ_{S_{0}}(π(γ)(g)) for g ∈ S_{0} by axiom (C) for the linking system L, π(γ) = Id_{S_{0}}. So by axiom (A) (and since S_{0} is fully centralized in F), γ = δ_{S_{0}}(h) for some h ∈ C_{S}(S_{0}).

Let H ≤ C_{S}(S_{0}) be the subgroup of all h such that the conjugation action of δ_{S_{0}}(h) on L_{0} is trivial. The p-group L/L_{0} = Aut_{L}(S_{0})/Aut_{L_{0}}(S_{0}) acts on δ_{S_{0}}(H) ≅ H by conjugation. Let H_{0} be the fixed subgroup of this action. Note that H_{0} ≤ Z(S) since H_{0} is fixed by δ_{S_{0}}(S) ≤ Aut_{L}(S_{0}). Fix h ∈ H_{0}, and set γ̄ = δ_{S}(h). Let γ̄(P) and γ̄ψγ̄^{−1} be as in Definition 1.27, but this time for all P ∈ Ob(L) and ψ ∈ Mor(L). For P ∈ Ob(L), γ̄(P) = hP h^{−1} = P.

Also, γ̄ψγ̄^{−1} = ψ for all ψ ∈ Aut_{L}(S_{0}) by definition of H_{0}, γ̄ψγ̄^{−1} = ψ for ψ ∈ Mor(L_{0}) by definition of H, and hence conjugation by γ̄ is the identity on morphisms in L between subgroups in H_{0} by condition (ii) in Definition 1.27. By Proposition 1.11(f), for each P, Q ∈ H, the restriction map from Mor_{L}(P, Q) to Mor_{L}(P ∩ S_{0}, Q ∩ S_{0}) is injective, and hence γ̄ψγ̄^{−1} = ψ for all ψ ∈ Mor_{L}(P, Q). Thus conjugation by γ̄ = δ_{S}(h) is the identity on L, and so h ∈ Z(F) = 1 by Lemma 1.14(a). Since H_{0} = 1 is the fixed subgroup of an action of the p-group L/L_{0} on the p-group H, H = 1, and so L_{0} is centric in L.

(b) Set F_{0} = O^{p′}(F). By [BCGLO2, Proposition 3.8(c)], a subgroup of S is F_{0}-centric if and only if it is F-centric. So upon letting H_{0} = H be the set of all F-centric subgroups of S, the hypotheses of Lemma 1.30 are satisfied. By the lemma, there is a normal pair L_{0} ⊴ L of linking systems associated to F_{0} ⊴ F with object set H_{0} = H; and for any such pair, L_{0} is centric in L. By the explicit description of L_{0} (formula (2) in Lemma 1.30), L_{0} is Aut_{typ}^{I}(L)-invariant.

(c) Fix Q ⊴ F and Inn(Q) ≤ K ⊴ Aut(Q), and set S_{0} = N_{S}^{K}(Q) and F_{0} = N_{F}^{K}(Q). Let H_{0} be the set of all F_{0}-centric subgroups of S_{0} which contain Q. We first check that all subgroups in H = {P ≤ S | P ∩S_{0} ∈ H_{0}} are F-centric; it suffices to show this for subgroups in H_{0}. By Lemma 1.20(c) (and since F_{0} ⊴ F), the set of F_{0}-centric subgroups, and hence also the set H_{0}, are closed under F-conjugacy. For each P ∈ H_{0}, C_{S}(P) ≤ C_{S}(Q) ≤ S_{0} since P ≥ Q, and hence C_{S}(P) = C_{S_{0}}(P) = Z(P) since P is F_{0}-centric. Since this holds for all subgroups F-conjugate to P, we conclude that P is F-centric.

We just saw that H_{0} is closed under F-conjugacy, and it is clearly closed under overgroups. Since Q ⊴ F_{0}, each subgroup of S_{0} which is F_{0}-centric and F_{0}-radical contains Q by Proposition 1.5, and thus lies in H_{0}. So by Lemma 1.30, there is a normal pair L_{0} ⊴ L of linking systems associated to F_{0} ⊴ F with object sets H_{0} and H, and for any such pair, L_{0} is centric in L. If Q = O_{p}(F) and K = Inn(Q), then Q = O_{p}(F_{0}) by Lemma 1.20(e), and so F_{0} is Aut(S, F)-invariant. Hence L_{0} is Aut_{typ}^{I}(L)-invariant by the explicit description of L_{0} in Lemma 1.30. □

2. Reduced fusion systems and tame fusion systems

Throughout this section, p denotes a fixed prime, and we work with fusion systems over finite p-groups. We first define reduced fusion systems and the reduction of a fusion system. We then define tame fusion systems, and prove that a reduced fusion system is tame if every saturated fusion system which reduces to it is realizable (Theorem B). We then make a digression to look at the existence of linking systems in certain situations, before proving that all fusion systems whose reduction is tame are realizable (Theorem A). We thus end up with a way to “detect” exotic fusion systems in general while looking only at reduced fusion systems.

2.1. Reduced fusion systems and reductions of fusion systems.

We begin with the definition of a reduced fusion system, and the reduction of an (arbitrary) fusion system. See Proposition 1.8 and the discussion before that for the definition and properties of quotient fusion systems.

Definition 2.1. A reduced fusion system is a saturated fusion system F such that

• F has no nontrivial normal p-subgroups,

• F has no proper normal subsystem of p-power index, and • F has no proper normal subsystem of index prime to p.

Equivalently, F is reduced if O_{p}(F) = 1, O^{p}(F) = F, and O^{p′}(F) = F.

For any saturated fusion system F, the reduction of F is the fusion system 𝔯𝔢𝔡(F) defined as follows. Set F_{0} = C_{F}(O_{p}(F))/Z(O_{p}(F)), and let F_{0} ⊇ F_{1} ⊇ F_{2} ⊇ ⋯ ⊇ F_{m} be such that F_{i} = O^{p}(F_{i−1}) if i is odd, F_{i} = O^{p′}(F_{i−1}) if i is even, and O^{p}(F_{m}) = O^{p′}(F_{m}) = F_{m}. Then 𝔯𝔢𝔡(F) = F_{m}.

Fix any F, and set Q = O_{p}(F) for short. By definition of centralizer fusion systems, every morphism in C_{F}(Q) extends to a morphism in F which is the identity on Q, and hence to a morphism in C_{F}(Q) which is the identity on Z(Q). This proves that Z(Q) is always central in C_{F}(Q), and hence that F_{0} = C_{F}(Q)/Z(Q) is well defined as a fusion system.

What is important in the last part of the definition of 𝔯𝔢𝔡(F) is that we give an explicit procedure for successively applying O^{p}(−) and O^{p′}(−), starting with F_{0}, until neither makes the fusion system any smaller. It seems likely that the final result 𝔯𝔢𝔡(F) is independent of the order in which we apply these reductions, but we have not shown this, and do not need to know it when proving the results in this section.

Clearly, for these definitions to make sense, we want 𝔯𝔢𝔡(F) to always be reduced.

Proposition 2.2. The reduction of any saturated fusion system is reduced.

For later reference, we also state the following, more technical result, which will be proven together with Proposition 2.2.

Lemma 2.3. Let F be a saturated fusion system. Set Q = O_{p}(F) and F_{0} = C_{F}(Q)/Z(Q). Let F_{0} ⊇ F_{1} ⊇ ⋯ ⊇ F_{m} = 𝔯𝔢𝔡(F) be such that for each i, F_{i} = O^{p}(F_{i−1}) or F_{i} = O^{p′}(F_{i−1}). Then O_{p}(F_{i}) = 1 for each 0 ≤ i ≤ m.

Proof. Fix F, and let Q ⊴ F and the F_{i} be as above. Since C_{F}(Q) ⊴ F by Proposition 1.25(c), O_{p}(C_{F}(Q)) ≤ O_{p}(F) = Q by Lemma 1.20(e). Hence O_{p}(C_{F}(Q)) = Z(Q). We just saw that Z(Q) is central in C_{F}(Q). So by Proposition 1.8, a subgroup P/Z(Q) ≤ C_{S}(Q)/Z(Q) is normal in C_{F}(Q)/Z(Q) only if P ⊴ C_{F}(Q). Thus O_{p}(F_{0}) = Z(Q)/Z(Q) = 1.

By definition, O^{p}(𝔯𝔢𝔡(F)) = O^{p′}(𝔯𝔢𝔡(F)) = 𝔯𝔢𝔡(F). By Proposition 1.25(a,b), F_{i} ⊴ F_{i−1} for each i ≥ 1. So by Lemma 1.20(e) again, O_{p}(F_{i}) = 1 if O_{p}(F_{i−1}) = 1. Since O_{p}(F_{0}) = 1, this proves that O_{p}(F_{i}) = 1 for each i. In particular, O_{p}(𝔯𝔢𝔡(F)) = 1, and hence 𝔯𝔢𝔡(F) is reduced. □

A saturated fusion system F is constrained if there is a normal subgroup Q ⊴ F which is F-centric (cf. [BCGLO1, § 4]).

Proposition 2.4. For any saturated fusion system F, 𝔯𝔢𝔡(F) = 1 (the fusion system over the trivial group) if and only if F is constrained.

Proof. If F is constrained, then clearly 𝔯𝔢𝔡(F) = 1. Conversely, assume F is a fusion system over a finite p-group S such that 𝔯𝔢𝔡(F) = 1. Set Q = O_{p}(F) and F_{0} = C_{F}(Q)/Z(Q). If F_{0} = 1, then C_{F}(Q) is a fusion system over Z(Q), and hence C_{S}(Q) = Z(Q). So Q is F-centric, and hence F is constrained in this case.

If F_{0} ≠ 1, then there is a sequence of fusion subsystems 1 = F_{m} ⫋ F_{m−1} ⫋ ⋯ ⫋ F_{0} such that for each i, F_{i+1} = O^{p}(F_{i}) or F_{i+1} = O^{p′}(F_{i}). By Lemma 2.3, O_{p}(F_{i}) = 1 for each 0 ≤ i ≤ m. Since F_{m−1} ≠ 1, it is a fusion system over a p-group S_{m−1} ≠ 1, so O^{p′}(F_{m−1}) ≠ 1 (it is over the same p-group), which implies O^{p}(F_{m−1}) = 1. Thus 𝔥𝔶𝔭(F_{m−1}) = 1 by Definition 1.21(a), so there are no nontrivial automorphisms of order prime to p in F_{m−1}, and F_{m−1} is the fusion system of the p-group S_{m−1}. This is impossible, since it would imply O_{p}(F_{m−1}) = S_{m−1} ≠ 1, and we conclude F_{0} = 1. □

2.2. Tame fusion systems and the proof of Theorem B.

Assume F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G). Let H be an Aut(G, S)-

invariant set of G-quasicentric subgroups of S such that L =^{def} L_{S}^{H}(G) is a linking system associated to F (i.e. H is closed under overgroups and contains all F-centric F-radical subgroups). Define the homomorphism

κ̃_{G}^{H}:Aut(G, S) ⟶ Aut_{typ}^{I}(L)

as follows. For β ∈ Aut(G, S), κ̃_{G}^{H}(β) sends P to β(P) and sends [a] ∈ Mor_{L}(P, Q) (for a ∈ N_{G}(P, Q)) to [β(a)].

For any g ∈ N_{G}(S), κ̃_{G}^{H} sends c_{g} ∈ Aut(G, S) to c_{[g]} ∈ Aut_{typ}^{I}(L), where [g] ∈ Aut_{L}(S) is the class of g. Thus by Lemma 1.14, κ̃_{G}^{H} induces a homomorphism

κ_{G}^{H}: Out(G) ⟶ Out_{typ}(L)

by sending the class of β to the class of κ̃_{G}^{H}(β). When L = L_{S}^{c}(G) is the centric linking system of G, we write κ̃_{G} = κ̃_{G}^{H} and κ_{G} = κ_{G}^{H} for short.

Note that when F = F_{S}(G) and L = L_{S}^{H}(G) as above, μ̃_{G}^{H} ∘κ̃_{G}^{H}: Aut(G, S) ⟶ Aut(S, F) is the restriction homomorphism.

Definition 2.5. A saturated fusion system F over S is tame if there is a finite group G which satisfies:

• S ∈ Syl_{p}(G) and F ≅ F_{S}(G); and

• κ_{G}: Out(G) ⟶ Out_{typ}(L_{S}^{c}(G)) is split surjective.

In this situation, we say F is tamely realized by G.

The condition that κ_{G} be split surjective was chosen since, as we will see shortly, that is what is needed in the proof of Theorem B. In contrast, Theorem A would still be true (with essentially the same proof) if we replaced “split surjective” by “an isomorphism” in the above definition.

By Lemma 1.17, Out_{typ}(L_{S}^{c}(G)) ≅ Out_{typ}(L_{S}^{H}(G)) for any Aut(S, F)-invariant set of objects H (which satisfies the conditions for L_{S}^{H}(G) to be a linking system). Hence Ker(κ_{G}^{H}) = Ker(κ_{G}), and κ_{G}^{H} is (split) surjective if and only if κ_{G} is.

By [BLO1, Theorem B], κ_{G} is split surjective if and only if the natural map from Out(G) to Out(BG_{p}^{∧}) is split surjective, where Out(BG_{p}^{∧}) is the group of homotopy classes of self equivalences of BG_{p}^{∧}. So this gives another way to formulate the definition of tameness.

It is natural to ask whether a tame fusion system F can always be realized by a finite group G such that κ_{G} is an isomorphism. We know of no counterexamples to this, but do not know how to prove it either.

We are now ready to prove Theorem B: every reduced fusion system which is not tame is the reduction of some exotic fusion system. This is basically a consequence of the definition of tameness, together with [O3, Theorem 9] which gives a general procedure for constructing extensions of linking systems.

Theorem 2.6. Let F be a reduced fusion system which is not tame. Then there is an exotic fusion system F whose reduction is isomorphic to F.

Proof. If F is itself exotic, then we take F = F. So assume F is the fusion system of a finite group, and hence that F has at least one associated centric linking system L. Assume L is chosen such that |Out_{typ}(L)| is maximal among all centric linking systems associated to F. (All such linking systems are isomorphic to each other by [BLO2, Proposition 3.1] and Theorem A in [O1, O2], but since those results use the classification of finite simple groups, we will not use them here.) Since F is not tame, it is not the fusion system of any finite group H for which κ_{H} is split surjective.

Since Z(F) = 1 (F is reduced), we can identify Aut_{L}(S) as a normal subgroup of Aut_{typ}^{I}(L) via its conjugation action on L (Lemma 1.14(a)). Thus Out_{typ}(L) = Aut_{typ}^{I}(L)/Aut_{L}(S). Let A be any finite abelian p-group on which Out_{typ}(L) acts faithfully, and let

ν: Aut_{typ}^{I}(L) ⟶ Aut(A)

denote the given action. Thus Ker(ν) = Aut_{L}(S).

Set S_{0} = A × S and F_{0} = A × F (= F_{A}(A) × F). We refer to the beginning of Section 3, or to [BLO2, § 1], for the definition of the product of two fusion systems. Set L_{0} = A × L: the centric linking system associated to F_{0} whose objects are the subgroups A × P ≤ S_{0} for P ∈ Ob(L), and where Mor_{L_{0}}(A × P, A × Q) = A × Mor_{L}(P, Q).

Set Γ_{0} = Aut_{L_{0}}(S_{0}) = A × Aut_{L}(S). Set Γ = A ⋊ Aut_{typ}^{I}(L): the semidirect product taken with respect to the action ν of Aut_{typ}^{I}(L) on A. Thus Γ_{0} embeds as a normal subgroup of Γ, and Γ/Γ_{0} ≅ Out_{typ}(L). To avoid confusion, an element ψ ∈ Aut_{L}(S) will be written c_{ψ} when regarded as an element of Γ_{0} ⊴ Γ.

We claim the given Γ-action on L_{0} satisfies the hypotheses of [O3, Theorem 9]. This means checking that the following diagram commutes:

\xymatrix{A × Aut_{L}(S) = Γ_{0} \ar[rr]^{conj} \ar[d]|{incl} &  & Aut_{typ}^{I}(L_{0}) = Aut_{typ}^{I}(A × L) \ar[dl]^{(α↦α_{A×S})} \\ A ⋊ Aut_{typ}^{I}(L) = Γ \ar[r]^{conj} \ar@{-->}[urr]|{τ} & Aut(Γ_{0}) ,}

where τ sends (a, γ) ∈ A ⋊ Aut_{typ}^{I}(L) to (ν(γ), γ) ∈ Aut_{typ}^{I}(A × L). For ψ ∈ Aut_{L}(S), ν(c_{ψ}) = Id_{A}, so τ(a, c_{ψ}) = (Id, c_{ψ}), which shows that the upper triangle commutes. As for the lower triangle, for (a, γ) ∈ Γ and (b, c_{ψ}) ∈ Γ_{0},

τ(a, γ)(b, c_{ψ}) = (ν(γ)(b), c_{γ(ψ)}) = (ν(γ)(b)·a, c_{γ(ψ)} ∘ γ)(a, γ)^{−1} = (a, γ)(b, c_{ψ})(a, γ)^{−1}

(since c_{γ(ψ)} = γ ∘ c_{ψ} ∘ γ^{−1}); and thus the lower triangle commutes.

Fix S ∈ Syl_{p}(Γ). We identify S_{0} = O_{p}(Γ_{0}) (via δ_{S_{0}}), and hence S_{0} ⊴ S. Since

C_{Γ}(S_{0}) = C_{Γ}(A × S) = C_{Γ_{0}}(A × S) = A × C_{Aut_{L}(S)}(S) = A × Z(S)

is a p-group, and since all objects in L_{0} are F_{0}-centric by construction, [O3, Theorem 9] shows that there exists a saturated fusion system F over S and an associated linking system L such that (S_{0}, F_{0}, L_{0}) ⊴ (S, F, L) and Aut_{L}(S_{0}) = Γ with the action on L_{0} given by τ. In particular,

Aut_{F}(S_{0}) = Aut_{Γ}(S_{0}) = {(ν(γ), μ̃_{L}(γ)) | γ ∈ Aut_{typ}^{I}(L)} .  (1)

Assume F is realizable: the fusion system of a finite group G. Since A is central in F_{0}, O_{p}(F_{0})/A is normal in F_{0}/A ≅ F by Proposition 1.8. Since O_{p}(F) = 1 (F is reduced), this shows that O_{p}(F_{0}) = A. By Lemma 1.20(e) we then get A ⊴ F. By Proposition 1.24 we have F ≅ F_{S}(G) = F_{S}(N_{G}(A)). Upon replacing G by N_{G}(A), we may assume A ⊴ G.

Set G_{0} = C_{G}(A) and G = G_{0}/A. Assume the following two statements hold:

(i) A = O_{p}(F) and C_{F}(A) = F_{0}.

(ii) The composite

ξ: Aut_{G}(A) ≅ G/G_{0} ⟶^{conj} Out(G) ⟶^{κ_{G}} Out_{typ}(L_{S}^{c}(G))

is injective.

We now finish the proof of the theorem, assuming (i) and (ii).

By (i), C_{F}(O_{p}(F))/Z(O_{p}(F)) = F_{0}/A ≅ F. Since O^{p}(F) = O^{p′}(F) = F, this shows 𝔯𝔢𝔡(F) ≅ F. Also, S_{0} = C_{S}(A) ∈ Syl_{p}(C_{G}(A)) since F_{0} = C_{F}(A). Hence by Proposition 1.24 (applied with K = 1), F_{0} = C_{F}(A) = F_{S_{0}}(C_{G}(A)), and so

F ≅ F_{0}/A ≅ F_{S_{0}/A}(G_{0}/A) ≅ F_{S}(G) .

By condition (ii) in Definition 1.18 (applied to F_{0} ⊴ F), and since A ⊴ F and A ≤ S_{0}, each φ ∈ Aut_{F}(A) has the form φ = φ_{0} ∘ α|_{A,A} for some α ∈ Aut_{F}(S_{0}) = Aut_{Γ}(S_{0}) and some φ_{0} ∈ Aut_{F_{0}}(A). Also, Aut_{F_{0}}(A) = 1 by definition, and thus

Aut_{G}(A) = Aut_{F}(A) = Aut_{Γ}(A) ≅ Γ/Γ_{0} ≅ Out_{typ}(L) .

So by (ii), there is a homomorphism s from Out_{typ}(L) to Out(G) such that κ_{G} ∘s is injective. Since L was chosen with |Out_{typ}(L)| maximal, κ_{G} ∘ s is an isomorphism, so κ_{G} is split surjective, contradicting the assumption that F is not tame. We conclude that F is exotic (and 𝔯𝔢𝔡(F) ≅ F).

It remains to prove (i) and (ii).

Proof of (i): For each P ∈ Ob(L), P_{0} = P ∩ S_{0} ∈ Ob(L_{0}), so P_{0} = A × Q for some Q ≤ S which is F-centric. Then C_{S}(P) ≤ C_{S}(A) = S ∩ C_{Γ}(A) = S ∩ Γ_{0} = S_{0}, so C_{S}(P) ≤ A × Z(Q) ≤ P. Since this holds for all subgroups F-conjugate to P, all objects in L are F-centric.

Set B = O_{p}(F). We already saw that A = O_{p}(F_{0}). Hence B ∩ S_{0} = A by Lemma 1.20(e). Since B ⊴ S and S_{0} ⊴ S, it follows that [B, S_{0}] ≤ A.

Since A ⊴ F_{0} and B ⊴ F, each φ ∈ Hom_{F_{0}}(P, Q) can be extended to a morphism φ ∈ Hom_{F}(P B, QB) such that φ|_{P A} ∈ Hom_{F_{0}}(P A, QA) and φ(B) = B. Then φ|_{A} = Id_{A}. Hence for each g ∈ B, g and φ(g) have the same conjugation action on A, and g^{−1}φ(g) ∈ C_{B}(A) = B ∩ C_{S}(A) = B ∩ S_{0} = A. By Lemma 1.29, c_{δ(g)} = τ(δ_{S_{0}}(g)) ∈ Aut_{typ}^{I}(L_{0}) is the identity modulo A, and thus g ∈ A by definition of τ. So B = O_{p}(F) = A.

Since C_{S}(A) = S_{0}, F_{0} and C_{F}(A) are both fusion systems over S_{0}. Also, F_{0} ⊆ C_{F}(A) since A is central in F_{0}. To see that C_{F}(A) = F_{0}, fix P, Q ≤ S_{0} and φ ∈ Hom_{C_{F} (A)}(P, Q). By definition, φ extends to φ ∈ Hom_{F}(AP, AQ) with φ(A) = A and φ|_{A} = Id_{A}. Since F_{0} ⊴ F, condition (ii) in Definition 1.18 shows that there are α ∈ Aut_{F}(S_{0}) and φ_{0} ∈ Hom_{F_{0}}(α(AP), AQ) such that φ = φ_{0} ∘ α|_{AP,α(AP)}. For each a ∈ A, α(a) = φ_{0}^{−1}(φ(a)) = a, and thus α|_{A} = Id_{A}. Hence

α ∈ {β ∈ Aut_{F}(S_{0}) | β|_{A} = Id_{A}} = {Id_{A}} × Aut_{F}(S) = Aut_{F_{0}}(S_{0}) ,

where the first equality holds by (1) (and since μ̃_{L}(Aut_{L}(S)) = Aut_{F}(S)). Thus α ∈ Aut_{F_{0}}(S_{0}), so φ ∈ Mor(F_{0}), and hence also φ ∈ Mor(F_{0}). This proves that C_{F}(A) = F_{0}.

Proof of (ii): Set

L^{∗} = L_{S}^{c}(G),  L_{0}^{∗} = L_{S_{0}}^{H_{0}}(G_{0}) = L_{S_{0}}^{c}(G_{0}),  and L^{∗} = L\begin{matrix}H \\ S \end{matrix}(G) ,

where H_{0} = Ob(L_{0}) and H = Ob(L). Note that (S_{0}, F_{0}, L_{0}^{∗}) ⊴ (S, F, L^{∗}) by Proposition 1.28. Let cj: G ⟶ Aut(G) denote the conjugation action of G on G. Set

H = {g ∈ N_{G}(S_{0}) | κ̃_{G}(cj(g)) = Id_{L^{∗}}} and T = H ∩ S .

We first claim T = A. By [BCGLO2, Theorem 6.8], L_{0}^{∗}/A is a centric linking system associated to F_{0}/A ≅ F. Hence the natural functor L_{0}^{∗}/A ⟶ L^{∗} (induced by the projection G_{0} ⟶ G) is an isomorphism, since it commutes with the structure functors. So for g ∈ S, g ∈ T if and only if c_{δ(g)} ∈ Aut_{typ}^{I}(L_{0}^{∗}) is the identity modulo A, in the sense of Lemma 1.29. We showed in the proof of (i) that each P ∈ Ob(L^{∗}) = Ob(L) is Fcentric. Hence by Lemma 1.29, applied to both normal pairs (S_{0}, F_{0}, L_{0}) ⊴ (S, F, L) and (S_{0}, F_{0}, L_{0}^{∗}) ⊴ (S, F, L^{∗}), g ∈ T if and only if c_{δ(g)} ∈ Aut_{typ}^{I}(L_{0}) is the identity modulo A; i.e., induces the identity on L. By definition of S ≤ Γ = A ⋊ Aut_{typ}^{I}(L), this is the case exactly when g ∈ A.

Thus H is a normal subgroup of N_{G}(S_{0}) whose intersection with its Sylow p-subgroup S is A. It follows that H/A has order prime to p. We claim that H ≤ G_{0}. Fix h ∈ H of order prime to p. Then cj(h) ∈ Aut(G) acts via the identity on S = S_{0}/A, so [h, S_{0}] ≤ A. Hence by (1), c_{h} = (ν(γ), Id_{S}) ∈ Aut_{G}(S_{0}) for some γ ∈ Aut_{typ}^{I}(L) such that γ ∈ Ker(μ̃_{L}). Since Ker(μ_{L}) is a p-group by Lemma 1.16 and h has order prime to p, γ ∈ Aut_{L}(S), so ν(γ) = 1 ∈ Aut(A), and h ∈ G_{0}. Thus H = O^{p}(H)·A ≤ G_{0}.

Fix g ∈ G such that c_{g} ∈ Ker(ξ). Recall we are only interested in g modulo C_{G}(A) = G_{0}. Since G = G_{0}·N_{G}(S_{0}) by the Frattini argument, we can assume g normalizes S_{0}. Thus κ_{G}([cj(g)]) = 1 in Out_{typ}(L^{∗}), so κ̃_{G}(cj(g)) = c_{γ} for some γ ∈ Aut_{L^{∗}}(S). Let h ∈ N_{G}(S) be such that γ = [h] and lift h to h̃ ∈ N_{G_{0}}(S_{0}). Upon replacing g by h̃^{−1}g, we can assume κ̃_{G}(cj(g)) = Id_{L^{∗}}, and thus g ∈ H ≤ G_{0}. Hence c_{g} = Id_{A}, ξ is injective, and this finishes the proof of (ii). □

2.3. Strongly tame fusion systems and linking systems for extensions.

We are now ready to start working on the proof of Theorem A. As stated in the introduction, this proof uses the vanishing of certain higher limit groups, and through that depends on the classification of finite simple groups. In order to have a clean statement which does not depend on the classification (Theorem 2.20), we first define a certain class of finite groups which in fact (using the classification) is shown to include all finite groups.

The obstruction groups for the existence and uniqueness of centric linking systems associated to a given saturated fusion system are higher derived functors for inverse limits taken over the centric orbit category of the fusion system. We begin by defining this category.

Definition 2.7. Let F be a fusion system over a finite p-group S, and let F^{c} ⊆ F be the full subcategory whose objects are the F-centric subgroups of S. The centric orbit category O(F^{c}) of F is the category with Ob(O(F^{c})) = Ob(F^{c}), and where

Mor_{O(F^{c})}(P, Q) = Inn(Q)\Hom_{F}(P, Q)

for any pair of objects P, Q ≤ S. In particular, Aut_{O(F^{c})}(P) = Out_{F}(P) for each P. If F_{0} ⊆ F^{c} is any full subcategory, then O(F_{0}) denotes the full subcategory of O(F^{c}) with the same objects as F_{0}.

We need the following technical result about higher limits over these orbit categories.

Lemma 2.8. Let F be a saturated fusion system over a finite p-group S. Let H ⊆ Ob(F^{c}) be any subset which is closed under F-conjugacy and overgroups, and let F^{H} ⊆ F^{c} be the full subcategory with object set H. Fix a functor F: O(F^{c})^{op} ⟶ ℤ_{(p)}-mod. Assume, for each P ∈ Ob(F^{c})∖H, that either O_{p}(Out_{F}(P)) ≠ 1, or some element of order p in Out_{F}(P) acts trivially on F (P). Let F_{0} : O(F^{c})^{op} ⟶ ℤ_{(p)}-mod be the functor where F_{0}(P) = F (P) if P ∈ H and F_{0}(P) = 0 otherwise. Then

\underleftarrow{lim}^{∗}_{O(F^{c})}(F) ≅ \underleftarrow{lim}^{∗}_{O(F^{c})}(F_{0}) ≅ \underleftarrow{lim}^{∗}_{O(F^{H})}(F |_{O(F^{H})^{op}}) .

Proof. Let F_{1} ⊆ F be the subfunctor defined by setting F_{1}(P) = F (P) if P ∉ H and F_{1}(P) = 0 otherwise. Thus F_{0} ≅ F/F_{1}. By [O1, Lemma 2.3], \underleftarrow{lim}_{O(F^{c})}^{∗}(F_{1}) = 0 if certain graded groups Λ^{∗}(Out_{F}(P); F (P)) vanish for each P ∈ Ob(F^{c})∖H. By [JMO, Proposition 6.1(ii)], this is the case whenever O_{p}(Out_{F}(P)) ≠ 1, or some element of order p in Out_{F}(P) acts trivially on F (P). This proves the first isomorphism.

