A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS

CARLES BROTO, RAN LEVI, AND BOB OLIVER

Abstract. A saturated fusion system consists of a finite p-group S, together with a category which encodes “conjugacy” relations among subgroups of S, and which satisfies certain axioms which are motivated by properties of the fusion in a Sylow p-subgroup of a finite group. We describe here new ways of constructing abstract saturated fusion systems, first as fusion systems of spaces with certain properties, and then via certain graphs.

A saturated fusion system consists of a finite p-group S, together with a category F whose objects are the subgroups of S, whose morphisms are group monomorphisms between those subgroups, and which satisfies certain axioms modelled on the fusion category for the p-subgroups of a finite group. The precise definition of a saturated fusion system is due to Puig [Pu], and our version of that definition is given in Section 1. Saturated fusion systems mimic in several ways the structure of finite groups and their classifying spaces. Examples have been known for some time of “exotic” saturated fusion systems — systems which do not arise from the fusion in any finite group — but the construction of such examples is very complicated, and we are looking for simpler and more systematic ways to construct them. One consequence of the main result in this paper is a way of constructing a variety of examples of saturated fusion systems. Of the examples constructed using this technique, some are then shown by other means to be exotic.

The definition of a fusion system over a p-group S is simple, and in most cases it is clear whether or not a given category satisfies it. In contrast, it is much harder to check whether a given fusion system is saturated. For example, for any map f: BS ⟶ X, where S is a finite p-group and X is a topological space, the fusion system of X over (S, f) is a category F_{S,f}(X) whose objects are the subgroups of S, and where Mor_{F_{S,f} (X)}(P, Q) is the set of all monomorphisms φ ∈ Hom(P, Q) such that (f|_{BP}) ∘ Bφ ≃ (f|_{BQ}). This is always a fusion system in the sense of Definition 1.1, but is not in general saturated.

The central result in this paper is Theorem 2.1, where we list some conditions on the map f which ensure that the fusion system F_{S,f}(X) is saturated. These conditions also ensure that F_{S,f}(X) has an associated linking system (see Definition 1.3), and hence that X, S, and f define a p-local finite group. Afterwards, we construct more concrete examples using that theorem, and show in many cases that they are “exotic” in the sense of not coming from any finite group. For example, in Theorem 4.2, in certain cases when G is an amalgamated free product of finite groups, we apply our theorem to BG_{p}^{∧} to show that the fusion system of G (taken over a maximal p-subgroup of G) is saturated. This result, which is stated in terms of trees of groups, was discovered and first proved as a special case of Theorem 2.1, but we also include a more elementary, purely graph theoretic proof here.

1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D20.

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

C. Broto is partially supported by MEC grant MTM2004-06686.

R. Levi is partially supported by EPSRC grant GR/M7831.

B. Oliver is partially supported by UMR 7539 of the CNRS.

All of the authors have been supported by the EU grant nr. HPRN-CT-1999-00119.

The paper is organized as follows. In Section 1, we give the definitions of abstract fusion and linking systems, as well as definitions of fusion and linking systems of groups and spaces and some background results about them. Our main theorem is proven in Section 2. In Section 3, we describe conditions under which the main theorem can be applied to the space BG_{p}^{∧} for an infinite discrete group G, to prove that the fusion system of G with respect to some finite p-subgroup is saturated (Theorem 3.3). A special case of this is then studied in Section 4 — the case where G acts on a tree with finite isotropy subgroups — and this in turn is applied in Section 5 to construct concrete examples of fusion systems, some of which are then shown to be “exotic”. We hope to find other applications of our main Theorem 2.1 in the future which allow us to construct a still wider variety of examples.

We would like to give our thanks to Michael Aschbacher and Andy Chermak, whose construction of the Solomon fusion systems in [AC] gave us the idea of restating the results in Section 3 terms of amalgamated free products.

1. A survey of fusion systems

We first recall some definitions, mostly from [BLO2].

Definition 1.1 ([Pu] and [BLO2, Definition 1.1]). A fusion system over a finite pgroup S is a category F, where Ob(F) is the set of all subgroups of S, and which satisfies the following two properties 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.

Fusion systems as defined above are too general for our purposes, and some additional definitions and conditions are needed so that they more closely model the fusion in finite groups. If F is a fusion system over a finite p-subgroup S, then two subgroups P, Q ≤ S are said to be F-conjugate if they are isomorphic as objects of the category F.

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

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

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

(I) For all 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 φP is fully centralized, and if we

set

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

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) defined in the introduction is a saturated fusion system (see [BLO2, Proposition 1.3]).

An alternative, simplified pair of axioms for a fusion system being saturated has been given by Radu Stancu [St].

We now turn to centric linking systems associated to abstract fusion systems. Whenever F is a fusion system over a finite p-group S, a subgroup P ≤ S is called F-centric if C_{S}(P′) = Z(P′) for all P′ ≤ S which are F-conjugate to P. We let F^{c} ⊆ F denote the full subcategory whose objects are the F-centric subgroups of S. If F = F_{S}(G) for some finite group G, then P ≤ S is F-centric if and only if P is p-centric in G; i.e., if and only if Z(P) ∈ Syl_{p}(C_{G}(P)).

Definition 1.3 ([BLO2, Definition 1.7]). Let F be a fusion system over the p-group S. A centric linking system associated to F is a category L whose objects are the Fcentric subgroups of S, together with a functor π: L ⟶ F^{c}, and “distinguished” monomorphisms P ⟶^{δ_{P}} Aut_{L}(P) for each F-centric subgroup P ≤ S, which satisfy the following conditions.

(A) π is the identity on objects. For each pair of objects P, Q ∈ Ob(L), Z(P) acts freely on Mor_{L}(P, Q) by composition (upon identifying Z(P) with δ_{P} (Z(P)) ≤ Aut_{L}(P)), and π induces a bijection

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

(B) For each F-centric subgroup P ≤ S and each x ∈ P, π(δ_{P} (x)) = c_{x} ∈ Aut_{F}(P).

(C) For each f ∈ Mor_{L}(P, Q) and each x ∈ P, f ∘ δ_{P} (x) = δ_{Q}(πf(x)) ∘ f.

A p-local finite group is defined to be a triple (S, F, L), where S is a finite p-group, F is a saturated fusion system over S, and L is a centric linking system associated to F. The classifying space of the triple (S, F, L) is the p-completed nerve |L|_{p}^{∧}.

In the following definition, recall that a (possibly infinite) group G is p-perfect if it has no normal subgroup of index p; or equivalently, if Hom(G, ℤ/p) contains only the trivial homomorphism. Clearly, if G is generated by p-perfect subgroups, then it is itself p-perfect. Hence any group G contains a maximal p-perfect subgroup, which is normal.

Definition 1.4. Fix any pair S ≤ G, where G is a (possibly infinite) group and S is a finite p-subgroup.

(a) Define F_{S}(G) to be the category whose objects are the 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}
≅ N_{G}(P, Q)/C_{G}(P).

Here c_{g} denotes the homomorphism conjugation by g (x ↦ gxg^{−1}), and N_{G}(P, Q) = {g ∈ G | gP g^{−1} ≤ Q} (the transporter set).

(b) For each P ≤ S, let C′_{G}(P) be the maximal p-perfect subgroup of C_{G}(P). Let L_{S}^{c}(G) be the category whose objects are the F_{S}(G)-centric subgroups of S, and where

Mor_{L_{S}^{c}(G)}(P, Q) = N_{G}(P, Q)/C′_{G}(P).

Let π: L_{S}^{c}(G) → F_{S}(G) be the functor which is the inclusion on objects and sends the class of g ∈ N_{G}(P, Q) to conjugation by g. For each F_{S}(G)-centric subgroup P ≤ G, let δ_{P} : P → Aut_{L_{S}^{c}(G)}(P) be the monomorphism induced by the inclusion P ≤ N_{G}(P).

It is clear from the definitions that F_{S}(G) is a fusion system for any S and G, and just as clear that it is not always saturated. When G is finite and S ∈ Syl_{p}(G), then F_{S}(G) is always saturated (see [Pu], or [BLO2, Proposition 1.3]), and L_{S}^{c}(G) is a centric linking system associated to F_{S}(G). Thus in this case, (S, F_{S}(G), L_{S}^{c}(G)) is a p-local finite group, with classifying space |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧} (see [BLO1, Proposition 1.1]).

When G is infinite, we note the following condition for L_{S}^{c}(G) to be a centric linking system.

Lemma 1.5. Fix any pair S ≤ G, where G is a (possibly infinite) group and S is a finite p-subgroup, and set F = F_{S}(G). Assume, for each F-centric subgroup P ≤ S, that H^{i}(C_{G}(P)/Z(P); 𝔽_{p}) = 0 for i = 1, 2. Then L_{S}^{c}(G) is a centric linking system associated to F.

Proof. Conditions (B) and (C) in Definition 1.3 hold by definition of L_{S}^{c}(G), the projection functor π, and the distinguished monomorphisms δ_{P}. Also, for each pair of objects P, Q, C_{G}(P) acts freely on N_{G}(P, Q) by right multiplication, so C_{G}(P)/C′_{G}(P) acts freely on Mor_{L_{S}^{c}(G)}(P, Q) with orbit set Hom_{F}(P, Q). So to prove that L_{S}^{c}(G) is a centric linking system associated to F, it remains only to show that for each F-centric subgroup P ≤ S, the inclusion Z(P) ≤ C_{G}(P) induces an isomorphism Z(P) ≅ C_{G}(P)/C′_{G}(P).

The assumption H^{1}(C_{G}(P)/Z(P); 𝔽_{p}) = 0 implies that C_{G}(P)/Z(P) is p-perfect. Since H^{2}(C_{G}(P)/Z(P); 𝔽_{p}) = 0 and Z(P) is a finite p-group, the exact sequences in group cohomology for extensions of modules show that H^{2}(C_{G}(P)/Z(P); Z(P)) = 0, and hence that C_{G}(P) splits as a product Z(P) × H for a normal subgroup H ◁ C_{G}(P). Thus H ≅ C_{G}(P)/Z(P) is the maximal p-perfect subgroup of C_{G}(P), and so C_{G}(P)/C′_{G}(P) = C_{G}(P)/H ≅ Z(P). □

Fusion systems and linking systems can also be defined for spaces. In the following definition, if H: X × I ⟶ Y is a homotopy (where I = [0, 1]), then [H] denotes its homotopy class among maps X ×I ⟶ Y whose restriction to X ×{0, 1} is the same as that of H. In other words, if we regard H as a path in Map(X, Y) by adjunction, then [H] denotes the homotopy class of that path rel endpoints.

For any p-group P and any g ∈ P, let H_{g}: BP × I ⟶ BP be the homotopy from Id_{BP} to Bc_{g} induced by the natural transformation of functors B(G) ⟶ B(G) which sends the unique object o_{G} in B(G) to the morphism ǧ corresponding to g ∈ G.

Definition 1.6. Fix a space X, a finite p-group S, and a map f: BS ⟶ X.

(a) Define F_{S,f}(X) to be the category whose objects are the subgroups of S, and whose morphisms are given by

Hom_{F_{S,f} (X)}(P, Q) = {φ ∈ Inj(P, Q) | f|_{BP} ≃ f|_{BQ} ∘ Bφ}

for each P, Q ≤ S.

(b) Define F′_{S,f}(X) ⊆ F_{S,f}(X) to be the subcategory with the same objects as F_{S,f}(X), and where Mor_{F′_{S,f} (X)}(P, Q) (for P, Q ≤ S) is the set of all composites of restrictions of morphisms in F_{S,f}(X) between F_{S,f}(X)-centric subgroups.

(c) Define L_{S,f}^{c}(X) to be the category whose objects are the F_{S,f}(X)-centric subgroups of S, and whose morphisms are defined by

Mor_{L_{S,f}^{c} (X)}(P, Q) = {(φ, [H]) | φ ∈ Inj(P, Q), H: BP × I ⟶ X,

H|_{BP ×0} = f|_{BP} , H|_{BP ×1} = f|_{BQ} ∘ Bφ} .

The composite in L_{S,f}^{c}(X) of morphisms

P ⟶^{(φ,[H])} Q ⟶^{(ψ,[K])} R ,

where H: BP × I → X and K: BQ × I → X are homotopies as described above, are defined by setting

(ψ, [K]) ∘ (φ, [H]) = (ψ ∘ φ, [(K ∘ (Bφ × Id)) · H]),

where · denotes composition (juxtaposition) of homotopies. Let

π: L_{S,f}^{c}(X) ⟶ F_{S,f}(X)

be the forgetful functor: it is the inclusion on objects, and sends a morphism (φ, [H]) to φ. For each F_{S,f}(X)-centric subgroup P ≤ S, let

δ_{P} : P ⟶ Aut_{L_{S,f}^{c} (X)}(P)

be the “distinguished homomorphism” which sends g ∈ P to (c_{g}, [f|_{BP} ∘ H_{g}]).

Equivalently, via adjunction, a morphism from P to Q in L_{S,f}^{c}(X) can be thought of as a pair (φ, [H]), where φ ∈ Hom(P, Q), H is a path in the mapping space Map(BP, X) from f|_{BP} to f|_{BQ} ∘ Bφ, and [H] is the homotopy class of the path H rel endpoints.

The categories F′_{S,f}(X) ⊆ F_{S,f}(X) are always fusion systems over S, but are not in general saturated. However, in certain situations we consider, F′_{S,f}(X) will be a saturated fusion system, even though F_{S,f}(X) might not be (see Example 3.4).

