SATURATED FUSION SYSTEMS WITH PARABOLIC FAMILIES

SILVIA ONOFREI

Abstract. Let G be group; a finite p-subgroup S of G is a Sylow p-subgroup if every finite p-subgroup of G is conjugate to a subgroup of S. In this paper, we examine the relations between the fusion system over S which is given by conjugation in G and a certain chamber system C, on which G acts chamber transitively with chamber stabilizer N_{G}(S).

Next, we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system. We determine some conditions the chamber system has to fulfill in order to assure the saturation of the underlying fusion system. We give an application to fusion systems with parabolic families of classical type.

1. Introduction

In the present paper we elaborate on the connections between fusion systems, chamber systems and parabolic systems. A fusion system F is a category whose objects are the subgroups of a finite p-group S; the morphisms are monomorphisms between these subgroups, such that all monomorphisms induced by conjugation in S are included. The saturated fusion systems satisfy extra conditions which model properties of the fusion in a finite group related to the Sylow p-subgroups. The first thorough study of fusion systems and saturated fusion systems is due to Puig; see the more recent [Pui06], for example. Applications to algebraic topology came from the introduction of the notion of p-local finite groups, having centric linking systems, by Broto, Levi and Oliver [BLO04]. The name saturated fusion system is also due to Broto, Levi and Oliver. Chamber systems were introduced by Tits [Tit81] in the study of local properties of buildings. All chamber systems used in this paper can be considered as simplicial complexes, with the chambers being simplices of maximal dimension. The codimension one faces are assigned types labeled by elements in an index set I, and correspond to the panels of the chamber system. Each chamber has |I| faces; two chambers are i-adjacent if they have a face of type i in common. Let G be a chamber transitive group of automorphisms of a chamber system, and let B denote the stabilizer of a chamber and the G_{i}, i ∈ I denote its panel stabilizers. If the chamber system is a building, then B is the Borel subgroup of G and G_{i} are the minimal parabolic subgroups. The family (B, G_{i} ; i ∈ I) forms a parabolic system of rank |I| if G = ⟨G_{i} ; i ∈ I⟩ and no proper subset {G_{j} ; j ∈ J ⊊ I} generates G.

Date: Wednesday, 31 August 2011.

1991 Mathematics Subject Classification. 20J15, 20E42.

This project was partially supported by the National Security Agency under grant number H98230-10-1-0174.

We now describe the approach and the results contained in this paper. Let A be a diagram of finite groups (B, G_{i}, G_{ij}; i, j ∈ I) together with injective group homomorphisms B ↪ G_{i} and G_{i} ↪ G_{ij}, for all i, j ∈ I, such that all squares commute. Assume that G is a faithful completion of A. Our first result Theorem 4.11 asserts that, under certain assumptions, the group G has a finite Sylow p-subgroup S and the fusion system F_{S}(G) associated to G is saturated. This theorem is in some sense a generalization of a result due to Broto, Levi and Oliver [BLO06, Theorem 4.2] regarding the fusion system associated to the colimit of a finite tree of finite groups. Although our proof is written in terms of chamber systems, the line of thought closely follows the one given in [BLO06]. We introduce the notion of a parabolic family for a fusion system F over a finite p-group S. Roughly speaking it consists of a collection of saturated constrained fusion subsystems {F_{i}, i ∈ I} which all contain B, the normalizer fusion subsystem N_{F}(S), and with the additional property that each subsystem F_{ij} = ⟨F_{i}, F_{j}⟩ is also saturated and constrained. We construct a certain group G, determined by p′-reduced p-constrained finite groups that realize the constrained fusion systems B, F_{i} and F_{ij}. A chamber system C over I can be associated to F and G. Our second main result is Theorem 5.8 which states that under a couple of assumptions, F is the saturated fusion system of G over S. For example, we assume that C^{P}, the fixed point set under the action of a p-subgroup P of G, is connected. The third achievement of the paper is Theorem 6.10. This is a reduction result which says that a fusion system F with a parabolic family contains a certain normal fusion subsystem denoted F̂, which also has a parabolic family. If the hypotheses from Theorem 5.8 hold in F̂, then this fusion system is saturated and realized by a normal subgroup of G. Also, there is a 2-covering Ĉ → C between the associated chamber systems. Assume now that the subsystems F_{i} and F_{ij} are realized by suitably chosen p′-reduced pconstrained extensions of finite groups of Lie type G_{i} and G_{ij} in characteristic p, of rank one and two respectively. In this case, a type (or diagram) ℳ can be associated to F. This is a graph whose vertices are labeled by the elements of I, the restriction to the nodes i and j is the Coxeter diagram corresponding to G_{ij}. Proposition 7.5 generalizes to fusion systems a well known result of Timmesfeld [Tim85, 3.1]. Specifically, if the type ℳ is classical, F is the fusion system of a group of Lie type extended by diagram and field automorphisms.