For any category C, let C-mod be the category of contravariant functors from C to abelian groups. Let E be the functor “extension by zero” from O(F^{H})-mod to O(F^{c})-mod. Since H ⊆ Ob(F^{c}) is closed under F-conjugacy and overgroups, E is right adjoint to the restriction functor. Thus E sends injectives to injectives. So for Φ in O(F^{H})-mod, \underleftarrow{lim}_{O(F^{H})}^{∗}(Φ) ≅ \underleftarrow{lim}_{O(F^{c})}^{∗}(E(Φ)). Since F_{0} = E(F |_{O(F^{H})^{op}}), the second isomorphism now follows. □

For any saturated fusion system F over a finite p-group S, let

Z_{F}: O(F^{c})^{op} ⟶ ℤ_{(p)}-mod

be the functor which sends an object P of F^{c} to Z(P) = C_{S}(P). For each φ ∈ Hom_{F}(P, Q) in F^{c},

Z_{F}([φ]) = φ^{−1}|_{Z(Q)} : Z(Q) ⟶ Z(P).

When F = F_{S}(G) for some finite group G and some S ∈ Syl_{p}(G), and H ⊴ G is a normal subgroup, we let Z_{F}^{H} ⊆ Z_{F} be the subfunctor which sends P to Z(P) ∩ H = Z_{F}(P) ∩ H.

The obstruction to the existence of a centric linking system associated to F lies in \underleftarrow{lim}_{O(F^{c})}^{3}(Z_{F}), and the obstruction to its uniqueness lies in \underleftarrow{lim}_{O(F^{c})}^{2}(Z_{F}) [BLO2, Proposition 3.1]. The main results in [O1] and [O2] state that these groups vanish whenever F is the fusion system of a finite group G.

Definition 2.9. Fix a prime p.

(a) Let 𝔏̂(p) be the class of all nonabelian finite simple groups L with the following property. For any finite group G, and any pair of subgroups H ⊴ K ⊴ G both normal in G such that K/H ≅ L^{m} for some m ≥ 1, if we set F = F_{S}(G) for some S ∈ Syl_{p}(G), then \underleftarrow{lim}_{O(F^{c})}^{i}(Z_{F}^{K}/Z_{F}^{H}) = 0 for i ≥ 2.

(b) Let 𝔊(p) be the class of all finite groups G all of whose nonabelian composition factors lie in 𝔏̂(p).

(c) A saturated fusion system F over a finite p-group S is strongly tame if it is tamely realizable by a group G ∈ 𝔊(p).

We first show that by results in [O1] and [O2], for each prime p, all nonabelian finite simple groups are in 𝔏̂(p). Hence all finite groups are in 𝔊(p), and all tame fusion systems are strongly tame.

Fix a finite group G with S ∈ Syl_{p}(G), and set F = F_{S}(G). Let O_{p}(G) be the p-subgroup orbit category of G as defined in [O2] and [O1]. Let Z_{G}: O_{p}(G)^{op} ⟶ Ab be the functor Z_{G}(P) = Z(P) if P is F-centric and Z_{G}(P) = 0 otherwise. For H ⊴ G, let Z_{G}^{H} ⊆ Z_{G} be the subfunctor Z_{G}^{H}(P) = Z(P) ∩ H if P is F-centric. By [O1, Lemma 2.1], for K ⊴ H ⊴ G both normal in G, \underleftarrow{lim}_{O(F^{c})}^{∗}(Z_{F}^{H}/Z_{F}^{K}) ≅ \underleftarrow{lim}_{O_{p}(G)}^{∗}(Z_{G}^{H}/Z_{G}^{K}). In particular, Z_{F} and Z_{G} have the same higher limits. This allows us, in the proofs of the next theorem and lemma, to apply the results in [O1] and [O2], which are all stated in terms of limits taken over O_{p}(G).

Theorem 2.10. For each prime p, the class 𝔏̂(p) contains all nonabelian finite simple groups, and the class 𝔊(p) contains all finite groups. Hence all tame fusion systems are strongly tame.

Proof. The last two statements follow immediately from the first one and the definitions. So we need only show that 𝔏̂(p) contains all nonabelian finite simple groups.

Assume p is odd. By [O1, Proposition 4.1] (and its proof), a nonabelian finite simple group L with S ∈ Syl_{p}(L) lies in 𝔏̂(p) if there is a subgroup Q ≤ 𝔛_{L}(S) which is centric in S (i.e., C_{S}(Q) ≤ Q) and not Aut(L)-conjugate to any other subgroup of S. Here, 𝔛_{L}(S) is a certain subgroup of S defined in [O1, §§ 3–4]. By [O1, Propositions 4.2–4.4] (and the classification theorem), all nonabelian finite simple groups have this property, and thus they all lie in 𝔏̂(p).

If p = 2, then by [O2, Proposition 2.7], a nonabelian finite simple group L is contained in 𝔏̂(2) if it is contained in the class 𝔏^{≥2}(2) defined in [O2, Definition 2.8]. By [O2, Theorems 5.1, 6.2, 7.5, 8.13, & 9.1] and the classification theorem, all nonabelian finite simple groups are contained in 𝔏^{≥2}(2). □

Theorem 2.10 together with Theorem 2.20 will imply Theorem A. From now on, for the rest of the section, we avoid using the classification theorem by assuming whenever necessary that our groups are in 𝔊(p) and applying the following lemma.

Lemma 2.11. Fix a finite group G with Sylow subgroup S ∈ Syl_{p}(G), and set F = F_{S}(G). Assume G ∈ 𝔊(p). Then the following hold.

(a) Assume Ĝ is a finite group with G ⊴ Ĝ. Fix Ŝ ∈ Syl_{p}(Ĝ) such that S = Ŝ ∩ G, and set F̂ = FŜ(Ĝ). Then \underleftarrow{lim}_{O(F̂^{c})}^{i}(ZG ; F̂) = 0 for i ≥ 2.

(b) If G = G_{1} × G_{2}, then G ∈ 𝔊(p) if and only if G_{1}, G_{2} ∈ 𝔊(p). If H ⊴ G and G/H is p-solvable, then H ∈ 𝔊(p) if and only if G ∈ 𝔊(p).

(c) Let L be a linking system associated to F such that all subgroups in H =^{def} Ob(L) are F-centric. Then L ≅ L_{S}^{H}(G).

(d) The homomorphism μ_{G}: Out_{typ}(L_{S}^{c}(G)) ⟶ Out(S, F) defined in Section 1.3 is surjective.

Proof. (a) Let 1 = G_{0} ⊴ G_{1} ⊴ ⋯ ⊴ G_{m} = G be a sequence of subgroups, all normal in Ĝ, such that for each r, G_{r+1}/G_{r} is a minimal nontrivial normal subgroup of Ĝ/G_{r}. By [G, Theorem 2.1.5], each quotient G_{r+1}/G_{r} is a product of simple groups isomorphic to each other. By [O2, Proposition 2.2], if G_{r+1}/G_{r} is abelian, then \underleftarrow{lim}_{O(F̂^{c})}^{i}(ZG_{r+1} ; F̂/ZG_{r} ; F̂) = 0 for

all i ≥ 1. Thus (a) follows immediately from the definition of 𝔏̂(p), together with the exact sequences

\underleftarrow{lim}^{i}_{O(F̂^{c})}(ZG_{s} ; F̂/ZG_{r} ; F̂) ⟶ \underleftarrow{lim}^{i}_{O(F̂^{c})}(ZG_{t} ; F̂/ZG_{r} ; F̂) ⟶ \underleftarrow{lim}^{i}_{O(F̂^{c})}(ZG_{t} ; F̂/ZG_{s} ; F̂)

for all 0 ≤ r < s < t ≤ m and all i ≥ 2.

(b) The first statement is immediate, since a simple group is a composition factor of G = G_{1} × G_{2} if and only if it is a composition factor of G_{1} or of G_{2}. When H ⊴ G and G/H is p-solvable, then the only simple groups which could be composition factors of G but not of H are C_{p} and simple groups of order prime to p. So we need only show that every nonabelian simple group of order prime to p lies in 𝔏̂(p).

Fix such a simple group L, and assume H ⊴ K ⊴ G (where H ⊴ G), and K/H ≅ L^{m} for some m. Set F = F_{S}(G) for some S ∈ Syl_{p}(G). Then Z_{F}^{H} = Z_{F}^{K} since K/H has order prime to p; and thus L ∈ 𝔏̂(p).

(c,d) By (a), applied with Ĝ = G, \underleftarrow{lim}_{O(F^{c})}^{2}(Z_{F}) = 0. So by [BLO1, Theorem E], μ_{G} is onto. This proves (d).

Now let L be a linking system associated to F, set H = Ob(L), and assume H ⊆ Ob(F^{c}). Since H contains all subgroups of S which are F-centric and F-radical, O_{p}(Out_{F}(P)) ≠ 1 for P ∈ Ob(F^{c})∖H. By Lemma 2.8, \underleftarrow{lim}_{O(F^{H})}^{2}(Z_{F}|_{O(F^{H})^{op}}) = 0 for i ≥ 2 since \underleftarrow{lim}_{O(F^{c})}^{2}(Z_{F}) = 0. So by the same argument as that used in the proof of [BLO2, Proposition 3.1], all linking systems associated to F with object set H are isomorphic. In particular, L ≅ L_{S}^{H}(G), and this proves (c). □

The following is the main technical result in this subsection, and will be needed in the proof of Theorem A. Given F_{0} ⊴ F satisfying certain technical assumptions, and given a linking system L_{0} associated to F_{0}, we want to find L associated to F such that L_{0} ⊴ L. It is natural to ask why this cannot be done using [O3, Theorem 9], where conditions are explicitly set up to construct extensions of fusion and linking systems. There seem to be two difficulties with that approach. First, the hypotheses of Proposition 2.12 are very different from those in [O3], and it is not clear how to convert from the one to the other. But more seriously, even if one does manage to do that and construct an extension (F′, L′) of (F_{0}, L_{0}), it is not clear how to prove that F′ ≅ F; i.e., that L′ really is a linking system associated to F.

Proposition 2.12. Let F be a saturated fusion system over a finite p-group S, and let F_{0} ⊴ F be a normal fusion subsystem over S_{0} ⊴ S. Assume F_{0} is strongly tame; and either

(a) F_{0} = O^{p}(F), or

(b) F_{0} = O^{p′}(F), or

(c) F_{0} = N_{F}^{K}(Q) for some Q ⊴ F and some K ⊴ Aut(Q) containing Inn(Q).

Then there is a centric linking system associated to F.

Proof. In all cases (a), (b), and (c), F_{0} ⊴ F by Proposition 1.25.

Since F_{0} is strongly tame, we can choose a finite group G_{0} ∈ 𝔊(p) such that S_{0} ∈ Syl_{p}(G_{0}), F_{0} = F_{S_{0}}(G_{0}), and κ_{G_{0}} is split surjective. We first claim that

μ̃_{G_{0}} ∘ κ̃_{G_{0}} : Aut(G_{0}, S_{0}) ⟶ Aut(S_{0}, F_{0}) is onto.  (2)

As noted in Section 2.2, this composite is defined by restriction. Since G_{0} ∈ 𝔊(p), μ_{G_{0}} is surjective by Lemma 2.11(d). Also, κ_{G_{0}} is split surjective by assumption. Thus every element of Out(S_{0}, F_{0}) = Aut(S_{0}, F_{0})/Aut_{G_{0}}(S_{0}) extends to an element of Out(G_{0}) = Aut(G_{0}, S_{0})/Aut_{N_{G_{0}}(S_{0})}(G_{0}), and hence the map in (2) is onto.

Define

Δ = {α ∈ Aut(G_{0}, S_{0}) | α|_{S_{0}} ∈ Aut_{F}(S_{0})} .

We just showed that every element in Aut_{F}(S_{0}) is the restriction of some element in Δ. Fix S_{Δ} ∈ Syl_{p}(Δ) which surjects onto Aut_{S}(S_{0}) under the restriction map to Aut_{F}(S_{0}). Set

Ĝ = G_{0} ⋊ Δ and Ŝ = S_{0} ⋊ S_{Δ}.

Thus Ŝ ∈ Syl_{p}(Ĝ).

Now set S_{1} = S, F_{1} = F, S_{2} = Ŝ, and F_{2} = FŜ(Ĝ). We claim, for each P_{0}, Q_{0} ≤ S_{0}, that

Hom_{F_{2}}(P_{0}, Q_{0}) = Hom_{F_{1}}(P_{0}, Q_{0}) =^{def} Hom_{F}(P_{0}, Q_{0})
Hom_{S_{2}}(P_{0}, Q_{0}) = Hom_{S_{1}}(P_{0}, Q_{0}) =^{def} Hom_{S}(P_{0}, Q_{0}) .

(3)

We have already remarked that F_{0} ⊴ F = F_{1}, and F_{0} ⊴ F_{2} by Proposition 1.28 since they are the fusion systems of G_{0} ⊴ Ĝ. Hence by condition (ii) in Definition 1.18, each φ ∈ Hom_{F_{i}}(P, Q) (for i = 1, 2 and P, Q ≤ S_{0}) is the composite of a morphism in F_{0} and the restriction of a morphism in Aut_{F_{i}}(S_{0}). Furthermore,

Aut_{F_{2}}(S_{0}) = AutĜ(S_{0}) = ⟨Aut_{G_{0}}(S_{0}) , Res_{S_{0}}^{G_{0}}(Δ)⟩ = Aut_{F_{1}}(S_{0})

by (2), and the first line in (3) now follows. The second holds since Aut_{S_{2}}(S_{0}) = Aut_{S}(S_{0}) by definition of S_{2} = S_{0} ⋊ S_{Δ}.

We next claim that for all P_{0} ≤ S_{0},

P_{0} is fully centralized in F_{2} ⟺ P_{0} is fully centralized in F_{1} = F .  (4)

By (3), the F_{1}- and F_{2}-conjugacy classes of P_{0} are the same, and Aut_{S_{2}}(S_{0}) = Aut_{S}(S_{0}). Hence for each Q_{0} which is F_{i}-conjugate to P_{0},

\frac{|C_{S_{1}}(Q_{0})|}{|C_{S_{1}}(S_{0})|} = |\{α ∈ Aut_{S}(S_{0}) | α|_{Q_{0}} = Id\}| = \frac{|C_{S_{2}}(Q_{0})|}{|C_{S_{2}}(S_{0})|} ,

and so |C_{S_{1}}(P_{0})| is maximal if and only if |C_{S_{2}}(P_{0})| is maximal.

We want to compute \underleftarrow{lim}_{O(F_{1}^{c})}^{∗}(Z_{F_{1}}) by comparing it with \underleftarrow{lim}_{O(F_{2}^{c})}^{∗}(Z_{F_{2}}). To do this, we first define in Step 1 certain full subcategories F_{i}^{∗} ⊆ F_{i}^{c}, and an intermediate category C which can be used to compare O(F_{1}^{∗}) with O(F_{2}^{∗}). Certain properties of the “comparison functors” Φ_{i}: O(F_{i}^{∗}) ⟶ C are stated and proven in Step 2. In Step 3, we define certain subfunctors Z_{i} ⊆ Z_{F_{i}} on O(F_{i}^{c}), and prove that \underleftarrow{lim}_{O(F_{1}^{c})}^{∗}(Z_{1}) ≅ \underleftarrow{lim}_{O(F_{2}^{c})}^{∗}(Z_{2}) using the intermediate categories O(F_{i}^{∗}) and C to compare them. Finally, in Step 4, we prove that \underleftarrow{lim}_{O(F_{2}^{c})}^{∗}(Z_{2}) = 0 for ∗ ≥ 2, and then show that \underleftarrow{lim}_{O(F^{c})}^{∗}(Z_{F}) ≅ \underleftarrow{lim}_{O(F_{1}^{c})}^{∗}(Z_{1}) for ∗ ≥ 1 by analyzing individually the three cases (a)–(c).

Throughout the rest of the proof, whenever P ≤ S_{1} or P ≤ S_{2}, we write P_{0} = P ∩ S_{0}.

Step 1: Let F_{i}^{∗} ⊆ F_{i} (i = 1, 2) be the full subcategories with objects

Ob(F_{i}^{∗}) = {P ≤ S_{i} | C_{S_{i}}(Q_{0}) ≤ Q for all Q F_{i}-conjugate to P }.

All objects in F_{i}^{∗} are F_{i}-centric; i.e., F_{i}^{∗} ⊆ F_{i}^{c}. Also, if P ≤ S_{i} is F_{i}-conjugate to an object in F_{i}^{∗}, then P ∈ Ob(F_{i}^{∗}).

We next construct a category C which acts as intermediary between the orbit categories of F_{1}^{∗} and F_{2}^{∗}. It will be a subcategory of a larger category Ĉ, defined by setting

Ob(Ĉ) = {(P_{0}, K) | P_{0} ≤ S_{0} is F_{0}-centric and fully centralized in F,

Inn(P_{0}) ≤ K ≤ Aut_{S}(P_{0})} and

MorĈ((P_{0}, K), (Q_{0}, L)) = L\{φ ∈ Hom_{F}(P_{0}, Q_{0}) | φK ⊆ Lφ}.

Here, we regard φK and Lφ as subsets of Hom_{F}(P_{0}, Q_{0}).

Define functors

O(F_{1}^{∗}) ⟶^{Φ_{1}} Ĉ ⟵^{Φ_{2}} O(F_{2}^{∗}) ,

by setting Φ_{i}(P) = (P_{0}, Aut_{P} (P_{0})) and Φ_{i}([φ]) = [φ|_{P_{0}}] for P, Q ∈ Ob(F_{i}^{∗}) and φ ∈ Hom_{F_{i}^{∗}}(P, Q). Since restriction sends Inn(Q) to Aut_{Q}(Q_{0}) for Q ∈ Ob(F_{i}^{∗}), Φ_{i} is well defined on morphisms if it is defined on objects.

To see that Φ_{i} is well defined on objects, fix P ∈ Ob(F_{i}^{∗}), and set K = Aut_{P} (P_{0}). Then Inn(P_{0}) ≤ Aut_{P} (P_{0}) ≤ Aut_{S}(P_{0}), since P ≥ P_{0}, and since Aut_{S}(P_{0}) = Aut_{S_{i}}(P_{0}) by (3). By Lemma 1.20(a), P is F_{i}-conjugate to some Q such that Q_{0} is fully normalized in F_{i} and in F_{0}. Hence P_{0} is F_{0}-centric by Lemma 1.20(c). Also, |C_{S_{i}}(P_{0})| = |C_{P} (P_{0})| = |C_{Q}(Q_{0})| = |C_{S_{i}}(Q_{0})| by definition of Ob(F_{i}^{∗}), and so P_{0} is fully centralized in F_{i} (hence in F by (4)) since Q_{0} is fully centralized in F_{i}. Thus (P_{0}, K) ∈ Ob(Ĉ).

We claim that Im(Φ_{1}) = Im(Φ_{2}). In what follows, when P_{0} ≤ S_{0} and K ≤ Aut(P_{0}), we set N_{S}^{K}(P_{0}) = {g ∈ N_{S}(P_{0}) | c_{g} ∈ K}. Then

P ∈ Ob(F_{i}^{∗}) and Φ_{i}(P) = (P_{0}, K) ⟹ P = N_{S_{i}}^{K}(P_{0})  (5)

since P ≥ C_{S_{i}}(P_{0}) by definition of Ob(F_{i}^{∗}).

Assume (P_{0}, K) ∈ Ob(Ĉ), and set P_{i} = N_{S_{i}}^{K}(P_{0}) for i = 1, 2. Then P_{i} ≥ P_{0} and K = Aut_{P_{i}}(P_{0}), since Inn(P_{0}) ≤ K ≤ Aut_{S}(P_{0}) by assumption. Also, P_{1} ∩ S_{0} = N_{S_{0}}^{K}(P_{0}) = P_{2} ∩ S_{0}, so P_{1} ∩ S_{0} = P_{0} if and only if P_{2} ∩ S_{0} = P_{0}, and we assume this is the case since otherwise (P_{0}, K) is in the image of neither functor Φ_{i} by (5). By assumption, P_{0} is fully centralized in F, and hence in F_{i} by (4). So for each Q which is F_{i}-conjugate to P_{i}, |C_{S_{i}}(Q_{0})| ≤ |C_{S_{i}}(P_{0})| = |C_{P_{i}}(P_{0})| = |C_{Q}(Q_{0})|, where the last equality holds since any φ ∈ Iso_{F_{i}}(P_{i}, Q) induces an isomorphism of pairs (P_{i}, P_{0}) ≅ (Q, Q_{0}). Thus C_{S_{i}}(Q_{0}) ≤ Q. This proves that P_{i} ∈ Ob(F_{i}^{∗}), and hence that Φ_{i}(P_{i}) = (P_{0}, K) for i = 1, 2.

Now fix objects (P_{0}, K) and (Q_{0}, L) in Im(Φ_{i}), and choose φ_{0} ∈ Hom_{F}(P_{0}, Q_{0}) such that φ_{0}K ⊆ Lφ_{0}. Thus [φ_{0}] ∈ MorĈ((P_{0}, K), (Q_{0}, L)). If [φ_{0}] = Φ_{i}([φ]) for some φ ∈ Hom_{F_{i}^{∗}}(P, Q) (i = 1 or 2), then φ(P) ∈ Ob(F_{i}^{∗}), so φ_{0}(P_{0}) is fully centralized in F, and hence in F_{1} and F_{2} by (4). So we assume this from now on.

Set P_{i} = N_{S_{i}}^{K}(P_{0}) and Q_{i} = N_{S_{i}}^{L}(Q_{0}): these are both in Ob(F_{i}^{∗}) by (5). Set R_{0} = φ_{0}(P_{0}), let φ̇_{0} ∈ Iso_{F}(P_{0}, R_{0}) be the restriction of φ_{0}, and set M = φ̇_{0}Kφ̇_{0}^{−1} ≤ Aut_{F}(R_{0}). Then φ_{0}Kφ̇_{0}^{−1} ⊆ L|_{R_{0}} ⊆ Hom_{S}(R_{0}, Q_{0}), and so M ≤ Aut_{S}(R_{0}) = Aut_{S_{i}}(R_{0}). By the extension axiom for F_{i}, φ_{0} extends to some φ_{i} ∈ Hom_{F_{i}}(P_{i}, S_{i}), and [φ_{0}] ∈ Im(Φ_{i}) if and only if φ_{i} can be chosen with φ_{i}(P_{i}) ≤ Q_{i}. Now, M = Aut_{φ_{i}(P_{i})}(R_{0}) since K = Aut_{P_{i}}(P_{0}), so Φ_{i}(φ_{i}(P_{i})) = (R_{0}, M), and φ_{i}(P_{i}) = N_{S_{i}}^{M}(R_{0}) by (5). Hence φ_{i}(P_{i}) ≤ Q_{i} if and only if for all α ∈ Aut_{S_{i}}(S_{0}),

α(R_{0}) = R_{0} and α|_{R_{0}} ∈ M ⟹ α(Q_{0}) = Q_{0} and α|_{Q_{0}} ∈ L .

Since Aut_{S_{1}}(S_{0}) = Aut_{S_{2}}(S_{0}) by (3), [φ_{0}] ∈ Im(Φ_{1}) if and only if [φ_{0}] ∈ Im(Φ_{2}).

Now set C = Im(Φ_{1}) = Im(Φ_{2}) ⊆ Ĉ. Since the Φ_{i} are injective on objects by (5), this is a subcategory of Ĉ. From now on, we regard the Φ_{i} as functors to C.

Step 2: For each i = 1, 2, and each P ∈ Ob(F_{i}^{∗}), set

Γ_{i}(P) = Ker[Out_{F_{i}}(P) ⟶^{Φ_{i}} Aut_{C}(P_{0}, Aut_{P} (P_{0}))]
= Ker[Out_{F_{i}}(P) ⟶^{R} N_{Aut_{F} (P_{0})}(Aut_{P} (P_{0}))/Aut_{P} (P_{0})]

where R is induced by restriction. We claim that, for each i = 1, 2,

(i) Φ_{i}: O(F_{i}^{∗}) ⟶ C is bijective on objects and surjective on morphism sets;

(ii) Γ_{i}(P) has order prime to p for all P; and

(iii) whenever ψ, ψ′ ∈ Mor_{O(F_{i}^{∗})}(P, Q) are such that Φ_{i}(ψ) = Φ_{i}(ψ′), there is χ ∈ Γ_{i}(P) such that ψ′ = ψ ∘ χ.

Point (i) follows from (5) and the definition of C in Step 1.

When proving (ii), it suffices to consider the case where P is fully normalized in F_{i}. If g ∈ N_{S_{i}}(P) is such that [c_{g}] ∈ Γ_{i}(P), then c_{g}|_{P_{0}} ∈ Aut_{P} (P_{0}); and since C_{S_{i}}(P_{0}) ≤ P, this implies g ∈ P and [c_{g}] = 1 ∈ Γ_{i}(P) ≤ Out_{F_{i}}(P). Thus Out_{S_{i}}(P) is a Sylow p-subgroup of Out_{F_{i}}(P) and intersects trivially with Γ_{i}(P) ⊴ Out_{F_{i}}(P), so |Γ_{i}(P)| is prime to p.

It remains to prove (iii). Assume ψ, ψ′ ∈ Mor_{O(F_{i}^{∗})}(P, Q) are such that Φ_{i}(ψ) = Φ_{i}(ψ′). Fix φ, φ′ ∈ Hom_{F_{i}^{∗}}(P, Q) such that ψ = [φ] and ψ′ = [φ′]. Then φ|_{P_{0}} = c_{g} ∘φ′|_{P_{0}} for some c_{g} ∈ Aut_{Q}(Q_{0}); i.e., for some g ∈ Q. So upon replacing φ′ by c_{g} ∘ φ′ (this time with c_{g} ∈ Inn(Q)), we can assume φ_{0} =^{def} φ|_{P_{0}} = φ′|_{P_{0}}. Since P ∈ Ob(F_{i}^{∗}), we have C_{S_{i}}(φ_{0}(P_{0})) ≤ φ(P), so

φ(P) = {g ∈ N_{S_{i}}(φ_{0}(P_{0})) | c_{g} ∈ φ_{0}Aut_{P} (P_{0})φ_{0}^{−1}} = φ′(P).

Hence there is a unique β ∈ Aut_{F_{i}}(P) such that φ′ = φ ∘β; and also β|_{P_{0}} = Id. So ψ′ = ψ ∘[β] in Mor_{O(F_{i}^{∗})}(P, Q), where [β] ∈ Γ_{i}(P).

Step 3: Define functors

Z_{i}: O(F_{i}^{c})^{op} ⟶ ℤ_{(p)}-mod and Z_{C}: C^{op} ⟶ ℤ_{(p)}-mod

by setting Z_{i}(P) = Z(P) ∩ S_{0} = C_{Z(P_{0})}(P) and Z_{C}(Q, K) = C_{Z(Q)}(K) (the subgroup of elements of Z(Q) fixed pointwise by K). Morphisms are sent in the obvious way. Set Z_{i∗} = Z_{i}|_{O(F_{i}^{∗})^{op}}.

We claim that

\underleftarrow{lim}^{∗}_{O(F_{1}^{c})}(Z_{1}) ≅ \underleftarrow{lim}^{∗}_{O(F_{1}^{∗})}(Z_{1∗}) ≅ \underleftarrow{lim}_{C}^{∗}(Z_{C}) ≅ \underleftarrow{lim}^{∗}_{O(F_{2}^{∗})}(Z_{2∗}) ≅ \underleftarrow{lim}^{∗}_{O(F_{2}^{c})}(Z_{2}).  (6)

Since Z_{i∗} = Z_{C} ∘ Φ_{i} by definition, the second and third isomorphisms follow from points (i–iii) in Step 2 and [BLO1, Lemma 1.3].

We prove the other isomorphisms in (6) using Lemma 2.8. Fix i = 1, 2. We already saw in Step 1 that Ob(F_{i}^{∗}) is closed (inside Ob(F_{i}^{c})) with respect to F_{i}-conjugacy. If P ≤ Q ≤ S_{i} and P ∈ Ob(F_{i}^{∗}), then for each Q^{∗} which is F_{i}-conjugate to Q, if we set P^{∗} = φ(P) ≤ Q^{∗} for some φ ∈ Iso_{F_{i}}(Q, Q^{∗}), then C_{S_{i}}(P_{0}^{∗}) ≤ P^{∗} implies C_{S_{i}}(Q_{0}^{∗}) ≤ Q^{∗}, and so Q ∈ Ob(F_{i}^{∗}). Thus Ob(F_{i}^{∗}) is closed with respect to overgroups.

For each object P in F_{i}^{c} not in F_{i}^{∗}, there is P^{∗} F_{i}-conjugate to P such that C_{S_{i}}(P_{0}^{∗}) ≰ P^{∗}. By Lemma 1.7, there is g ∈ N_{S_{i}}(P^{∗})∖P^{∗} which centralizes P_{0}^{∗}. Thus [c_{g}] ∈ Out_{F_{i}}(P^{∗}) is a nontrivial element of p-power order which acts trivially on Z_{i}(P^{∗}). So by Lemma 2.8, \underleftarrow{lim}_{O(F_{i}^{c})}^{∗}(Z_{i}) ≅ \underleftarrow{lim}_{O(F_{i}^{∗})}^{∗}(Z_{i∗}) for each i = 1, 2; and this finishes the proof of (6).