Theorem A of [5a1] says that if all morphisms in a fusion system are obtained as composites of restrictions of morphisms between centric subgroups, then it is saturated if the saturation conditions (I) and (II) hold on centric subgroups. Thus it makes sense, for a general abstract fusion system F, to define the subsystem F′ ⊆ F over the same p-group S to be the subcategory with the same objects, but with only those morphisms which are obtained as composites of restrictions of morphisms in F between F-centric subgroups. One particularly well behaved situation is that in which F′ has no more centric subgroups than those already centric in F. In this case, it clearly follows that the full subcategories of centric objects in F and in F′ are equal, and hence that one can check conditions (I) and (II) in either subcategory.

These arguments are collected in the following proposition.

Proposition 1.7. Fix a space X, a finite p-group S, and a map f: BS ⟶ X. If all F_{S,f}(X)-centric subgroups P ≤ S satisfy conditions (I) and (II) in Definition 1.2, and if all F′_{S,f}(X)-centric subgroups of S are F_{S,f}(X)-centric, then F′_{S,f}(X) is a saturated fusion system.

In the situation of Proposition 1.7, there could possibly be a F′_{S,f}(X)-centric subgroup P ≤ S which is not F_{S,f}(X)-centric, because it is F_{S,f}(X)-conjugate to a subgroup which is not centric in S. When this is the case, [5a1, Theorem 2.2] cannot be applied to prove the above proposition, since we’ve changed the set of centric subgroups in question. This is why we need to assume that the two fusion systems have the same centric subgroups.

Our main theorem will give some conditions on a map BS ⟶^{f} X which ensure that a triple (S, F′_{S,f}(X), L_{S,f}^{c}(X)) is a p-local finite group. More generally, however, without any extra hypotheses, the category L_{S,f}^{c}(X) does satisfy most of the axioms for being a centric linking system associated to F_{S,f}(X).

In the following lemma, for any f: BS ⟶ X as above, and any P ≤ S, we let

ω_{P} : BZ(P) ⟶ Map(BP, X)_{f|_{BP}}

be the map which is adjoint to the composite

BZ(P) × BP ⟶^{Bμ} BS ⟶^{f} X,

where μ: Z(P) × P ⟶ S is multiplication.

Lemma 1.8. Fix a space X, a finite p-group S, and a map f: BS ⟶ X. Then the category L_{S,f}^{c}(X), together with the functor

π: L_{S,f}^{c}(X) ⟶ F_{S,f}(X)

and the distinguished homomorphisms δ_{P} : P ⟶ Aut_{L_{S,f}^{c} (X)}(P), satisfy axioms (B) and (C) in Definition 1.3. If in addition,

Z(P) ^{π_{1}(ω_{P})} → π_{1}(Map(BP, X), f|_{BP} ) is an isomorph. ∀ F_{S,f}(X)-centric P ≤ S, (∗)

then L_{S,f}^{c}(X) is a linking system associated to F_{S,f}(X).

Proof. Proving this means essentially repeating the proof of [BLO2, Theorem 7.5]. Set F = F_{S,f}(X) and L = L_{S,f}^{c}(X) for short. Condition (B) in Definition 1.3 clearly holds.

For g ∈ P ≤ S, set Ĥ_{g} = f|_{BP} ∘ H_{g}: a homotopy BP × I → X from f|_{BP} to f|_{BP} ∘ Bc_{g}. Thus the distinguished homomorphism P ⟶^{δ_{P}} Aut_{L}(P) is defined by sending g ∈ P to (c_{g}, [Ĥ_{g}]).

Condition (C) means showing, for each (φ, [H]) ∈ Mor_{L}(P, Q) and each g ∈ P, that the following square commutes:

\begin{matrix}P & \overline{^{(φ,[H])} →} & Q \\ P & ↓^{(c_{g},[Ĥ_{g}])} & \overline{^{(φ,[H])} →} & Q. & ↓^{(c_{φ(g)},[Ĥ_{φ(g)}])} \end{matrix}

Here, H: BP × I ⟶ X is a homotopy from f|_{BP} to f|_{BQ} ∘ Bφ. Clearly, φ ∘ c_{g} = c_{φ(g)} ∘ φ. It remains to check that the two juxtaposed homotopies described in the following diagram are homotopic among homotopies from f|_{BP} to f|_{BQ} ∘ B(φ ∘ c_{g}):

\begin{matrix}f|_{BP} & \overline{H} & f|_{BQ} ∘ Bφ \\ f|_{BP} ∘ Bc_{g} & Ĥ_{g} & \overline{H∘(Bc_{g}×Id)} & f|_{BQ} ∘ B(φ ∘ c_{g}). & Ĥ_{φ(g)}∘(Bφ×Id) \end{matrix}

The map

F: BP × I × I ⟶ X defined by F (x, s, t) = H(H_{g}(x, t), s)

defines a homotopy between them, since

F (x, s, 0) = H(x, s),
F (x, 1, t) = (f|_{BQ} ∘ Bφ ∘ H_{g})(x, t) = Ĥ_{φ(g)}(Bφ(x), t),
F (x, 0, t) = f|_{BP} ∘ H_{g}(x, t) = Ĥ_{g}(x, t),  and
F (x, s, 1) = H(Bc_{g}(x), s).

It remains to prove (A) while assuming that (∗) holds. For any F-centric subgroup P ≤ S, we identify π_{1}(Map(BP, X), f|_{BP}) as a subgroup of Aut_{L}(P): the subgroup of elements of the form (Id, [H]) when H is a homotopy from f|_{BP} to itself. Under this identification, δ_{P} restricts to the homomorphism from Z(P) to π_{1}(Map(BP, X), f|_{BP}) which sends g ∈ Z(P) to [Ĥ_{g}], where Ĥ_{g} is now regarded as a loop in Map(BP, X). By definition of F and L, for any other F-centric subgroup Q ≤ S, π_{1}(Map(BP, X), f|_{BP}) acts freely on Mor_{L}(P, Q) with orbit set Hom_{F}(P, Q). So to prove (A), we must show that the isomorphism π_{1}(ω_{P}) of (∗) sends g ∈ Z(P) to [Ĥ_{g}].

Let [1] be the category with two objects 0, 1, and one nonidentity morphism 0 → 1. Fix g ∈ Z(P), and consider the composite functor

Ψ: B(P) × [1] ⟶^{Id ×ψ} B(P) × B(Z(P)) ⟶^{B(μ)} B(P),

where ψ: [1] → B(Z(P)) sends 0 → 1 to the morphism g, and where B(μ) is induced by multiplication. Then

|Ψ|: BP × I ⟶ BP

is induced by the natural homomorphism of functors from Id_{B(P)} to itself defined by sending the object o_{P} to the morphism corresponding to g, and is thus the homotopy H_{g} of Definition 1.6. By definition, π_{1}(ω_{P})(g) is the homotopy class of f ∘ |Ψ|, when regarded as a loop in Map(BP, X), and is thus equal to [f ∘ H_{g}] = [Ĥ_{g}]. This finishes the proof of (A), and hence of the lemma. □

We will refer several times to the following classical result.

Proposition 1.9. For any pair of discrete groups H and G, the natural map

Rep(H, G) =^{def} Hom(H, G)/ Inn(G) ⟶ [BH, BG]

is a bijection. For each ρ ∈ Hom(H, G), the homomorphism C_{G}(ρ(H)) × H → G is adjoint to a homotopy equivalence

BC_{G}(ρ(H)) ⟶^{≃} Map(BH, BG)_{Bρ}.

Proof. See, for example, [BrK, Proposition 7.1]. □

2. A new topological characterization of fusion systems

In this section, we show, for a p-complete space X, a p-group S, and a map f: BS → X, that the triple (S, F′_{S,f}(X), L_{S,f}^{c}(X)) is a p-local finite group if X, S, and f satisfy certain conditions listed in Theorem 2.1 below.

When S is a p-group, a map f: BS ⟶ X will be called Sylow if every map BP ⟶ X, for a p-group P, factors through f up to homotopy. A map f: X ⟶ Y between arbitrary spaces is called centric if the induced map

Map(X, X)_{Id} ⟶^{f ∘−} Map(X, Y)_{f}

is a homotopy equivalence.

In [BLO2, Theorem 7.5], we showed that a p-complete space X is the classifying space of some p-local finite group if and only if there is a pair (S, f), where S is a p-group and f: BS → X is a map, such that (a) F_{S,f}(X) is saturated, (b) X ≃ |L_{S,f}^{c}(X)|_{p}^{∧}, and (c) f|_{BP} is a centric map for each F_{S,f}(X)-centric subgroup P ≤ S. The following theorem is similar in nature, although aimed at finding conditions for (S, F_{S,f}(X), L_{S,f}^{c}(X)) to be a p-local finite group rather than for X to be the classifying space of a p-local finite group. The main new result here is the geometric condition for the fusion system F_{S,f}(X) to be saturated.

Theorem 2.1. Fix a space X, a p-group S, and a map f: BS ⟶ X. Assume that

(a) f is Sylow;

(b) f|_{BP} is a centric map for each F_{S,f}(X)-centric subgroup P ≤ S; and

(c) every F′_{S,f}(X)-centric subgroup of S is also F_{S,f}(X)-centric.

Then the triple (S, F′_{S,f}(X), L_{S,f}^{c}(X)) is a p-local finite group.

Proof. For each F_{S,f}(X)-centric subgroup P ≤ S, (b) implies that composition with f|_{BP} induces a homotopy equivalence Map(BP, BP)_{Id} → Map(BP, X)_{f}. Also, by Proposition 1.8, Map(BP, BP)_{Id} ≃ BZ(P), and the resulting homotopy equivalence BZ(P) \begin{matrix}η \\ ≃ \end{matrix}→ Map(BP, X)_{f} is adjoint to the composite

BZ(P) × BP ⟶^{Bμ} BS ⟶^{f} X,

where μ: Z(P) × P ⟶ S is multiplication. Thus ω_{P} = η, where ω_{P} is the map of Lemma 1.8, and hence is a homotopy equivalence.

Condition (∗) of Lemma 1.8 thus holds, and so L_{S,f}^{c}(X) is a linking system associated to F_{S,f}(X) by that lemma. It remains only to prove that F′_{S,f}(X) is saturated. This proof is based on two lemmas which will be stated and proven later in this section.

Write L = L_{S,f}^{c}(X), F = F_{S,f}(X), and F′ = F′_{S,f}(X) for short. By Proposition 1.7 and (c), in order to prove that F′ is saturated, it suffices to show that conditions (I) and (II) in Definition 1.2 hold for all F′-centric subgroups P ≤ S. If P ≤ S is F′-centric, then it is also F-centric by (c), and hence f|_{BP} is a centric map by (b).

We first prove condition (I). Assume P ≤ S is F′-centric and fully normalized in F′. Since P is F′-centric, it is fully centralized, and it remains only to show that Aut_{S}(P) ∈ Syl_{p}(Aut_{F′}(P)). We identify P with δ_{P} (P) ≤ Aut_{L}(P). Since L is a centric linking system associated to F or F′, the homomorphism

π_{P,P} : Aut_{L}(P) ⟶ Aut_{F}(P) = Aut_{F′}(P)

induced by the functor π: L → F is surjective with kernel Z(P). Also, π_{P,P} (P) = Inn(P) is normal in Aut_{F}(P), and thus P ◁ Aut_{L}(P). By axiom (B) for a linking system, π_{P,P} sends g ∈ Aut_{L}(P) to c_{g} ∈ Aut(P), and thus

C_{Aut_{L}(P)}(P) = Ker(π_{P,P}) = Z(P).  (1)

By Lemma 2.2, f|_{BP} extends up to homotopy to a map f̂: B Aut_{L}(P) ⟶ X (by definition, Aut_{L}(P) = Aut_{L_{P,f}^{c} (X)}(P)). If T is any Sylow p-subgroup of Aut_{L}(P), then T ≥ P since P ◁ Aut_{L}(P), f̂|_{BT} factors through BS by condition (a), and thus there is a homomorphism φ: T ⟶ S such that φ|_{P} ∈ Hom_{F′}(P, S). In particular, φ|_{P} is a monomorphism. Hence since Z(T) ≤ C_{T} (P) ≤ P by (1), Ker(φ) ∩ Z(T) ≤ Ker(φ) ∩ P = 1; and (since a nontrivial normal subgroup intersects nontrivially with the center) this implies that φ is a monomorphism. Hence |N_{S}(φ(P))| ≥ |T | since N_{S}(φ(P)) ≥ φ(T); and also |N_{S}(P)| ≥ |N_{S}(φ(P))| since P is fully normalized. Since P is centric in S,

| Aut_{S}(P)| = |N_{S}(P)|/|Z(P)| ≥ |N_{S}(φ(P))|/|Z(P)| ≥ |T/Z(P)|;

and thus Aut_{S}(P) ∈ Syl_{p}(Aut_{F′}(P)) since Aut_{F′}(P) ≅ Aut_{L}(P)/Z(P) and T ∈ Syl_{p}(Aut_{L}(P)).