Outline of the paper. We start with a review of some basic results on fusion systems. In Section 3, we recall the standard terminology on chamber systems and parabolic systems. In Section 4 we prove Theorem 4.11, while in Section 5 we prove Theorem 5.8. Section 6 is dedicated to the proof of Theorem 6.10. We finish our paper with an application to chamber systems of type ℳ and their generalization to fusion systems.

Acknowledgements. We would like to thank the anonymous referee for a very thorough reading of the paper and for numerous suggestions leading to significant improvements in the paper.

2. Recollections on fusion systems

For an introduction to fusion systems see [BLO03] and [Lin07]. We review here the basic definitions and a few results needed in the paper.

Background and terminology. Given two groups P and Q, let Hom(P, Q) denote the set of group homomorphisms from P to Q and let Inj(P, Q) denote the subset of monomorphisms. If P and Q are subgroups of a group G, then c_{g} : P → Q denotes the map x ↦ gxg^{−1} induced by conjugation by g ∈ G. We write ^{g}P = gP g^{−1} and P^{g} = g^{−1}P g. The transporter set of P into Q is N_{G}(P, Q) = {g ∈ G| ^{g}P ≤ Q}. Also let Hom_{G}(P, Q) = N_{G}(P, Q)/C_{G}(P) denote the set of group homomorphisms from P into Q induced by conjugation in G. Set Aut_{G}(P) = N_{G}(P)/C_{G}(P) and Out_{G}(P) = Aut_{G}(P)/Inn(P).

Definition 2.1. A fusion system F over a finite p-group S is a category whose objects are the subgroups of S and whose morphism sets Hom_{F}(P, Q) satisfy the following two conditions:

(i). Hom_{S}(P, Q) ⊆ Hom_{F}(P, Q) ⊆ Inj(P, Q); (ii). Every F-morphism factors as an F-isomorphism followed by an inclusion.

Definition 2.2. Let F be a fusion system over a finite p-group S. A subgroup P of S is (i). fully F-centralized if |C_{S}(P)| ≥ |C_{S}(φ(P))|, for all φ ∈ Hom_{F}(P, S); (ii). fully F-normalized if |N_{S}(P)| ≥ |N_{S}(φ(P))|, for all φ ∈ Hom_{F}(P, S); (iii). F-centric if C_{S}(φ(P)) = Z(φ(P)) for all φ ∈ Hom_{F}(P, S); (iv). F-radical if Inn(P) = O_{p}(Aut_{F}(P)); (v). F-essential if Q is F-centric and Out_{F}(P) has a strongly p-embedded^{1} subgroup.

2.3. For P ≤ S and φ ∈ Hom_{F}(P, S) define: N_{φ} = {x ∈ N_{S}(P) | φ∘c_{x}∘φ^{−1} ∈ Aut_{S}(φ(P))}. It is always the case that P C_{S}(P) ≤ N_{φ} ≤ N_{S}(P).

Definition 2.4. [BLO06] The fusion system F over a finite p-group S is saturated if the following two conditions hold. (I). For all P ≤ S which are fully F-normalized, P is fully F-centralized and Aut_{S}(P) is a

Sylow p-subgroup of Aut_{F}(P). (II). If P ≤ S and φ ∈ Hom_{F}(P, S) are such that φ(P) is fully F-centralized, then there is

a morphism φ̂ ∈ Hom_{F}(N_{φ}, S) such that φ̂|_{P} = φ.

2.5. Let F_{i}, i = 1, 2 be fusion systems over subgroups S_{i} of a p-group S. We say F_{1} is a fusion subsystem of F_{2} and write F_{1} ⊆ F_{2} if S_{1} ≤ S_{2} and Hom_{F_{1}}(P, Q) ⊆ Hom_{F_{2}}(P, Q) for all subgroups P and Q of S.

2.6. Let E and F be fusion systems over S with E ⊆ F. We say that E is normal in F and write E ⊴ F, if for every isomorphism φ : P → P′ in F and subgroups Q, Q′ of P we have φ_{|Q′}∘Hom_{E}(Q, Q′)∘φ_{|φ(Q)}^{−1} ⊆ Hom_{E}(φ(Q), φ(Q′)).

1A proper subgroup M of Out_{F}(P) is strongly p-embedded if M contains a Sylow p-subgroup Q of Out_{F}(P)

such that Q ≠ 1 and ^{φ}Q ∩ Q = {1} for every φ ∈ Out_{F}(P) \ M.