Step 4: By Lemma 2.11(a), \underleftarrow{lim}_{O(F_{2}^{c})}^{j}(Z_{2}) = \underleftarrow{lim}_{O(F_{2}^{c})}^{j}(Z_{F_{2}}^{G_{0}}) = 0 for j ≥ 2. Hence by (6), \underleftarrow{lim}_{O(F^{c})}^{j}(Z_{1}) = 0 for j ≥ 2 (recall F = F_{1}).

We claim that for j ≥ 2,

\underleftarrow{lim}^{j}_{O(F^{c})}(Z_{F}) ≅ \underleftarrow{lim}^{j}_{O(F^{c})}(Z_{1}).  (7)

Set Ẑ = Z_{F}/Z_{1} for short; thus Ẑ(P) = Z(P)/(Z(P) ∩ S_{0}) for each P. If F_{0} = O^{p′}(F), then (7) holds since S_{0} = S and hence Z_{F} = Z_{1}. If F_{0} = N_{F}^{K}(Q) for some Q ⊴ F and some K ⊴ Aut(Q), then Ẑ(P) = 0 for each P ∈ Ob(F^{c}) which contains Q, in particular for each subgroup which is F-centric and F-radical (Proposition 1.5); and (7) holds by Lemma 2.8.

Assume F_{0} = O^{p}(F). For each P ∈ Ob(F^{c}), let H_{P} ⊴ Out_{F}(P) be the kernel of the Out_{F}(P)-action on Ẑ(P) = Z(P)/(Z(P) ∩ S_{0}). By definition of S_{0} = 𝔥𝔶𝔭(F), H_{P} contains O^{p}(Out_{F}(P)), and thus Out_{F}(P)/H_{P} is a p-group. So for j ≥ 1,

Λ^{j}(Out_{F}(P); Ẑ(P)) ≅ \begin{cases}0 & if p||H_{P} | \\ 0 & if p ∤ |Out_{F}(P)| \\ Λ^{j}(Out_{F}(P)/H_{P}; Ẑ(P)) = 0 & otherwise \end{cases}

by [JMO, Proposition 6.1]: by point (ii) of the proposition in the first case, by point (i) in the second, and by points (iii) and (ii) in the third. So by [O1, Lemma 2.3], \underleftarrow{lim}_{O(F^{c})}^{j}(Ẑ) = 0 for all j ≥ 1, and (7) also holds in this case.

We now conclude that \underleftarrow{lim}_{O(F^{c})}^{j}(Z_{F}) = 0 for all j ≥ 2. So by [BLO2, Proposition 3.1], there is a (unique) centric linking system L^{c} associated to F. □

2.4. Proof of Theorem A.

We want to show that if 𝔯𝔢𝔡(F) is tame, then so is F. The proof splits naturally into two parts. We first show, under certain additional hypotheses, that if F_{0} ⊴ F and F_{0} is tame, then F is tame. Afterwards, we show (again under additional hypotheses) that F is tame if F/Z(F) is tame. In both cases, this means proving that certain homomorphisms are split surjective, by first constructing an appropriate pullback square of automorphism groups, and then applying the following elementary lemma.

Lemma 2.13. If the following square of groups and homomorphisms

\xymatrix{A_{1} \ar[r]^{α} \ar[d] & A_{2} \ar[d] \\ B_{1} \ar[r]^{β} & B_{2}}

is a pullback square, and β is split surjective, then α is split surjective. □

We first work with normal subsystems. We first recall some convenient notation. When P is a p-centric subgroup of a finite group G (i.e., an F_{S}(G)-centric subgroup when P ≤ S ∈ Syl_{p}(G)), we set C′_{G}(P) = O^{p}(C_{G}(P)). Thus C′_{G}(P) has order prime to p, and C_{G}(P) = Z(P) × C′_{G}(P).

For any normal pair (S_{0}, F_{0}, L_{0}) ⊴ (S, F, L), let

ρ̃ = ρ̃_{L_{0}}^{L}: Aut_{L}(S_{0}) ⟶ Aut_{typ}^{I}(L_{0})

be the homomorphism which sends γ ∈ Aut_{L}(S_{0}) to c_{γ}. Here, c_{γ} ∈ Aut_{typ}^{I}(L_{0}) sends an object P to π(γ)(P) and sends ψ ∈ Mor_{L_{0}}(P, Q) to (γ|_{Q,π(γ)(Q)}) ∘ ψ ∘ (γ|_{P,π(γ)(P)})^{−1} (well defined by Definition 1.27). Let

ρ = ρ_{L_{0}}^{L}: L/L_{0}_{=Aut_{L}(S_{0})/Aut_{L_{0}}(S_{0})} ⟶ Out_{typ}(L_{0})_{=Aut_{typ}^{I}(L_{0})/{c_{γ} | γ∈Aut_{L_{0}}(S_{0})}}

be the homomorphism induced by ρ̃, which sends [γ] to the class of c_{γ}. This is analogous to the conjugation homomorphism G/G_{0} ⟶ Out(G_{0}) for a pair of groups G_{0} ⊴ G. For example, L_{0} is centric in L (see Definition 1.27) if and only if ρ_{L_{0}}^{L} is injective.

We next show that when F_{0} ⊴ F have associated linking systems L_{0} ⊴ L, where L_{0} is centric in L and F_{0} is realizable, then under some extra conditions, F is also realizable.

Lemma 2.14. Fix a normal pair (S_{0}, F_{0}, L_{0}) ⊴ (S, F, L) such that L_{0} is centric in L. Set H_{0} = Ob(L_{0}) and H = Ob(L), and assume H_{0} is Aut(S_{0}, F_{0})-invariant. Assume there is a finite group G_{0} such that

(a) S_{0} ∈ Syl_{p}(G_{0}), F_{0} = F_{S_{0}}(G_{0}), and L_{0} ≅ L_{S_{0}}^{H_{0}}(G_{0});

(b) Z(G_{0}) = Z(F_{0}); and

(c) there is a homomorphism ρ̂: L/L_{0} ⟶ Out(G_{0}) such that

κ_{G_{0}}^{H_{0}} ∘ ρ̂ = ρ_{L_{0}}^{L}: L/L_{0} ⟶ Out_{typ}(L_{0}) .

Then F = F_{S}(G) and L ≅ L_{S}^{H}(G) for some finite group G such that S ∈ Syl_{p}(G), G_{0} ⊴ G, G/G_{0} ≅ L/L_{0}, and such that the extension realizes the given outer action ρ̂ of G/G_{0} ≅ L/L_{0} on G_{0}.

Proof. We construct the group G in Step 1, and prove that L_{S}^{H}(G) ≅ L and F_{S}(G) = F in Step 2. Throughout the proof, we identify L_{0} with L_{S_{0}}^{H_{0}}(G_{0}).

Step 1: Consider the following diagram whose rows are exact by Lemma 1.14:

[figure: 1; Z(G_{0}); N_{G_{0}}(S_{0}); conj; λ_{0}; Aut(G_{0}, S_{0}); ^{pr^{1}}; κ̃; =κ_{G_{0}}^{H_{0}}; Out(G_{0}); κ; =κ_{G_{0}}^{H_{0}}; 1; 1; Z(F_{0}); Aut_{L_{0}}(S_{0}); conj; Aut_{typ}^{I}(L_{0}); _{pr_{2}}; Out_{typ}(L_{0}); 1 .] (8)

Here, λ_{0} sends g ∈ N_{G_{0}}(S_{0}) to its class in Aut_{L_{0}}(S_{0}) = N_{G_{0}}(S_{0})/C′_{G_{0}}(S_{0}). The first and third squares clearly commute. The second square commutes since for g ∈ N_{G_{0}}(S_{0}), κ̃ sends c_{g} to the automorphism [a] ↦ [gag^{−1}] = c_{λ_{0}(g)}(a). By definition of κ̃ = κ̃_{G_{0}}^{H_{0}},

κ̃(β)(λ_{0}(g)) = λ_{0}(β(g)) for all β ∈ Aut(G_{0}, S_{0}), g ∈ N_{G_{0}}(S_{0}) .  (9)

Set Aut(G_{0}, S_{0})ρ̂ = pr_{1}^{−1}(ρ̂(L/L_{0})). Since L_{0} is centric in L, ρ = ρ_{L_{0}}^{L} sends L/L_{0} injectively into Out_{typ}(L_{0}). Hence κ sends ρ̂(L/L_{0}) injectively into Out_{typ}(L_{0}). So by a diagram chase in (8), N_{G_{0}}(S_{0}) is the pullback of Aut(G_{0}, S_{0})ρ̂ and Aut_{L_{0}}(S_{0}) over Aut_{typ}^{I}(L_{0}).

Let H be the group which makes the following square a pullback:

\xymatrix{H \ar[r]^{φ} \ar@{->>}[d]^{λ} & Aut(G_{0}, S_{0})ρ̂ \ar[d]^{κ̃} \\ Aut_{L}(S_{0}) \ar[r]^{ρ̃} & Aut_{typ}^{I}(L_{0}) .}  (10)

For each α ∈ Aut_{L}(S_{0}), ρ̃(α) ∈ pr_{2}^{−1}(ρ(L/L_{0})) by definition, and hence lifts to an element of Aut(G_{0}, S_{0})ρ̂. This proves that λ is onto. By comparison with the middle square in (8), we can identify N_{G_{0}}(S_{0}) with λ^{−1}(Aut_{L_{0}}(S_{0})) ⊴ H. Thus

H/N_{G_{0}}(S_{0}) = H/H_{0} ≅ Aut_{L}(S_{0})/Aut_{L_{0}}(S_{0}) = L/L_{0} ,  (11)

where we set H_{0} = N_{G_{0}}(S_{0}), regarded as a subgroup of G_{0} and of H.

We claim that for all h ∈ H and a ∈ H_{0},

φ(h)(a) = hah^{−1} ∈ H_{0} .  (12)

Since (10) is a pullback, it suffices to prove (12) after applying φ and after applying λ. It holds after applying λ (or λ_{0}) since

λ_{0}(φ(h)(a)) = κ̃(φ(h))(λ_{0}(a)) = ρ̃(λ(h))(λ_{0}(a)) = λ(h)λ_{0}(a)λ(h)^{−1} = λ_{0}(hah^{−1}) :

the first equality by (9), the second by the commutativity of (10), and the third since ρ̃ is defined by conjugation in L. Since φ|_{H_{0}} is also defined to be conjugation,

φ(φ(h)(a)) = c_{φ(h)(a)} = φ(h) ∘ c_{a} ∘ φ(h)^{−1} = φ(h) ∘ φ(a) ∘ φ(h)^{−1} = φ(hah^{−1}) .

This finishes the proof of (12).

We want to construct a group G with G_{0} ⊴ G, G/G_{0} ≅ L/L_{0}, and N_{G}(S_{0}) = H. To do this, first set Γ = G_{0} ⋊ H: the semidirect product with the action of H on G_{0} given by φ as defined in (10). Elements of Γ are written as pairs (g, h) for g ∈ G_{0} and h ∈ H. Thus (g, h)(g′, h′) = (g·φ(h)(g′), hh′). Set N = {(a, a^{−1}) | a ∈ H_{0}}. For a, b ∈ H_{0},

(a, a^{−1})(b, b^{−1}) = (a·φ(a^{−1})(b), a^{−1}b^{−1}) = (a·a^{−1}ba, a^{−1}b^{−1}) = (ba, (ba)^{−1}) ∈ N,

where the second equality holds by (12). Thus N is a subgroup. For g ∈ G_{0} and a ∈ H_{0},

(g, 1)(a, a^{−1})(g, 1)^{−1} = (ga, a^{−1})(g^{−1}, 1) = (ga·φ(a^{−1})(g^{−1}), a^{−1})
= (ga·a^{−1}g^{−1}a, a^{−1}) = (a, a^{−1}) ;

where φ(a^{−1})(g^{−1}) = a^{−1}g^{−1}a since by construction, φ|_{H_{0}} is the conjugation homomorphism of (8). Thus (g, 1) normalizes (centralizes) N. For h ∈ H and a ∈ H_{0},

(1, h)(a, a^{−1})(1, h)^{−1} = (φ(h)(a), ha^{−1})(1, h^{−1}) = (φ(h)(a), (hah^{−1})^{−1}) ∈ N

by (12), and thus (1, h) also normalizes N. This proves that N ⊴ Γ.

Now set G = Γ/N, and regard G_{0} and H as subgroups of G. By construction, G = G_{0}H, G_{0} ∩ H = H_{0} = N_{G_{0}}(S_{0}), G_{0} ⊴ G, and G/G_{0} ≅ H/H_{0} ≅ L/L_{0} (the last isomorphism by (11)). Also, H ≤ N_{G}(S_{0}), and since [H:N_{G_{0}}(S_{0})] = [G:G_{0}] ≥ [N_{G}(S_{0}):N_{G_{0}}(S_{0})], we have H = N_{G}(S_{0}). The outer conjugation action of G/G_{0} on G_{0} is induced by φ. So by (10) and the definition of Aut(G_{0}, S_{0})ρ̂, and since κ sends Im(ρ̂) isomorphically to Im(ρ_{L_{0}}^{L}) (since ρ_{L_{0}}^{L} is injective), this outer action is equal to ρ̂ via our identification G/G_{0} ≅ L/L_{0}.

By comparison of (8) and (10), we see that

Ker(λ) = Ker(λ_{0}) = C′_{G_{0}}(S_{0}) .

In particular, Ker(λ) has order prime to p. Also, δ_{S_{0}}(S) is a Sylow p-subgroup of Aut_{L}(S_{0}) by Proposition 1.11(d). Fix any Sylow p-subgroup of λ^{−1}(δ_{S_{0}}(S)), and identify it with S via δ_{S_{0}}^{−1} ∘ λ. Since [G:H] = [G_{0}:H_{0}] is prime to p, we also have S ∈ Syl_{p}(G).

Step 2: Set F′ = F_{S}(G) for short. By Proposition 1.28, F_{0} = F_{S_{0}}(G_{0}) is normal in F′. So by Lemma 1.20(d), H = Ob(L) contains all subgroups of S which are F′-centric and F′-radical.

We next show that all subgroups in H are G-quasicentric. Since overgroups of quasicentric subgroups are quasicentric, it suffices to prove this for P ∈ H_{0}. Fix such P, and assume it is fully centralized in F′. We must show that O_{p′}(C_{G}(P)) = O^{p}(C_{G}(P)); i.e., that C_{G}(P) contains a normal subgroup of order prime to p and of p-power index. Define

Φ_{P} : N_{G}(P) ⟶ Aut_{L}(P)

as follows. Fix g ∈ N_{G}(P), write g = g_{0}h for some g_{0} ∈ G_{0} and h ∈ H = N_{G}(S_{0}), and set Φ_{P} (g) = [g_{0}] ∘ λ(h)|_{P,hP h^{−1}}, where [g_{0}] ∈ Mor_{L_{0}}(hP h^{−1}, P) is induced by the identification

L_{0} = L_{S_{0}}^{H_{0}}(G_{0}). If g = g_{0}h = g′_{0}h′ where g_{0}, g′_{0} ∈ G_{0} and h, h′ ∈ H, then a =^{def} g_{0}^{−1}g′_{0} = hh′^{−1} ∈

H_{0}, so g′_{0} = g_{0}a, h = ah′, and

[g_{0}] ∘ λ(h)|_{P,hP h^{−1}} = [g_{0}] ∘ λ(a)|_{h′P h′^{−1},hP h^{−1}} ∘ λ(h′)|_{P,h′P h′^{−1}} = [g_{0}a] ∘ λ(h′)|_{P,h′P h′^{−1}}.

Thus Φ_{P} (g) is well defined, independently of the choice of g_{0} and h, and Φ_{S_{0}} = λ. Moreover, Φ_{P} |_{N_{S}(P)} = δ_{P}, since λ|_{S} = δ_{S_{0}} : S ⟶ Aut_{L}(P) by the identification of S as a subgroup of H. To see that Φ_{P} is a homomorphism, it suffices to check that

[hg_{0}h^{−1}] = λ(h) ∘ [g_{0}] ∘ λ(h)^{−1}  (13)

for each g_{0} ∈ G_{0} and h ∈ H, and this follows from the commutativity of (10).

We next claim that the composite π_{P} ∘ Φ_{P} : N_{G}(P) ⟶ Aut_{F}(P) sends g ∈ N_{G}(P) to c_{g} ∈ Aut(P). Set g = g_{0}h as above. By definition of the linking system L_{S_{0}}^{H_{0}}(G_{0}), π_{P} (Φ_{P} (g_{0})) = π_{P} ([g_{0}]) = c_{g_{0}}. By (13) and axiom (C) for the linking system L, π_{S_{0}}(λ(h)) ∈ Aut_{F}(S_{0}) is conjugation by h, and hence it is also conjugation by h on P ≤ S_{0} ⊴ H. This proves the claim.

Since F_{0} ⊴ F, F_{0} ⊴ F′, and Aut_{F}(S_{0}) = Aut_{F′}(S_{0}), the F- and F′-conjugacy classes of any subgroup Q ≤ S_{0} are the same. It follows that H is closed under F′-conjugacy, and that P is fully centralized in F. Hence

• Ker[Aut_{L}(P) ⟶^{π_{P}} Aut_{F}(P)] = δ_{P} (C_{S}(P));

• Φ_{P} |_{N_{S}(P)} = δ_{P} is injective by Proposition 1.11(c);

• Ker(π_{P} ∘ Φ_{P}) = C_{G}(P) since π_{P} ∘ Φ_{P} (g) = c_{g}; and

• C_{S}(P) ∈ Syl_{p}(C_{G}(P)) by [BLO2, Proposition 1.3].

Hence Ker(Φ_{P}) is a normal subgroup of C_{G}(P) of order prime to p, and C_{G}(P)/Ker(Φ_{P}) ≅ C_{S}(P) is a p-group. It follows that Ker(Φ_{P}) = O^{p}(C_{G}(P)), and thus that P is G-quasicentric.

Set L′ = L_{S}^{H}(G). We have now shown that H satisfies the conditions which ensure that L′ is a linking system associated to F_{S}(G). By Proposition 1.28 again, L′ contains L_{0} as a normal linking subsystem. Also, Aut_{L′}(S_{0}) = H/Ker(λ) ≅ Aut_{L}(S_{0}) since O^{p}(C_{G}(S_{0})) = Ker(Φ_{S_{0}}) = Ker(λ), and they have the same action on L_{0} (under this identification) by the commutativity of (10).

By the remarks at the end of [O3] (after the proof of Theorem 9), the existence of the linking systems L and L′ imply that conditions (2) and (3) in the statement of [O3, Theorem 9] hold. So by the uniqueness statement in that theorem, F = F′ and L ≅ L′. □

In order to compare tameness in F_{0} and in F when (S_{0}, F_{0}, L_{0}) ⊴ (S, F, L), we need to compare the automorphisms of L_{0} with those of L. This is done in the following lemma. For any normal pair L_{0} ⊴ L of linking systems, we set

Aut_{L}(L_{0}) = ρ̃_{L_{0}}^{L}(Aut_{L}(S_{0})) = {c_{γ} | γ ∈ Aut_{L}(S_{0})} ≤ Aut_{typ}^{I}(L_{0})
Out_{L}(L_{0}) = ρ_{L_{0}}^{L}(L/L_{0}) = Aut_{L}(L_{0})/Aut_{L_{0}}(L_{0}) ≤ Out_{typ}(L_{0}) .

Lemma 2.15. Fix a pair of finite groups G_{0} ⊴ G, let S_{0} ⊴ S be Sylow p-subgroups of G_{0} ⊴ G, and set F_{0} = F_{S_{0}}(G_{0}) and F = F_{S}(G). Assume Z(G_{0}) = Z(F_{0}). Let H_{0} and H be sets of subgroups such that

L_{0} =^{def} L_{S_{0}}^{H_{0}}(G_{0})  and  L =^{def} L_{S}^{H}(G)

are linking systems associated to F_{0} and F, respectively. Assume

L_{0} ⊴ L ,  L_{0} is centric in L ,  and  L/L_{0} ≅ G/G_{0} .

Assume also H_{0} is Aut(S_{0}, F_{0})-invariant, and H is Aut(S, F)-invariant. Then the following square

\xymatrix{Out(G, G_{0}) \ar[r]^{κ} \ar[d]_{R_{1}} & Out_{typ}(L, L_{0}) \ar[d]_{R_{2}} \\ N_{Out(G_{0})}(Out_{G}(G_{0}))/Out_{G}(G_{0}) \ar[r]^{κ^{∗}} & N_{Out_{typ}(L_{0})}(Out_{L}(L_{0}))/Out_{L}(L_{0})}  (14)

is a pullback. Here, Out(G, G_{0}) ≤ Out(G) and Out_{typ}(L, L_{0}) ≤ Out_{typ}(L) are the subgroups of classes of automorphisms which leave G_{0} and L_{0} invariant, respectively, κ is the restriction of κ_{G}^{H}, κ^{∗} is induced by κ_{G_{0}}^{H_{0}}, and R_{1} and R_{2} are induced by restriction.

Proof. By the Frattini argument, G = G_{0}·N_{G}(S_{0}) (all subgroups G-conjugate to S_{0} are G_{0}-conjugate to S_{0}). Hence G/G_{0} ≅ N_{G}(S_{0})/N_{G_{0}}(S_{0}), while

L/L_{0} =^{def} Aut_{L}(S_{0})/Aut_{L_{0}}(S_{0}) = (N_{G}(S_{0})/O^{p}(C_{G}(S_{0})))/(N_{G_{0}}(S_{0})/C′_{G_{0}}(S_{0}))

(and S_{0} is G-quasicentric since it is an object of the linking system L = L_{S}^{H}(G)). Since G/G_{0} ≅ L/L_{0}, it follows that O^{p}(C_{G}(S_{0})) = C′_{G_{0}}(S_{0}). Also, for each g ∈ C_{G}(G_{0}) ≤ N_{G}(S_{0}),

[g] ∈ Aut_{L}(S_{0}) acts trivially on L_{0} under conjugation, so [g] ∈ Aut_{L_{0}}(S_{0}) since L_{0} is centric in L, and hence g ∈ G_{0}. We have now shown that

O^{p}(C_{G}(S_{0})) = C′_{G_{0}}(S_{0}) and C_{G}(G_{0}) = Z(G_{0}) .  (15)

Step 1: We first show the following square is a pullback:

\xymatrix{Aut(G, G_{0}, S·C′_{G_{0}}(S_{0})) \ar[r]^{κ̃} \ar[d]_{_{Res_{1}}} & Aut_{typ}^{I}(L, L_{0}) \ar[d]_{_{Res_{2}}} \\ N_{Aut(G_{0},S_{0})}(Aut_{G}(G_{0}, S_{0})) \ar[r]^{κ̃_{0}} & N_{Aut_{typ}^{I}(L_{0})}(Aut_{L}(L_{0})) .}  (16)

Here, Aut(G, G_{0}, S·C′_{G_{0}}(S_{0})) is the group of automorphisms of G which send both G_{0} and S·C′_{G_{0}}(S_{0}) to themselves and Aut_{typ}^{I}(L, L_{0}) ≤ Aut_{typ}^{I}(L) is the subgroup of elements which leave L_{0} invariant.

Both Res_{1} and Res_{2} are defined by restriction. Each α ∈ Aut(G, G_{0}, S·C′_{G_{0}}(S_{0})) leaves S_{0} × C′_{G_{0}}(S_{0}) = G_{0} ∩ (S·C′_{G_{0}}(S_{0})) invariant, and hence also leaves S_{0} invariant. Clearly, α|_{Aut(G_{0},S_{0})} normalizes Aut_{G}(G_{0}, S_{0}). To see that Res_{2} maps to the normalizer, fix σ ∈ Aut_{typ}^{I}(L, L_{0}) and γ ∈ Aut_{L}(S_{0}), and set σ_{0} = σ|_{L_{0}} ∈ Aut_{typ}^{I}(L_{0}). Then

σ_{0}c_{γ}σ_{0}^{−1} = c_{σ(γ)},  (17)

(using Lemma 1.15 to show this holds on objects), and thus σ_{0} normalizes Aut_{L}(L_{0}).

The homomorphism κ̃_{0} is the restriction of κ̃_{G_{0}}^{H_{0}}, which is defined since H_{0} is Aut(S_{0}, F_{0})- invariant. Since κ̃_{G_{0}}^{H_{0}} maps Aut_{G}(G_{0}, S_{0}) onto Aut_{L}(L_{0}), it sends the normalizer of Aut_{G}(G_{0}, S_{0}) into the normalizer of Aut_{L}(L_{0}).

Defining κ̃ requires more explanation. For α ∈ Aut(G, G_{0}, S·C′_{G_{0}}(S_{0})), α(S) is a Sylow p-subgroup of S·C′_{G_{0}}(S_{0}), so α(S) = hSh^{−1} for some h ∈ C′_{G_{0}}(S_{0}). Hence c_{h}^{−1} ∘ α ∈ Aut(G, G_{0}, S) and we define κ̃(α) = κ̃_{G}^{H}(c_{h}^{−1} ∘ α) ∈ Aut_{typ}^{I}(L, L_{0}). If h′ ∈ C′_{G_{0}}(S_{0}) with α(S) = h′Sh′^{−1}, then h^{−1}h′ ∈ C′_{G_{0}}(S_{0}) ∩ N_{G}(S). Since S_{0} is strongly closed in F, the restriction homomorphism

N_{G}(S)/C′_{G}(S) = Aut_{L}(S) ⟶ Aut_{L}(S_{0}) = N_{G}(S_{0})/C′_{G_{0}}(S_{0})

is injective by Proposition 1.11(f). It follows that h^{−1}h′ ∈ C′_{G}(S), so κ̃_{G}^{H}(c_{h^{−1}h′}) = 1 since C′_{G}(S) ≤ O^{p}(C_{G}(P)) for each P ≤ S. Thus κ̃ is well defined, and it is easily seen to be a homomorphism. Since conjugation by any element of C′_{G_{0}}(S_{0}) induces the identity in Aut_{typ}^{I}(L_{0}) (and since Res_{2} ∘ κ̃_{G}^{H} = κ̃_{G_{0}}^{H_{0}} ∘ Res_{1} as maps from Aut(G, G_{0}, S) to Aut_{typ}^{I}(L_{0})), square (16) commutes.

Next consider the following commutative diagram:

[figure: 1; Z(G_{0}); N_{G}(S_{0}); ^{cj^{1}}; λ_{0}; g↦[g]; Aut_{G}(G_{0}, S_{0}); κ̃_{1}; 1; 1; Z(F_{0}); δ_{S_{0}}; Aut_{L}(S_{0}); _{cj_{2}}; Aut_{L}(L_{0}); 1 .]  (18)

Here, cj_{1} and cj_{2} are induced by conjugation, and κ̃_{1} is the restriction of κ̃_{0}. Both rows in (18) are exact: the first since Ker(cj_{1}) = C_{G}(G_{0}) = Z(G_{0}) by (15); and the second since Ker(cj_{2}) ≤ Aut_{L_{0}}(S_{0}) (L_{0} is centric in L) and hence Ker(cj_{2}) = Z(F_{0}) by Lemma 1.14(a). Thus the right hand square in (18) is a pullback square.

Fix automorphisms

α ∈ N_{Aut(G_{0},S_{0})}(Aut_{G}(G_{0}, S_{0})) and χ ∈ Aut_{typ}^{I}(L, L_{0})

such that χ|_{L_{0}} = κ̃_{0}(α). Then χ(S_{0}) = S_{0}, so χ_{S_{0}} is an automorphism of Aut_{L}(S_{0}) = N_{G}(S_{0})/C′_{G_{0}}(S_{0}) by (15).

We first construct β ∈ Aut(N_{G}(S_{0})) such that for each g ∈ N_{G}(S_{0}), c_{β(g)} = αc_{g}α^{−1} in Aut(G_{0}) and χ_{S_{0}}([g]) = [β(g)] in Aut_{L}(S_{0}). Consider the following automorphisms

c_{α} ∈ Aut(Aut_{G}(G_{0}, S_{0})),  χ_{S_{0}} ∈ Aut(Aut_{L}(S_{0})),  c_{κ̃_{0}(α)} = c_{χ} ∈ Aut(Aut_{L}(L_{0}))

of groups in the pullback square in (18). We want to define β as the pullback of c_{α} and χ_{S_{0}} over c_{χ}. For γ ∈ Aut_{L}(S_{0}), c_{χ}(cj_{2}(γ)) = χc_{γ}χ^{−1} = c_{χ(γ)} = cj_{2}(χ_{S_{0}}(γ)) (using (17)) and thus cj_{2} ∘ χ_{S_{0}} = c_{χ} ∘ cj_{2}. By a similar (but simpler) computation, κ̃_{1} ∘ c_{α} = c_{κ̃_{0}(α)} ∘ κ̃_{1}; and hence these three automorphisms pull back (via the pullback square in (18)) to a unique β ∈ Aut(N_{G}(S_{0})). Thus for g ∈ N_{G}(S_{0}),

[β(g)] = χ_{S_{0}}([g]) ∈ Aut_{L}(S_{0}) and cj_{1}(β(g)) = c_{α} ∘ cj_{1}(g) = αc_{g}α^{−1} ∈ Aut(G_{0}) . (19)

Now, χ_{S_{0}}(δ_{S_{0}}(S_{0})) = δ_{S_{0}}(S_{0}) and χ_{S_{0}}(δ_{S_{0}}(S)) = δ_{S_{0}}(S) since χ is isotypical and sends inclusions to inclusions (and hence restrictions to restrictions). Since Aut_{L}(S_{0}) = N_{G}(S_{0})/C′_{G_{0}}(S_{0}) by (15), (19) implies that β sends S_{0} × C′_{G_{0}}(S_{0}) to itself and sends S·C′_{G_{0}}(S_{0}) to itself. In particular, β(S_{0}) = S_{0}.