It remains to prove condition (II). Fix a morphism φ ∈ Hom_{F′}(P, S), and set

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

as usual. Consider the diagram

\xymatrix{BP \ar[r]^{Bφ} \ar[d]_{incl} & BS \ar[d]_{f} \\ BN \ar[r]^{f|_{BN}} \ar@{-->}[ur]^{Bφ′} & X.}

The square commutes up to homotopy since φ is a morphism in F′, and condition (1) in Lemma 2.3 holds by definition of N. Thus, by Lemma 2.3, there is a homomorphism φ′ ∈ Hom(N, S) such that Bφ′ makes both triangles in the above diagram commute up to homotopy. The commutativity of the lower triangle means that φ′ ∈ Hom_{F′}(N, S). The commutativity of the upper triangle implies that φ′|_{P} = φ ∘ c_{g} for some g ∈ P (Proposition 1.9), and thus φ =^{def} φ′ ∘c_{g}^{−1} is an extension of φ which lies in Hom_{F′}(N, S). This finishes the proof of (II). □

It remains to state and prove the technical lemmas used in the proof of Theorem 2.1.

Lemma 2.2. Fix a space X, a p-group P, and a centric map f: BP ⟶ X. Set L = L_{P,f}^{c}(X) for short. Then f extends (up to homotopy) to a map

f: B Aut_{L}(P) ⟶ X.

Proof. We first consider the following abstract situation. Fix a space Y, a basepoint y_{0} ∈ Y, and a finite group G with a right action on Y. Consider the following commutative diagram

\begin{aligned}Ω(Y ×_{G} EG) & \overline{→} & G & \overline{^{ι_{1}} →} & Y × EG & \overline{→} & Y ×_{G} EG \\ F & ↓^{pr_{1} ≃} & \overline{→} & G & ‖ & \overline{^{ι_{2}} →} & Y . & ↓^{≃} \end{aligned}

Here, ι_{1} and ι_{2} are defined by the action at the basepoints: ι_{1}(g) = (y_{0}g, g^{−1}) and ι_{2}(g) = y_{0}g. Also, F is the “standard” homotopy fiber of ι_{2}:

F = {(g, H) | g ∈ G, H: I → Y, H(0) = y_{0}, H(1) = y_{0}g}.  (I = [0, 1])

This is an H-space, via the product (g′, H′)(g, H) = (g′g, (R_{g} ∘ H′) · H), where R_{g} denotes the right action of g on Y and “·” denotes composition of paths in Y. We can also regard Ω(Y ×_{G} EG) as the standard homotopy fiber of ι_{1}. Then pr_{1} is defined by projecting a path in Y × EG to the first factor, and is a map of H-spaces and a homotopy equivalence. In particular, it induces an isomorphism of groups

π_{1}(Y ×_{G} EG) ≅ π_{0}(Ω(Y ×_{G} EG)) ⟶^{π_{0}(pr_{1})} π_{0}(F) .  (1)

Set F = F_{P,f}(X) for short. We apply the above remarks to the space Y = Map(BP, X)_{f}, the point y_{0} = f, and the group G = Aut_{F}(P), where α ∈ Aut_{F}(P) acts on Y via right composition by Bα. Thus after replacing paths in Y by homotopies,

F = {(φ, H) | φ ∈ Aut_{F}(P), H: BP × I ⟶ X, H|_{BP ×0} = f, H|_{BP ×1} = f ∘ Bφ},

and Aut_{L}(P) = π_{0}(F) by definition. Also, since f is centric,

Y = Map(BP, X)_{f} ≃ Map(BP, BP)_{Id} ≃ BZ(P),

where the last equivalence follows from Proposition 1.9. Then Y ×_{G} EG is also aspherical, and so Y ×_{G} EG ≃ B Aut_{L}(P) by (1).

Since Bφ fixes the base point of BP for all φ ∈ Aut(P), the evaluation map

Y = Map(BP, X)_{f} ⟶^{eval} X

is Aut_{F}(P)-equivariant (with respect to the trivial action on X). It thus factors through the orbit space, or alternatively through the Borel construction:

f: B Aut_{L}(P) ≃ Map(BP, X)_{f} ×_{Aut_{F} (P)} E Aut_{F}(P) ⟶^{eval} X .

It remains to show that f|_{BP} ≃ f, where BP is included into B Aut_{L}(P) via the distinguished monomorphism δ_{P}. By the naturality of these maps, it suffices to do this when X = BP and f = Id. In this case, that means showing that π_{1}(f|_{BP}) = Id_{P}. Fix g ∈ P, and let H_{g}: BP × I ⟶ BP be as in Definition 1.6. We also regard H_{g} as a path in Map(BP, BP)_{Id} from Id_{BP} to Bc_{g}, whose restriction to the basepoint of BP is by definition the loop in BP representing g. By the above construction, g ∈ π_{1}(BP) corresponds to the class

[H_{g}, ϕ] ∈ π_{1}(Map(BP, X)_{f} ×_{Aut_{F} (P)} E Aut_{F}(P)),

where ϕ is any path in EP ⊆ E Aut_{F}(P) from the vertex Id to the vertex c_{g}^{−1}. Hence upon evaluating this at the basepoint of BP, we see that eval([H_{g}, ϕ]) is the loop in BP representing g, and thus that π_{1}(f)(g) = g. □

It remains to prove the existence of certain homotopy liftings.

Lemma 2.3. Fix a finite group H, a normal p-subgroup P ◁ H, a p-group S, and a monomorphism φ: P ⟶ S such that C_{S}(φ(P)) = Z(φ(P)). Let X be a space, and let f: BS ⟶ X be such that f ∘ Bφ is centric. Assume that

for each x ∈ H, φc_{x}φ^{−1} ∈ Aut_{S}(φ(P)).  (1)

Let s: BH ⟶ X be such that the square in the following diagram commutes up to homotopy:

\xymatrix{BP \ar[r]^{Bφ} \ar[d]_{incl} & BS \ar[d]_{f} \\ BH \ar[r]^{s} \ar@{-->}[ur]^{Bφ′} & X.}  (2)

Then there is a homomorphism φ′ ∈ Hom(H, S) such that the two triangles in diagram (2) commute up to homotopy.

Proof. We identify BP with EH/P and BH with

(E(H/P) × EH)/H = E(H/P) ×_{H/P} EH/P.

Thus the inclusion BP ⊆ BH is induced by the inclusion of an orbit H/P ⊆ E(H/P). Let

Φ̂: EH/P ⟶ BS and ŝ: E(H/P) ×_{H/P} EH/P ⟶ X

be maps homotopic to Bφ and s under these identifications.

By (1), the connected component Map(EH/P, BS)_{Bφ} is invariant under the action of H/P induced by the action of the group on EH/P. We thus get the following square of equivariant maps between spaces with (H/P)-action

\xymatrix{H/P \ar[r]^{v} \ar[d]_{incl} & Map(EH/P, BS)Φ̂ \ar[d]^{f ∘−}_{≃} \\ E(H/P) \ar[r]^{u} \ar@{-->}[ur]^{ũ} & Map(EH/P, X)_{f ∘Φ̂} .}  (3)

Here, u is adjoint to ŝ (when regarded as a map defined on E(H/P) × EH/P); and v is defined by setting v(gP)(xP) = Φ̂(xgP) for x ∈ EH. The square in (3) commutes up to equivariant homotopy by the commutativity of the square in (2).

Now, Map(EH/P, BS)_{Bφ} ≃ BC_{S}(φ(P)) ≃ BZ(P) by Proposition 1.9, and since φ(P) is centric in S by assumption. Also, Map(EH/P, X)_{f ∘Φ̂} ≃ BZ(P) since f ∘ Bφ is a centric map by assumption. Thus the map (f ∘−) is H/P -equivariant and a homotopy equivalence. Since the H/P -action on E(H/P) is free, there is an equivariant lifting ũ of u as in the above diagram which makes both triangles in (3) commute up to equivariant homotopy. This is adjoint to an H/P -equivariant map from E(H/P) × EH/P to BS, which (since H/P acts trivially on BS) factors through

s̃: BH = E(H/P) ×_{H/P} EH/P ⟶ BS

which makes the two triangles in (2) commute up to homotopy. Finally, s̃ ≃ Bφ′ for some φ′ ∈ Hom(H, S) by Proposition 1.9 again, and this finishes the proof. □

3. Fusion systems of completed classifying spaces of groups

In order to apply Theorem 2.1 to a space X, we must have good control over the mapping spaces Map(BP, X) for finite p-groups P. One interesting case where we can do this is when X = BG_{p}^{∧} for certain infinite groups G. This is based on a theorem of Broto and Kitchloo [BrK].

When G is an infinite group, we say that a subgroup S ≤ G is a Sylow p-subgroup if S is a finite p-subgroup, and if all other finite p-subgroups of G are conjugate to subgroups of S.

Proposition 3.1. Fix a prime p and a discrete group G. Assume there is an 𝔽_{p}-acyclic G-complex X with finitely many orbits of cells and with finite isotropy subgroups. Let S ≤ G be any finite p-subgroup, and let f: BS ⟶ BG_{p}^{∧} be the inclusion. Then the following hold.

(a) F_{S,f}(BG_{p}^{∧}) = F_{S}(G).

(b) If S is a Sylow p-subgroup of G, then the map f is Sylow.

(c) For any P ≤ S, f|_{BP} is a centric map if and only if H^{i}(C_{G}(P)/Z(P); 𝔽_{p}) = 0 for all i > 0.

Proof. In the notation of [BrK], K_{1}X is a class of topological groups which includes all discrete groups which act on 𝔽_{p}-acyclic complexes with finitely many orbits of cells and with finite isotropy subgroups. (The definition in [BrK] also requires that the fixed point set of any finite p-group be 𝔽_{p}-acyclic, but this follows from Smith theory, since the complex is finite dimensional.) In particular, this class includes G. Hence by [BrK, Corollary 3.3], for any finite p-group P, the natural map

Rep(P, G) =^{def} Hom(P, G)/ Inn(G) ⟶^{≅} [BP, BG_{p}^{∧}]  (1)

is a bijection. Also, for each ρ ∈ Hom(P, G), the homomorphism P ×C_{G}(ρ(P)) ⟶^{(ρ,incl)} G induces a homotopy equivalence

BC_{G}(P)_{p}^{∧} ⟶^{≃} Map(BP, BG_{p}^{∧})_{Bρ}.  (2)

Point (a) follows immediately from (1).

Assume S is a Sylow p-subgroup of G. If P is any finite p-group and s: BP → BG_{p}^{∧} is a map, then s ≃ Bφ for some φ ∈ Hom(P, G) by (1), φ(P) is G-conjugate to some Q ≤ S since S is Sylow, and thus Bφ ≃ f ∘ Bφ′ for some φ′ ∈ Hom(P, S). Thus the map f is Sylow, and this proves (b).

By (2), for any P ≤ S, f|_{BP} is a centric map if and only if the inclusion of BZ(P) into BC_{G}(P)_{p}^{∧} is a homotopy equivalence, or equivalently, if the inclusion of BZ(P) into BC_{G}(P) is an 𝔽_{p}-homology isomorphism. Since Z(P) is central in C_{G}(P), this last condition is equivalent to requiring that H^{i}(C_{G}(P)/Z(P); 𝔽_{p}) = 0 for all i > 0, and this proves (c). □

Before stating our theorem, we need one more definition.

Definition 3.2. Fix a prime p.

(a) If H ≤ G are finite groups, then H is strongly embedded in G at p if p||H|, but H ∩ gHg^{−1} has order prime to p for all g ∈ G∖N_{G}(H).

(b) If G is a finite group and S ∈ Syl_{p}(G), a subgroup P ≤ S is essential if either P = S, or P is p-centric in G and Out_{G}(P) has a strongly embedded subgroup at p.

By Goldschmidt’s version of Alperin’s fusion theorem [Gd, Theorem 3.3], for any finite group G and any S ∈ Syl_{p}(G), each morphism in F_{S}(G) is a composite of restrictions of morphisms between subgroups of S which are essential in G. Note that each essential subgroup is also radical — Out_{G}(P) has no strongly embedded subgroup if O_{p}(Out_{G}(P)) ≠ 1.

A finite group G has a strongly embedded subgroup H if and only if the poset of nontrivial p-subgroups of G is disconnected (cf. [As, 46.6]), in which case the stabilizer of a connected component is strongly embedded.

The following theorem is a first application of Theorem 2.1.

Theorem 3.3. Fix a prime p and a discrete group G. Let X be an 𝔽_{p}-acyclic Gcomplex with finitely many orbits of cells and with finite isotropy subgroups. Fix a vertex x_{∗} ∈ X, let G_{∗} be the isotropy subgroup of x_{∗}, choose S ∈ Syl_{p}(G_{∗}), and set F = F_{S}(G) and L = L_{S}^{c}(G). Assume the following hold:

(a) For each finite p-subgroup P ≤ G, X^{P} contains at least one point in the orbit Gx_{∗}.

(b) If P ≤ S is F-centric, then X^{P} /C_{G}(P) is 𝔽_{p}-acyclic.

(c) If P ≤ S is a Sylow p-subgroup of the isotropy subgroup of an edge of X, or an essential p-subgroup of the isotropy subgroup of a vertex, then P is F-centric.

Then F is a saturated fusion system over S, and L is a centric linking system associated to F.

Proof. If f: BS ⟶ BG_{p}^{∧} is the map induced by the inclusion, then F = F_{S,f}(BG_{p}^{∧}) by Proposition 3.1(a). For any finite p-subgroup P ≤ G, there is g ∈ G such that gx_{∗} ∈ X^{P} by point (a) above, and hence P is contained in the isotropy subgroup gG_{∗}g^{−1} of gx_{∗}. Since gSg^{−1} ∈ Syl_{p}(gG_{∗}g^{−1}), this shows that P is G-conjugate to a subgroup of S. Thus S is a Sylow p-subgroup of G; and hence by Proposition 3.1(b), the map f is Sylow.