Now, for all g ∈ N_{G_{0}}(S_{0}),

λ_{0}(α(g)) = [α(g)] = κ̃_{0}(α)([g]) = χ_{S_{0}}([g]) ∈ Aut_{L}(S_{0})  (κ̃_{0}(α) = χ|_{L_{0}})

and

cj_{1} ∘ α(g) = c_{α(g)} = αc_{g}α^{−1} ∈ Aut_{G}(G_{0}, S_{0}) .

Thus λ_{0}(α(g)) = λ_{0}(β(g)) and cj_{1}(α(g)) = cj_{1}(β(g)) by comparison with (19); and hence α(g) = β(g) by the pullback square in (18). This proves that α|_{N_{G_{0}}(S_{0})} = β|_{N_{G_{0}}(S_{0})}.

We already saw that G = G_{0}·N_{G}(S_{0}). Define α̂ ∈ Aut(G, G_{0}, S·C′_{G_{0}}(S_{0})) by setting α̂(g_{0}h) = α(g_{0})β(h) for g_{0} ∈ G_{0} and h ∈ N_{G}(S_{0}). Since α|_{N_{G_{0}}(S_{0})} = β|_{N_{G_{0}}(S_{0})}, this is well defined as a bijective map of sets. For all g_{0}, g′_{0} ∈ G_{0} and h, h′ ∈ N_{G}(S_{0}),

α̂(g_{0}h·g′_{0}h′) = α̂(g_{0}·c_{h}(g′_{0})·hh′) = α(g_{0})α(c_{h}(g′_{0}))β(hh′)
= α(g_{0})c_{β(h)}(α(g′_{0}))β(hh′) = α(g_{0})β(h)α(g′_{0})β(h′) = α̂(g_{0}h)α̂(g′_{0}h′),

where the third equality follows from the condition c_{β(h)} = αc_{h}α^{−1}. It now follows that α̂ ∈ Aut(G, G_{0}). Also, α̂ sends S·C′_{G_{0}}(S_{0}) to itself since β does.

By construction, Res_{1}(α̂) = α̂|_{G_{0}} = α. We claim that κ̃(α̂) = χ. Since α̂|_{G_{0}} = α and χ|_{L_{0}} = κ̃_{0}(α), κ̃(α̂) and χ define the same action on L_{0} (by the commutativity of (16)). Choose h ∈ C′_{G_{0}}(S_{0}) = O^{p}(C_{G}(S_{0})) with α̂(S) = hSh^{−1} and let τ ∈ Aut(S) be given by τ(s) = c_{h}^{−1}(α̂(s)). For g ∈ N_{G}(S_{0}), κ̃(α̂)([g]) = [h^{−1}α̂(g)h] = [α̂(g)] = [β(g)] = χ([g]) in Aut_{L}(S_{0}) by (19) and since α̂|_{N_{G}(S_{0})} = β. Hence κ̃(α̂) and χ define the same action on Aut_{L}(S_{0}). Since L_{0} and Aut_{L}(S_{0}) generate the full subcategory L|_{≤S_{0}}, κ̃(α̂) and χ are equal after restriction to this subcategory.

We just showed that χ_{S_{0}}([s]) = κ̃(α̂)_{S_{0}}([s]) for s ∈ S. Hence χ_{S}([s]) = κ̃(α̂)_{S}([s]) in Aut_{L}(S). Lemma 1.15 now implies that χ(P) = κ̃(α̂)(P) for P ∈ Ob(L). Since both κ̃(α̂) and χ send inclusions to inclusions, and since the restriction map from Mor_{L}(P, Q) to Mor_{L}(P ∩ S_{0}, Q ∩ S_{0}) is injective for all P, Q ∈ H by Proposition 1.11(f), it now follows that κ̃(α̂) = χ.

To prove (16) is a pullback, it remains to show κ̃ × Res_{1} is injective. So assume α̂ ∈ Aut(G, G_{0}, S·C′_{G_{0}}(S_{0})) is such that α̂|_{G_{0}} = Id_{G_{0}} and κ̃(α̂) = Id_{L}. For each g ∈ G, c_{α̂(g)} = c_{g} ∈ Aut(G_{0}), and hence g^{−1}α̂(g) ∈ C_{G}(G_{0}) = Z(G_{0}) by (15). Since κ̃(α̂) = Id_{L}, α̂ induces the identity on Aut_{L}(S_{0}) = N_{G}(S_{0})/C′_{G_{0}}(S_{0}) (see (15) again). Since G = G_{0}·N_{G}(S_{0}) and α̂|_{G_{0}} = Id, g^{−1}α̂(g) ∈ C′_{G_{0}}(S_{0}) for all g ∈ G. Finally, C′_{G_{0}}(S_{0}) ∩ Z(G_{0}) = 1 because Z(G_{0}) = Z(F_{0}) ≤ S_{0} is a p-group, and we conclude that α̂ = Id_{G}.

Step 2: We are now ready to prove (14) is a pullback. Fix elements

[α] ∈ N_{Out(G_{0})}(Out_{G}(G_{0}))/Out_{G}(G_{0}) and [χ] ∈ Out_{typ}(L, L_{0})

such that κ^{∗}([α]) = R_{2}([χ]), and choose liftings α ∈ Aut(G_{0}, S_{0}) and χ ∈ Aut_{typ}^{I}(L, L_{0}). Then α normalizes Aut_{G}(G_{0}), and hence also normalizes Aut_{G}(G_{0}, S_{0}).

Since κ^{∗}([α]) = R_{2}([χ]), χ|_{L_{0}} = κ̃_{0}(α) ∘ c_{[x]} for some element x ∈ N_{G}(S_{0}) (where [x] ∈ Aut_{L}(S_{0}) is the class of x). Upon replacing α by α ∘ c_{x} ∈ Aut(G_{0}), we can arrange that χ|_{L_{0}} = κ̃_{0}(α). Hence α and χ pull back to an element of Aut(G, G_{0}, S·C′_{G_{0}}(S_{0})) by Step 1, and so [α] and [χ] pull back to an element of Out(G, G_{0}).

To see that this pullback is unique, fix [γ] ∈ Out(G, G_{0}) such that R_{1}([γ]) = 1 and κ([γ]) = 1, and choose γ ∈ Aut(G, G_{0}) which represents [γ]. Then γ(S) = gSg^{−1} for some g ∈ G, and upon replacing γ by c_{g}^{−1} ∘ γ, we can assume γ(S) = S. Also, κ̃(γ) = c_{[y]} for some y ∈ N_{G}(S); and upon replacing γ by γ ∘ c_{y}^{−1}, we can assume κ̃(γ) = Id_{L}. Now, γ|_{G_{0}} = c_{h} for some h ∈ N_{G}(S_{0}), and c_{[h]} = Id_{L_{0}}. Hence h ∈ G_{0} since L_{0} is centric in L, and so h ∈ C_{G_{0}}(S_{0}) = Z(S_{0}) × C′_{G_{0}}(S_{0}).

Write h = h_{1}h_{2}, where h_{1} ∈ Z(S_{0}) and h_{2} ∈ C′_{G_{0}}(S_{0}). Thus [h] = [h_{1}] ∈ Aut_{L_{0}}(S_{0}), and h_{1} ∈ Z(F_{0}) = Z(G_{0}) since c_{[h]} = Id_{L_{0}} (see Lemma 1.14(a)). Thus γ|_{G_{0}} = c_{h} = c_{h_{2}} in Aut(G_{0}). Since [S, h_{2}] ≤ [S, C′_{G_{0}}(S_{0})] ≤ C′_{G_{0}}(S_{0}), c_{h_{2}} ∈ Aut(G, G_{0}, S·C′_{G_{0}}(S_{0})). Also, κ̃(c_{h_{2}}) = Id by definition of κ̃ (and since h_{2} ∈ C′_{G_{0}}(S_{0})). Thus γ = c_{h_{2}} since (16) is a pullback, and so [γ] = 1 in Out(G, G_{0}). □

We are finally ready to prove:

Proposition 2.16. Let (S_{0}, F_{0}, L_{0}) ⊴ (S, F, L) be a normal pair such that L_{0} is centric in L, Ob(L_{0}) and Ob(L) are Aut(S_{0}, F_{0})- and Aut(S, F)-invariant, respectively, and L_{0} is Aut_{typ}^{I}(L)-invariant. Assume F_{0} is tamely realized by some finite group G_{0} such that S_{0} ∈ Syl_{p}(G_{0}), Z(G_{0}) = Z(F_{0}), and L_{0} ≅ L_{S_{0}}^{Ob(L_{0})}(G_{0}). Then F is tamely realized by a finite group G such that S ∈ Syl_{p}(G), G_{0} ⊴ G and G/G_{0} ≅ L/L_{0}.

Proof. Set H = Ob(L) and H_{0} = Ob(L_{0}). By assumption, F_{0} = F_{S_{0}}(G_{0}), and κ_{G_{0}} is split surjective. Also, L_{0} ≅ L_{S_{0}}^{H_{0}}(G_{0}) by assumption, and we identify these two linking systems. By Lemma 1.17, Out_{typ}(L_{0}) ≅ Out_{typ}(L_{S_{0}}^{H_{0}^{c}}(G_{0})) ≅ Out_{typ}(L_{S_{0}}^{c}(G_{0})), where H_{0}^{c} is the set of F_{0}-centric subgroups in H_{0}. Choose a splitting

s: Out_{typ}(L_{0}) ≅ Out_{typ}(L_{S_{0}}^{c}(G_{0})) ⟶ Out(G_{0})

for κ_{G_{0}}^{H_{0}}, and set

ρ̂ = s ∘ ρ_{L_{0}}^{L}: L/L_{0} ⟶ Out_{typ}(L_{0}) ⟶ Out(G_{0}) .

By Lemma 2.14, there is a finite group G such that S ∈ Syl_{p}(G), G_{0} ⊴ G, F = F_{S}(G), L ≅ L_{S}^{H}(G), G/G_{0} ≅ L/L_{0}, and such that the outer action of G/G_{0} on G_{0} is equal to ρ̂ via this last isomorphism. In particular, s sends Out_{L}(L_{0}) = Im(ρ_{L_{0}}^{L}) isomorphically to Out_{G}(G_{0}) = Im(ρ̂).

Since L_{0} is Aut_{typ}^{I}(L)-invariant by assumption, Out_{typ}(L, L_{0}) = Out_{typ}(L). So by Lemma 2.15, the following is a pullback square:

\xymatrix{Out(G, G_{0}) \ar[r]^{κ} \ar[d]_{R_{1}} & Out_{typ}(L) \ar[d]_{R_{2}} \\ N_{Out(G_{0})}(Out_{G}(G_{0}))/Out_{G}(G_{0}) \ar[r]^{κ^{∗}} & N_{Out_{typ}(L_{0})}(Out_{L}(L_{0}))/Out_{L}(L_{0})}  (20)

where κ^{∗} is induced by κ_{G_{0}}^{H_{0}}. Since the splitting s of κ_{G_{0}}^{H_{0}} sends Out_{L}(L_{0}) isomorphically to Out_{G}(G_{0}), it induces a splitting s^{∗} of κ^{∗}. Since (20) is a pullback, s^{∗} induces a splitting of κ = κ_{G}^{H}|_{Out(G,G_{0})} (Lemma 2.13). By Lemma 1.17, Out_{typ}(L) ≅ Out_{typ}(L_{S}^{c}(G)), and so F is tamely realized by G. □

We next turn to central extensions of fusion and linking systems. In the following lemma, when L is a linking system associated to F over the p-group S, and A ≤ S, we set

Aut_{typ}^{I}(L, A) = {α ∈ Aut_{typ}^{I}(L) | α_{S}(δ_{S}(A)) = δ_{S}(A)} ,

and let Out_{typ}(L, A) be its image in Out_{typ}(L).

Lemma 2.17. Fix a finite group G and a central p-subgroup A ≤ Z(G). Choose S ∈ Syl_{p}(G), and set G = G/A and S = S/A ∈ Syl_{p}(G). Set F = F_{S}(G), F = F_{S}(G) and

H = {P ≤ S | P ≥ A, P/A is F-centric} .

Then H contains all subgroups of S which are F-centric and F-radical, all subgroups in H are F-centric, and hence L =^{def} L_{S}^{H}(G) is a linking system associated to F. If, furthermore, O_{p′}(G) = 1 and Z(G) = Z(F), then the following square is a pullback:

\xymatrix{Out(G, A) \ar[d]^{ν_{1}} \ar[r]^{κ_{G,A}^{H}} & Out_{typ}(L, A) \ar[d]^{ν_{2}} \\ Out(G) \ar[r]^{κ_{G}} & Out_{typ}(L) ,}  (21)

where L = L_{S}^{c}(G), κ_{G,A}^{H} is defined analogously to κ_{G}, and ν_{1} and ν_{2} are induced by the

projections G −↠ G and L −↠ L.

Proof. We first prove the statements about H = Ob(L). If P ∈ H, then P is F-centric since A ≤ P and P/A is F-centric (cf. [BCGLO2, Lemma 6.4(a)]). Now assume P ≤ S is F-centric and F-radical; we must show P ∈ H. It suffices to do this when P is fully normalized in F. Since P is F-centric, A ≤ C_{S}(P) ≤ P. For x ∈ S with xA ∈ C_{S}(P/A), c_{x} induces the identity on A and on P/A. Hence c_{x} ∈ O_{p}(Aut_{F}(P)) by Lemma 1.6, so x ∈ P by Lemma 1.4. This proves C_{S}(P/A) ≤ P/A. By Proposition 1.8, P/A is fully normalized and hence fully centralized in F. We conclude that P/A is F-centric, so P ∈ H.

Consider the following diagram (with homomorphisms defined below):

\xymatrix{1 \ar[r] & Hom(G, A) \ar[r]^{λ_{1}} \ar[d]^{τ}_{≅} & Aut(G, S, A) \ar[r]^{(ν̃_{1},r_{1})} \ar[d]^{κ̃_{1}} & Aut(G, S) × Aut(A) \ar[d]^{κ̃_{2}×Id} \\ 1 \ar[r] & Hom(π_{1}(|L|), A) \ar[r]^{λ_{2}} & Aut_{typ}^{I}(L, A) \ar[r]^{(ν̃_{2},r_{2})} & Aut_{typ}^{I}(L) × Aut(A) .} (22)

Here, ν̃_{1} and ν̃_{2} are induced by the projection G ⟶ G and r_{1} and r_{2} by restriction to A, and Aut(G, S, A) ≤ Aut(G) is the subgroup of automorphisms which leave both S and A invariant. Also, κ̃_{1} = κ̃_{G,A}^{H} (defined analogously to κ̃_{G}), and κ̃_{2} = κ̃_{G}. The right hand square clearly commutes.

For β ∈ Hom(G, A) and g ∈ G, λ_{1}(β)(g) = g·β(gA). For any morphism ψ ∈ Mor_{L}(P, Q), let [ψ] ∈ π_{1}(|L|) be the class of the loop based at the vertex S, formed by the edges ι_{P}^{S}, ψ, and ι_{Q}^{S} (in that order). For β ∈ Hom(π_{1}(|L|), A), λ_{2}(β) is the automorphism of L which is the identity on objects, and sends ψ ∈ Mor_{L}(P, Q) (with image ψ ∈ Mor_{L}(P/A, Q/A)) to ψ ∘ δ_{P} (β([ψ])). It follows immediately from these definitions that for i = 1, 2, λ_{i} is injective and (ν̃_{i}, r_{i}) ∘ λ_{i} is trivial.

Since A is a finite abelian p-group, Hom(π_{1}(X), A) ≅ H^{1}(X; A) ≅ H^{1}(X_{p}^{∧}; A) for any “p-good” space X (the second isomorphism by [BK, Definition I.5.1]). Also, |L| is p-good by [BLO2, Proposition 1.12], BG is p-good since it has finite fundamental group (cf. [BK, Proposition VII.5.1]), and BG_{p}^{∧} ≃ |L|_{p}^{∧} by [BLO1, Proposition 1.1]. We thus get an isomorphism

τ: Hom(G, A) ⟶^{≅} H^{1}(BG_{p}^{∧}; A) ⟶^{≅} H^{1}(|L|_{p}^{∧}; A) ⟶^{≅} Hom(π_{1}(|L|), A) .

Alternatively, by [BCGLO2, Theorem B], π_{1}(|L|)/O^{p}(π_{1}(|L|)) ≅ S/𝔥𝔶𝔭(F), where for an infinite group Γ, O^{p}(Γ) denotes the intersection of all normal subgroups of p-power index. By the hyperfocal subgroup theorem for groups [Pg1, § 1.1], G/O^{p}(G) ≅ S/𝔥𝔶𝔭(F); and these isomorphisms induce an isomorphism

τ: Hom(G, A) ⟶^{≅} Hom(S/𝔥𝔶𝔭(F), A) ⟶^{≅} Hom(π_{1}(|L|), A) .

By either construction, τ makes the left hand square in (22) commute.

An element α ∈ Ker(ν̃_{1}, r_{1}) is an automorphism of G which induces the identity on A and on G = G/A, and since A ≤ Z(G), any such automorphism has the form α(g) = g·β(gA) for some unique β ∈ Hom(G, A). Thus the top row in (22) is exact.

Similarly, an element α ∈ Ker(ν̃_{2}, r_{2}) is an isotypical automorphism of L which sends inclusions to inclusions and induces the identity on L and on A. Since L ⟶ L is bijective on objects (by definition), α induces the identity on objects in L, and on morphisms it has the form α(ψ) = ψ ∘ β(ψ) for some β: Mor(L) ⟶ A which preserves composition and sends inclusions to the identity. Such a β is equivalent to a homomorphism from π_{1}(|L|) to A (cf. [OV1, Proposition A.3(a)]), so α = λ_{2}(β), and thus the second row in (22) is exact.

We are now ready to prove that (21) is a pullback. Fix automorphisms α ∈ Aut(G, S) and β ∈ Aut_{typ}^{I}(L, A) such that κ_{G}([α]) = ν_{2}([β]). Then ν̃_{2}(β) = κ̃_{2}(α) ∘ c_{[x]} for some x ∈ N_{G}(S) which induces [x] ∈ Aut_{L}(S). So upon replacing α by α ∘ c_{x}, we can assume κ̃_{2}(α) = ν̃_{2}(β). Consider the following diagram:

\xymatrix{1 \ar[r] & A \ar[r] \ar[d]_{≅}^{r_{2}(β)} & G \ar[r] \ar@{-->}[d]^{α̂} & G \ar[r] \ar[d]_{≅}^{α} & 1 \\ 1 \ar[r] & A \ar[r] & G \ar[r] & G \ar[r] & 1}

We want to find α̂ ∈ Aut(G) which makes the two squares commute. This means showing that the class [G] ∈ H^{2}(G; A) is invariant under the automorphism of H^{2}(G; A) induced by r_{2}(β) and α. But β ∈ Aut_{typ}^{I}(L) induces an automorphism γ = β_{S}|_{δ_{S}(S)} ∈ Aut(S, F) (see Lemma 1.15). Also, γ|_{A} = β_{S}|_{A} = r_{2}(β), γ induces the automorphism (ν̃_{2}(β))_{S}|_{S} = α|_{S} on S, and thus [S] ∈ H^{2}(S; A) is invariant under these automorphisms of S and A. Since H^{2}(G; A) injects into H^{2}(S; A) under restriction, this proves that [G] is also invariant, and hence that there is an automorphism α̂ ∈ Aut(G, S, A) as desired.

Thus (ν̃_{1}, r_{1})(α̂) = (α, r_{2}(β)). By the commutativity of (22),

(ν̃_{2}, r_{2})(κ̃_{1}(α̂)) = (κ̃_{2}(α), r_{2}(β)) = (ν̃_{2}, r_{2})(β).

Hence there is χ ∈ Hom(G, A) such that λ_{2}(τ(χ)) = κ̃_{1}(α̂)^{−1} ∘ β, and the element α̂ ∘ λ_{1}(χ) ∈ Aut(G, S, A) pulls back α ∈ Aut(G, S) and β ∈ Aut_{typ}^{I}(L, A).

This proves that Out(G) surjects onto the pullback in square (21). To prove that it injects into the pullback, fix α̂ ∈ Aut(G, S, A) such that κ_{G}^{H}([α̂]) = 1 and ν_{1}([α̂]) = 1. Upon composing α̂ by an appropriate inner automorphism, we can assume it induces the identity on G. Thus κ̃_{1}(α̂) = c_{[x]} ∈ Aut_{typ}^{I}(L) for some x ∈ N_{G}(S) inducing [x] ∈ Aut_{L}(S), where c_{[x]} induces the identity on L. This means that xA ∈ Z(F) (Lemma 1.14(a)), and hence xA ∈ Z(G) by assumption. So upon replacing α̂ by α̂ ∘ c_{x}^{−1} ∈ Aut(G) we have an automorphism which induces the identity on L and on G. By the exactness of the rows in (22) again, α̂ = Id, and this finishes the proof. □

Lemma 2.17 now implies the result we need about tameness.

Proposition 2.18. Fix a saturated fusion system F over a finite p-group S. Assume F/Z(F) is tamely realized by the finite group G such that O_{p′}(G) = 1 and Z(G) = Z(F/Z(F)). Then F is tamely realized by a finite group G such that Z(G) = Z(F) and G/Z(G) ≅ G, and hence O_{p′}(G) = 1. If G ∈ 𝔊(p), then G ∈ 𝔊(p).

Proof. Set A = Z(F) and S = S/A for short. By assumption, S ∈ Syl_{p}(G), F/A ≅ F_{S}(G), κ_{G} is split surjective, O_{p′}(G) = 1, and Z(G) = Z(F/A).

By [BCGLO2, Corollary 6.14], the fusion system F is realizable, and by the proof of that corollary, it is realizable by a finite group G such that S ∈ Syl_{p}(G), A ≤ Z(G), and G/A ≅ G. Hence O_{p′}(G) = 1, so Z(G) is a p-group which is central in F. Thus Z(G) = Z(F).

Let L ⊆ L_{S}^{c}(G) be the full subcategory whose objects are the subgroups P ≤ S such that P ≥ A and P/A is F/A-centric, and set L = L_{S}^{c}(G). Then L is a linking system associated to F by Lemma 2.17, and A = Z(F) is invariant under all automorphisms in Aut_{typ}^{I}(L) by Lemma 1.15. Lemma 2.17 now implies that the following is a pullback square:

\xymatrix{Out(G, A) \ar[r]^{κ} \ar[d] & Out_{typ}(L) \ar[d] \\ Out(G) \ar[r]^{κ_{G}} & Out_{typ}(L) .}

By assumption, κ_{G} is split surjective. Hence κ = κ_{G}^{H}|_{Out(G,A)} (H = Ob(L)) is also split surjective by Lemma 2.13, so κ_{G}^{H} is split surjective. Since Out_{typ}(L) ≅ Out_{typ}(L_{S}^{c}(G)) by Lemma 1.17, this finishes the proof that F is tame.

By construction, G and G have the same nonabelian composition factors. Hence G ∈ 𝔊(p) if G ∈ 𝔊(p). □

One more technical lemma is needed before we can prove Theorem A.

Lemma 2.19. Let F be a saturated fusion system over a finite p-group S. If F is tame, then there is a finite group G such that O_{p′}(G) = 1 and F is tamely realized by G. If F is strongly tame, then G can be chosen such that in addition, G ∈ 𝔊(p).

Proof. Fix any Ĝ which tamely realizes F. If F is strongly tame, we assume Ĝ ∈ 𝔊(p). Thus S ∈ Syl_{p}(Ĝ), F ≅ F_{S}(Ĝ), and κĜ is split surjective. Set G = Ĝ/O_{p′}(Ĝ), and identify S with its image in G. Since G is a quotient group of Ĝ, G ∈ 𝔊(p) if Ĝ ∈ 𝔊(p).

By construction, F_{S}(G) ≅ F_{S}(Ĝ) ≅ F, and O_{p′}(G) = 1. The natural homomorphism from Ĝ onto G induces a homomorphism between their outer automorphism groups and an isomorphism between their linking systems, and the resulting square

\xymatrix{Out(Ĝ) \ar[r] \ar[d]_{κĜ} & Out(G) \ar[d]^{κ_{G}} \\ Out_{typ}(L_{S}^{c}(Ĝ)) \ar[r]^{≅} & Out_{typ}(L_{S}^{c}(G))}

commutes. Since κĜ is split surjective, so is κ_{G}. □

We are now ready to prove Theorem A. Recall that 𝔯𝔢𝔡(F) denotes the reduction of a fusion system F (see Definition 2.1).

Theorem 2.20. For any saturated fusion system F over a finite p-group S, if 𝔯𝔢𝔡(F) is strongly tame, then F is tame.

Proof. Set Q = O_{p}(F), S_{0} = C_{S}(Q)/Z(Q), and F_{0} = C_{F}(Q)/Z(Q). Let 𝔯𝔢𝔡(F) = F_{m} ⊆ F_{m−1} ⊆ ⋯ ⊆ F_{0} be a sequence of fusion subsystems, where for each i, F_{i} = O^{p}(F_{i−1}) or F_{i} = O^{p′}(F_{i−1}). Let S_{m} ⊴ ⋯ ⊴ S_{0} be the corresponding sequence of p-groups: each F_{i} is a fusion system over S_{i}. By Lemma 2.3, O_{p}(F_{i}) = 1 for each i, and hence Z(F_{i}) = 1 for each i.

We first show inductively that each of the F_{i} is strongly tame. Fix 1 ≤ i ≤ m, and assume F_{i} is tamely realized by G_{i} ∈ 𝔊(p). By Lemma 2.19, we can assume O_{p′}(G_{i}) = 1. Thus Z(G_{i}) is a p-group central in the fusion system F_{i}, and hence Z(G_{i}) = 1 since Z(F_{i}) = 1. By Proposition 2.12(a,b), there is a centric linking system associated to F_{i−1}.

Hence by Proposition 1.31(a,b), there are linking systems L_{i} ⊴ L_{i−1} associated to F_{i} ⊴ F_{i−1} such that L_{i} is a centric linking system (so Ob(L_{i}) is Aut(S_{i}, F_{i})-invariant), Ob(L_{i−1}) is Aut(S_{i−1}, F_{i−1})-invariant, and L_{i} is Aut_{typ}^{I}(L_{i−1})-invariant. Also, L_{i} is centric in L_{i−1} by Proposition 1.31(a,b) again (and since Z(F_{i−1}) = 1). By Lemma 2.11(c), L_{i} ≅ L_{S_{i}}^{c}(G_{i}). The hypotheses of Proposition 2.16 are thus satisfied, and hence F_{i−1} is tamely realized by some G_{i−1} such that G_{i} ⊴ G_{i−1} and G_{i−1}/G_{i} ≅ L_{i−1}/L_{i}. In particular, G_{i−1}/G_{i} is p-solvable, and so G_{i−1} ∈ 𝔊(p) by Lemma 2.11(b).

Since F_{m} was assumed to be tamely realized by some G_{m} ∈ 𝔊(p), we now conclude that F_{0} is tamely realized by G_{0} ∈ 𝔊(p). By Lemma 2.19 again, we can assume O_{p′}(G_{0}) = 1, and Z(G_{0}) = 1 since Z(F_{0}) = 1. Next consider the saturated fusion system F^{∗} =^{def} N_{F}^{Inn(Q)}(Q) over S^{∗} =^{def} Q·C_{S}(Q). Since F^{∗} ⊴ F by Proposition 1.25(c), O_{p}(F^{∗}) = Q by Lemma 1.20(e). Let Z(Q) = Z_{1}(Q) ≤ Z_{2}(Q) ≤ ⋯ ≤ Q be the upper central series for Q. Since Aut_{F^{∗}}(Q) = Inn(Q), Z_{i+1}(Q)/Z_{i}(Q) is central in F^{∗}/Z_{i}(Q) for each i. Also, by repeated application of Proposition 1.8, if P/Z_{i}(Q) = Z(F^{∗}/Z_{i}(Q)), then P ⊴ F^{∗}, and hence P ≤ Q. Thus Z(F^{∗}/Z_{i}(Q)) ≤ Z(Q/Z_{i}(Q)) = Z_{i+1}(Q)/Z_{i}(Q), and these two subgroups are equal.

Thus F_{0} = C_{F}(Q)/Z(Q) ≅ F^{∗}/Q is obtained from F^{∗} by sequentially dividing out by its center until the fusion system is centerfree. By repeated application of Proposition 2.18, F^{∗} is tamely realizable by some finite group G^{∗} ∈ 𝔊(p) such that O_{p′}(G^{∗}) = 1 and Z(G^{∗}) = Z(F^{∗}).

By Proposition 2.12(c), there is a centric linking system associated to F. Hence by Proposition 1.31(c), there are linking systems L^{∗} ⊴ L associated to F^{∗} ⊴ F, where all objects in L^{∗} are F^{∗}-centric, Ob(L^{∗}) is Aut(S^{∗}, F^{∗})-invariant, Ob(L) is Aut(S, F)-invariant, L^{∗} is Aut_{typ}^{I}(L)-invariant, and L^{∗} is centric in L. By Lemma 2.11(c) (and since G^{∗} ∈ 𝔊(p)), L^{∗} ≅ L_{S^{∗}}^{H^{∗}}(G^{∗}). Hence by Proposition 2.16, F is tamely realized by a finite group G. □

3. Decomposing reduced fusion systems as products

If F_{1} and F_{2} are fusion systems over finite p-groups S_{1} and S_{2}, respectively, then F_{1} × F_{2} is the fusion system over S_{1} × S_{2} defined as follows. For all P, Q ≤ S_{1} × S_{2}, if P_{i}, Q_{i} ≤ S_{i} denote the images of P and Q under projection to S_{i}, then

Hom_{F_{1}×F_{2}}(P, Q) = {(φ_{1}, φ_{2})|_{P} | φ_{i} ∈ Hom_{F_{i}}(P_{i}, Q_{i}), (φ_{1}, φ_{2})(P) ≤ Q} .

Here, we regard P and Q as subgroups of P_{1} × P_{2} and Q_{1} × Q_{2}, respectively. Thus F_{1} × F_{2} is the smallest fusion system over S_{1} × S_{2} for which

Hom_{F_{1}×F_{2}}(P_{1} × P_{2}, Q_{1} × Q_{2}) = Hom_{F_{1}}(P_{1}, Q_{1}) × Hom_{F_{2}}(P_{2}, Q_{2})

for each P_{i}, Q_{i} ≤ S_{i}. By [BLO2, Lemma 1.5], F_{1} × F_{2} is saturated if F_{1} and F_{2} are saturated. We leave it as an easy exercise to check, for any pair of finite groups G_{1}, G_{2} with Sylow subgroups S_{i} ∈ Syl_{p}(G_{i}), that F_{S_{1}×S_{2}}(G_{1} × G_{2}) = F_{S_{1}}(G_{1}) × F_{S_{2}}(G_{2}).

We say that a nontrivial fusion system F is indecomposable if it has no decomposition as a product of fusion systems over nontrivial p-groups. The main result in this section is Theorem C: every reduced fusion system has a unique decomposition as a product of reduced indecomposable fusion systems, and the product is tame if each of the indecomposable factors is tame. The first statement will be proven as Proposition 3.6, and the second as Theorem 3.7.

We first prove the following easy lemma about fusion systems over products of finite p-groups.

Lemma 3.1. Let S_{1}, S_{2} be a pair of finite p-groups, and set S = S_{1} × S_{2}. For each subgroup P ≤ S which does not split as a product P = P_{1} × P_{2} for P_{i} ≤ S_{i}, there is x ∈ N_{S}(P)∖P such that c_{x} ∈ O_{p}(Aut(P)). Hence for each saturated fusion system F over S, and each subgroup P ≤ S which is F-centric and F-radical, P = P_{1} × P_{2} for some pair of subgroups P_{i} ≤ S_{i}.

Proof. We prove the first statement; the last then follows by Lemma 1.4.

Fix P ≤ S. For i = 1, 2, let P_{i} ≤ S_{i} be the image of P under projection. Thus P ≤ P_{1}×P_{2}. Let Z_{k}(P) and Z_{k}(P_{i}) be the k-th terms in the upper central series for P and P_{i}; i.e., Z_{1}(P) = Z(P) and Z_{k+1}(P)/Z_{k}(P) = Z(P/Z_{k}(P)). We claim that for each k,

Z_{k}(P) = P ∩ (Z_{k}(P_{1}) × Z_{k}(P_{2})) .  (1)

This is clear for k = 1: an element of P is central only if it commutes with all elements in P_{1} and all elements in P_{2}. If (1) holds for k, then P/Z_{k}(P) can be identified as a subgroup of (P_{1}/Z_{k}(P_{1})) × (P_{2}/Z_{k}(P_{2})) (a subgroup which projects onto each factor), and the result for Z_{k+1}(P) then follows immediately.

If P ≨ P_{1} × P_{2}, then choose x ∈ N_{P_{1}×P_{2}}(P)∖P (see [Sz1, Theorem 2.1.6]). By (1), conjugation by x acts via the identity on each quotient Z_{k+1}(P)/Z_{k}(P). So c_{x} ∈ O_{p}(Aut(P)) by Lemma 1.6. □

The next lemma gives some basic properties of product fusion systems.

Lemma 3.2. Assume F_{1} and F_{2} are saturated fusion systems over finite p-groups S_{1} and S_{2}. For each i = 1, 2, let F′_{i} ⊆ F_{i} be a saturated fusion subsystem over S′_{i} ≤ S_{i}.

(a) If F′_{i} ⊴ F_{i} for i = 1, 2, then F′_{1} × F′_{2} is normal in F_{1} × F_{2}.

(b) If F′_{i} has index prime to p in F_{i} for i = 1, 2, then F′_{1} × F′_{2} has index prime to p in F_{1} × F_{2}.

Proof. Set S = S_{1} × S_{2}, S′ = S′_{1} × S′_{2}, F = F_{1} × F_{2}, and F′ = F′_{1} × F′_{2}.

(a) Since S′_{i} is strongly closed in F_{i}, S′ is strongly closed in F.

Fix P, Q ≤ S′ and φ ∈ Hom_{F}(P, Q). Let P_{i}, Q_{i} ≤ S′_{i} be the images of P and Q under projection to S′_{i}. Then φ = (φ_{1}, φ_{2})|_{P} for some φ_{i} ∈ Hom_{F_{i}}(P_{i}, Q_{i}). By condition (ii) in Definition 1.18, there are morphisms α_{i} ∈ Aut_{F_{i}}(S′_{i}) and φ′_{i} ∈ Hom_{F′_{i}}(α_{i}(P_{i}), Q_{i}) such that φ_{i} = φ′_{i} ∘ α_{i}|_{P_{i},α_{i}(P_{i})}. Set α = (α_{1}, α_{2}) ∈ Aut_{F}(S′), and set φ′ = (φ′_{1}, φ′_{2})|_{α(P)}. Then φ′(α(P)) ≤ Q, so φ′ ∈ Hom_{F′}(α(P), Q) and φ = φ′ ∘ α|_{P,α(P)}. This proves condition (ii) for the pair F′ ⊆ F.

Let P, Q ≤ S′ and P_{i}, Q_{i} ≤ S′_{i} be as above, and fix φ = (φ_{1}, φ_{2})|_{P} ∈ Hom_{F′}(P, Q) and β = (β_{1}, β_{2}) ∈ Aut_{F}(S′). Then β_{i}φ_{i}β_{i}^{−1} ∈ Hom_{F′_{i}}(β_{i}(P_{i}), β_{i}(Q_{i})) by condition (iii) for the normal pair F′_{i} ⊴ F_{i}. Also, βφβ^{−1}(β(P)) ≤ β(Q), and hence βφβ^{−1} ∈ Hom_{F′}(β(P), β(Q)). This proves condition (iii) for the pair F′ ⊆ F, and finishes the proof that F′ is normal in F.

(b) Note that S′_{i} = S_{i}, since F′_{i} has index prime to p in F_{i}. Since F′_{i} ⊇ O^{p′}(F_{i}), it suffices to prove this point when F′_{i} = O^{p′}(F_{i}), and thus when F′_{i} ⊴ F_{i} (Proposition 1.25(b)). Hence F′_{1} × F′_{2} is normal in F_{1} × F_{2} by (a). Since they are fusion systems over the same p-group, the result now follows by Lemma 1.26. □

We next prove the following criterion for a reduced fusion system to decompose: F factors as a product of fusion subsystems whenever S factors as a product of subgroups which are strongly closed in F.

Proposition 3.3. Let F be a saturated fusion system over a finite p-group S = S_{1}×⋯×S_{m}, where S_{1}, …, S_{m} are all strongly closed in F. Set F_{i} = F|_{S_{i}} (i = 1, …, m): the full subcategory of F with objects the subgroups of S_{i}, regarded as a fusion system over S_{i}. For each i, let S_{i}^{∗} = ∏_{j≠i} S_{j}, identify S = S_{i} × S_{i}^{∗}, and let F′_{i} ⊆ F_{i} be the fusion subsystem over S_{i} where for P, Q ≤ S_{i},

Hom_{F′_{i}}(P, Q) = {φ ∈ Hom_{F_{i}}(P, Q) | (φ, Id_{S_{i}^{∗}}) ∈ Hom_{F}(P × S_{i}^{∗}, Q × S_{i}^{∗})} .

Then F′_{i} and F_{i} are saturated fusion systems for each i, O^{p′}(F_{i}) ⊆ F′_{i}; and

F′_{1} × ⋯ × F′_{m} ⊆ F ⊆ F_{1} × ⋯ × F_{m} .

If O^{p′}(F) = F, then F′_{i} = F_{i} for each i, and hence F = F_{1} × ⋯ × F_{m}.

Proof. Fix i ∈ {1, …, m}. We first claim that

∀ P, Q ≤ S_{i} and φ ∈ Hom_{F_{i}}(P, Q), there are ψ ∈ Aut_{F}(S_{i}^{∗}) and χ ∈ Aut_{F_{i}}(S_{i})

such that (φ, ψ) ∈ Hom_{F}(P × S_{i}^{∗}, Q × S_{i}^{∗}) and χ|_{Q} ∘ φ ∈ Hom_{F′_{i}}(P, S_{i}) . (2)

If φ(P) is fully centralized in F, the existence of ψ follows by the extension axiom, and since the S_{i} are all strongly closed in F. The general case then follows upon choosing α ∈ Iso_{F}(φ(P), R) where R ≤ S_{i} is fully centralized in F, and applying the extension axiom to α ∘ φ and to α. By the extension axiom again, this time applied to ψ, there is χ such that (χ^{−1}, ψ) ∈ Aut_{F}(S), and hence χ|_{Q} ∘ φ ∈ Hom_{F′_{i}}(P, S_{i}). This finishes the proof of (2).

Two subgroups of S_{i} are F_{i}-conjugate if and only if they are F-conjugate; and they cannot be F-conjugate to any other subgroups of S since S_{i} is strongly closed. Also, for P ≤ S_{i}, |N_{S}(P)| = |N_{S_{i}}(P)|·|S_{i}^{∗}| and |C_{S}(P)| = |C_{S_{i}}(P)|·|S_{i}^{∗}|. Hence P is fully normalized (centralized) in F_{i} if and only if it is fully normalized (centralized) in F. By (2), P, Q ≤ S_{i} are F_{i}-conjugate only if P is F′_{i}-conjugate to a subgroup in the Aut_{F_{i}}(S_{i})-orbit of Q, and hence P is fully normalized (centralized) in F_{i} if and only if it is fully normalized (centralized) in F′_{i}. Also, in the context of axiom (II), N_{φ}^{F} = N_{φ}^{F_{i}} × S_{i}^{∗} for all φ ∈ Mor(F_{i}), and N_{φ}^{F′_{i}} = N_{φ}^{F_{i}} for all φ ∈ Mor(F′_{i}). Axioms (I) and (II) for F_{i} and for F′_{i} now follow easily from the same axioms applied to F; and thus F_{i} and F′_{i} are saturated.

Fix P ≤ S_{i}, and choose φ ∈ Aut_{F_{i}}(P) and α ∈ Aut_{F′_{i}}(P). By (2), there is ψ ∈ Aut_{F}(S_{i}^{∗}) such that (φ, ψ), (α, Id) ∈ Aut_{F}(P × S_{i}^{∗}). Hence (φαφ^{−1}, Id) ∈ Aut_{F}(P × S_{i}^{∗}), φαφ^{−1} ∈ Aut_{F′_{i}}(P), and so Aut_{F′_{i}}(P) is normal in Aut_{F_{i}}(P). When P is fully normalized, Aut_{F′_{i}}(P) contains Aut_{S_{i}}(P) ∈ Syl_{p}(Aut_{F_{i}}(P)), and thus Aut_{F′_{i}}(P) ≥ O^{p′}(Aut_{F_{i}}(P)). Hence F′_{i} has index prime to p in F_{i} (see Definition 1.21), and so F′_{i} ⊇ O^{p′}(F_{i}).

Clearly, F contains F′_{1} × ⋯ × F′_{m}. By Lemma 3.1 together with Alperin’s fusion theorem (Theorem 1.3), each morphism in F is a composite of restrictions of automorphisms of subgroups of the form P_{1} × … × P_{m} for P_{i} ≤ S_{i}. Since the S_{i} are strongly closed in F, each such automorphism has the form (φ_{1}, …, φ_{m}) for some φ_{i} ∈ Aut_{F}(P_{i}) = Aut_{F_{i}}(P_{i}). Hence for arbitrary P, Q ≤ S, if P_{i}, Q_{i} ≤ S_{i} denote the images of P and Q under projection, then each φ ∈ Hom_{F}(P, Q) extends to some morphism (φ_{1}, …, φ_{m}) where φ_{i} ∈ Hom_{F}(P_{i}, Q_{i}). Since Hom_{F}(P_{i}, Q_{i}) = Hom_{F_{i}}(P_{i}, Q_{i}), this shows that F ⊆ F_{1} × ⋯ × F_{m}.

Since F′_{i} has index prime to p in F_{i} for each i, F′_{1} × ⋯ × F′_{m} has index prime to p in F_{1} × ⋯ × F_{m} by Lemma 3.2(b), and hence has index prime to p in F. So if O^{p′}(F) = F, then F = F′_{1} × ⋯ × F′_{m}; and F_{i} = F′_{i} for each i by definition of F_{i}. □

Note that if F is any fusion system (saturated or not) over a finite p-group S = S_{1} × S_{2}, and F factors as a product of fusion systems over S_{1} and S_{2}, then the factors must be the fusion subsystems F_{i} = F′_{i} as defined in Proposition 3.3. In other words, if there is any such factorization, it must be unique.

We next show that a product of reduced fusion systems is reduced.

Proposition 3.4. Fix finite p-groups S_{1} and S_{2} and saturated fusion systems F_{i} over S_{i}. Set F = F_{1} × F_{2}. Then

O_{p}(F) = O_{p}(F_{1}) × O_{p}(F_{2}), O^{p}(F) = O^{p}(F_{1}) × O^{p}(F_{2}), O^{p′}(F) = O^{p′}(F_{1}) × O^{p′}(F_{2}) .

In particular, F is reduced if and only if F_{1} and F_{2} are both reduced.

Proof. Set S = S_{1} × S_{2}. The decomposition of O_{p}(F) is clear: if P ≤ S is normal in F, then so are its projections into S_{1} and S_{2}, and P_{i} ⊴ F_{i} implies P_{1} × P_{2} ⊴ F.

The relation “of index prime to p” among fusion systems is transitive (see Definition 1.21), and hence O^{p′}(O^{p′}(F)) = O^{p′}(F). So by Proposition 3.3, O^{p′}(F) = F′_{1} × F′_{2} for some pair of fusion systems F′_{i} over S_{i}. Also, O^{p′}(F) ⊆ O^{p′}(F_{1}) × O^{p′}(F_{2}) by Lemma 3.2(b), so F′_{i} ⊆ O^{p′}(F_{i}), and F′_{i} has index prime to p in F_{i} since F′_{1} × F′_{2} has index prime to p in F. Thus F′_{i} = O^{p′}(F_{i}).

By definition,

𝔥𝔶𝔭(F) = ⟨s^{−1}α(s) | s ∈ P ≤ S, α ∈ O^{p}(Aut_{F}(P))⟩ = 𝔥𝔶𝔭(F_{1}) × 𝔥𝔶𝔭(F_{2}) .

Since O^{p}(F) is the unique fusion subsystem over 𝔥𝔶𝔭(F) of p-power index in F (Theorem 1.22(a)), we have O^{p}(F) = O^{p}(F_{1}) × O^{p}(F_{2}).

The last statement is now immediate. □

By definition, every fusion system F factors as a product of indecomposable fusion systems. The following lemma is the key step when showing that this factorization is unique (not only up to isomorphism) when F is reduced.

Lemma 3.5. Let F be a reduced fusion system over a finite p-group S. Assume F = F_{1} × F_{2} = F_{3} × F_{4}, where each F_{i} is a saturated fusion system over some S_{i} ≤ S. Set S_{ij} = S_{i} ∩ S_{j} for i = 1, 2 and j = 3, 4. Then F = F_{13} × F_{14} × F_{23} × F_{24}, where F_{ij} is a reduced fusion system over S_{ij}.

Proof. By assumption, the subgroups S_{i} for i ∈ {1, 2, 3, 4} are all strongly closed in F, and S = S_{1} ×S_{2} = S_{3} ×S_{4}. Fix x, y ∈ S_{1} which are F-conjugate, and choose φ ∈ Hom_{F}(⟨x⟩, ⟨y⟩) which sends x to y. Write x = x_{3}x_{4} and y = y_{3}y_{4}, where x_{3}, y_{3} ∈ S_{3} and x_{4}, y_{4} ∈ S_{4}. There are homomorphisms φ_{i} ∈ Hom_{F_{i}}(⟨x_{i}⟩, ⟨y_{i}⟩) for i = 3, 4 which send x_{i} to y_{i}, and such that φ is the restriction of (φ_{3}, φ_{4}). Hence (φ_{3}, Id_{S_{4}})(x) = y_{3}x_{4}, y_{3}x_{4} ∈ S_{1} since S_{1} is strongly closed, and thus x_{3}^{−1}y_{3} ∈ S_{13}. By a similar argument, x_{4}^{−1}y_{4} ∈ S_{14}, and thus x^{−1}y ∈ S_{13} × S_{14}. This proves that 𝔣𝔬𝔠(F_{1}) ≤ S_{13} × S_{14}.

By a similar argument, 𝔣𝔬𝔠(F_{2}) ≤ S_{23} × S_{24}. Since F = F_{1} × F_{2}, it follows that

𝔣𝔬𝔠(F) = 𝔣𝔬𝔠(F_{1}) × 𝔣𝔬𝔠(F_{2}) ≤ S_{13} × S_{14} × S_{23} × S_{24} ≤ S .

Also, 𝔣𝔬𝔠(F) = S since F is reduced (Theorem 1.22(a)), so S is the product of the S_{ij}. Since the intersection of two subgroups which are strongly closed in F is strongly closed in F, F splits as a product of reduced fusion systems F_{ij} over S_{ij} by Propositions 3.3 and 3.4 (recall O^{p′}(F) = F since F is reduced). □

This now implies the uniqueness of any decomposition of a reduced fusion system as a product of indecomposables.

Proposition 3.6. Each reduced fusion system F over a finite p-group S has a unique factorization F = F_{1} × ⋯ × F_{m} as a product of indecomposable fusion systems F_{i} over subgroups S_{i} ⊴ S. Moreover, the F_{i} are all reduced, and each fusion preserving automorphism α ∈ Aut(S, F) permutes the factors S_{i}.

Proof. Let F = F_{1} × ⋯ × F_{m} = F′_{1} × ⋯ × F′_{n} be two decompositions as products of indecomposable fusion systems. By Lemma 3.5 applied to the decompositions F = F_{1} × ∏_{i≥2} F_{i} = F′_{1} × ∏_{i≥2} F′_{i}, and since F_{1} and F′_{1} are indecomposable, either F_{1} = F′_{1} and ∏_{i≥2} F_{i} = ∏_{i≥2} F′_{i}, or F_{1} is a direct factor in ∏_{i≥2} F′_{i}. In the latter case, we can assume by induction on |S| that the decomposition of ∏_{i≥2} F′_{i} is unique, and hence that for some j, F_{1} = F′_{j} and so ∏_{i≠1} F_{i} = ∏_{i≠j} F′_{i} (Lemma 3.5 again). By the same induction hypothesis, this proves that the two decompositions are equal up to permutation of the factors. The factors F_{i} are all reduced by Proposition 3.4.

Fix α ∈ Aut(S, F). Since S = ∏_{i=1}^{m} α(S_{i}) is a product of subgroups which are strongly closed in F, F factors as a product of saturated fusion systems over the α(S_{i}) by Proposition 3.3 (and since O^{p′}(F) = F). So α permutes the factors S_{i} by the uniqueness of the decomposition. □

We are now ready to prove that a product of reduced, indecomposable, tame fusion systems is tame. Together with Theorem 2.20, this shows that any “minimal” exotic fusion system is indecomposable as well as reduced.

Theorem 3.7. Fix a reduced fusion system F over a finite p-group S, and let F = F_{1} × ⋯ × F_{m} be its unique factorization as a product of indecomposable fusion systems. If F_{i} is tame (strongly tame) for each i, then F is tame (strongly tame).

Proof. Let S = S_{1} × ⋯ × S_{m} be the corresponding decomposition of p-groups; i.e., F_{i} is a fusion system over S_{i}. Assume each F_{i} is tame, and let G_{i} be a finite group which tamely realizes F_{i}. Assume also that these are chosen so that G_{i} ≅ G_{j} if F_{i} ≅ F_{j}. Set L_{i} = L_{S_{i}}^{c}(G_{i}). Set G = G_{1} × ⋯ × G_{m}, L = L_{S}^{c}(G), and L̂ = L_{1} × ⋯ × L_{m}. We identify L̂ with the full subcategory of L having as objects those P = P_{1} × ⋯ × P_{m} where P_{i} ∈ Ob(L_{i}). Note that L̂ is not a linking system, since Ob(L̂) is not closed under overgroups.

Set m = {1, …, m}. Define

Aut_{typ}^{0}(L) = {α ∈ Aut_{typ}^{I}(L) | α_{S}(δ_{S}(S_{i})) = δ_{S}(S_{i}) for each i ∈ m} .

We first construct a monomorphism

Ψ: Aut_{typ}^{0}(L) ⟶ Aut_{typ}^{I}(L_{1}) × ⋯ × Aut_{typ}^{I}(L_{m})

such that for each α ∈ Aut_{typ}^{0}(L), if Ψ(α) = (α_{1}, …, α_{m}), then α|L̂ = ∏_{i∈m} α_{i}.

To define Ψ, fix α ∈ Aut_{typ}^{0}(L), and let β ∈ Aut(S, F) be the induced automorphism of Lemma 1.15 (i.e., δ_{S}(β(g)) = α(δ_{S}(g)) for g ∈ S). Then β(S_{i}) = S_{i} for each i since δ_{S} is injective. Also, by Lemma 1.15, α(P) = β(P) for each P ∈ Ob(L), and π ∘ α = c_{β} ∘ π, where c_{β} ∈ Aut(F) is conjugation by β (and its restrictions).

Fix i ∈ m, set S_{i}^{∗} = ∏_{j≠i} S_{j} and L_{i}^{∗} = ∏_{j≠i} L_{j}, and identify S = S_{i} ×S_{i}^{∗} and L̂ = L_{i} ×L_{i}^{∗}. We claim the following:

∀ ψ ∈ Mor(L_{i}), ∃ α_{i}(ψ) ∈ Mor(L_{i}) such that α(ψ, Id_{S_{i}^{∗}}) = (α_{i}(ψ), Id_{S_{i}^{∗}}).  (3)

For each ψ ∈ Mor(L_{i}),

π(α(ψ, Id_{S_{i}^{∗}})) = c_{β}(π(ψ), Id_{S_{i}^{∗}}) = (c_{β}(π(ψ)), Id_{S_{i}^{∗}}) ∈ Mor(F)

since β(S_{j}) = S_{j} for all j. Hence by axiom (A), α(ψ, Id_{S_{i}^{∗}}) = (α_{i}(ψ), δ_{S_{i}^{∗}}(z)) for some α_{i}(ψ) ∈ Mor(L_{i}) and some z ∈ Z(S_{i}^{∗}). In particular, (3) holds when ψ is an automorphism of order prime to p. Since α(δ_{P} (x)) = δ_{β(P)}(β(x)) for all P ∈ Ob(L) and all x ∈ N_{S}(P) (and since β(S_{i}) = S_{i}), (3) also holds when ψ = δ_{P} (x) for P ≤ S_{i} and x ∈ N_{S_{i}}(P). When P ∈ Ob(L_{i}) is fully normalized in F_{i}, Aut_{L_{i}}(P) is generated by elements of order prime to p and by its Sylow p-subgroup δ_{P} (N_{S_{i}}(P)) (Proposition 1.11(d)), and hence (3) holds for all ψ ∈ Aut_{L_{i}}(P). Finally, by Theorem 1.12, all morphisms in L_{i} are composites of restrictions of automorphisms of fully normalized subgroups, and hence (3) holds for all ψ ∈ Mor(L_{i}).

Now let α_{i} ∈ Aut(L_{i}) be the automorphism defined by sending P ∈ Ob(L_{i}) to β(P), and ψ ∈ Mor(L_{i}) to α_{i}(ψ) as defined in (3). This is clearly a functor, it is isotypical since α is, and it preserves inclusions since α does. Set Ψ(α) = (α_{1}, …, α_{m}). Since each morphism in L̂ is a composite of restrictions of morphisms of the form (ψ_{i}, Id_{S_{i}^{∗}}) for ψ_{i} ∈ Mor(L_{i}), the restriction of α to L̂ is ∏_{i∈m} α_{i}.

By construction, Ψ is a homomorphism. If Ψ(α) = (Id_{L_{1}}, …, Id_{L_{m}}), then α|L̂ = Id by the above remarks, α is the identity on objects since α_{S} = Id_{Aut_{L}(S)} (Lemma 1.15), and so α = Id_{L} by Theorem 1.12 and since all F-centric F-radical subgroups are objects in L̂ (Lemma 3.1). Hence Ψ is injective. Finally, since Aut_{L}(S) ≅ ∏_{i∈m} Aut_{L_{i}}(S_{i}), Ψ induces a monomorphism

Ψ: Out_{typ}^{0}(L) =^{def} Aut_{typ}^{0}(L)/{c_{ζ} | ζ ∈ Aut_{L}(S)} ⟶ Out_{typ}(L_{1}) × ⋯ × Out_{typ}(L_{m}) .

Next consider the equivalence relation ∼ on m, where i ∼ j if G_{i} ≅ G_{j} (equivalently, F_{i} ≅ F_{j}). Fix isomorphisms τ_{ij} ∈ Iso(G_{i}, G_{j}) for all pairs i ∼ j of elements in m, such that

τ_{ij}(S_{i}) = S_{j}, τ_{ii} = Id_{G_{i}}, τ_{ji} = τ_{ij}^{−1}, and τ_{ik} = τ_{jk} ∘τ_{ij} whenever i ∼ j ∼ k. Let τ̂_{ij}: L_{i} ⟶^{≅} L_{j}

be the induced isomorphism of linking systems. Then conjugation by τ̂_{ij} sends Out_{typ}(L_{i}) to Out_{typ}(L_{j}). For each i, fix a splitting s_{i}: Out_{typ}(L_{i}) ⟶ Out(G_{i}) of κ_{G_{i}}, chosen so that c_{τ_{ij}} ∘ s_{i} = s_{j} ∘ cτ̂_{ij} if i ∼ j.

Let Σ ≤ Σ_{m} be the group of permutations σ of m such that σ(i) ∼ i for each i. For each σ ∈ Σ, let σ̂_{G} ∈ Aut(G) be the automorphism which sends G_{i} to G_{σ(i)} via τ_{i,σ(i)}, and set σ̂_{L} = κ̃_{G}(σ̂_{G}). Thus σ̂_{L} ∈ Aut_{typ}^{I}(L) sends each L_{i} to L_{σ(i)} via τ̂_{i,σ(i)}.

Fix α ∈ Aut_{typ}^{I}(L), and let β ∈ Aut(S, F) be the restriction of α_{S} ∈ Aut(Aut_{L}(S)) to S ≅ δ_{S}(S). By Proposition 3.6, there is σ ∈ Σ_{m} such that β(S_{i}) = S_{σ(i)} for each i. Since β is fusion preserving, F_{i} ≅ F_{σ(i)}, and hence i ∼ σ(i), for each i. Thus σ ∈ Σ, and σ̂_{L}^{−1} ∘ α ∈ Aut_{typ}^{0}(L). So Aut_{typ}^{I}(L) is generated by Aut_{typ}^{0}(L) and the σ̂_{L}.

Now let s: Out_{typ}(L) ⟶ Out(G) be the composite

Out_{typ}(L) = Out_{typ}^{0}(L) ⋊ {[σ̂_{L}] | σ ∈ Σ} ⟶_{([σ̂_{L}]↦σ)}^{Ψ⋊} (Out_{typ}(L_{1}) × ⋯ × Out_{typ}(L_{m})) ⋊ Σ

⟶_{(σ↦[σ̂_{G}])}^{(s_{1},...,s_{m})⋊} (Out(G_{1}) × ⋯ × Out(G_{m})) ⋊ {[σ̂_{G}] | σ ∈ Σ} ⟶^{incl} Out(G) .

We must show κ_{G} ∘ s = Id. Since κ_{G}(s([σ̂_{L}])) = κ_{G}([σ̂_{G}]) = [σ̂_{L}] for σ ∈ Σ, it will suffice to show κ_{G}(s([α])) = [α] for α ∈ Aut_{typ}^{0}(L). Let Out^{0}(G) ≤ Out(G) be the subgroup of classes of automorphisms which leave each G_{i} invariant, and consider the following composite:

Out_{typ}^{0}(L) ⟶^{Ψ} ∏_{i=1}^{m} Out_{typ}(L_{i}) \underrightarrow{∏ s_{i}} ∏_{i=1}^{m} Out(G_{i}) ≅ Out^{0}(G)

⟶^{κ_{G}|_{Out^{0}(G)}} Out_{typ}^{0}(L) ⟶^{Ψ} ∏_{i=1}^{m} Out_{typ}(L_{i}) .

Here, (∏ s_{i}) ∘ Ψ = s|_{Out_{typ}^{0}(L)}, Ψ ∘ κ_{G}|_{Out^{0}(G)} = ∏ κ_{G_{i}}, and (∏ κ_{G_{i}}) ∘ (∏ s_{i}) = Id. This proves

that Ψ ∘ κ_{G}|_{Out^{0}(G)} ∘ s|_{Out_{typ}^{0}(L)} = Ψ. Since Ψ is injective, κ_{G} ∘ s = Id on Out_{typ}^{0}(L). Thus s is a splitting for κ_{G}, and this finishes the proof that F is tame.