By Theorem 2.1, to prove that F = F_{S}(G) is saturated, it remains only to check condition 2.1(b), and to show that F = F′_{S,f}(BG_{p}^{∧}). (This last claim also implies condition 2.1(c).) In Step 1, we prove condition 2.1(b), and also prove that L = L_{S}^{c}(G) is a centric linking system associated to F. In Step 2, we prove that F = F′_{S,f}(BG_{p}^{∧}); i.e., that F is generated by morphisms between F-centric subgroups.

Step 1: Fix an F-centric subgroup P ≤ S. Thus C_{S}(P′) = Z(P′) for each P′ ≤ S which is G-conjugate to P. If Q ≤ C_{G}(P) is a finite p-subgroup, then P Q is a finite p-group, so gP Qg^{−1} ≤ S for some g ∈ G, and Q ≤ Z(P) by the above remark applied to P′ = gP g^{−1}. Thus Z(P) is maximal among finite p-subgroups of C_{G}(P).

Set C_{G}(P) = C_{G}(P)/Z(P) for short. For each x ∈ X^{P}, G_{x} is a finite group which contains P, so C_{G_{x}}(P)/Z(P) is finite of order prime to p, and hence its classifying space is 𝔽_{p}-acyclic. Consider the projection maps

BC_{G}(P) ⟵^{pr_{1}} EC_{G}(P) ×_{C_{G}(P)} X^{P} ⟶^{pr_{2}} X^{P} /C_{G}(P) = X^{P} /C_{G}(P)

associated to the Borel construction on X^{P}. All fibers (point inverses) of pr_{1} are homeomorphic to X^{P}, and thus 𝔽_{p}-acyclic by Smith theory (X is 𝔽_{p}-acyclic and finite dimensional). For each x ∈ X^{P} with orbit x̄ ∈ X^{P} /C_{G}(P) and with stabilizer subgroup G_{x},

pr_{2}^{−1}(x̄) ≅ EC_{G}(P)/C_{G_{x}}(P) ≃ B(C_{G_{x}}(P)/Z(P))

is 𝔽_{p}-acyclic. Hence by a spectral sequence argument (or by an appropriate version of the Vietoris mapping theorem), pr_{1} and pr_{2} are both 𝔽_{p}-homology equivalences. Since X^{P} /C_{G}(P) is 𝔽_{p}-acyclic by assumption, this implies that BC_{G}(P) is 𝔽_{p}-acyclic.

Thus H^{i}(C_{G}(P); 𝔽_{p}) = 0 for all i > 0. Hence by Proposition 3.1(c), the map f|_{BP} is centric, and this finishes the proof of condition 2.1(b). By Lemma 1.5, this also shows that L = L_{S}^{c}(G) is a centric linking system associated to F.

Step 2: Fix any φ = c_{g} ∈ Hom_{F}(P, Q) in F. Then P ≤ S and gP g^{−1} ≤ S, so P is contained in the isotropy subgroups of both x_{∗} and g^{−1}(x_{∗}). Choose a path ϕ in the 1skeleton of X^{P} from x_{∗} to g^{−1}(x_{∗}). Let v_{0} = x_{∗}, v_{1}, …, v_{m} = g^{−1}(x_{∗}) be the successive vertices in the path ϕ, let e_{i} be the edge connecting v_{i−1} to v_{i}, and set H_{i} = G_{e_{i}} and K_{i} = G_{v_{i}}. Thus by construction, P ≤ H_{i}, and K_{i−1} ≥ H_{i} ≤ K_{i} for all 1 ≤ i ≤ m. Also, S ∈ Syl_{p}(K_{0}) and K_{m} = g^{−1}K_{0}g.

Fix Sylow subgroups P_{i} ∈ Syl_{p}(H_{i}) such that P ≤ P_{i}. Choose Q_{i}, Q′_{i} ∈ Syl_{p}(K_{i}) such that Q′_{i−1} ≥ P_{i} ≤ Q_{i}, and let k_{i} ∈ K_{i} be such that Q′_{i} = k_{i}Q_{i}k_{i}^{−1}. We also assume that Q_{0} = S and Q′_{m} = g^{−1}Sg. Finally, since S is Sylow in G, there are elements g_{i} ∈ G such that Q_{i} ≤ g_{i}Sg_{i}^{−1}. In particular, when i = m, since

g^{−1}Sg = Q′_{m} = k_{m}Q_{m}k_{m}^{−1},

we can choose g_{m} = k_{m}^{−1}g^{−1}.

Consider the following diagram, where all subgroups are contained in S:

P \overline{→^{ψ_{0}}} P^{k_{0}} \overline{→^{φ_{1}}} P^{g_{1}} \overline{→^{ψ_{1}}} P^{k_{1}g_{1}} \overline{→^{φ_{2}}} P^{g_{2}}  P^{k_{m−1}g_{m−1}} →^{φ_{m}} P^{g_{m}} →^{ψ_{m}} P^{k_{m}g_{m}}
⋯
P_{1}^{k_{0}} ↓ \overline{→^{φ_{1}}} P_{1}^{g_{1}} ↓  P_{2}^{k_{1}g_{1}} ↓ \overline{→^{φ_{2}}} P_{2}^{g_{2}} ↓  P_{m}^{k_{m−1}g_{m−1}} ↓  →^{φ_{m}} P_{m}^{g_{m}}. ↓

Here, we use the standard notation H^{g} = g^{−1}Hg. Also, ψ_{i} is conjugation by g_{i}^{−1}k_{i}^{−1}g_{i} ∈ K_{i}^{g_{i}} (where g_{0} = 1), and φ_{i} and φ_{i} are conjugation by g_{i}^{−1}k_{i−1}g_{i−1}. All of these subgroups are contained in S by construction. Also, by the above choice of g_{m}, ψ_{m} is conjugation by gg_{m}. Thus the composite of these morphisms ψ_{i} and φ_{i} is conjugation by

(gg_{m})(g_{m}^{−1}k_{m−1}g_{m−1})(g_{m−1}^{−1}k_{m−1}^{−1}g_{m−1}) ⋯ (g_{2}^{−1}k_{1}g_{1})(g_{1}^{−1}k_{1}^{−1}g_{1})(g_{1}^{−1}k_{0}g_{0})(g_{0}^{−1}k_{0}^{−1}g_{0}) = g.

(Recall that g_{0} = 1.)

By (c), each subgroup of S which is conjugate to any P_{i} is F-centric. Each ψ_{i} is a morphism in the fusion system of K_{i}^{g_{i}}, and hence a composite of restrictions of morphisms between essential p-subgroups of this group [Gd, Theorem 3.3]. Since all such subgroups are F-centric by (c), this finishes the proof. □

We finish the section with two very simple examples which illustrate why some of these assumptions are needed. The first example shows why condition (c) is needed in Theorem 3.3. It also shows why we cannot take F′ = F in Theorem 2.1.

Example 3.4. Let S be an abelian p-group, T ≨ S a proper subgroup of order > 2, and H ≤ Aut(T) a nontrivial subgroup of order prime to p. Set G = S ∗_{T}(T ⋊H). Then G acts on a tree with isotropy subgroups all G-conjugate to S, T, or T ⋊H. This action satisfies conditions (a) and (b) in Theorem 3.3, but not condition (c); and the inclusion map f: BS ⟶ BG_{p}^{∧} satisfies all of the hypotheses (a)–(c) in Theorem 2.1. The fusion system F_{S}(G) = F_{S,f}(BG_{p}^{∧}) is not saturated. The fusion system F′_{S,f}(BG_{p}^{∧}) is equal to F_{S}(S), and is thus a proper subsystem of F_{S}(G) and is saturated.

Proof. By [Se, Theorem I.9], G acts freely on a tree X with isotropy subgroups conjugate to S and T ⋊H on vertices, and to T on edges, and with fundamental domain an interval. Since S does not fix any edges (and X^{S} must be a tree), X^{S} is a point, and hence X^{S}/C_{G}(S) is also a point. Since T has index prime to p in T ⋊H, X^{P} contains elements in the orbit G/S for each finite p-subgroup P ≤ G. Thus the action of G on X satisfies conditions 3.3(a) and 3.3(b). Conditions (a)–(c) in Theorem 2.1 then follow using Proposition 3.1.

The fusion system F = F_{S}(G) is not saturated, since the automorphisms in the group Aut_{F}(T) ≅ H do not extend to automorphisms in Aut_{F}(S). □

The next example shows that condition (b) in Theorem 2.1 and condition (b) in Theorem 3.3 must be assumed (in each theorem) for all F-centric subgroups: it does not suffice to assume them when P = S.

Example 3.5. Set p = 2, S = D_{8} × C_{2}^{2}, T = C_{4} × C_{2}^{2}, and H = C_{4} × A_{4}, with the obvious inclusions of T in S and H. Set G = S ∗_{T} H. Then G acts on a tree with all isotropy subgroups G-conjugate to S, T, or H. This action satisfies conditions (a) and (c) in Theorem 3.3 as well as conditions (a) and (c) in Theorem 2.1, but does not satisfy condition (b) in either theorem. The inclusion map f: BS ⟶ BG_{p}^{∧} is Sylow and centric, but the inclusion map f|_{BT} : BT ⟶ BG_{p}^{∧} is not centric. Neither fusion system F_{S}(G) = F_{S,f}(BG_{p}^{∧}) nor F′_{S,f}(BG_{p}^{∧}) is saturated.

Proof. Set F = F_{S,f}(BG_{p}^{∧}) and F′ = F′_{S,f}(BG_{p}^{∧}) for short. As in the last example, G acts on a tree X with isotropy subgroups as described, and with fundamental domain an interval, by [Se, Theorem I.9]. Since any action of a finite group on a tree has a fixed point, every finite subgroup of G is contained in an isotropy subgroup, and thus in a subgroup G-conjugate to S or H. This shows that S is a Sylow 2-subgroup of G, and hence (by Proposition 3.1(b)) that f is Sylow. Thus both conditions 2.1(a) and 3.3(a) hold. Also, f is centric by Proposition 3.1(c), since C_{G}(S) = Z(S).

Since S is a 2-group and H has a normal Sylow 2-subgroup, their only radical 2subgroups (hence their only essential 2-subgroups) are S and T, respectively. Since both are centric in S, condition 3.3(c) holds. As seen in Step 2 of the proof of Theorem 3.3, this implies that F′ = F, and thus that condition 2.1(c) also holds.

Now, T is normal in G, since it is normal in S and H, and G/T ≅ (S/T)∗(H/T) ≅ C_{2} ∗ C_{3}. Hence

C_{G}(T)/T ≅ Ker[C_{2} ∗ C_{3} ⟶ C_{2} × C_{3}],

and this is a free group (since it acts freely on a tree). In particular, H^{1}(C_{G}(T)/T; 𝔽_{2}) ≠ 0; and (since C_{G}(T)/T acts freely on the tree X^{T}) X^{T} /C_{G}(T) ≃ B(C_{G}(T)/T) is not 𝔽_{2}-acyclic. So by Proposition 3.1(c), f|_{BT} is not a centric map; and this shows that conditions 2.1(b) and 3.3(b) both fail.

Now, Aut_{F}(C_{4} × C_{2}^{2}) ≅ C_{2} × C_{3}, but Out_{F}(D_{8} × C_{2}^{2}) ≅ 1 (and the same for F′). The automorphism of C_{4} × C_{2}^{2} of order 3 thus fails to extend to S, so axiom (II) fails, and F is not saturated. □

4. Fusion systems of trees of groups

Let (G, T ) be a tree of groups in the sense of [Se, §I.4.4]. Thus T is a tree; and G assigns groups G(v) and G(e) to each vertex v ∈ T^{0} and each edge e ∈ T^{1}, and a monomorphism G(e) → G(v) for each pair (e, v) where v is an endpoint of e. For any such tree of groups (G, T ), we let G_{T} denote the amalgamated free product of the groups G(v) over the G(e), as described in [Se, §I.4.4]. Thus G_{T} is the free product of the groups G(v) for all vertices v ∈ T^{0}, modulo the relations given by the inclusions of groups G(e) for e ∈ T^{1} into the groups of the endpoints of e.

Alternatively, one can regard T as a category whose set of objects is the disjoint union of T^{0} and T^{1}, and with a pair of morphisms w ← e → v for each edge e ∈ T^{1} with endpoints v, w. Then G is a functor from T to the category Gr^{+} of groups and monomorphisms, and G_{T} = \underrightarrow{colim}_{T} (G).

In this paper, we will be considering only finite trees of finite groups; i.e., pairs (G, T ) where T is a finite tree, and G(v) is a finite group for each v ∈ T^{0}. Our goal in this section is to find some conditions on G and T which ensure that the group G_{T} gives rise to a saturated fusion system and associated centric linking system.