If each F_{i} is strongly tame, then we can choose the G_{i} to all be in the class 𝔊(p). Hence G ∈ 𝔊(p) by Lemma 2.11(b), and F is strongly tame. □

Theorem 3.7 does not say that an arbitrary product of reduced, tame fusion systems is tame: such a product could conceivably have an indecomposable factor which is not tame. However, at least when p = 2, a theorem of Goldschmidt implies this is not possible.

Theorem 3.8. Assume p = 2, and let F be a reduced fusion system over a 2-group S. Assume F = F_{1} × F_{2}, where F_{i} is a fusion system over S_{i} and S = S_{1} × S_{2}. Then F is realizable, tame, or strongly tame if and only if F_{1} and F_{2} are both realizable, tame, or strongly tame, respectively.

Proof. Assume F = F_{S}(G), where G is a finite group and S ∈ Syl_{2}(G). If F is tame, we also assume κ_{G} is split surjective, and if F is strongly tame, we also assume G ∈ 𝔊(2). By Lemma 2.19, we can assume O_{2′}(G) = 1.

Let G_{i} ⊴ G be the normal closure of S_{i} in G. Since F factors as a product F_{1} × F_{2}, the subgroups S_{1} and S_{2} are strongly closed in F, and hence strongly closed in G in the sense of [Gd]. So by Goldschmidt’s theorem [Gd, Corollary A1], G_{1} ∩ G_{2} = 1. Thus G_{1} × G_{2} is a normal subgroup of odd index in G. Since F = F_{S}(G) has no proper normal subsystem of odd index (since it is reduced), F_{S}(G) = F_{S}(G_{1} × G_{2}) = F_{S_{1}}(G_{1}) × F_{S_{2}}(G_{2}). Hence F_{i} = F_{S_{i}}(G_{i}) for i = 1, 2 (there can be at most one way to factor F as a product of fusion systems over the S_{i}), and thus each F_{i} is realizable.

Set L = L_{S}^{c}(G) and L_{i} = L_{S_{i}}^{c}(G_{i}). Define Φ: Aut_{typ}^{I}(L_{1}) × Aut_{typ}^{I}(L_{2}) ⟶ Aut_{typ}^{I}(L) as follows. Fix α_{i} ∈ Aut_{typ}^{I}(L_{i}) (i = 1, 2). Let β_{i} ∈ Aut(S_{i}, F_{i}) be the corresponding automorphisms (see Lemma 1.15), and set β = (β_{1}, β_{2}) ∈ Aut(S, F). Thus α_{i}(P_{i}) = β_{i}(P_{i}) for each P_{i} ∈ Ob(L_{i}) and π(α_{i}(ψ_{i})) = β_{i}π(ψ_{i})β_{i}^{−1} for ψ_{i} ∈ Mor(L_{i}). Define α ∈ Aut_{typ}^{I}(L) on objects by setting α(P) = β(P) for P ∈ Ob(L). Fix ψ ∈ Mor_{L}(P, Q), let P_{i}, Q_{i} ≤ S_{i} be the images of P and Q under projection, and set P̂ = P_{1} × P_{2} and Q̂ = Q_{1} × Q_{2}. Since G and G_{1} × G_{2} have the same fusion system over S, ψ = [g] for some g = (g_{1}, g_{2}) ∈ N_{G}(P, Q), where g_{i} ∈ G_{i}. Then g_{i} ∈ N_{G_{i}}(P_{i}, Q_{i}), and hence ψ extends to ψ̂ = (ψ_{1}, ψ_{2}) ∈ Mor_{L}(P̂, Q̂) where ψ_{i} = [g_{i}] ∈ Mor_{L_{i}}(P_{i}, Q_{i}). Also, π(α_{1}(ψ_{1}), α_{2}(ψ_{2})) = β(π(ψ_{1}), π(ψ_{2}))β^{−1} sends β(P) into β(Q), and we define α(ψ) = (α_{1}(ψ_{1}), α_{2}(ψ_{2}))|_{β(P),β(Q)}. Finally, α ∈ Aut_{typ}^{I}(L) since α_{i} ∈ Aut_{typ}^{I}(L_{i}), and we set Φ(α_{1}, α_{2}) = α.

Assume F is tamely realized by G, and let s: Out_{typ}(L) ⟶ Out(G) be a splitting for κ_{G}. For each α_{1} ∈ Aut_{typ}^{I}(L_{1}), s([Φ(α_{1}, Id_{L_{2}})]) = [γ] for some γ ∈ Aut(G, S) such that γ|_{S_{2}} = Id. Also, γ(G_{2}) = G_{2} since G_{2} is the normal closure of S_{2} in G, and so γ induces γ ∈ Aut(G/G_{2}, S_{1}). The class [γ] ∈ Out(G/G_{2}) is independent of the choice of γ modulo Inn(G), and hence this gives a well defined homomorphism s_{1} from Out_{typ}^{I}(L_{1}) to Out(G/G_{2}).

Also, F_{S_{1}}(G/G_{2}) ≅ F/S_{2} ≅ F_{1}, so L_{S_{1}}^{c}(G/G_{2}) ≅ L_{1}; and s_{1} is a splitting for κ_{G/G_{2}} since s is a splitting for κ_{G}. Thus F_{1} is tame, and F_{2} is tame by a similar argument. If F is strongly tame, then we can choose G ∈ 𝔊(2), so G/G_{i} ∈ 𝔊(2) (i = 1, 2) by Lemma 2.11(b), and hence F_{1} and F_{2} are strongly tame.

This proves the “only if” part of the theorem. Clearly, F is realizable if both factors are. If F_{1} and F_{2} are both (strongly) tame, then we have just shown that each of the indecomposable factors of F_{1} and F_{2} is (strongly) tame, and so F is (strongly) tame by Theorem 3.7. □

4. Examples

We now give three families of examples, to illustrate some of the techniques which can be used to prove tameness of reduced fusion systems. As an introduction to these techniques, we first list the reduced fusion systems over dihedral and semidihedral groups and prove they are all tame. Next, we prove that certain fusion systems studied in [OV2, § 4–5] are reduced and tame; as a way of explaining how the information about these fusion systems given in [OV2] is just what is needed to prove tameness. As a third example, we prove that the fusion systems of all alternating groups are tame, and that they are reduced with certain obvious exceptions.

In general, tameness is shown by examining, for a p-local finite group (S, F, L) realized by G, the homomorphisms

Out(G) ⟶^{κ_{G}} Out_{typ}(L) ⟶^{μ_{G}} Out(S, F)

defined in Sections 2.2 and 1.3. By definition, F is tame if κ_{G} is split surjective (for some choice of G). However, the group Out(S, F) is usually much easier to describe than Out_{typ}(L), and the composite μ_{G} ∘ κ_{G} is induced by restriction to S. So we need some way of describing Ker(μ_{G}).

We first recall some definitions. A proper subgroup H ≨ G of a finite group G is strongly p-embedded if p||H|, and for each g ∈ G∖H, H ∩ gHg^{−1} has order prime to p. It is not hard to see that G has a strongly p-embedded subgroup if and only if the poset S_{p}(G) of nontrivial p-subgroups is disconnected (cf. [HB3, Theorem X.4.11(b)]), but we will not be using that here.

When F is a saturated fusion system over a finite p-group S, then a proper subgroup P ≨ S is F-essential if it is F-centric and fully normalized, and Out_{F}(P) contains a strongly pembedded subgroup. Thus each F-essential subgroup is fully normalized and F-centric by definition, and is F-radical since O_{p}(Γ) = 1 for any group Γ which has a strongly p-embedded subgroup. See, e.g., [Sz2, Theorem 6.4.3] for a proof of this last statement (it is shown there only for p = 2, but the same proof works for odd primes). The following proposition is a stronger version of Theorems 1.3 and 1.12, and helps show the importance of essential subgroups when working with fusion systems.

Theorem 4.1. Let F be any saturated fusion system over a finite p-group S. Let E be the set of F-essential subgroups of S, and set E_{+} = E ∪ {S}. Then each morphism in F is a composite of restrictions of elements of Aut_{F}(P) for P ∈ E_{+}. If L is a linking system associated to F, then each morphism in L is a composite of restrictions of elements of Aut_{L}(P) for P ∈ E_{+}.

Proof. The statement about morphisms in F is shown in [Pg2, § 5], and also in [OV2, Corollary 2.6]. The second statement follows from this together with Proposition 1.11(a) (and since Ob(L) is closed under overgroups). □

The following proposition will be useful when describing Ker(μ_{G}), and for determining whether or not explicit elements in this group vanish. In fact, it applies to help describe Ker(μ_{L}), when L is an arbitrary linking system (not necessarily induced by a finite group). For any fusion system F over S and any P ≤ S, we write

C_{Z(P)}(Aut_{F}(P)) = {g ∈ Z(P) | α(g) = g for all α ∈ Aut_{F}(P)}

and similarly for C_{Z(P)}(Aut_{S}(P)) and C_{Z(P)}(Aut_{L}(P)).

Proposition 4.2. Let F be a saturated fusion system over the finite p-group S, and let L be a linking system associated to F. Let L^{c} ⊆ L be the full subcategory whose objects are the F-centric objects in L. Each element in Ker(μ_{L}) is represented by some α ∈ Aut_{typ}^{I}(L) such that α_{S} = Id_{Aut_{L}(S)}. For each such α, there are elements g_{P} ∈ C_{Z(P)}(Aut_{S}(P)), defined for each fully normalized subgroup P ∈ Ob(L^{c}), for which the following hold:

(a) α_{P} ∈ Aut(Aut_{L}(P)) is conjugation by δ_{P} (g_{P}), and g_{P} is uniquely determined by α modulo C_{Z(P)}(Aut_{F}(P)). In particular, α_{P} = Id_{Aut_{L}(P)} if and only if g_{P} ∈ C_{Z(P)}(Aut_{F}(P)).

(b) Assume P, Q ∈ Ob(L^{c}) are both fully normalized in F. If Q = aP a^{−1} for some a ∈ S, then we can choose g_{Q} = ag_{P} a^{−1}. More generally, if Q is F-conjugate to P, and there is ζ ∈ Iso_{L}(P, Q) such that α(ζ) = ζ, then we can choose g_{Q} = π(ζ)(g_{P}). In either case, α_{P} = Id_{Aut_{L}(P)} if and only if α_{Q} = Id_{Aut_{L}(Q)}.

(c) If Q ≤ P are both fully normalized objects in L^{c}, then g_{P} ≡ g_{Q} (mod C_{Z(Q)}(Aut_{F}(P, Q))), where Aut_{F}(P, Q) is the group of those φ ∈ Aut_{F}(P) such that φ(Q) = Q.

(d) Let E be the set of all F-essential subgroups P ≨ S and let E_{0} ⊆ E be the subset of those P ∈ E such that C_{Z(P)}(Aut_{F}(P)) ≨ C_{Z(P)}(Aut_{S}(P)). Then [α] = 1 in Out_{typ}(L) if and only if there is g ∈ C_{Z(S)}(Aut_{F}(S)) such that g_{P} ∈ g·C_{Z(P)}(Aut_{F}(P)) for all P ∈ E_{0}.

(e) Let E_{0} be as in (d), and let Ê_{0} be the set of all P ∈ E_{0} such that P = C_{S}(E) for some elementary abelian p-subgroup E ≤ S which is fully centralized in F. Then [α] = 1 in Out_{typ}(L) if and only if there is g ∈ C_{Z(S)}(Aut_{F}(S)) such that g_{P} ∈ g·C_{Z(P)}(Aut_{F}(P)) for all P ∈ Ê_{0}. In fact, it suffices that this hold for at least one subgroup P ∈ Ê_{0} in each F-conjugacy class intersecting Ê_{0} nontrivially.

Proof. We identify S with δ_{S}(S) ≤ Aut_{L}(S) for short. Fix α ∈ Aut_{typ}^{I}(L) such that [α] ∈ Ker(μ_{L}). Set β = μ̃_{L}(α); thus β ∈ Aut_{F}(S). Choose ζ ∈ Aut_{L}(S) such that π(ζ) = β. Then μ̃_{L}(c_{ζ}) = β by axiom (C) for the linking system L, and so upon replacing α by α ∘ c_{ζ}^{−1}, we can arrange that α_{S} is the identity on δ_{S}(S) ⊴ Aut_{L}(S). We will show in the proof of (a) how to arrange that α_{S} = Id_{Aut_{L}(S)}.

(a) Fix a fully normalized subgroup P ∈ Ob(L^{c}). Set Γ = Aut_{L}(P) for short, and identify P with δ_{P} (P) ⊴ Γ. Set Out(Γ, P) = Aut(Γ, P)/Inn(Γ), where Aut(Γ, P) ≤ Aut(Γ) is the subgroup of automorphisms leaving P invariant. By [OV2, Lemma 1.2], there is an exact sequence

1 ⟶ H^{1}(Γ/P; Z(P)) ⟶^{η} Out(Γ, P) ⟶^{R} N_{Out(P)}(Out_{Γ}(P))/Out_{Γ}(P),

where R is induced by restriction. Since α_{P} ∈ Aut(Γ) and α_{P} |_{δ_{P} (N_{S}(P))} = Id, [α_{P}] ∈ Ker(R), and η^{−1}([α_{P}]) is trivial after restriction to H^{1}(N_{S}(P)/P; Z(P)). The restriction map from H^{1}(Γ/P; Z(P)) to H^{1}(N_{S}(P)/P; Z(P)) is injective since δ_{P} (N_{S}(P)) ∈ Syl_{p}(Γ) (Proposition 1.11(d)), and hence [α_{P}] = 1. Thus α_{P} = c_{δ_{P} (g_{P})} for some g_{P} ∈ Z(P) which is uniquely determined modulo C_{Z(P)}(Γ) = C_{Z(P)}(Aut_{F}(P)). Also, g_{P} ∈ C_{Z(P)}(Aut_{S}(P)), since α_{P} is the identity on δ_{P} (N_{S}(P)).

Set γ = δ_{S}(g_{S}) ∈ Aut_{L}(S). Upon replacing α by α ∘ c_{γ}^{−1}, we can arrange that α_{S} = Id.

(b) Assume ζ ∈ Iso_{L}(P, Q) and α(ζ) = ζ. Fix ψ ∈ Aut_{L}(Q), and set φ = ζ^{−1}ψζ ∈ Aut_{L}(P). Set g = π(ζ)(g_{P}); then ζ ∘ δ_{P} (g_{P}) ∘ ζ^{−1} = δ_{Q}(g) by axiom (C) for a linking system. Hence

α_{Q}(ψ) = α_{Q}(ζφζ^{−1}) = ζα_{P} (φ)ζ^{−1} = ζδ_{P} (g_{P})φδ_{P} (g_{P})^{−1}ζ^{−1} = δ_{Q}(g)ψδ_{Q}(g)^{−1} ,

and we can choose g_{Q} = g.

If Q = aP a^{−1} and ζ = δ_{P,Q}(a), then α(ζ) = ζ since α_{S} = Id (and since α sends inclusions to inclusions). So again we can choose g_{Q} = c_{a}(g_{P}).

In either case, g_{P} ∈ C_{Z(P)}(Aut_{F}(P)) if and only if g_{Q} ∈ C_{Z(Q)}(Aut_{F}(Q)), and hence α_{P} = Id if and only if α_{Q} = Id.

(c) Assume Q ≤ P, and let Aut_{L}(P, Q) be the group of elements ψ ∈ Aut_{L}(P) such that π(ψ)(Q) = Q. Then α commutes with the restriction map

Res_{Q}^{P}: Aut_{L}(P, Q) ⟶ Aut_{L}(Q)

which is injective by Proposition 1.11(f). So if α acts on Aut_{L}(P, Q) via conjugation by δ_{P} (g_{P}) and on Aut_{L}(Q) via conjugation by δ_{Q}(g_{Q}), they must have the same action on Aut_{L}(P, Q). Since g_{Q} and g_{P} both lie in Z(Q) ≥ Z(P), we conclude g_{Q} ≡ g_{P} (mod C_{Z(Q)}(Aut_{F}(P, Q))).

(d) By Theorem 4.1, all morphisms in L are composites of restrictions of elements in Aut_{L}(P) for P F-essential or P = S. Hence if α ≠ Id_{L}, then since α_{S} = Id by assumption, α_{P} ≠ Id for some P ∈ E. By (a), g_{P} ∈ C_{Z(P)}(Aut_{S}(P)) but g_{P} ∉ C_{Z(P)}(Aut_{F}(P)), and so P ∈ E_{0}. The converse is clear: if α = Id_{L}, then α_{P} = Id and hence g_{P} ∈ C_{Z(P)}(Aut_{F}(P)) for all P ∈ E_{0}.

By definition, [α] = 1 in Out_{typ}(L) if and only if α = c_{β} for some β ∈ Aut_{L}(S). Since α_{S} = Id, β ∈ Z(Aut_{L}(S)), and hence β = δ_{S}(g) for some g ∈ C_{Z(S)}(Aut_{F}(S)). Thus [α] = 1 if and only if α = c_{δ_{S}(g)} for some g ∈ C_{Z(S)}(Aut_{F}(S)), which we just saw is the case exactly when g^{−1}g_{P} ∈ C_{Z(P)}(Aut_{F}(P)) for all P ∈ E_{0}.

(e) We first prove the following statement:

α = Id_{L} ⟺ g_{P} ∈ C_{Z(P)}(Aut_{F}(P)) for all P ∈ Ê_{0}
⟺ ∀ P ∈ Ê_{0} ∃ Q ∈ Ê_{0} F-conjugate to P with g_{Q} ∈ C_{Z(Q)}(Aut_{F}(Q)) .

(1)

The first statement implies the second by (a), and the second implies the third tautologically.

Now assume α ≠ Id_{L}. As was just seen in the proof of (d), there is P ∈ E_{0} such that g_{P} ∉ C_{Z(P)}(Aut_{F}(P)). Assume P is such that |P | is maximal among orders of all such subgroups. We will show that P ∈ Ê_{0} (possibly after replacing P by another subgroup in its F-conjugacy class), and that g_{Q} ∉ C_{Z(Q)}(Aut_{F}(Q)) for each Q ∈ Ê_{0} which is F-conjugate to P. This will prove the remaining implication in (1).

We first check that

T ∈ Ob(L) and |T | > |P | ⟹ α_{T} = Id .  (2)

If T = S or T ∈ E_{0}, this follows by assumption. If T ∈ E∖E_{0}, then g_{T} ∈ C_{Z(T)}(Aut_{S}(T)) = C_{Z(T)}(Aut_{F}(T)), and hence α_{T} = Id by definition of g_{T}. Otherwise, each ψ ∈ Aut_{L}(T) is a composite of restrictions of automorphisms of subgroups in E ∪ {S} (Theorem 4.1), each of those automorphisms and its restrictions are sent to themselves by α, and hence α_{T} (ψ) = ψ.

We next claim that

for all Q F-conjugate to P, there is ζ ∈ Iso_{L}(P, Q) such that α_{P,Q}(ζ) = ζ.  (3)

Choose any ζ_{0} ∈ Iso_{L}(P, Q). By Theorem 4.1 again, ζ_{0} is the composite of restrictions of automorphisms ψ_{i} ∈ Aut_{L}(R_{i}) for subgroups R_{i} ≤ S with |R_{i}| ≥ |P |. If we remove from this composite all ψ_{i} for which |R_{i}| = |P |, we get an isomorphism ζ ∈ Iso_{L}(P, Q) which is a composite of restrictions of automorphisms of strictly larger subgroups. We just showed that α_{R_{i}}(ψ_{i}) = ψ_{i} whenever |R_{i}| > |P |, and thus α_{P,Q}(ζ) = ζ.

Set E = Ω_{1}(Z(P)): the p-torsion subgroup of the center Z(P). If E is not fully normalized in F, then choose φ ∈ Hom_{F}(N_{S}(E), S) such that φ(E) is fully normalized (using [BLO2, Proposition A.2(b)]). Then φ(P) is fully normalized since N_{S}(φ(P)) ≥ φ(N_{S}(P)). By (3), there is ζ ∈ Iso_{L}(P, φ(P)) such that α_{P,φ(P)}(ζ) = ζ. So α_{φ(P)} ≠ Id by (b). Upon replacing P by φ(P) and E by φ(E), we can now assume E and P are both fully normalized.

Set P^{∗} = N_{C_{S}(E)}(P) ≥ P and Γ = Aut_{L}(P) for short. To simplify notation, we identify N_{S}(P) with δ_{P} (N_{S}(P)). Then E ⊴ Γ, so C_{Γ}(E) ⊴ Γ; and P^{∗} ∈ Syl_{p}(C_{Γ}(E)) since N_{S}(P) ∈ Syl_{p}(Γ) (Proposition 1.11(d)). Also, C_{Γ}(Z(P)) ⊴ Γ, and has p-power index in C_{Γ}(E) since each automorphism of Z(P) which is the identity on its p-torsion subgroup E has p-power order (cf. [G, Theorem 5.2.4]). Hence each Sylow p-subgroup of C_{Γ}(E) is C_{Γ}(Z(P))-conjugate to P^{∗} = C_{N_{S}(P)}(E). By the Frattini argument,

Γ = N_{Γ}(P^{∗})·C_{Γ}(Z(P)) .  (4)

Since α_{P} ≠ Id_{Γ} is conjugation by g_{P} ∈ Z(P), α_{P} is the identity on C_{Γ}(Z(P)). Hence by (4), α_{P} is not the identity on N_{Γ}(P^{∗}). By Proposition 1.11(e), each α ∈ N_{Γ}(P^{∗}) extends to α ∈ Aut_{L}(P^{∗}), and thus α_{P^{∗}} ≠ Id_{Aut_{L}(P^{∗})}. If C_{S}(E) ≩ P, then P^{∗} = N_{C_{S}(E)}(P) ≩ P (cf. [Sz1, Theorem 2.1.6]), which would imply α_{P^{∗}} = Id by (2). We now conclude that C_{S}(E) = P, and hence that P ∈ Ê_{0}.

Assume Q ∈ Ê_{0} is F-conjugate to P. By (3), there is ζ ∈ Iso_{L}(P, Q) such that α(ζ) = ζ. So by (b), α_{Q} ≠ Id since α_{P} ≠ Id, and this finishes the proof of (1).

The rest of the proof of (e) is identical to that of (d). □

As one simple application of Proposition 4.2, consider the group G = A_{6} ≅ P SL_{2}(9). Set

T_{1} = ⟨(1 2)(3 4), (1 3)(2 4)⟩ ≅ C_{2}^{2} , T_{2} = ⟨(1 2)(3 4), (3 4)(5 6)⟩ ≅ C_{2}^{2} ,

and S = ⟨T_{1}, T_{2}⟩ ∈ Syl_{2}(G), and let F = F_{S}(G) and L = L_{S}^{c}(G). Then E = E_{0} = Ê_{0} = {T_{1}, T_{2}}. Set g = (5 6), and consider the automorphism α = κ̃_{G}(c_{g}) ∈ Aut_{typ}^{I}(L). Then α ∈ Ker(μ̃_{G}), since [g, S] = 1 (and since μ̃_{G} ∘ κ̃_{G} sends β ∈ Aut(G, S) to β|_{S}). Since [g, N_{G}(T_{1})] = 1, α_{T_{1}} = Id_{Aut_{L}(T_{1})}. Since (1 2)(3 4)(5 6) commutes with N_{G}(T_{2}) = ⟨T_{2}, (1 3)(2 4), (1 3 5)(2 4 6)⟩, α_{T_{2}} acts on Aut_{L}(T_{2}) ≅ N_{G}(T_{2}) as conjugation by x = (1 2)(3 4) ∈ Z(S). So in the notation of Proposition 4.2, g_{T_{1}} = 1 and g_{T_{2}} = x. In both cases, C_{Z(T_{i})}(Aut_{F}(T_{i})) = 1, so the g_{T_{i}} are uniquely determined. Hence by Proposition 4.2(d), [α] = κ_{G}([c_{g}]) represents a nontrivial element in Ker(μ_{G}).

If [α] ∈ Ker(μ_{G}) is arbitrary, represented by α ∈ Aut_{typ}^{I}(L) such that α_{S} = Id_{Aut_{L}(S)}, then by Proposition 4.2 again, g_{T_{i}} ∈ Z(S) for i = 1, 2, and [α] = 1 if and only if g_{T_{1}} = g_{T_{2}}. Thus Ker(μ_{G}) ≅ C_{2} is generated by κ_{G}([c_{g}]) as described above. Using this, and the well known description of Out(A_{6}) ≅ C_{2}^{2}, it is not hard to see that κ_{G} is an isomorphism from Out(G) to Out_{typ}(L).

This example will be generalized in two different ways below: to other groups P SL_{2}(q) for q ≡ ±1 (mod 8) in Proposition 4.3, and to other alternating groups in Proposition 4.8.

4.1. Dihedral and semidihedral 2-groups.

As our first examples, we list all reduced fusion systems over dihedral and semidihedral 2-groups, and prove they are all tame. The list of all fusion systems over such groups is well known; it turns out that each of them supports exactly one fusion system which is reduced.

As usual, v_{p}(−) denotes the p-adic valuation: v_{p}(n) = k if p^{k}|n but p^{k+1}∤n.

Proposition 4.3. Let S be a dihedral group of order 2^{k} (k ≥ 3). Then there is a unique reduced fusion system F over S, and it is tame. Let q be a prime power such that v_{2}(q^{2}−1) = k + 1, set G = P SL_{2}(q), and fix S^{∗} ∈ Syl_{2}(G). Then S ≅ S^{∗} and F ≅ F_{S^{∗}}(G); and κ_{G} is an isomorphism if q = p^{2^{k−2}} for some prime p ≡ 5 (mod 8).

Proof. Fix a, b ∈ S such that ⟨a⟩ has index two and S = ⟨a, b⟩. For each i ∈ ℤ, set T_{i} = ⟨a^{2^{k−2}}, a^{i}b⟩ ≅ C_{2}^{2}. Two subgroups T_{i} and T_{j} are S-conjugate if and only if i ≡ j (mod 2). Set P = {T_{i} | i ∈ ℤ}.

If P ≤ S is cyclic of order 2^{m}, then Aut(P) ≅ (ℤ/2^{m})^{×} is a 2-group. If P ≤ S is dihedral of order 2^{m} ≥ 8, then there is a unique cyclic subgroup of index two in P, and Aut(P) is a 2-group by Lemma 1.6. Thus the only subgroups P ≤ S for which Aut(P) is not a 2-group are the T_{i}.

Define F to be the fusion system over S generated by the automorphisms in Inn(S), Aut(P) for P ∈ P, and their restrictions. Assume F is saturated (this will be shown later). Then 𝔣𝔬𝔠(F) = ⟨[S, S], P⟩ = S, and hence O^{2}(F) = F (Theorem 1.22(a)). Also, O^{2′}(F) = F since any normal subsystem of odd index would have to contain the same automorphism groups, and O_{2}(F) = 1 by inspection. Thus F is reduced.

Let F^{∗} be an arbitrary saturated fusion system over S such that 𝔣𝔬𝔠(F^{∗}) = S. Let E be the set of all F^{∗}-essential subgroups of S. If P ∈ E, then Aut(P) must have elements of odd order, and hence P ∈ P. For each T_{i} ∈ P, Aut(T_{i}) ≅ Σ_{3} and Aut_{S}(T_{i}) ≅ C_{2}. Hence Aut_{F^{∗}}(T_{i}) = Aut(T_{i}) if T_{i} ∈ E. Since Aut(S) is a 2-group, Theorem 4.1 implies F^{∗} is generated by automorphisms in Aut_{F^{∗}}(S) = Inn(S), the Aut(P) for P ∈ E ⊆ P, and their restrictions. In particular, 𝔣𝔬𝔠(F^{∗}) ≤ ⟨[S, S], E⟩, and this has index at least two in S if E ⫋ P. Hence E = P, and so F^{∗} = F.

Set G = P SL_{2}(q) for any prime power q ≡ ±1 (mod 8), and fix S^{∗} ∈ Syl_{2}(G). As is well known, S^{∗} is a dihedral group and |G| = \frac{1}{2}q(q^{2} − 1), so S^{∗} ≅ D_{2^{k}} where k = v_{2}(q^{2} − 1) − 1. So we identify S^{∗} = S for S as above. Since G is simple, 𝔣𝔬𝔠(F_{S}(G)) = S ∩ [G, G] = S by the focal subgroup theorem (cf. [G, Theorem 7.3.4]), and we have just seen this implies F_{S}(G) = F. In particular, F is saturated, and hence reduced.

Now assume q = p^{2^{k−2}}, where p ≡ 5 (mod 8) (and k ≥ 3). The homomorphism κ_{G} is an isomorphism in this case by [BLO1, Proposition 7.9], where it is shown more generally for p ≡ ±3 (mod 8). But we give a different proof here to illustrate how Proposition 4.2 can be applied.

Set G̃ = SL_{2}(q). Fix u ∈ 𝔽_{q}^{×} of order 2^{k}. Set ã = ( \begin{matrix}u & 0 \\ 0 & u^{−1} \end{matrix} ) and b̃ = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ), and let a, b ∈ G

be their images in the quotient. Then S =^{def} ⟨a, b⟩ ∈ Syl_{2}(G). Let δ ∈ Aut(G) be conjugation by ( \begin{matrix}u & 0 \\ 0 & 1 \end{matrix} ); then δ(a) = a and δ(b) = ab. Since u is not a square in 𝔽_{q}^{×}, [δ] generates the subgroup (of order 2) of diagonal automorphisms in Out(G).