If (G, T ) is a tree of groups, and G = G_{T}, then we let T̃ denote the graph with vertex and edge sets

T̃^{0} = {(gG(v), v) | v ∈ T^{0}, g ∈ G} = ∐_{v∈T^{0}}(G/G(v) × {v})
T̃^{1} = {(gG(e), e) | e ∈ T^{1}, g ∈ G} = ∐_{e∈T^{1}}(G/G(e) × {e})

(with the obvious choices of endpoints). Equivalently, T̃ = \underrightarrow{hocolim}_{T} (G/−). By [Se, Theorem I.9, p. 38], T̃ is a tree upon which G acts with orbit graph T , with fundamental domain which can be identified with T (the subtree spanned by vertices (1G(v), v)), and with isotropy subgroups on the fundamental domain given by G.

For any pair of groups H, G, let Rep(H, G) = Hom(H, G)/ Inn(G), and let [α] ∈ Rep(H, G) be the class of α ∈ Hom(H, G). If (G, T ) is a tree of groups, and H is any finite group, we let Rep(H, G) be the graph with vertex and edge sets

Rep(H, G)^{0} = {(v, [α]) | v ∈ T^{0}, [α] ∈ Rep(H, G(v))}
Rep(H, G)^{1} = {(e, [α]) | e ∈ T^{1}, [α] ∈ Rep(H, G(e))}.

When α ∈ Hom(H, G(x)), where x is a vertex or edge in T , we write (x, [α]) for the pair (x, [α]), where [α] is the class of α in Rep(H, G(x)). Alternatively, if we regard T as a category, then

Rep(H, G) = \underrightarrow{hocolim}_{T} Rep(H, G(−)).

Lemma 4.1. Fix a finite tree of finite groups (G, T ). Set G = G_{T} = \underrightarrow{colim}(G), and let T̃ be as above. Then the following hold for any vertex v_{∗} of T and any subgroup H ≤ G(v_{∗}):

(a) The connected component of Rep(H, G) which contains (v_{∗}, [incl_{H}^{G(v_{∗})}]) is isomorphic (as a graph) to T̃^{H}/C_{G}(H).

(b) The natural map

Φ_{H}: π_{0}(Rep(H, G)) ⟶^{≅} Rep(H, G)

is a bijection. In particular, for x a vertex or edge of T and α ∈ Hom(H, G(x)), (x, [α]) lies in the connected component of the vertex (v_{∗}, [incl_{H}^{G(v_{∗})}]) if and only if α ∈ Hom_{G}(H, G(x)).

Proof. When x is a vertex or edge of T , we write G_{x} = G(x) for short: the isotropy subgroup at x of the G-action, when we regard T as a subtree (a fundamental domain) of T̃.

If K ≤ G and gK ∈ (G/K)^{H}, then H ≤ gKg^{−1}, and so we can regard c_{g}^{−1} : x ↦ g^{−1}xg as a homomorphism from H to K whose class in Rep(H, K) depends only on the coset gK. Hence it makes sense to define

f_{H}: T̃^{H} ⟶ Rep(H, G)

by sending each vertex (gG_{v}, v) to the pair (v, [c_{g}^{−1}]) and each edge (gG_{e}, e) to the pair (e, [c_{g}^{−1}]). We claim the following hold:

(i) Im(f_{H}) is the connected component of (v_{∗}, [incl_{H}^{G_{v∗}}]) in Rep(H, G).

(ii) A vertex (v, [α]), for α ∈ Hom(H, G_{v}), lies in Im(f_{H}) if and only if the composite H ^{α}→ G_{v} ≤ G is G-conjugate to the inclusion.

(iii) f_{H} induces an isomorphism of graphs (T̃^{H})/C_{G}(H) ≅ Im(f_{H}).

Point (ii) is immediate.

If (e, [α]) is an edge of Rep(H, G) with endpoint f_{H}(gG_{v}, v) = (v, [c_{g}^{−1}]), then v is an endpoint of e, and [α] = [c_{g}^{−1}] ∈ Rep(H, G_{v}). Thus α = c_{h}^{−1}c_{g}^{−1} = c_{gh}^{−1} for some h ∈ G_{v}, and (e, [α]) = f_{H}(ghG_{e}, e). So an edge of Rep(H, G) lies in Im(f_{H}) if one of its endpoints lies in Im(f_{H}), and thus Im(f_{H}) is a union of connected components of Rep(H, G). Since T̃^{H} is a tree by [Se, §I.6.1], Im(f_{H}) is nonempty and connected, hence is a connected component of Rep(H, G), and this finishes the proof of (i).

Two vertices (gG_{v}, v) and (hG_{w}, w) are sent to the same vertex of Rep(H, G) if and only if v = w and [c_{g}^{−1}] = [c_{h}^{−1}] in Rep(H, G_{v}). This last condition is equivalent to saying that h ∈ C_{G}(H)gG_{v}; i.e., that (gG_{v}, v) and (hG_{w}, w) are in the same C_{G}(H)-orbit; and thus (iii) holds. Points (i) and (iii) together imply (a).

By (ii), for any α ∈ Hom(H, G), Φ_{H} sends the connected component of a vertex (v, [β]) in Rep(H, G) to [α] if and only if (v, [βα^{−1}]) ∈ Im(f_{α(H)}). Hence by (i), Φ_{H}^{−1}([α]) contains exactly one connected component. This shows that Φ_{H} is a bijection, and proves (b). □

We originally discovered the following theorem as a special case of Theorem 2.1 (and of Theorem 3.3), and it was certainly motivated by those results. However, since it also has a more elementary proof which does not use certain deep theorems in homotopy theory, we give both proofs here.

Theorem 4.2. Fix a prime p and a finite tree of finite groups (G, T ). Fix a vertex v_{∗} of T , set G_{∗} = G(v_{∗}) for short, and choose S ∈ Syl_{p}(G_{∗}). Set G = G_{T} = \underrightarrow{colim}(G), F = F_{S}(G), and L = L_{S}^{c}(G). Assume the following hold:

(a) For each vertex v ≠ v_{∗} of T , if e is the edge adjacent to v in the (unique) minimal path from v to v_{∗}, then [G(v) : G(e)] is prime to p.

(b) If P ≤ S is F-centric (equivalently, if Z(P) is a maximal p-subgroup of C_{G}(P)), then the component of (v_{∗}, [incl_{P}^{G_{∗}}]) in the graph Rep(P, G) is a tree.

(c) If some P ≤ S is G-conjugate to an essential p-subgroup of G(v) for any vertex v, then P is F-centric.

Then F is a saturated fusion system over S, and L is a centric linking system associated to F.

Proof. Let T̃ be as defined above: the tree upon which G acts with orbit space and fundamental domain T . When x is a vertex or edge of T , we write G_{x} = G(x) for short.

We first show how the theorem follows as a special case of Theorem 3.3, applied to the action of G on T̃. Condition 3.3(b) follows from condition (b) here, together with Lemma 4.1(a); while condition 3.3(c) follows from condition (c) here.

It remains to describe how condition 3.3(a) follows from condition (a) here. Let P ≤ G be any finite p-subgroup; we must show that T̃^{P} contains some vertex in the orbit of (1G_{∗}, v_{∗}) in T̃. Since T̃ is a tree, the fixed point set of the P -action is also a tree, and hence its image in the orbit tree T is nonempty and connected. Let v be the vertex in that image which is closest to v_{∗}. If v ≠ v_{∗}, then there is some g ∈ G such that gG_{v} ∈ (G/G_{v})^{P}, and hence g^{−1}P g ≤ G_{v}. Let e be the edge adjacent to v on the minimal path from v to v_{∗}; then [G_{v} : G_{e}] is prime to p by (a), and hence there is g′ ∈ G such that g′^{−1}P g′ ≤ G_{e}. Then the edge (g′G_{e}, e) is in T̃^{P}, which contradicts the original assumption about v. This shows that v = v_{∗}, and thus that some vertex of the form (gG_{∗}, v_{∗}) is in T̃^{P}.

Since this theorem also has a more elementary algebraic proof, we give that here. We first note that the argument just given also shows:

(a′) For each vertex v in T and each p-subgroup P ≤ G_{v}, P is G-conjugate to a subgroup of S.

By a proof identical to Step 2 in the proof of Theorem 3.3, we show (using (c)) that every morphism in F is a composite of restrictions of morphisms between F-centric subgroups. Hence by [5a1, Theorem 2.3], F is saturated if it satisfies axioms (I) and (II) in Definition 1.2 for all F-centric subgroups P ≤ S. So it remains to prove (I) and (II) for F-centric subgroups, and to prove that L is a centric linking system associated to F.

For any H ≤ G_{∗}, let Rep(H, G)_{∗} be the connected component of (v_{∗}, [incl_{H}^{G_{∗}}]) in Rep(H, G). By Lemma 4.1(b), if x is any vertex or edge in T , and φ ∈ Hom(H, G_{x}), then (x, [φ]) lies in Rep(H, G)_{∗} if and only if φ ∈ Hom_{G}(H, G_{x}).

Proof of (I) for F-centric subgroups. Let P ≤ S be any subgroup which is Fcentric and fully normalized in F. By Lemma 4.1, there is a bijection π_{0}(Rep(P, G)) ≅ Rep(P, G) which sends Rep(P, G)_{∗} to [incl_{P}^{G}], and which is equivariant with respect to the Aut(P)-action on both sets. Thus Aut_{F}(P) = Aut_{G}(P) is the isotropy subgroup of Rep(P, G)_{∗} ∈ π_{0}(Rep(P, G)) under the Aut(P)-action. In particular, Aut_{F}(P) leaves Rep(P, G)_{∗} invariant. Since Rep(P, G)_{∗} is a tree by (b), and since every action of a finite group on a tree has a fixed point, there is a vertex (v, [α]) in Rep(P, G)_{∗} which is fixed by Aut_{F}(P). Thus α ∈ Hom_{G}(P, G_{v}). Set P′ = α(P) ≤ G_{v}, so that Aut_{G_{v}}(P′) = α Aut_{F}(P)α^{−1}. Fix Q′ ∈ Syl_{p}(N_{G_{v}}(P′)). By (a′), there is Q ≤ S which is G-conjugate to Q′. Fix β ∈ Iso_{G}(Q′, Q), and set P′′ = β(P′). Then

| Aut_{S}(P)| = \frac{|N_{S}(P)|}{|Z(P)|} ≥ \frac{|N_{S}(P′′)|}{|Z(P′′)|} ≥ \frac{|Q′|}{|Z(P′)|} = | Aut_{Q′}(P′)|;

where the first inequality holds since P′′ is G-conjugate (hence F-conjugate) to P and P is fully centralized in F. Since Aut_{Q′}(P′) ∈ Syl_{p}(Aut_{G}(P′)) and Aut_{G}(P′) ≅ Aut_{G}(P), this proves that Aut_{S}(P) ∈ Syl_{p}(Aut_{G}(P)), and finishes the proof of (I).

Proof of (II) for F-centric subgroups. Fix φ ∈ Hom_{F}(P, S) where P is F-centric, and set

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

We claim that

Im[Rep(N_{φ}, G)_{∗} ⟶^{Res} Rep(P, G)_{∗}] = Rep(P, G)_{∗}^{K}.  (1)

Clearly, if x is a vertex or edge in T , then the restriction of any β ∈ Rep(N_{φ}, G_{x}) lies in Rep(P, G_{x})^{K}, so the problem is to prove that Rep(P, G)_{∗}^{K} lies in the image. By (b), Rep(P, G) is a tree, and hence the fixed point set of the finite group K is also a tree. So to prove (1), it suffices to show, for any edge (e, [α]) in Rep(P, G)_{∗}^{K} and any vertex (v, [β]) in Rep(N_{φ}, G)_{∗} such that (v, [β|_{P}]) is an endpoint of (e, [α]), that (e, [α]) also lies in the image of the restriction map.

In this situation, v is an endpoint of e, and we regard G_{e} as a subgroup of G_{v} as usual. Then β|_{P} = c_{g} ∘ α for some g ∈ G_{v}. Set P′ = α(P) and K′ = αKα^{−1} ≤ Aut(P′) for short, and consider the subgroup

N_{e} = {a ∈ N_{G_{e}}(P′) | c_{a} ∈ K′}.

Fix Q ∈ Syl_{p}(N_{e}). Then Aut_{N_{e}}(P′) = K′ since (e, [α]) is fixed by K, and Aut_{Q}(P′) = K′ since K′ is a p-group. Also, since β|_{P} = c_{g} ∘ α, gQg^{−1} ≤ β(N_{φ})·C_{G_{v}}(β(P)). Since β(P) is p-centric in G_{v}, gQg^{−1} and β(N_{φ}) are both Sylow p-subgroups of this last group. Hence there is h ∈ C_{G_{v}}(β(P)) such that hgQg^{−1}h^{−1} = β(N_{φ}). Set α = c_{hg}^{−1} ∘ β ∈ Iso(N_{φ}, Q). Then α|_{P} = c_{g}^{−1} ∘ β|_{P} = α since h centralizes β(P), so Res((e, [α])) = (e, [α]). Also, (v, [α]) = (v, [β]) is an endpoint of (e, [α]) since hg ∈ G_{v}, so [e, α] is an edge in Rep(N_{φ}, G), and this finishes the proof of (1).

Now, (v_{∗}, [φ]) ∈ Rep(P, G)_{∗} by Lemma 4.1(b), and is fixed by the K action by definition of N_{φ}. So by (1), there is ψ ∈ Hom(N_{φ}, G_{∗}) such that (v_{∗}, [ψ]) is in Rep(N_{φ}, G)_{∗} and [ψ|_{P}] = [φ] in Rep(P, G_{∗}). Thus ψ ∈ Hom_{G}(N_{φ}, G_{∗}), and ψ|_{P} = c_{g} ∘ φ for some g ∈ G_{∗}. By axiom (II) for the saturated fusion system F_{G_{∗}}(S), c_{g}^{−1} = φ ∘ (ψ|_{P})^{−1}

extends to some χ ∈ Hom_{G_{∗}}(ψ(N_{φ}), S), and hence φ =^{def} χ ∘ ψ ∈ Hom_{G}(N_{φ}, S) extends φ. This finishes the proof of (II) for F-centric subgroups.

L is a centric linking system. If P is F-centric, then by point (b) and Lemma 4.1, C_{G}(P)/Z(P) acts on the tree T̃^{P} with orbit space a tree. Furthermore, since Z(P) is maximal among finite p-subgroups of C_{G}(P), all isotropy subgroups of this action are finite of order prime to p. Hence by I.10, p.39]Serre, C_{G}(P)/Z(P) is an amalgamated product of finite groups of order prime to p taken over a finite tree. Such a group is clearly p-perfect (it is generated by elements of order prime to p); and Mayer-Vietoris sequences for the homology of amalgamated products (cf. [Bw, §VII.9]) show that H^{i}(C_{G}(P)/Z(P); 𝔽_{p}) = 0 for all i > 0. So by Lemma 1.5, L = L_{S}^{c}(G) is a centric linking system associated to F. □

The most difficult hypothesis to check in the above theorem is (b). For this reason, we give here some equivalent formulations. The equivalence of the first three conditions is implicit in the above proof, but we make them more explicit here.

Lemma 4.3. Fix a finite tree of finite groups (G, T ), and set G = G_{T} = \underrightarrow{colim}(G). Choose a vertex v_{∗} of T , and a subgroup H ≤ G(v_{∗}). Then the following three conditions are equivalent:

(1) The abelianization of C_{G}(H) is finite.

(2) C_{G}(H) is a finite amalgamated product of finite groups.

(3) The component of (v_{∗}, [incl_{H}^{G(v_{∗})}]) in the graph Rep(H, G) is a tree.

Furthermore, if (1)–(3) hold, then:

(4) There is a vertex v ∈ T^{0} and an element x ∈ G, such that xHx^{−1} ≤ G(v) and Aut_{G}(xHx^{−1}) = Aut_{G(v)}(xHx^{−1}).

Proof. Let T̃ be the tree upon which G acts with orbit space and fundamental domain T , as in Lemma 4.1. Again, when x is a vertex or edge of T , we write G_{x} = G(x) for short. By Lemma 4.1, the component of Rep(H, G) which contains (v_{∗}, [incl_{H}^{G_{v∗}}]) can be identified with T̃^{H}/C_{G}(H). If this orbit graph is a tree, then by [Se, Theorem I.10, p.39], C_{G}(H) is an amalgamated product of finite groups taken over a finite tree; and in particular, its abelianization is finite. If the orbit graph T̃^{H}/C_{G}(H) is not a tree, then by [Se, Corollary 1, p. 55], there is a surjection of C_{G}(H) onto its fundamental group, an infinite free group, and hence the abelianization of C_{G}(H) is not finite and C_{G}(H) is not a finite amalgamated product of free groups. This proves the equivalence of (1), (2), and (3).

Now assume that (3) holds, and thus (by Lemma 4.1) that T̃^{H}/C_{G}(H) is a tree. The finite group Aut_{G}(H) ≅ N_{G}(H)/C_{G}(H) acts on this tree, and hence fixes some vertex (cf. [Se, §I.6.1]). Assume the orbit of the vertex (a^{−1}G_{v}, v) is fixed by Aut_{G}(H); in particular, aHa^{−1} ≤ G_{v} since aG_{v} ∈ (G/G_{v})^{H}. Also, each α ∈ Aut_{G}(H) is of the form α = c_{g} for some g ∈ N_{G}(H) which fixes the vertex (a^{−1}G_{v}, v) in T̃^{H}, which implies that aga^{−1} ∈ G_{v}. This shows that Aut_{G}(aHa^{−1}) = Aut_{G_{v}}(aHa^{−1}), and thus that (4) holds. □

As shown in [AC], the fusion systems F_{Sol}(q) constructed in [LO] by the second and third authors are the fusion systems of certain amalgamated products Spin_{7}(q) ∗_{B} K, where B is the normalizer in Spin_{7}(q) of a certain elementary abelian 2-subgroup of rank 2, and K contains B with index 3. The proof in [LO] that these fusion systems are saturated is very long and technical, and so it is natural to wonder whether or not this could be shown as an application of Theorem 4.2. As seen in [LO] or [AC], when F = F_{Sol}(q) and S ∈ Syl_{p}(Spin_{7}(q)), then there is an elementary abelian 2-subgroup E ≤ S of rank 4 such that Aut_{F}(E) = Aut(E) ≅ GL_{4}(2). Hence if the saturation of F could be proven using Theorem 4.2, then by Lemma 4.3(4), some vertex of the tree defining the amalgamated product would be fixed by an extension of E by Aut(E), and this is not the case. More precisely, this shows that condition (b) in Theorem 4.2 fails to hold for this amalgamated product. So Theorem 4.2 cannot be applied in this case.

5. Examples

We now look at some applications of Theorem 4.2, to produce explicit exotic fusion systems. These examples will all be based on Proposition 5.1 below, which in turn is a special case of Threorem 4.2.

For any fusion system F_{0} over a p-group S, any collection of subgroups Q_{1}, …, Q_{m} ≤ S, and outer automorphism groups Δ_{i} ≤ Out(Q_{i}) containing Out_{F_{0}}(Q_{i}), let

⟨F_{0}; Δ_{1}, …, Δ_{m}⟩

denote the fusion system over S generated by F_{0} and restrictions of automorphisms in the Δ_{i} to subgroups of Q_{i}. In other words, F = ⟨F_{0}; Δ_{i}, …, Δ_{m}⟩ is the fusion system over S such that for all P, Q ≤ S, Hom_{F}(P, Q) is the set of composites

P = P_{0} ⟶^{φ_{1}} P_{1} ⟶^{φ_{2}} P_{2} ⟶ ⋯ ⟶ P_{k−2} ⟶^{φ_{k−1}} P_{k−1} ⟶^{φ_{k}} P_{k} = Q

such that each j, either φ_{j} lies in Hom_{F_{0}}(P_{j−1}, P_{j}), or for some 1 ≤ i ≤ m, P_{j−1}, P_{j} ≤ Q_{i} and φ_{j} is the restriction of some α_{i} ∈ Aut(Q_{i}) such that [α_{i}] ∈ Δ_{i}.

The following proposition is also a generalization of [BLO2, Proposition 9.1].

Proposition 5.1. Fix a finite group G, a Sylow p-subgroup S ≤ G, and subgroups Q_{1}, …, Q_{m} ≤ S such that no Q_{i} is G-conjugate to a subgroup of Q_{j} for i ≠ j. For each i, set K_{i} = Out_{G}(Q_{i}), and fix subgroups Δ_{i} ≤ Out(Q_{i}) which contain K_{i}. Set F = ⟨F_{S}(G); Δ_{1}, …, Δ_{m}⟩. Assume for each i that

(1) p ∤ [Δ_{i}:K_{i}];

(2) Q_{i} is p-centric in G, but no proper subgroup P ≨ Q_{i} is F-centric or an essential p-subgroup of G; and

(3) for all α ∈ Δ_{i}∖K_{i}, K_{i} ∩ αK_{i}α^{−1} has order prime to p.

Then F is a saturated fusion system over S, and has an associated centric linking system.

Proof. For each i, set H_{i} = N_{G}(Q_{i}) and T_{i} = O^{p}(C_{G}(Q_{i})). Then T_{i} has order prime to p since Q_{i} is p-centric in H_{i}; and thus Q_{i}·C_{G}(Q_{i}) = Q_{i}T_{i} ≅ Q_{i} × T_{i} and K_{i} ≅ H_{i}/Q_{i}T_{i}.

We first construct a finite group G_{i} ≥ H_{i} such that H_{i} has index prime to p in G_{i}, Q_{i} ◁ G_{i}, and Out_{G_{i}}(Q_{i}) = Δ_{i}. By (3), K_{i} ∩ αK_{i}α^{−1} has order prime to p for all α ∈ Δ_{i}∖K_{i}; and hence the restriction homomorphism

H^{j}(Δ_{i}; Z(Q_{i})) ≅ H^{j}(K_{i}; Z(Q_{i}))

is an isomorphism for all j > 0 by the description in of the image in terms of stable (or G-invariant) elements (cf. [AM, Theorem II.6.6] or [Bw, Theorem III.10.3]). When j = 3, the injectivity of the restriction map tells us that the obstruction to the existence of an extension

1 ⟶ Q_{i} ⟶ G′_{i} ⟶ Δ_{i} ⟶ 1

vanishes [McL, Theorem IV.8.7], since its restriction to K_{i} ≤ Δ_{i} vanishes. When j = 2, the group H^{2}(Δ_{i}; Z(Q_{i})) ≅ H^{2}(K_{i}; Z(Q_{i})) acts freely and transitively on the sets of all such extensions of Q_{i} by Δ_{i} or by K_{i} [McL, Theorem IV.8.8], and thus G′_{i} can be chosen to contain the group H_{i}/T_{i}.

Now let T_{i} ≀K_{i} and T_{i} ≀Δ_{i} be the “regular” wreath products: the semidirect products T_{i}^{|K_{i}|}⋊K_{i} and T_{i}^{|Δ_{i}|}⋊Δ_{i} where K_{i} and Δ_{i} permute the factors T_{i} freely and transitively. There is an obvious embedding of T_{i} ≀ K_{i} into T_{i} ≀ Δ_{i}; and by [Hu, I.15.9], there is an embedding of H_{i}/Q_{i} (as an extension of T_{i} by K_{i}) into T_{i} ≀ K_{i}. We can thus regard H_{i}/Q_{i} as a subgroup of T_{i} ≀ Δ_{i}. So if we define G_{i} to be the pullback of the maps

G′_{i} ⟶ Δ_{i} ⟵ T_{i} ≀ Δ_{i},

then G_{i} sits in an extension

1 ⟶ Q_{i} × T_{i}^{|Δ_{i}|} ⟶ G_{i} ⟶ Δ_{i} ⟶ 1;

and we can identify H_{i} (regarded as a pullback of H_{i}/T_{i} and H_{i}/Q_{i} over K_{i}) as a subgroup of G_{i} with index prime to p. Also, Q_{i} ◁ G_{i} and Δ_{i} = Out_{G_{i}}(Q_{i}).

We will apply Theorem 4.2 to the tree which has m + 1 vertices v_{∗}, v_{1}, …, v_{m} and edges e_{i} connecting v_{∗} to v_{i}, and to the functor G(v_{∗}) = G, G(v_{i}) = G_{i}, and G(e_{i}) = H_{i}.

Let Ĝ denote the amalgamated product of this tree of groups. Then F = F_{S}(Ĝ), and it remains only to check that conditions (a), (b), and (c) in Theorem 4.2 hold. Condition (a) holds by (1).

We next check condition 4.2(c). By definition of F, for any subgroup P ≤ S which pcentric in G, P is F-centric unless there is some P′ ≤ S which is F-conjugate to P but not G-conjugate, in which case P must be G-conjugate to a proper subgroup P′ ≨ Q_{i} for some i. For each i, any Sylow p-subgroup of H_{i}, or any essential p-subgroup (hence radical p-subgroup) of G_{i}, must contain O_{p}(H_{i}) ≥ Q_{i}; and the Q_{i} are all p-centric in G (hence F-centric) by assumption. Any essential p-subgroup P of G is p-centric in G; and hence is F-centric since by (b) again, no essential p-subgroup of G is properly contained in any Q_{i}. This finishes the proof of (c) in Theorem 4.2.

It remains to check condition 4.2(b). Fix an F-centric subgroup P ≤ S; we must show that the component Γ of (v_{∗}, [incl_{P}^{G}]) in Rep(P, G) is a tree. The edges in Rep(P, G) adjacent to (v_{∗}, [incl_{P}^{G}]) are of the form (e_{i}, [α]), for α ∈ Hom_{G}(P, H_{i}). For any such edge, its other vertex (v_{i}, [α]) is the endpoint of a second edge (e_{i}, [β]) only if β = c_{g} ∘ α for some g ∈ G_{i}∖H_{i}. In particular, α(P) and gα(P)g^{−1} are both contained in H_{i}, and thus α(P) ≤ H_{i} ∩ g^{−1}H_{i}g. By (3), (H_{i} ∩ g^{−1}H_{i}g)/Q_{i} has order prime to p, and hence α(P) ≤ Q_{i}. Since Q_{i} is a minimal F-centric subgroup, this implies that α(P) = Q_{i}. Since no two of the Q_{i} are G-conjugate, this can occur for at most one i. We thus have two possibilities:

• P is not G-conjugate to any Q_{i}. In this case, every edge in Γ is adjacent to the vertex (v_{∗}, [incl_{P}^{G}]), and Γ is a tree.