By [St, § 3], Out(G) = ⟨[δ]⟩ × ⟨[ψ^{p}]⟩ ≅ C_{2} × C_{2^{k−2}}, where ψ^{p} is the field automorphism which acts via x ↦ x^{p} on matrix elements. Also, ψ^{p}(a) = a^{p} and ψ^{p}(b) = b. Since p ≡ 5 (mod 8), [δ|_{S}] and [ψ^{p}|_{S}] generate Out(S). Thus

μ_{G} ∘ κ_{G}: Out(G) ⟶ Out(S, F) = Out(S)

is surjective with kernel generated by [α], where α = (ψ^{p})^{2^{k−3}} is the field automorphism of order 2.

To prove that κ_{G} is an isomorphism, it remains to show that Ker(μ_{G}) has order 2 and is generated by κ_{G}([α]). Set w = a^{2^{k−2}} ∈ Z(S). We refer to Proposition 4.2. Since α is the identity on S = N_{G}(S) (α(a) = a since the field automorphism of order two sends u to −u), there are elements g_{T_{i}} ∈ C_{Z(T_{i})}(Aut_{S}(T_{i})) = ⟨w⟩ for each i such that κ̃_{G}(α) acts on Aut_{L}(T_{i}) via conjugation by g_{T_{i}}. These elements are uniquely defined since C_{Z(T_{i})}(Aut_{F}(T_{i})) = 1.

When i is even, T_{i} ≤ G_{0} =^{def} P SL_{2}(\sqrt{q}) (recall T_{i} = ⟨a^{2^{k−2}}, a^{i}b⟩), and N_{G_{0}}(T_{i}) has index at most two in N_{G}(T_{i}). Since α|_{G_{0}} = Id and α|_{S} = Id, α is the identity on N_{G}(T_{i}) ≅ Σ_{4} in this case, and so g_{T_{i}} = 1.

Now consider T_{i} for odd i. Let T̃_{i} ≅ Q_{8} be the inverse image in G̃ of T_{i} ≤ G, let w̃ be any lifting of w to G̃, and set z = w̃^{2} = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) ∈ Z(G̃). Since the field automorphism of order two sends u to −u, it sends ã^{i}b̃ to zã^{i}b̃. If α acted on N_{G}(T_{i}) ≅ Σ_{4} via the identity, then its action on NG̃(T̃_{i}) would be the identity on a subgroup of index two, which necessarily would include T̃_{i}. Since this is not the case, we conclude that α acts via conjugation by w, and thus that g_{T_{i}} = w for i odd.

By Proposition 4.2(d), since g_{T_{0}} = 1 and g_{T_{1}} = w (and C_{Z(T_{i})}(Aut_{F}(T_{i})) = 1), κ_{G}([α]) ≠ 1 in Out_{typ}(L), and it is the only nontrivial element in Ker(μ_{G}). Thus Ker(μ_{G}) ≅ C_{2}, which is what was left to prove. □

We now consider the semidihedral case.

Proposition 4.4. Let S be a semidihedral group of order 2^{k} (k ≥ 4). Then there is a unique reduced fusion system F over S, and it is tame. Let q be a prime power such that v_{2}(q − 1) = k − 2, set G = P SU_{3}(q), and fix S^{∗} ∈ Syl_{2}(G). Then S ≅ S^{∗} and F ≅ F_{S^{∗}}(G), and κ_{G} is an isomorphism if 3∤(q + 1) and q = p^{2^{k−4}} for some prime p ≡ 5 (mod 8).

Proof. Fix a, b ∈ S such that ⟨a⟩ has index two, b^{2} = 1, and S = ⟨a, b⟩. Then |a^{i}b| = 2 for i even and |a^{i}b| = 4 for i odd. For each i ∈ ℤ, set T_{i} = ⟨a^{2^{k−2}}, a^{2i}b⟩ ≅ C_{2}^{2}, and R_{i} = ⟨a^{2^{k−3}}, a^{2i+1}b⟩ ≅ Q_{8}. The T_{i} are all S-conjugate to each other, and similarly for the R_{i}. Set P = {T_{i}, R_{i} | i ∈ ℤ}.

As shown in the proof of Proposition 4.3, Aut(P) is a 2-group for each P ≤ S which is cyclic, or dihedral of order ≥ 8. The same argument applies when P is quaternion of order ≥ 16, and also to S itself. Thus the only subgroups P ≤ S for which Aut(P) is not a 2-group are those in P.

Define F to be the fusion system over S generated by the automorphisms in Inn(S), Aut(P) for P ∈ P, and their restrictions. Assume F is saturated (to be shown later). Then 𝔣𝔬𝔠(F) = ⟨[S, S], P⟩ = S, and hence O^{2}(F) = F (Theorem 1.22(a)). Also, O^{2′}(F) = F since any normal subsystem of odd index would have to contain the same automorphism groups, and O_{2}(F) = 1 by inspection. Thus F is reduced.

Let F^{∗} be an arbitrary saturated fusion system over S such that 𝔣𝔬𝔠(F^{∗}) = S. Let E be the set of all F^{∗}-essential subgroups of S. If P ∈ E, then Aut(P) must have elements of odd order, and hence P ∈ P. For all P ∈ P, [Aut(P):Aut_{S}(P)] = 3, and hence Aut_{F^{∗}}(P) = Aut(P) if P ∈ E. Since Aut(S) is a 2-group, Theorem 4.1 implies F^{∗} is generated by automorphisms in Aut_{F^{∗}}(S) = Inn(S), the Aut(P) for P ∈ E, and their restrictions. In particular, 𝔣𝔬𝔠(F^{∗}) ≤ ⟨[S, S], E⟩, and this has index at least two in S if E ⫋ P. Hence E = P, and so F^{∗} = F.

Fix a prime power q ≡ 1 (mod 4), set G = P SU_{3}(q), and fix S^{∗} ∈ Syl_{2}(G). Then |G| = \frac{1}{d}q^{3}(q^{2} − 1)(q^{3} + 1) where d = gcd(3, q + 1) [Ta, p. 118], and hence |S^{∗}| = 2^{k} where k = v_{2}(q − 1) + 2. Since GU_{2}(q) has odd index in SU_{3}(q), and the Sylow 2-subgroups of GU_{2}(q) are semidihedral by [CF, p.143], the Sylow 2-subgroups of SU_{3}(q) and of G are also semidihedral. Thus S^{∗} ≅ SD_{2^{k}}, and we identify S^{∗} = S as above. Since G is simple, 𝔣𝔬𝔠(F_{S}(G)) = S (cf. [G, Theorem 7.3.4]), and we just saw this implies F_{S}(G) = F. In particular, F is saturated.

Now assume 3∤(q + 1) and q = p^{2^{k−4}} for some prime p ≡ 5 (mod 8). By [St, § 3], Out(G) is generated by diagonal and field automorphisms; where the group of diagonal automorphisms has order gcd(3, q + 1) = 1. Thus Out(G) = ⟨[ψ^{p}]⟩, generated by the class of the field automorphism (x ↦ x^{p}). Since G = P SU_{3}(q) is defined via matrices over 𝔽_{q^{2}}, ψ^{p} has order 2^{k−3}.

More explicitly, regard G = P SU_{3}(q) = SU_{3}(q) as the group of matrices M ∈ SL_{3}(q^{2}) such that ψ^{q}(M^{t}) = M^{−1}, where M^{t} is the transpose (a_{ij}) ↦ (a_{4−j,4−i}). Fix u ∈ 𝔽_{q^{2}}^{×} of order 2^{k−1} (recall v_{2}(q − 1) = k − 2), and set a = diag(u, −1, u^{−q}). Since u^{q−1} = −1, a ∈ SU_{3}(q). Set b = ( \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ). Then bab^{−1} = a^{−q} = a^{2^{k−2}−1}, and so S = ⟨a, b⟩ is semidihedral. Also, ψ^{p}(a) = a^{p}, ψ^{p}(b) = b, so [ψ^{p}|_{S}] generates Out(S), and we conclude that

μ_{G} ∘ κ_{G}: Out(G) ⟶^{≅} Out(S, F) = Out(S)

is an isomorphism.

It remains to prove that Ker(μ_{G}) = 1. Fix [α] ∈ Ker(μ_{G}), and choose a representative α ∈ Aut_{typ}^{I}(L_{S}^{c}(G)) for the class [α] such that α_{S} is the identity on Aut_{L_{S}^{c}(G)}(S). In the notation of Proposition 4.2, E_{0} contains only the subgroups T_{i}, since Z(R_{i}) ≅ C_{2} (and hence C_{Z(R_{i})}(Aut_{S}(R_{i})) = C_{Z(R_{i})}(Aut_{F}(R_{i}))). If α is represented by elements g_{P}, then g_{T_{i}} ∈ C_{Z(T_{i})}(Aut_{S}(T_{i})) = Z(S) for each i, and is uniquely determined since C_{Z(T_{i})}(Aut_{F}(T_{i})) = 1. All of the g_{T_{i}} are equal by point (b) in the proposition, and hence [α] = 1 by point (d). □

4.2. Tameness of some fusion systems studied in [OV2].

We next consider some fusion systems studied in [OV2, §4–5], and prove they are reduced and tame using the lists of essential subgroups and other information determined there.

Proposition 4.5. The fusion systems at the prime 2 of the group P SL_{4}(5), and of the sporadic simple groups M_{22}, M_{23}, McL, J_{2}, and J_{3}, are all reduced and tame. Moreover, if G is any of these groups, then κ_{G} is an isomorphism.

Proof. By [GL, §1.5], Out(G) ≅ C_{2} when G ≅ M_{22}, McL, J_{2}, or J_{3}, while Out(M_{23}) = 1. By [St, (3.2)], when G = P SL_{4}(5), Out(G) is generated by diagonal automorphisms (induced by conjugation by diagonal matrices in GL_{4}(5)) and a graph automorphism (induced by transpose inverse). Since all multiples of the identity in GL_{4}(5) have determinant one, the group of diagonal outer automorphisms is isomorphic to 𝔽_{5}^{×} ≅ C_{4}. Since the graph automorphism inverts all diagonal matrices, we get Out(G) ≅ D_{8}.

Now let G be any of the above six groups, fix S ∈ Syl_{2}(G), and set F = F_{S}(G). We prove below in each case that among the homomorphisms

Out(G) ⟶^{κ_{G}} Out_{typ}(L_{S}^{c}(G)) ⟶^{μ_{G}} Out(S, F) ,

μ_{G} ∘ κ_{G} is an isomorphism and μ_{G} is injective. It then follows that κ_{G} is an isomorphism.

We show that μ_{G} ∘κ_{G} is injective for each of these groups, using arguments suggested to us by Richard Lyons. These are based on the following statement, applied to certain subgroups H ≤ G:

\begin{matrix}α ∈ Aut(H), S ∈ Syl_{2}(H), α|_{S} = Id_{S} \\ Q = O_{2}(H), C_{H}(Q) ≤ Q, \end{matrix} \} ⟹ α ∈ Aut_{Z(S)}(H).  (5)

This follows, for example, from [OV2, Lemma 1.2]: α ∈ Inn(H) if a certain element in H^{1}(H/Q; Z(Q)) vanishes, and this element does vanish since its restriction to the Sylow subgroup S/Q vanishes. Thus α ∈ Inn(H) and is the identity on S, so it must be conjugation by an element of C_{H}(S) = Z(S).

As in [OV2], we let S_{0} = UT_{3}(4) denote the group of upper triangular 3 × 3 matrices over 𝔽_{4} with 1’s on the diagonal. For x ∈ 𝔽_{4} and 1 ≤ i < j ≤ 3, e_{ij}^{x} ∈ UT_{3}(4) is the matrix with entry x in position (i, j), 1’s on the diagonal, and 0’s elsewhere. Set

E_{ij} = {e_{ij}^{x} | x ∈ 𝔽_{4}} , A_{1} = ⟨E_{12}, E_{13}⟩ , and A_{2} = ⟨E_{13}, E_{23}⟩ .

The field automorphism of 𝔽_{4} is denoted x ↦ x̄, and we write 𝔽_{4} = {0, 1, ω, ω̄}. Also, τ, ρ_{1}^{∗}, ρ_{2}^{∗}, γ_{0}, γ_{1}, c_{ϕ} ∈ Aut(S_{0}) are the automorphisms

τ (( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & c & b \\ 0 & 1 & a \\ 0 & 0 & 1 \end{matrix} )^{−1} , ρ_{1}^{∗} (( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & a & b+ā \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} ) , ρ_{2}^{∗} (( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & a & b+c̄ \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}) ,
γ_{0}(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & ωa & ω̄b \\ 0 & 1 & ωc \\ 0 & 0 & 1 \end{matrix} ) , γ_{1}(( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & ωa & b \\ 0 & 1 & ω̄c \\ 0 & 0 & 1 \end{matrix} ) , c_{ϕ} (( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} )) = ( \begin{matrix}1 & ā & b̄ \\ 0 & 1 & c̄ \\ 0 & 0 & 1 \end{matrix}) .

The group Out(S_{0}) ≅ C_{2}^{4} ⋊ (Σ_{3} × Σ_{3}) is described precisely by [OV2, Lemma 4.5]. In particular, the subgroups ⟨γ_{0}, c_{ϕ} ∘ τ⟩ and ⟨γ_{1}, τ⟩ are isomorphic to Σ_{3} and commute with each other.

By the focal subgroup theorem (cf. [G, Theorem 7.3.4]), 𝔣𝔬𝔠(F) = S ∩ [G, G] = S in each case, and hence O^{2}(F) = F. Assume O_{2}(F) = 1, and set F_{0} = O^{2′}(F). By Lemma 1.20(e), O_{2}(F_{0}) = 1, so F_{0} is a centerfree, nonconstrained fusion system over S, and is included in the list of such fusion systems over S in [OV2, Theorems 4.8 & 5.11]. By inspection of those lists, we see that F_{0} = F in all cases. So to prove F is reduced, it remains only to show that O_{2}(F) = 1.

In each of Cases 1 and 2 below, we prove successively that (i) μ_{G} ∘ κ_{G} is an isomorphism, (ii) μ_{G} is injective, and (iii) O_{2}(F) = 1.

Case 1: Assume first that S = S_{ϕ} = S_{0} ⋊ ⟨ϕ⟩: the extension of UT_{3}(4) by a field automorphism of 𝔽_{4}. Then S_{0} = ⟨A_{1}, A_{2}⟩ is characteristic in S, since A_{1} and A_{2} are the unique subgroups of S isomorphic to C_{2}^{4} (cf. [OV2, Lemma 5.1(b)]). Since c_{ϕ} permutes freely a basis of Z(S_{0}) = E_{13}, [OV2, Corollary 1.3 & Lemma 4.5(a)] imply there is an isomorphism

Out(S) ⟶_{≅}^{Res} C_{Out(S_{0})}(⟨[c_{ϕ}]⟩)/⟨[c_{ϕ}]⟩ = ⟨[ρ_{1}^{∗}], [ρ_{2}^{∗}], [τ]⟩ ≅ D_{8} .

Let τ̇, ρ̇_{1}^{∗}, ρ̇_{2}^{∗} ∈ Aut(S) be the extensions of τ, ρ_{1}^{∗}, ρ_{2}^{∗} ∈ Aut(S_{0}) which send ϕ to itself.

Set H_{i} = ⟨A_{i}, ϕ⟩, and N_{i} = ⟨H_{i}, e_{12}^{1}e_{23}^{1}⟩. By [OV2, Theorem 5.11] and Table 5.2 in its proof, in all cases, S_{0} is F-essential, and for i = 1, 2 either H_{i} or N_{i} is F-essential but not both. Also, Out_{F}(S) = 1 (since Out(S) is a 2-group), and Out_{F}(S_{0}) = ⟨[γ_{0}], [c_{ϕ}]⟩ or ⟨[γ_{0}], [γ_{1}], [c_{ϕ}]⟩. By [OV2, Lemma 5.8], there is a unique possibility for Out_{F}(N_{i}) if N_{i} is essential, and hence this group is normalized by [ρ̇_{1}^{∗}] and [ρ̇_{2}^{∗}]. By [OV2, Lemma 5.7], there are two possibilities for Out_{F}(H_{i}) (if H_{i} is essential) which are exchanged under conjugation by [ρ̇_{i}^{∗}] and invariant under conjugation by [ρ̇_{3−i}^{∗}].

By inspection, for i = 1, 2, [ρ_{i}^{∗}, γ_{0}] = 1 but [ρ_{i}^{∗}, γ_{1}] ≠ 1. Together with the above observations about the action of ρ̇_{i}^{∗} on the possibilities for Out_{F}(H_{i}) and Out_{F}(N_{i}), this shows that ρ̇_{i}^{∗} is fusion preserving (contained in Aut(S, F)) exactly when N_{i} is F-essential and [γ_{1}] ∉ Out_{F}(S_{0}) (and ρ̇_{1}^{∗}ρ̇_{2}^{∗} ∈ Aut(S, F) only if N_{1} and N_{2} are both essential). Also, τ̇ is fusion preserving if either the N_{i} are both essential or the H_{i} are both essential (and the Out_{F}(H_{i}) are chosen appropriately in the latter case), and otherwise Out(S, F) ≤ ⟨[ρ̇_{1}^{∗}], [ρ̇_{2}^{∗}]⟩. Thus Out(S, F) is as described in Table 4.1, where we refer to [OV2, Table 5.2] for the information about the fusion systems.

G  F-essential Out_{F}(S_{0})  Out(S, F) Out(G)
M_{22}  S_{0}, H_{1}, N_{2}  ⟨[γ_{0}], [c_{ϕ}]⟩  ⟨[ρ̇_{2}^{∗}]⟩ ≅ C_{2}  C_{2}
M_{23}  S_{0}, H_{1}, N_{2} ⟨[γ_{0}], [γ_{1}], [c_{ϕ}]⟩  1  1
P SL_{4}(5) S_{0}, N_{1}, N_{2}  ⟨[γ_{0}], [c_{ϕ}]⟩ Out(S) ≅ D_{8} D_{8}
McL  S_{0}, N_{1}, N_{2} ⟨[γ_{0}], [γ_{1}], [c_{ϕ}]⟩ ⟨[τ̇]⟩ ≅ C_{2}  C_{2}

Table 4.1

(i) Since |Out(G)| = |Out(S, F)|, it suffices to prove μ_{G} ∘ κ_{G} is injective. Fix α ∈ Aut(G, S) such that μ_{G}(κ_{G}([α])) = 1; thus α|_{S} = c_{g} for some g ∈ N_{G}(S). Upon replacing α by c_{g}^{−1} ∘ α, we can assume α|_{S} = Id_{S}. When G is one of the three sporadic groups, then by [GL, §1.5], A_{i} is centric in N_{G}(A_{i}) (i = 1, 2) and G = ⟨N_{G}(A_{1}), N_{G}(A_{2})⟩. When G ≅ P SL_{4}(5) ≅ P Ω_{6}^{+}(5), this is easily checked by identifying S_{0} ≤ P Ω_{6}^{+}(5) as the subgroup generated by classes of diagonal matrices (with respect to an orthonormal basis), together with permutation matrices for the permutations (1 2)(3 4) and (3 4)(5 6). So by (5), there are elements z_{1}, z_{2} ∈ Z(S) = ⟨e_{13}^{1}⟩ such that α|_{N_{G}(A_{i})} = c_{z_{i}} for i = 1, 2. Let g ∈ N_{G}(S_{0}) be such that c_{g} = γ_{0} ∈ Aut_{F}(S_{0}). Then g ∈ N_{G}(A_{i}) for i = 1, 2 since γ_{0} leaves the A_{i} invariant, so α(g) = c_{z_{1}}(g) = c_{z_{2}}(g), and hence z_{1} = z_{2} since [g, Z(S)] ≠ 1. Thus α ∈ Aut_{Z(S)}(G). (ii) Set L = L_{S}^{c}(G). By Proposition 4.2, each element of Ker(μ_{G}) is represented by some α ∈ Aut_{typ}^{I}(L) which is the identity on objects and on Aut_{L}(S), and such that for each fully normalized P ∈ Ob(L), α_{P} ∈ Aut(Aut_{L}(P)) is conjugation by some element g_{P} ∈ C_{Z(P)}(Aut_{S}(P)). Since Z(N_{i}) ≅ C_{2} (so C_{Z(N_{i})}(Aut_{S}(N_{i})) = C_{Z(N_{i})}(Aut_{F}(N_{i}))), the only F-essential subgroups which could be in the set E_{0} defined in Proposition 4.2(d) are S_{0}, and H_{1} and its S-conjugates if they are essential.

When P = S_{0},

g_{P} ∈ C_{Z(P)}(Aut_{S}(P)) = ⟨e_{13}^{1}⟩ = Z(S) = C_{Z(S)}(Aut_{F}(S)) .  (6)

So if H_{1} is not F-essential, then [α] = 1 in Out_{typ}(L) by Proposition 4.2(d).

Assume now that H_{1} is F-essential. Then H_{1} and A_{1} are both F-centric and fully normalized in F, and (6) holds when P is either of these subgroups. By the description of Aut_{F}(S_{0}) and Aut_{F}(H_{1}) in Table 4.1 and [OV2, Lemma 5.7(a)], A_{1} is invariant under all F-automorphisms of S_{0} and of H_{1}, and hence Aut_{F}(S_{0}, A_{1}) = Aut_{F}(S_{0}) and Aut_{F}(H_{1}, A_{1}) = Aut_{F}(H_{1}). Also, C_{A_{1}}(Aut_{F}(S_{0})) = C_{A_{1}}(Aut_{F}(H_{1})) = 1. Proposition 4.2(c) now implies g_{H_{1}} = g_{A_{1}} = g_{S_{0}}. So [α] = 1 by Proposition 4.2(d) again; and thus μ_{G} is injective.

(iii) By [OV2, Table 5.2], for each i = 1, 2, Out_{F}(A_{i}) is isomorphic to one of the groups Σ_{5}, (C_{3} × A_{5})⋊C_{2}, A_{6}, or A_{7}. Hence A_{1} and A_{2} are F-radical and F-centric, and O_{2}(F) ≤ A_{1} ∩ A_{2} = E_{13} by Proposition 1.5. Since no proper nontrivial subgroup of E_{13} is invariant under the action of γ_{0} ∈ Aut_{F}(S_{0}), and E_{13} itself is not invariant under the action of ν_{2} ∈ Aut_{F}(N_{2}) (see [OV2, Lemma 5.8]), we conclude that O_{2}(F) = 1.

Case 2: Now assume S = S_{θ} = S_{0} ⋊ ⟨θ⟩, where c_{θ} = τ ∘ c_{ϕ} ∈ Aut(S_{0}). Thus G = J_{2} or J_{3}. Again in this case, S_{0} is characteristic in S (cf. [OV2, Lemma 4.1(d)]). Since c_{θ} permutes freely a basis of Z(S_{0}) = E_{13}, [OV2, Corollary 1.3] together with the description of Out(S_{0}) in [OV2, Lemma 4.5], imply there is an isomorphism

Out(S) ⟶_{≅}^{Res} C_{Out(S_{0})}(⟨[c_{θ}]⟩)/⟨[c_{θ}]⟩ ≅ Σ_{4}.

Set Q = ⟨E_{13}, e_{12}^{1}e_{23}^{1}, e_{12}^{ω}eω̄ ; 23, θ⟩, an extraspecial group of type D_{8}×_{C_{2}} Q_{8} with Out(Q) ≅ Σ_{5}. Let γ̇_{1} ∈ Aut(S) be the extension of γ_{1} ∈ Aut(S_{0}) which sends θ to itself. By results in [OV2, §4.2–3], F = F_{S}(G) is isomorphic to the fusion system generated by automorphisms

Out_{F}(S) = ⟨[γ̇_{1}]⟩,  Out_{F}(S_{0}) = ⟨[γ_{0}], [γ_{1}], [c_{θ}]⟩ ≅ C_{3} × Σ_{3},  Out_{F}(Q) ≅ A_{5};

and by Out_{F}(A_{i}) ≅ GL_{2}(4) if G = J_{3}. Since Aut(S, F)/Inn(S) normalizes Out_{F}(S) ≅ C_{3}, and the normalizer in Σ_{4} of a subgroup of order 3 has order 6, |Out(S, F)| ≤ 2.

(i) In both cases (G ≅ J_{2} or J_{3}), Out(G) ≅ C_{2}. So to prove μ_{G} ∘ κ_{G} is an isomorphism, it suffices to show it is injective. Fix α ∈ Aut(G, S) such that μ_{G}(κ_{G}([α])) = 1; as before, we can assume α|_{S} = Id_{S}. By [GL, §1.5], N_{G}(Z(S)) and N_{G}(E_{13}) satisfy the hypotheses of (5), and they generate G since both are maximal proper subgroups. By (5), α|_{N_{G}(Z(S))} = Id, and α|_{N_{G}(E_{13})} = c_{z} for some z ∈ Z(S). Thus α ∈ Aut_{Z(S)}(G). (ii) Set L = L_{S}^{c}(G). By Proposition 4.2, each element of Ker(μ_{G}) is represented by some α ∈ Aut_{typ}^{I}(L) which is the identity on objects, and such that for each fully normalized P ∈ Ob(L), α_{P} ∈ Aut(Aut_{L}(P)) is conjugation by some element g_{P} ∈ C_{Z(P)}(Aut_{S}(P)). Since Z(Q) ≅ C_{2} (hence C_{Z(Q)}(Aut_{S}(Q)) = C_{Z(Q)}(Aut_{F}(Q))), E_{0} contains at most the subgroups S_{0}, A_{1}, and A_{2}. Note that in both cases, A_{1} and A_{2} are F-centric and fully normalized in F.

In both cases, γ_{0} ∈ Aut_{F}(S_{0}) leaves A_{1} and A_{2} invariant, and acts on each of the groups Z(S_{0}) = E_{13}, A_{1}, and A_{2} with trivial fixed subgroup. Hence

C_{Z(A_{1})}(Aut_{F}(S_{0}, A_{1})) = C_{Z(A_{2})}(Aut_{F}(S_{0}, A_{2})) = 1 ,

and so g_{A_{1}} = g_{S_{0}} = g_{A_{2}} by Proposition 4.2(c). Also, g_{S_{0}} ∈ C_{Z(S_{0})}(Aut_{S}(S_{0})) = ⟨e_{13}^{1}⟩ = Z(S) = C_{Z(S)}(Aut_{F}(S)), and Proposition 4.2(d) applies (with g = g_{S_{0}}) to show that [α] = 1 in Out_{typ}(L). Thus μ_{G} is injective. (iii) Since S_{0} and Q are F-centric and F-radical, O_{2}(F) ≤ S_{0} ∩ Q. Also, Aut_{F}(Q) acts transitively on the set of elements of order four in Q, and on the set of noncentral elements of order two. Since each of those sets contains elements in S_{0} and elements not in S_{0}, this implies O_{2}(F) ≤ Z(Q) = ⟨e_{13}^{1}⟩. Since γ_{0} ∈ Aut_{F}(S_{0}) and γ_{0}(e_{13}^{1}) ≠ e_{13}^{1}, it follows that O_{2}(F) = 1. □

4.3. Alternating groups.

We prove here that all fusion systems of alternating groups are tame, and are also (with the obvious exceptions) reduced. Unlike the other examples given in this paper, we prove tameness without first determining the list of essential subgroups.

We first fix some notation when working with alternating and symmetric groups. We always regard A_{n} ≤ Σ_{n} as groups of permutations of the set n = {1, …, n}. For σ ∈ Σ_{n}, we set supp(σ) = {i ∈ n | σ(i) ≠ i} (the support of σ). Likewise, for H ≤ Σ_{n}, supp(H) is defined to be the union of the supports of its elements.

Lemma 4.6. Fix a prime p and n ≥ p^{2}. Assume n ≥ 8 if p = 2. Set G = A_{n}, and fix S ∈ Syl_{p}(G). Set q = p if p is odd, and q = 4 if p = 2. Then

Out(S, F_{S}(G)) ≅ \begin{cases}C_{2} & if n ≡ 0, 1 (mod q) \\ 1 & otherwise. \end{cases}

In all cases, μ_{G} ∘ κ_{G} sends Out(G) = Out_{Σ_{n}}(G) ≅ C_{2} onto Out(S, F_{S}(G)).

Proof. Set F = F_{S}(G) for short. Set E_{∗} = ⟨(1 2 ⋯ p)⟩ ≅ C_{p} if p is odd, and E_{∗} = ⟨(1 2)(3 4), (1 3)(2 4)⟩ ≅ C_{2}^{2} if p = 2. Let Q ≤ S be the subgroup generated by all subgroups of S which are G-conjugate to E_{∗}. If E_{1} and E_{2} are G-conjugate to E_{∗}, then either E_{1} = E_{2}, or supp(E_{1}) ∩ supp(E_{2}) = ∅ and [E_{1}, E_{2}] = 1, or ⟨E_{1}, E_{2}⟩ is not a p-group. Since this last case is impossible when E_{1}, E_{2} ≤ S, we conclude that Q = Q_{1} × ⋯ × Q_{k}, where k = [n/q] and the Q_{i} are pairwise commuting subgroups conjugate to E_{∗}.