• P is G-conjugate to Q_{j} for some fixed j ∈ {1, …, m}. In this case, let Γ_{0} ⊆ Γ be the subgraph of all vertices sitting over v_{∗} or v_{j}, and all edges sitting over e_{j}. Each vertex of Γ not in Γ_{0} sits over v_{i} for some i ≠ j, and by the above remarks is connected to Γ_{0} by a unique edge. Thus Γ_{0} is a deformation retract of Γ. Each edge in Γ_{0} has the form (e_{j}, [α]) for some α ∈ Iso(P, Q_{i}). If two edges (e_{j}, [α]) and (e_{j}, [α′]) have the same vertex (v_{∗}, [α]) = (v_{∗}, [α′]), then α′ = c_{g} ∘ α for some g ∈ G, so g ∈ N_{G}(Q_{i}) = H_{i}, and the edges (e_{j}, [α]) and (e_{j}, [α′]) are equal. Thus no vertex in Γ_{0} over v_{∗} can be attached to two edges, and this proves that Γ_{0} (and hence Γ) is a tree.

This finishes the proof of condition 4.2(b), and hence of the proposition. □

Note that (3) implies (1) in the above proposition; condition (1) has been kept for emphasis.

Condition (3) means that the subgroup K_{i} is strongly embedded in Δ_{i} at the prime p (see Definition 3.2). This puts fairly restrictive conditions on K_{i} and Δ_{i}, especially when p = 2. By a theorem of Bender [Be], if Δ has a strongly embedded subgroup at p = 2, then either its Sylow 2-subgroups are cyclic or quaternion, or there is a normal series A ◁ B ◁ Δ where A and Δ/B have odd order, and B/A is isomorphic to P SL_{2}(q), Sz(q), or P SU_{3}(q) for q some power of 2. The severe restrictions which this places on the groups involved when applying Proposition 5.1 with p = 2 help to explain why it seems unlikely that we could construct an exotic fusion system at the prime 2 using this proposition, although we are not yet able to completely exclude that possibility.

The following lemma is a refinement of [BLO2, Lemma 9.2], and will be used to show that certain fusion systems are not fusion systems of finite groups. When F is a fusion system over the p-group S, a subgroup P ≤ S is strongly closed if no element of P is F-conjugate to an element of S∖P. The subgroup P is normal in F if each morphism in F extends to a morphism between subgroups containing P which sends P to itself. If P is normal in F, then it is strongly closed, but not conversely.

As usual, a finite group G is almost simple if it contains a normal, nonabelian simple subgroup L ◁ G such that C_{G}(L) = 1. In otherwords, G can be identified with a subgroup of Aut(L), and G/L with a subgroup of Out(L).

Lemma 5.2. Let F be a fusion system over a nonabelian p-group S. Assume, for each subgroup 1 ≠ P ≤ S which is strongly closed in F, that

(a) P is centric in S (i.e., C_{S}(P) = Z(P));

(b) P is not normal in F; and

(c) P does not factorize as a product of two or more subgroups which are permuted transitively by Aut_{F}(P).

Then if F is the fusion system of a finite group, it is the fusion system of a finite almost simple group.

Proof. Assume that F = F_{S}(G) for some finite group G with S ∈ Syl_{p}(G), and that G is a subgroup of minimal order with this property. Let 1 ≠ L ◁ G be a minimal nontrivial normal subgroup. Set P = L ∩ S ∈ Syl_{p}(L); then P is strongly closed in F. If P = 1 (i.e., L has order prime to p), then F is also the fusion system of G/L, which contradicts the minimality assumption. If L = P is an abelian p-group, then it is normal in F, which contradicts (b). Thus, since L is minimal, it is a product of nonabelian simple groups isomorphic to each other (cf. [Go, Theorem 2.1.5]); and these must be permuted transitively by N_{G}(L) = G since otherwise L is not minimal. Then L must be simple by (c). Also, C_{G}(L) ∩ S ≤ C_{S}(P) ≤ P by (a). Since C_{G}(L) ◁ G, this means it must have order prime to p (otherwise it would intersect every Sylow p-subgroup nontrivially); and this implies C_{G}(L) = 1 by the minimality assumption (again since C_{G}(L) ◁ G). Thus G is almost simple; i.e., L ◁ G ≤ Aut(L). □

We next focus attention on cases where Proposition 5.1 can be applied with Q_{i} ≅ C_{p}^{2} (for p an odd prime). By [Hu, Satz III.14.23], a p-group S contains a centric subgroup of order p^{2} if and only if it has maximal class; i.e., if and only if it has nilpotence class n−1 when |G| = p^{n}. For odd p, the structure of p-groups of maximal class is described in detail in [Hu, §III.14], and include the following examples when p = 3.

Example 5.3. Set p = 3, and let S be one of the following groups of order 3^{4}:

S′ = ⟨a, b, x | a^{9} = b^{3} = x^{3} = [a, b] = 1, xax^{−1} = ab, xbx^{−1} = ba^{−3}⟩.
S′′ = ⟨a, b, x | a^{9} = b^{3} = x^{3} = [a, b] = 1, xax^{−1} = ab, xbx^{−1} = ba^{3}⟩.

Let ω, η ∈ Aut(S) be the automorphisms

ω(a) = a^{−1}, ω(b) = x^{−1}bx = \begin{cases}ba^{3} & if S = S′ \\ ba^{−3} & if S = S′′, \end{cases} ω(x) = x^{−1}

η(a) = a^{−1}, η(b) = b^{−1}, η(x) = x.

For i = 0, 1, 2, set

R_{i} = ⟨xa^{i}b^{i}, a^{3}⟩  and  Q_{i} = ⟨R_{i}, b⟩.

Then R_{i} ≅ C_{3}^{2} if S = S′ or if i = 0, and R_{i} ≅ C_{9} if S = S′′ and i = 1, 2. Also, Q_{i} is extraspecial of order 27, and has exponent 3 if S = S′ or i = 0 and exponent 9 otherwise. All of these subgroups are invariant under ω, while η leaves R_{0} and Q_{0} invariant and switches R_{1} and R_{2}.

When S = S′′, then the following fusion systems over S

⟨F_{S}(S⋊⟨ω⟩); SL(R_{0})⟩ and ⟨F_{S}(S⋊⟨ω, η⟩); GL(R_{0})⟩

are both saturated, and not fusion systems of any finite group. When S = S′, then the following table describes different fusion systems over S via the automorphism groups Out_{F}(P) for P = S, R_{i}, or Q_{i}, and where an asterisk marks those which are not fusion systems of any finite group:

Out_{F}(S) Aut_{F}(R_{0}) Out_{F}(Q_{0}) Aut_{F}(R_{1}) Aut_{F}(R_{2})  group
⟨ω⟩  SL_{2}(3)  —  SL_{2}(3)  SL_{2}(3)  L_{3}^{±}(q) (v_{3}(q ∓ 1) = 2)
⟨ω⟩  —  —  SL_{2}(3)  SL_{2}(3)  ∗
⟨ω⟩  SL_{2}(3)  —  —  —  ∗
⟨η, ω⟩  GL_{2}(3)  —  SL_{2}(3)  L_{3}^{±}(q)⋊C_{2} (v_{3}(q ∓ 1) = 2)
⟨η, ω⟩  —  GL_{2}(3)  SL_{2}(3)  ^{3}D_{4}(q) (v_{3}(q^{2} − 1) = 1)
⟨η, ω⟩  —  —  SL_{2}(3)  ∗
⟨η, ω⟩  GL_{2}(3)  —  —  ∗

Here, L_{n}^{+}(q) = P SL_{n}(q) and L_{n}^{−}(q) = P SU_{n}(q). Also, for any prime power q, ^{3}D_{4}(q) is the fixed subgroup of a certain “triality” graph automorphism of order 3 on Spin_{8}(q^{3}).

Proof. That these fusion systems are all saturated is a special case of Proposition 5.1, applied with G = S⋊⟨ω⟩ or G = S⋊⟨η, ω⟩ as appropriate.

Let F be any of these fusion systems, and assume P ◁ S is a proper strongly closed subgroup. Then P ≥ ⟨a^{3}⟩ (any normal subgroup contains the center). If F contains SL(R_{i}) for some i, then P ≥ R_{i} since ⟨a^{3}⟩ is F-conjugate to the other subgroups of order 3 in R_{i}; and hence P ≥ Q_{i} (the normal closure of R_{i} in S). By similar reasoning, if F contains SL(Q_{i}) for some i, then either P = ⟨a^{3}⟩, or P ≥ Q_{i}. Thus in all cases listed above, the only nontrivial subgroups strongly closed in F are S, and possibly one of the Q_{i}. Hence by Lemma 5.2, if F is the fusion system of a finite group, then it is the fusion system of a finite almost simple group G, which contains a normal simple group L ◁ G with Sylow 3-subgroup S or Q_{i}. If L contains a Sylow 3-subgroup Q_{i}, then 3||G/L||| Out(L)|, and this is impossible by Lemma 5.4 below.

It remains to consider the case where [G : L] is prime to 3, and thus where S ∈ Syl_{3}(L). By [GLS, Tables 5.3 & 5.6.1], none of the sporadic simple groups has Sylow 3-subgroup of order 3^{4} and rank 2. If v_{3}(|A_{n}|) = 4, then n = 9, 10, 11, and rk_{3}(A_{n}) = 3. By [GLS, Table 2.2], the only simple groups of Lie type in characteristic 3 whose Sylow 3-subgroups have order 3^{4} are the groups B_{2}(3), and these also have 3-rank equal to 3. Finally, using [GLS, Table 2.2] and [GL, 10-1 & 10-2], one checks that the only simple groups of Lie type whose Sylow 3-subgroups have order 3^{4} and rank 2 are the groups L_{3}(q) when v_{3}(q − 1) = 2, U_{3}(q) when v_{3}(q + 1) = 2, and ^{3}D_{4}(q) when v_{3}(q^{2} − 1) = 1. The precise fusion systems of these groups (and the fact that their Sylow subgroups are isomorphic to S′) is determined directly, or with the help of the lists of maximal subgroups in [GLS, Theorem 6.5.3] (for L_{3}^{±}(q)) and [Kl] (for ^{3}D_{4}(q)). For example, by [Kl], there are subgroups

3^{2}·SL_{2}(3) ≤ ^{3}D_{4}(q) and 3_{+}^{1+2}·GL_{2}(3) ≤ SL_{3}^{±}(q)·S_{3} ≤ ^{3}D_{4}(q)

(when q ≡ ±1 (mod 3)), and this determines the structure of the fusion system of 3D_{4}(q). □

It remains to prove the following lemma, which will also be used later.

Lemma 5.4. There is no pair (L, p), where L is a finite simple group, p is an odd prime, the Sylow p-subgroups of L are extraspecial of order p^{3}, and p|| Out(L)|.

Proof. If L is a sporadic or alternating group, then | Out(L)| is a power of 2, so this is impossible. Thus L is of Lie type, and hence by [Ca, Theorem 12.5.1], Out(L) is generated by field, graph, and diagonal automorphisms. We refer to [Ca, 9.4.10, 10.2.4– 5, 14.3.2] for the orders of the simple groups of Lie type. The only simple groups with graph automorphisms of odd order are the groups D_{4}(q) (with graph automorphisms of order 3), and |D_{4}(q)| = q^{12}(q^{8} + q^{4} + 1)(q^{6} − 1)(q^{2} − 1) is a multiple of 3^{4} for all q. If L has a field automorphism of order p, where p is an odd prime, then L is defined over a field of order q^{p} for some prime power q; if q^{pn} ± 1 is divisible by p then it is divisible by p^{2}, and the list of orders of groups of Lie type makes it clear that this case is impossible. So if there is a pair (L, p) as above, then L must have a diagonal automorphism of order p.

The only simple groups of Lie type with diagonal automorphisms of order p ≥ 3 are P SL_{n}(q) (for p|(n, q − 1)), P SU_{n}(q) (for p|(n, q + 1)), E_{6}(q) (for p = 3|q − 1), and 2E_{6}(q) (for p = 3|q + 1). Of these, the only cases where the simple group has p-rank ≤ 2 occur when p = 3, and L = P SL_{3}(q) (where 3|(q −1) and |L| = \frac{1}{3}q^{3}(q^{2} −1)(q^{3} −1)) or P SU_{3}(q) (where 3|(q + 1) and |L| = \frac{1}{3}q^{3}(q^{2} − 1)(q^{3} + 1)). In both of these cases, v_{3}(|L|) = 2v_{3}(q ± 1) ≠ 3. □

For the rest of the section, we let p be any odd prime, and consider the group

S = ⟨a, b, c, x | a^{p} = b^{p} = c^{p} = x^{p} = [a, b] = [a, c] = [b, c] = 1,

xax^{−1} = a, xbx^{−1} = ab, xcx^{−1} = bc⟩.

Set A = ⟨a, b, c⟩, Q = ⟨a, b, x⟩, and R = ⟨a, x⟩. Set

Ω = (GL_{2}(p) × 𝔽_{p}^{×})/{(uI, u^{−2}) | u ∈ 𝔽_{p}^{×}},

and let [B, u] denote the class of the pair (B, u) for B ∈ GL_{2}(p) and u ∈ 𝔽_{p}^{×}. Define an action χ: Ω → Aut(A) as follows. Identify A with the additive group S_{2}(p) of symmetric 2 × 2 matrices by setting a = ( \begin{matrix}2 & 0 \\ 0 & 0 \end{matrix} ), b = ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ), and c = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ); and let [B, u] ∈ Ω act by sending M to u·BMB^{t}. These identifications are chosen so that the action of X =^{def} [( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ), 1] on A is precisely the action of x ∈ S by conjugation. We can thus identify S as a Sylow p-subgroup of G =^{def} A⋊Ω. Then Ω is the normalizer in Aut(A) ≅ GL_{3}(p) of the orthogonal group GO_{3}(p), for an appropriate choice of quadratic form on A.

For a given pair of fusion systems F′ ⊆ F over the same p-group S, we say that F′ has index prime to p in F if Aut_{F′}(P) ≥ O^{p′}(Aut_{F}(P)) for all P ≤ S [5a2, Definition 3.1]. In [5a2, §5], we prove that for any saturated fusion system F, there is a unique minimal fusion subsystem O^{p′}(F) ⊆ F of index prime to p. This terminology provides a convenient framework for describing the next result.

Example 5.5. Fix an odd prime p, and let S be the group of order p^{4} defined above, with subgroups A, Q, R ≤ S. Then the following hold.

(a) There are unique saturated fusion systems F_{Q} and F_{R} over S such that

Out_{F_{Q}}(S) ≅ C_{p−1} × C_{p−1}, Aut_{F_{Q}}(A) = Ω, Out_{F_{Q}}(Q) = Out(Q) ≅ GL_{2}(p)
Out_{F_{R}}(S) ≅ C_{p−1} × C_{p−1}, Aut_{F_{R}}(A) = Ω, Aut_{F_{R}}(R) = Aut(R) ≅ GL_{2}(p) ;

and Q is not F_{R}-radical.

(b) O^{p′}(F_{Q}) has index 2 in F_{Q}, Out_{O^{p′}(F_{Q})}(Q) is the unique subgroup of index 2 in Out(Q) ≅ GL_{2}(p), and

Aut_{O^{p′}(F_{Q})}(A) = {[B, 1] | B ∈ GL_{2}(p)}.

For all p, F_{Q} is the fusion system of Aut(P Sp_{4}(p)) = P Sp_{4}(p)⋊C_{2} (the extension by diagonal automorphisms), and O^{p′}(F_{Q}) is the fusion system of P Sp_{4}(p).

(c) O^{p′}(F_{R}) has index (4, p−1) in F_{R}, Aut_{O^{p′}(F_{R})}(R) is the unique subgroup of index (4, p−1) in Aut(Q) ≅ GL_{2}(p), and

Aut_{O^{p′}(F_{R})}(A) = {[B, u] ∈ Ω | det(B)·u^{−1} ∈ 𝔽_{p}^{×4}}.

When p = 3, F_{R} is the fusion system of Σ_{9} and O^{3′}(F_{R}) is the fusion system of A_{9}. When p = 5, F_{R} is the fusion system of P ΣL_{5}(16) ≅ P SL_{5}(16)⋊C_{4} (the extension by field automorphisms), and O^{5′}(F_{R}) is the fusion system of P SL_{5}(16). When p ≥ 7, no fusion subsystem of index prime to p in F_{R} is the fusion system of a finite group.

Proof. Set G = A⋊Ω, and identify S with A⋊⟨X⟩ ≤ G. Since N_{Ω}(⟨X⟩) is generated by X = [( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ), 1] together with elements [( \begin{matrix}u & 0 \\ 0 & v \end{matrix} ), w] for u, v, w ∈ 𝔽_{p}^{×}, and since

[( \begin{matrix}u & 0 \\ 0 & u \end{matrix} ), u^{−2}] = 1,

Out_{F_{Q}}(S) = Out_{F_{R}}(S) = Out_{G}(S) = {[η_{u}ω_{v}] | u, v ∈ 𝔽_{p}^{×}} ≅ C_{p−1} × C_{p−1},

where [η_{u}] and [ω_{v}] are the classes modulo Inn(S) of the automorphisms

η_{u} = c([( \begin{matrix}1 & 0 \\ 0 & u \end{matrix} ), 1]) : a ↦ a  b ↦ b^{u}a^{−(u−1)/2} c ↦ c^{u^{2}}  x ↦ x^{1/u}
ω_{v} = c([I, v]) :  a ↦ a^{v}  b ↦ b^{v}  c ↦ c^{v}  x ↦ x

for all u ∈ 𝔽_{p}^{×}. Here, c(g) denotes conjugation by g (a ↦ gag^{−1}). Note also the relation

c([( \begin{matrix}u & 0 \\ 0 & u^{−1} \end{matrix} ), 1]) = c([( \begin{matrix}1 & 0 \\ 0 & u^{−2} \end{matrix} ), u^{2}]) = ω_{u^{2}}η_{1/u^{2}} = ω_{u}^{2}η_{u}^{−2}.

It follows that

Out_{G}(Q) = ⟨η_{u}|_{Q}, ω_{u}|_{Q}⟩ = N_{Out(Q)}(Out_{G}(Q)).

So we can apply Proposition 5.1 with m = 1 and Q_{1} = Q (and with G as above), to prove that the fusion system F_{Q} is saturated. Similarly,

Aut_{G}(R) = ⟨η_{u}|_{R}, ω_{u}|_{R}⟩ = N_{Aut(R)}(Aut_{G}(R)),

and so F_{R} is saturated by Proposition 5.1 again.

We next calculate O^{p′}(F_{Q}). Let O_{∗}^{p′}(F_{Q}) ⊆ F_{Q} be the fusion subsystem generated by the automorphism groups O^{p′}(Aut_{F}(P)) for P ≤ S. Consider the subgroup Out_{F_{Q}}^{0}(S) ◁ Out_{F_{Q}}(S) as defined in [5a2, §5.1]:

Out_{F_{Q}}^{0}(S) = ⟨α ∈ Out_{F_{Q}}(S) | α|_{P} ∈ Mor_{O_{∗}^{p′}(F_{Q})}(P, S), some F_{Q}-centric P ≤ S⟩.

For u, v ∈ 𝔽_{p}^{×}, (η_{u}ω_{v})|_{Q} ∈ O^{p′}(Aut_{F_{Q}}(Q)) ≅ SL_{2}(p) if and only if v = 1; while (η_{u}ω_{v})|_{A} ∈ O^{p′}(Aut_{F_{Q}}(A)) = Ω_{0} if and only if v = 1/u ∈ 𝔽_{p}^{×2}. Thus Out_{F_{Q}}^{0}(S) = {η_{u}ω_{v^{2}}}. This shows that O^{p′}(F_{Q}) has index 2 in F_{Q}, and has the form described in (b).

A similar argument shows that O^{p′}(F_{R}) has index (4, p−1) in F_{R}, and has the form described in (c).

Thus for all F-centric subgroups P′ ≤ S, Aut_{F}(P′) contains O^{p′}(Aut_{F_{P}} (P′)) (P = Q or R). Hence F has index prime to p in F_{P}, and by [5a2, Theorem 5.4], O^{p′}(F_{P}) ⊆ F ⊆ F_{P}.

It is straightforward to check that the finite groups listed in (b) and (c) have the automorphism groups as indicated, and we have seen that this determines their fusion system. So it remains to show that the fusion systems in (c) are not fusion systems of finite groups for p ≥ 7.

Let F be any of these fusion systems. If 1 ≠ P ◁ S is strongly closed in F, then it must contain Z(S) = ⟨a⟩ (any nontrivial normal subgroup intersects nontrivially with Z(S)); hence contains A (since the subgroups Aut_{F}(A)-conjugate to ⟨a⟩ generate A in all cases); and hence is equal to S since either Q or R is F-radical. So by [BLO2, Lemma 9.2], if F is the fusion system of a finite group, it must be the fusion system of a finite almost simple group. More precisely, F = F_{S}(G) for some G with normal simple subgroup L ◁ G of index prime to p such that C_{G}(L) = 1. By a direct check through the list of finite simple groups, one sees that the following are the only simple groups which have Sylow p-subgroup isomorphic to S:

• (any p) P Sp_{4}(p)

• (p = 3) P SL_{4}(q) (q ≡ 4, 7 (mod 9)), P SU_{4}(q) (q ≡ 2, 5 (mod 9)), P Sp_{6}(q) (q ≡ ±2, ±4 (mod 9)), Ω_{7}(q) (q ≡ ±2, ±4 (mod 9)), A_{n} (n = 9, 10, 11).

• (p = 5) P SL_{5}(q) (q ≡ 6, 11, 16, 21 (mod 5)), P SU_{5}(q) (q ≡ 4, 9, 14, 19 (mod 5)), Co_{1}.

By elimination, none of the F_{R,i} for p ≥ 7 is the fusion system of a finite group. □

In fact, in the above situation, if F is any saturated fusion system over S such that A is F-radical but not normal in F, then F is isomorphic to a fusion system F′ over S which has index prime to p in one of the fusion systems F_{Q} or F_{R}. To see this, set Γ = Aut_{F}(A) ≤ GL_{3}(p) for short, and let Γ′ be the image of Γ ∩ SL_{3}(p) in P SL_{3}(p). If Γ′ has a nontrivial normal subgroup of order prime to p, then either the action on A is decomposable (which is impossible since the action of Aut_{S}(A) is indecomposable), or A splits as a sum of three subspaces which are permuted by Γ. The latter case implies that Γ ≤ C_{p−1} ≀ Σ_{3}, and thus is possible only if p = 3 and Γ ◁ C_{2} ≀ Σ_{3} = Ω. Otherwise, if Γ′ has no nontrivial normal subgroups of prime power order, then by [Bl, Theorem 1.1], Γ′ must be isomorphic to P SL_{2}(p) or P GL_{2}(p); and conjugate to the indecomposable representation of these groups described in [Bl, Lemma 6.3]. Hence up to conjugacy, Γ ∩ SL_{3}(p) contains Ω_{0} as a normal subgroup, and hence that Γ ≤ N_{GL_{3}(p)}(P SL_{2}(p)) = Ω.

In particular, every proper subgroup P ≨ S not contained in A is either F-conjugate to Q or R; or else p = 3 and P is cyclic or extraspecial of exponent 9 (in which case P cannot be F-radical). Hence since A is not normal in F, one of the subgroups Q or R must be F-radical. If P = Q or R is F-radical, then Out_{F}(P) ≤ Out(P) ≅ GL_{2}(p) contains at least two subgroups of order p; any two such subgroups generate SL_{2}(p); and thus Out_{F}(P) ≥ SL_{2}(p).

When p ≥ 5, there are also saturated fusion systems over S where A is not radical, but is not normal either. Fix any subset I ⊆ {0, …, p − 1} with |I| ≥ 2, and choose P_{i} = ⟨a, b, c^{i}x⟩ or ⟨a, c^{i}x⟩ for each i ∈ I. Let F_{I} be the fusion system over S generated by Out_{F_{I}}(S) = ⟨η_{u}, ω_{u}⟩ ≅ C_{p−1}^{2} (where η_{u} and ω_{u} are defined as above), and Out_{F_{I}}(P_{i}) = Out(P_{i}) for all i ∈ I. (This depends not just on I but also on the choice of the P_{i}.) These are saturated fusion systems by Proposition 5.1, and have no proper strongly closed subgroups. So by the list of simple groups with Sylow subgroup S given above, these systems are all exotic.

We look more closely only at the case where |I| = 1 and P_{i} ≅ R. In this case, there is a proper strongly closed subgroup.

Example 5.6. Fix an odd prime p, and let S be the group of order p^{4} defined above. For u = 1, …, p − 1, let ω_{u}, η_{u} ∈ Aut(S) be the automorphisms

ω_{u}(a) = a^{u}, ω_{u}(b) = b^{u}, ω_{u}(c) = c^{u}, ω_{u}(x) = x;
η_{u}(a) = a, η_{u}(b) = b^{u}a^{−(u−1)/2}, η_{u}(c) = c^{u^{2}}, η_{u}(x) = x^{1/u}.

Set

Γ = {ω_{u}η_{v} | u, v = 1, …, p − 1}  and  Γ_{0} = {ω_{u}η_{u} | u = 1, …, p − 1}.

Set R = ⟨x, a⟩ ≅ C_{p}^{2}; a subgroup invariant under each ω_{i} and η_{i}. Then the fusion systems

F = ⟨F_{S}(S⋊Γ); GL(R)⟩  and  F_{0} = ⟨F_{S}(S⋊Γ_{0}); SL(R)⟩

are both saturated, and neither is the fusion system of a finite group.

Proof. The fusion systems F and F_{0} are saturated by Proposition 5.1, applied with G = S⋊Γ or S⋊Γ_{0}, respectively.

If P ≠ 1 is strongly closed in F or F_{0}, then P ≥ ⟨a⟩ = Z(S) (since any nontrivial normal subgroup intersects nontrivially with the center), hence P ≥ R, and hence P ≥ Q = ⟨a, b, x⟩ since bx is S-conjugate to x. Thus P = Q or S. Hence by Lemma 5.2, if F or F_{0} is the fusion system of a finite group G, then we can assume that there is a normal simple subgroup L ◁ G such that C_{G}(L) = 1 (so G/L ≤ Out(L)), and such that L ≥ S or L ∩ S = Q. If L ∩ S = Q, then p||G/L||| Out(L)|, and this is impossible by Lemma 5.4. Thus S ∈ Syl_{p}(L).

In the proof of Example 5.5, we listed all simple groups L with Sylow p-subgroup isomorphic to S for some odd p. In all cases, the (unique) abelian subgroup of index p in S is radical in L, and thus F_{S}(L) is not contained in F. □

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Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain

  E-mail address: broto@mat.uab.es

Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, U.K.

  E-mail address: ran@maths.abdn.ac.uk

LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

  E-mail address: bob@math.univ-paris13.fr