Fix α ∈ Aut(S, F), and set R = α(Q). We first show that R = Q. For i ≥ 1, let r_{i} be the number of orbits of length p^{i} under the action of R on n. Thus

∑_{i≥1} p^{i}r_{i} = |supp(R)| ≤ \begin{cases}q·[n/q] & if p is odd or r_{1} = 0 \\ 2·[n/2] & if p = 2 and r_{1} ≥ 1 \end{cases}  (7)

since supp(R) has order a multiple of p, and a multiple of 4 when p = 2 and r_{1} = 0. Since R ≅ Q is elementary abelian, R is contained in a product ∏_{i≥1}(B_{i})^{r_{i}}, where B_{i} ≅ C_{p}^{i} acts freely on a subset of n of order p^{i}, and hence

rk(Q) = rk(R) ≤ \begin{cases}∑_{i≥1} ir_{i} & if p is odd or r_{1} = 0 \\ ∑_{i≥1} ir_{i} − 1 & if p = 2 and r_{1} ≥ 1 . \end{cases}  (8)

In the last case, “−1” appears since R contains only even permutations, and since the only factors B_{i} which act via odd permutations are those for i = 1.

Thus if p is odd or r_{1} = 0, then by (7) and (8),

∑_{i≥1} p^{i}r_{i} ≤ q·[n/q] = qk = p·rk(Q) ≤ ∑_{i≥1} pir_{i} .  (9)

Also, p^{i} ≥ pi, with equality only when i = 1 or p^{i} = 4. Hence (9) is possible only when p is odd, r_{1} = k, and r_{i} = 0 for i > 1; or when p = 2, r_{2} = k, and r_{i} = 0 for i > 2. In both cases, R is a product of subgroups conjugate to E_{∗}, and thus R = Q.

Now assume p = 2 and r_{1} ≠ 0. By (7) and (8) again,

∑_{i≥1} 2^{i}r_{i} − 2 ≤ 2·([n/2] − 1) ≤ 4·[n/4] = 4k = 2·rk(Q) ≤ ∑_{i≥1} 2ir_{i} − 2 ,

so r_{i} = 0 for i ≥ 3, and the inequalities are equalities. In particular, r_{1}+2r_{2} = [n/2] = 2k+1, so r_{1} and [n/2] are both odd. Hence R ≅ (C_{2}^{2})^{r_{2}} × C_{2}^{r_{1}−1} (and r_{1} ≥ 3), where each element in Aut_{G}(R) permutes the C_{2}^{2}-factors and the C_{2}-factors. It follows that Aut_{G}(R) ≅ (Σ_{3} ≀ Σ_{r_{2}}) × Σ_{r_{1}}. Since α is fusion preserving, we have Aut_{G}(R) ≅ Aut_{G}(Q), where Aut_{G}(Q) = Aut_{Σ_{n}}(Q) ≅ Σ_{3} ≀ Σ_{k} since [n/2] is odd (n − 4k ≥ 2 where 4k = |supp(Q)|, so there is a transposition which centralizes Q). Thus Σ_{3} ≀ Σ_{k} ≅ (Σ_{3} ≀ Σ_{r_{2}}) × Σ_{r_{1}}. Since (Σ_{3} ≀ Σ_{ℓ})^{ab} ≅ C_{2}^{2} for all ℓ ≥ 2, we get r_{2} = 1, Σ_{3} ≀ Σ_{k} ≅ Σ_{3} × Σ_{r_{1}}, and this is clearly impossible.

Thus α(Q) = Q. Since α is fusion preserving, it permutes the G-conjugacy classes in Q. For each 1 ≤ r ≤ k, there are \binom{k}{r}·(q − 1)^{r} products of r disjoint p-cycles in Q if p is odd, and \binom{k}{r}·(q − 1)^{r} products of 2r disjoint 2-cycles in Q if p = 2. Clearly, k(q − 1) < \binom{k}{r}·(q − 1)^{r} for 1 < r < k, and k(q − 1) < (q − 1)^{k} since k > 1 and (k, q) ≠ (2, 3) by assumption. Hence α sends the set of p-cycles in Q (products of two 2-cycles in Q) to itself. Since the p-cycles (products of two 2-cycles) are precisely the nonidentity elements in ⋃_{i=1}^{k} Q_{i}, and since Q_{1}, …, Q_{k} are the maximal subgroups in this set, α permutes the Q_{i}.

Thus there is g ∈ N_{Σ_{n}}(Q) such that c_{g}|_{Q} = α|_{Q}, and hence

Aut_{gSg^{−1}}(Q) = (α|_{Q})Aut_{S}(Q)(α|_{Q})^{−1} = Aut_{S}(Q)

since α ∈ Aut(S). Since Q ⊴ S by construction, this implies gSg^{−1} ≤ S·C_{Σ_{n}}(Q), where S normalizes C_{Σ_{n}}(Q) since it normalizes Q. Hence there is h ∈ C_{Σ_{n}}(Q) such that hg ∈ N_{Σ_{n}}(S) (and α|_{Q} = c_{hg}|_{Q}). Upon replacing α by α ∘ c_{hg}^{−1}, we can now assume α|_{Q} = Id.

Set F_{0} = N_{F}(Q). Since Q is fully normalized in F and Q ⊴ S, this is a saturated fusion system over S. Also, C_{S}(Q) ≤ Q: any permutation which centralizes Q must leave each set supp(Q_{i}) invariant, and hence C_{G}(Q) ≅ (C_{A_{q}}(E_{∗}))^{k} × A_{n−qk} ≅ Q × A_{n−qk}. Thus Q is normal and centric in F_{0}, so F_{0} is constrained in the sense of [BCGLO1, Definition 4.1]. By [BCGLO1, Proposition 4.3], there is a finite group G_{0}, unique up to isomorphism, such that O_{p′}(G_{0}) = 1, Q ⊴ G_{0}, C_{G_{0}}(Q) ≤ Q, S ∈ Syl_{p}(G_{0}), and F_{0} = F_{S}(G_{0}). Thus Q ⊴ G_{0} and G_{0}/Q ≅ Aut_{F}(Q). The fusion preserving automorphism α induces an automorphism of F_{0} = N_{F}(Q), and hence by the uniqueness of G_{0} induces an automorphism β ∈ Aut(G_{0}) such that β|_{S} = α. Let H ⊴ G_{0} be the group of those g ∈ G_{0} such that c_{g} sends each Q_{i} to itself via an automorphism of order prime to p. Thus H/Q ≤ (C_{p−1})^{k} (with index 1 or 2) when p is odd, and H/Q ≅ C_{3}^{k} when p = 2. Since β|_{Q} = Id_{Q} and H/Q has order prime to p, β|_{H} is conjugation by an element a ∈ Q. Upon replacing α and β by α ∘ c_{a}^{−1} and β ∘ c_{a}^{−1}, we can assume β|_{H} = Id_{H}. But now, Z(H) = 1, so distinct elements of G_{0} have distinct conjugation actions on H, and hence β = Id_{G_{0}}. Thus α = β|_{S} = Id_{S}.

We have now shown that each element of Aut(S, F) is conjugation by some element of Σ_{n}. Since n > 6, Out(G) = Out_{Σ_{n}}(G) by, e.g., [Sz1, Theorem 3.2.17]. Thus μ_{G} ∘ κ_{G} sends Out(G) ≅ C_{2} onto Out(S, F). This last group is trivial exactly when there is g ∈ N_{Σ_{n}}(S)∖A_{n} such that c_{g}|_{S} ∈ Aut_{F}(S); i.e., when c_{g}|_{S} = c_{h}|_{S} for some h ∈ N_{G}(S). Upon replacing g by gh^{−1}, we see that Out(S, F) = 1 if and only if some odd permutation g ∈ Σ_{n}∖A_{n} centralizes S.

If n ≢ 0, 1 (mod q), then there is a transposition (i j) which centralizes S: when p is odd because one can choose i, j ∈ n∖supp(S), and when p = 2 because the S-action on n has an orbit {i, j} of order 2. Thus Out(S, F) = 1 in this case. If n ≡ 0, 1 (mod q), then |n∖supp(Q)| ≤ 1, and so

C_{Σ_{n}}(S) ≤ C_{Σ_{n}}(Q) = Q ≤ A_{n} .

Thus Out(S, F) = Out_{Σ_{n}}(S) has order two in this case. □

The following well known lemma will be needed when working with elementary abelian subgroups of symmetric groups.

Lemma 4.7. Fix n ≥ 1 and an abelian subgroup G ≤ Σ_{n}. Let H_{1}, …, H_{m} ≤ G be the distinct stabilizer subgroups for the action of G on n, and let X_{i} ⊆ n be the set of elements with stabilizer subgroup H_{i} (so n is the disjoint union of the X_{i}). Then each X_{i} is G-invariant.

Let k_{i} be the number of G-orbits in X_{i}. Then

C_{Σ_{n}}(G) ≅ ∏_{i=1}^{m}(G/H_{i}) ≀ Σ_{k_{i}} ,

where each factor (G/H_{i}) ≀ Σ_{k_{i}} has support X_{i}, Σ_{k_{i}} permutes the k_{i} G-orbits in X_{i}, and each factor G/H_{i} in (G/H_{i})^{k_{i}} has as support one of those G-orbits.

Proof. Let Y_{1}, …, Y_{t} be the G-orbits in n, and let C_{0} ≤ C_{Σ_{n}}(G) be the subgroup of elements which leave each of the Y_{i} invariant. Since G is abelian, y and g(y) have the same stabilizer subgroup for each g ∈ G and each y ∈ n. Let K_{i} be the stabilizer subgroup of the elements in Y_{i}. Then the homomorphism

χ: ∏_{i=1}^{t}(G/K_{i}) ⟶ C_{0} ,

defined by setting χ(g_{1}K_{1}, …, g_{t}K_{t})(y) = g_{i}(y) for y ∈ Y_{i}, is an isomorphism.

Since all elements in each orbit have the same stabilizer subgroup, each set X_{i} is a union of orbits Y_{j} (i.e., is G-invariant). Also, C_{0} is normal in C_{Σ_{n}}(G): it is the kernel of the homomorphism to Σ_{t} which describes how an element σ permutes the orbits. Each σ ∈ C_{Σ_{n}}(G) sends each orbit in n to another orbit with the same stabilizer subgroup, and thus leaves each X_{i} invariant. Since X_{i} contains k_{i} orbits, C_{Σ_{n}}(G)/C_{0} ≅ ∏_{i=1}^{m} Σ_{k_{i}}, and C_{Σ_{n}}(G) is isomorphic to the product of the wreath products (G/H_{i}) ≀ Σ_{k_{i}}. □

We are now ready to prove that all fusion systems of alternating groups are tame.

Proposition 4.8. Fix a prime p and n ≥ 2, set G = A_{n}, and choose S ∈ Syl_{p}(G). Then F_{S}(G) is tame. If p = 2 and n ≥ 8; or if p is odd, n ≥ p^{2} and n ≡ 0, 1 (mod p); then

κ_{G}: Out(G) ⟶ Out_{typ}(L_{S}^{c}(G)) ≅ C_{2}

is an isomorphism.

Proof. Set F = F_{S}(G) and L = L_{S}^{c}(G). If n < p^{2}, or if p = 2 and n < 6, then the Sylow p-subgroups of A_{n} are abelian, so F is constrained, 𝔯𝔢𝔡(F) = 1 by Proposition 2.4, and so F is tame by Theorem A. If p = 2 and n = 6, 7, then since A_{6} ≅ P SL_{2}(9) and A_{7} has the same fusion system as A_{6}, F is tame by Proposition 4.3.

If p is odd and n ≥ p^{2}, then μ_{G}: Out_{typ}(L) ⟶^{≅} Out(S, F) is an isomorphism by [BLO1, Theorem E] and [O1, Theorem A]. So by Lemma 4.6, either n ≡ 0, 1 (mod p) and κ_{G} is an isomorphism, or Out_{typ}(L) = 1 and hence κ_{G} is split surjective. Thus F is tame in these cases.

It remains to handle the case p = 2 and n ≥ 8. By Lemma 4.6 again, it suffices to prove

Ker(μ_{G}) = 1 if n ≡ 0, 1 (mod 4) and |Ker(μ_{G})| ≤ 2 if n ≡ 2, 3 (mod 4) ,  (10)

and also

n ≡ 2, 3 (mod 4) ⟹ there is x ∈ C_{Σ_{n}}(S)∖G such that κ_{G}([c_{x}]) ≠ 1 .  (11)

Let Q ≤ S be as in the proof of Lemma 4.6: the subgroup generated by all subgroups of S G-conjugate to E_{∗} = ⟨(1 2)(3 4), (1 3)(2 4)⟩. We saw in the proof of the lemma that Q = Q_{1} × ⋯ × Q_{k}, where k = [n/4], the Q_{i} are the only subgroups of S G-conjugate to E_{∗}, and they have pairwise disjoint support. Thus Q is weakly closed: the unique subgroup of S in its G-conjugacy class.

Fix [α] ∈ Ker(μ_{G}). By Proposition 4.2, we can assume [α] is the class of α ∈ Aut_{typ}^{I}(L) for which α_{S} = Id_{Aut_{L}(S)}. Let g_{P} ∈ C_{Z(P)}(Aut_{S}(P)), for P ≤ S F-centric and fully normalized, be the elements defined in Proposition 4.2. Set g = g_{Q} ∈ C_{Q}(Aut_{S}(Q)) = Z(S) (the last equality since Q is normal and centric in S). For each fully normalized P ≥ Q (including P = S), all automorphisms in Aut_{F}(P) leave Q invariant since it is weakly closed, so g_{P} ≡ g_{Q} = g (mod C_{Z(Q)}(Aut_{F}(P)) = C_{Z(P)}(Aut_{F}(P))) by Proposition 4.2(c). So upon replacing α by α ∘ c_{[g]}^{−1}, we can assume g = 1, and α_{P} = Id_{Aut_{L}(P)} (and g_{P} = 1) for all fully normalized P ≥ Q.

For each 1 ≤ i ≤ k, there is a 3-cycle h_{i} ∈ N_{G}(Q) which permutes transitively the involutions in Q_{i} and centralizes the other Q_{j}. Thus C_{Q}(Aut_{F}(Q)) ≤ ⋂_{i=1}^{k} C_{Q}(h_{i}) = 1. So by Proposition 4.2(e), [α] = 1 if and only if for each F-conjugacy class P of subgroups which do not contain Q, if P ∩ Ê_{0} ≠ ∅, then there is at least one subgroup P ∈ P ∩ Ê_{0} such that g_{P} ∈ C_{Z(P)}(Aut_{F}(P)). Recall P ∈ Ê_{0} if P is F-essential, P = C_{S}(E) for some elementary abelian subgroup E fully centralized in F, and C_{Z(P)}(Aut_{S}(P)) ≩ C_{Z(P)}(Aut_{F}(P)). Let Ê_{0}^{≱Q} be the set of subgroups P ∈ Ê_{0} which do not contain Q.

Fix P = C_{S}(E) ∈ Ê_{0}^{≱Q}. Since E is fully centralized, P ∈ Syl_{2}(C_{G}(E)). Since P is Fessential, Out_{F}(P) has a strongly 2-embedded subgroup, and hence all involutions in any Sylow 2-subgroup of Out_{F}(P) are in its center (cf. [OV2, Propositions 3.3(a) & 3.2]). In particular, Out_{F}(P) contains no subgroup isomorphic to D_{8}.

Fix P ∈ Syl_{2}(C_{Σ_{n}}(E)) which contains P. Thus P = P ∩ A_{n}. Also, E ⊴ P, so P ≤ P ·C_{Σ_{n}}(P) ≤ C_{Σ_{n}}(E), and hence

P ·C_{Σ_{n}}(P)/P has odd order.  (12)

By Lemma 4.7, each union of m E-orbits of order q = 2^{i} which have the same stabilizer subgroup contributes a factor E_{q} ≀ Σ_{m} to C_{Σ_{n}}(E), where E_{q} ≅ (C_{2})^{i} is acting freely on an orbit of order q in n. Since a Sylow 2-subgroup of Σ_{m} is a product of wreath products C_{2} ≀ ⋯ ≀ C_{2}, P ∈ Syl_{2}(C_{Σ_{n}}(E)) is a product of subgroups of the form E_{q} ≀ C_{2} ≀ ⋯ ≀ C_{2} (or E_{q}) with pairwise disjoint support. If P contains a factor E_{q} ≀ C_{2} ≀ ⋯ ≀ C_{2} for q = 2^{r} ≥ 8, then Out_{F}(P) contains GL_{r}(2) ≥ D_{8}, which we just saw is impossible.

Write n = X_{0} ⨿ X_{1} ⨿ X_{2}, where X_{0} is the set of points fixed by P, X_{1} is the union of P -orbits of length 2, and X_{2} is the union of P -orbits of length ≥ 4. By the above description of P, P = P_{1} × P_{2}, where supp(P_{i}) = X_{i} for i = 1, 2, P_{1} ≅ C_{2}^{m} where 2m = |X_{1}|, and P_{2} is a product of subgroups E_{4} ≀ C_{2} ≀ ⋯ ≀ C_{2} and C_{2} ≀ ⋯ ≀ C_{2} (the latter of order ≥ 8). By (12), |X_{0}| ≤ 1, since otherwise there would be a 2-cycle in C_{Σ_{n}}(P) not in P.

Each factor E_{4} or C_{2} ≀ C_{2} (with support of order 4) contains a subgroup conjugate to E_{∗} (thus one of the factors Q_{i} in Q). Thus X_{2} ⊆ supp(Q ∩ P). If n − |X_{2}| ≤ 3, then X_{2} = supp(Q), so Q ≤ P ∩ A_{n} = P, contradicting the original assumption on P. Thus |X_{0} ∪ X_{1}| > 3. Since |X_{0}| ≤ 1 and |X_{1}| = 2m is even, we have m ≥ 2.

If {i, j} is any of the m orbits of order 2 in X_{1}, then (i j) ∈ C_{Σ_{n}}(P)∖A_{n} and P = ⟨P, (i j)⟩. Thus N_{Σ_{n}}(P) = N_{Σ_{n}}(P), C_{Σ_{n}}(P) = C_{Σ_{n}}(P), P ·C_{Σ_{n}}(P) = P ·C_{Σ_{n}}(P), and so N_{Σ_{n}}(P)/P ·C_{Σ_{n}}(P) ≅ N_{G}(P)/P ·C_{G}(P). This proves that

Out_{G}(P) = Out_{Σ_{n}}(P) ≅ Out_{Σ_{n}}(P) ≅ Σ_{m} × Out_{Σ_{X_{2}}}(P_{2}),

where the first isomorphism is induced by restriction. Here, Σ_{X_{2}} is the group of permutations of X_{2}.

If m = 2, then O_{2}(Out_{G}(P)) ≠ 1, and if m ≥ 4, then Out_{G}(P) ≥ D_{8}. Either of these would contradict the assumption that Out_{G}(P) contains a strongly 2-embedded subgroup. Thus m = 3, and X_{1} = supp(P_{1}) has order 6. A group with a strongly 2-embedded subgroup cannot split as a product of two groups of even order, so |Out_{Σ_{X_{2}}}(P_{2})| is odd. Since P_{2}·C_{Σ_{X_{2}}}(P_{2})/P_{2} is isomorphic to a subgroup of P ·C_{Σ_{n}}(P)/P, it has odd order by (12), and hence

|N_{Σ_{X_{2}}}(P_{2})/P_{2}| = |\frac{N_{Σ_{X_{2}}}(P_{2})}{P_{2}·C_{Σ_{X_{2}}}(P_{2})}| · |\frac{P_{2}·C_{Σ_{X_{2}}}(P_{2})}{P_{2}} | = |Out_{Σ_{X_{2}}}(P_{2})|·|P_{2}·C_{Σ_{X_{2}}}(P_{2})/P_{2}|

is also odd. If P_{2} ≤ T ∈ Syl_{2}(Σ_{X_{2}}), then N_{T} (P_{2})/P_{2} has odd order, so P_{2} = T (cf. [Sz1, Theorem 2.1.6]), and thus P_{2} ∈ Syl_{2}(Σ_{X_{2}}).

Since P_{2} is a Sylow 2-subgroup of a symmetric group and has no orbits of order 2, it is a product of subgroups C_{2} ≀ ⋯ ≀ C_{2} of order ≥ 8. Since 4||X_{2}| (a union of orbits of order 2^{i} ≥ 4) and |X_{0}| ≤ 1,

n = |X_{0}| + 6 + |X_{2}| ≡ 2, 3 (mod 4) .

If R is any other subgroup in Ê_{0}^{≱Q}, then R = R ∩ G, n = Y_{0} ⨿ Y_{1} ⨿ Y_{2} where Y_{0} is the set of elements fixed by R and Y_{1} is the union of R-orbits of order 2, R = R_{1} × R_{2} where supp(R_{i}) = Y_{i}, R_{2} ∈ Syl_{2}(Σ_{Y_{2}}), |Y_{1}| = 6 = |X_{1}|, and |Y_{2}| = |X_{2}| (the largest multiple of 4 which is ≤ n−6). Thus R is Σ_{n}-conjugate to P, and is A_{n}-conjugate to P since there are odd permutations which centralize P (the transpositions in P_{1}).

Now, Z(P) = P_{1} × Z(P_{2}), where Z(P_{2}) is a product of one copy of C_{2} for each factor C_{2} ≀ ⋯ ≀ C_{2} in P_{2} (equivalently, for each P_{2}-orbit in X_{2}). Also, each of these factors C_{2} has support the corresponding P_{2}-orbit, hence of order a multiple of 4, and hence contained in A_{n}. Thus Z(P_{2}) ≤ G = A_{n}. Also, Aut_{F}(P) acts via the identity on Z(P_{2}), since all of the factors C_{2} ≀ ⋯ ≀ C_{2} in P_{2} have different orders (hence their supports have different orders). Since Aut_{A_{n}}(P_{1} ∩ A_{n}) ≅ Σ_{3} acts on P_{1} by permuting the three transpositions, Aut_{F}(P) acts on P_{1} ∩ A_{n} ≅ C_{2}^{2} with trivial fixed set. Since Z(P) = (P_{1} ∩ A_{n}) × Z(P_{2}), it now follows that C_{Z(P)}(Aut_{S}(P))/C_{Z(P)}(Aut_{F}(P)) has order two.

To summarize, every class in Ker(μ_{G}) is represented by some α such that α_{P} = Id when P ≥ Q, and for such α, [α] = 1 if and only if g_{P} ∈ C_{Z(P)}(Aut_{F}(P)) for some representative in each F-conjugacy class in Ê_{0}^{≱Q}. When n ≡ 0, 1 (mod 4), Ê_{0}^{≱Q} = ∅, so Ker(μ_{G}) = 1. When n ≡ 2, 3 (mod 4), all subgroups in Ê_{0}^{≱Q} are F-conjugate to some fixed P, and so |Ker(μ_{G})| ≤ |C_{Z(P)}(Aut_{S}(P))/C_{Z(P)}(Aut_{F}(P))| = 2. This proves (10).

Assume n ≡ 2, 3 (mod 4), and P = P ∩ A_{n} ∈ Ê_{0}^{≱Q} as above. Set k = [n/4] as before. Assume S was chosen so that supp(S) = X_{1} ⨿ X_{2} = {1, …, 4k + 2} and supp(Q) = {3, …, 4k + 2}. We have shown that |X_{2}| = |supp(S)| − 6 = 4k − 4 and X_{2} ⊆ supp(Q). Thus, after possibly rearranging the elements of n, we can assume X_{2} = {7, 8, …, 4k + 2} and P_{1} = ⟨(1 2), (3 4), (5 6)⟩.

Set x = (1 2). Then Out(G) = ⟨[c_{x}]⟩ ≅ C_{2}, [x, S] = 1, and c_{x} is the identity on N_{G}(Q)/C′_{G}(Q) and hence on Aut_{L}(Q). (Note that if n = 4k + 3, then C′_{G}(Q) = ⟨(1 2 n)⟩ does not commute with x.) Also, (1 2)(3 4)(5 6) centralizes N_{G}(P), and hence c_{x} acts on Aut_{L}(P) (or on N_{G}(P)) via conjugation by g_{P} =^{def} (3 4)(5 6) ∈ C_{Z(P)}(Aut_{S}(P)). Since g_{P} ∉ C_{Z(P)}(Aut_{F}(P)), [c_{x}] is sent to a nontrivial element in Ker(μ_{G}). This proves (11), and finishes the proof of the proposition. □

We finish by proving that with the obvious exceptions, most fusion systems of alternating groups are reduced.

Proposition 4.9. Fix a prime p and n ≥ p^{2} such that n ≡ 0, 1 (mod p). Assume n ≥ 8 if p = 2. Set G = A_{n}, and choose S ∈ Syl_{p}(G). Then the fusion system F_{S}(G) is reduced.

Proof. Set F = F_{S}(G). By the focal subgroup theorem (cf. [G, Theorem 7.3.4]), 𝔣𝔬𝔠(F) = S ∩ [G, G] = S, so O^{p}(F) = F.

Let Q ≤ S be as in the proof of Lemma 4.6: the subgroup generated by all subgroups of S G-conjugate to E_{∗}, where E_{∗} = ⟨(1 2 ⋯ p)⟩ ≅ C_{p} if p is odd, and E_{∗} = ⟨(1 2)(3 4), (1 3)(2 4)⟩ ≅ C_{2}^{2} if p = 2. We saw in the proof of the lemma that Q = Q_{1}×⋯×Q_{k}, where k = [n/p] (p > 2) or [n/4] (p = 2), the Q_{i} are the only subgroups of S G-conjugate to E_{∗}, and they have pairwise disjoint support. Thus Q is Aut_{F}(S)-invariant. We also saw that C_{S}(Q) = Q, and hence Q is F-centric (since it is the only subgroup in its F-conjugacy class by construction). Finally,

Aut_{Σ_{n}}(Q) ≅ Aut(E_{∗}) ≀ Σ_{k}  where Aut(E_{∗}) ≅ \begin{cases}C_{p−1} & if p > 2 \\ Σ_{3} & if p = 2 , \end{cases}  (13)

and hence Aut_{F}(Q) has index at most two in this wreath product. When p = 2, since Σ_{k} ≤ Aut_{Σ_{n}}(Q) permutes the Q_{i} with support of order 4, it is contained in Aut_{F}(Q).

Set R = O_{p}(F). Since Q is F-centric, and is F-radical by (13), R ≤ Q by Proposition 1.5. Assume R ≠ 1, and fix g ∈ R of order p. There is h ∈ Q which is G-conjugate to g (a product of the same number of p-cycles) such that gh is a p-cycle (or a product of two 2-cycles if p = 2). Then h ∈ R since R ⊴ F, and so gh ∈ R. Since each Q_{i} is generated by elements G-conjugate to gh, this would imply that R = Q. But in all cases, there are elements both in Q and in S∖Q which are products of p disjoint p-cycles, so Q is not strongly closed in F. We conclude that R = O_{p}(F) = 1.

Now set F_{0} = O^{p′}(F); we must show F_{0} = F. By [BCGLO2, Theorem 5.4], it suffices to show that Aut_{F_{0}}(S) = Aut_{F}(S). Also, by the same theorem,

Aut_{F_{0}}(S) = Aut_{F}^{0}(S) ≥ ⟨α ∈ Aut_{F}(S) | α|_{P} ∈ O^{p′}(Aut_{F}(P)),

some F-centric subgroup P ≤ S with α(P) = P ⟩ .

For α ∈ Aut_{F}(S), if α|_{Q} ∈ O^{p′}(Aut_{F}(Q)), then α ∈ Aut_{F_{0}}(S). If p = 2, then O^{2′}(Aut_{F}(Q)) = Aut_{F}(Q) by the description in (13), so F_{0} = F in this case.

Assume p is odd. Let p^{ℓ} be the largest power of p such that p^{ℓ} ≤ n. Write S = S_{1} × S_{2}, where supp(S_{1}) ∩ supp(S_{2}) = ∅ and |supp(S_{1})| = p^{ℓ}. Fix T ∈ Syl_{p}(Σ_{p}), and identify

S_{1} = T ≀ T ≀ ⋯ ≀ T ≤ Σ_{p} ≀ Σ_{p} ≀ ⋯ ≀ Σ_{p} ≤ Σ_{p^{ℓ}} ≤ Σ_{n}.

Let Φ: (Σ_{p})^{ℓ} ⟶ Σ_{p} ≀ ⋯ ≀ Σ_{p} ≤ Σ_{p^{ℓ}} be the monomorphism which sends the first factor diagonally to (Σ_{p})^{p^{ℓ−1}}, the second factor diagonally to (1 ≀ Σ_{p})^{p^{ℓ−2}}, etc. Set P_{1} = Φ(T^{ℓ}) and P = P_{1} × S_{2} ≤ S. Fix u ∈ 𝔽_{p}^{×} of order p − 1, and choose h ∈ N_{Σ_{p}}(T) such that hgh^{−1} = g^{u} for g ∈ T. Let α ∈ Aut_{F}(S) be conjugation by Φ(h, h^{−1}, 1, …, 1). Then α|_{P_{1}} has matrix diag(u, u^{−1}, 1, …, 1) ∈ SL_{ℓ}(p) with respect to the canonical basis. Since Aut_{F}(P_{1}) has index at most two in Aut_{Σ_{n}}(P_{1}) ≅ GL_{ℓ}(p), we get α|_{P} ∈ O^{p′}(Aut_{F}(P)), and so α ∈ Aut_{F_{0}}(S) since P is F-centric. Also, α|_{Q} represents a generator of Aut_{F}(Q)/O^{p′}(Aut_{F}(Q)) ≅ 𝔽_{p}^{×}, so this finishes the proof that Aut_{F_{0}}(S) = Aut_{F}(S) and hence that F_{0} = F. Thus F is reduced. □

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Matematisk Institut, Ny Munkegade, 8000 Aarhus C, Denmark

  E-mail address: kksa@imf.au.dk

LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

  E-mail address: bobol@math.univ-paris13.fr

Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049–001 Lisboa, Portugal

  E-mail address: jventura@math.ist.utl.pt