2.7. Assume that S is a finite p-group and for each i = 1, … n we are given subgroups S_{i} ≤ S and fusion systems F_{i} over S_{i}. Define F = ⟨F_{i} ; i ∈ I⟩ to be the fusion system generated by {F_{i} ; i ∈ I}, which is the smallest fusion system over S containing each member of the given collection. The fusion system generated by two saturated fusion systems need not be saturated.

Definition 2.8. Let F be a fusion system over a finite p-group S and let P be a subgroup of S. The normalizer of P in F is the fusion system N_{F}(P) on N_{S}(P) having as morphisms all group homomorphisms φ : Q → R, for Q, R subgroups of N_{S}(P), for which there exists a morphism φ̂ : P Q → P R in F satisfying φ̂(P) = P and φ̂|_{Q} = φ. If F = N_{F}(P) then we say that P is normal in F and we write P ⊴ F. In a saturated fusion system F, if P is fully F-normalized then N_{F}(P) is a saturated fusion system on N_{S}(P); for a proof of this statement see [Lin07, Theorem 3.2] for example.

Notation 2.9. Let O_{p}(F) denote the largest normal p-subgroup in F. It is a standard result that O_{p}(F) is contained in every F-centric F-radical subgroup of S.

2.10. Alperin’s fusion theorem and its refinement Alperin-Goldschmidt theorem hold for saturated fusion systems. The latter is the statement that every morphism in F is a composite of restrictions of automorphisms of S, and of automorphisms of fully F-normalized F-essential subgroups of S; see [Sta06, Theorem 2.8] for a proof. Note that an F-essential subgroup has to be a proper subgroup of S because Out_{F}(S) = Aut_{F}(S)/Aut_{S}(S) is a p′-group.

Fusion systems realized by finite groups. It is known, due to the work of Leary and Stancu [LS07] and Robinson [Rob07] that every fusion system on a finite p-group S is equal to the fusion system associated to a group G with Sylow p-subgroup S. In what follows we discuss the relationship between fusion systems and their associated groups.

2.11. We say that a group G has a (finite) Sylow p-subgroup S, if S is a finite p-subgroup of G and if every finite p-subgroup of G is conjugate to a subgroup of S. To a group G with Sylow p-subgroup S we can associate a category F_{S}(G) whose objects are the subgroups of S and whose morphisms are Hom_{F}(P, Q) = Hom_{G}(P, Q) whenever P and Q are subgroups of S.

2.12. Notice that in general if G has a Sylow p-subgroup, it does not follow that a subgroup H of G also has a Sylow p-subgroup, even if H is normal in G. However, every subgroup H of G contains a maximal normal p-subgroup, which we shall denote O_{p}(H).

Definition 2.13. A fusion system F is realized by a group G (not necessarily finite) if G contains S as a Sylow p-subgroup and F = F_{S}(G).

2.14. The classical examples of saturated fusion systems are the ones coming from finite groups. If G is a finite group and S is a Sylow p-subgroup of G, then F_{S}(G) is a saturated fusion system. A subgroup P ≤ S is fully F_{S}(G)-centralized if and only if C_{S}(P) ∈ Syl_{p}(C_{G}(P)), while P is fully F_{S}(G)-normalized if and only if N_{S}(P) ∈ Syl_{p}(N_{G}(P)). For proofs see [BLO03, Proposition 1.3]. Further, a subgroup P ≤ S is F-centric if and only if P is pcentric in G, and P is F-radical if and only if O_{p}(N_{G}(P)/P C_{G}(P)) = 1.

There are examples of fusion systems that cannot be realized by a finite group, they are called exotic. On the other side, there are examples where one can always construct a finite group with p-local structure equivalent to the given fusion system. This is the case for constrained fusion systems. The fusion system F is said to be constrained if O_{p}(F) is F-centric. We first mention a useful property of the saturated constrained fusion systems.

Proposition 2.15. Let F be a saturated fusion system over a p-group S, and let P be a fully F-normalized subgroup of S. If N_{F}(P) is constrained then O_{p}(N_{F}(P)) is F-centric.

Proof. Let P be a fully F-normalized subgroup of S. Set Q = O_{p}(N_{F}(P)) and let φ ∈ Hom_{F}(Q, S). As P is fully F-normalized, there exists an F-morphism ϕ : N_{S}(φ(P)) → N_{S}(P); see [Lin07, Lemma 2.6]. Then ϕ∘φ maps P to P and maps Q into N_{S}(P), since φ(Q) ≤ N_{S}(φ(P)). But Q = O_{p}(N_{F}(P)) and ϕ∘φ fixes Q. Since C_{S}(φ(Q)) ≤ N_{S}(φ(P)), we get ϕ(C_{S}(φ(Q))) ≤ C_{S}(Q). As N_{F}(P) is constrained, Q is centric in N_{F}(P), so ϕ(C_{S}(φ(Q))) ≤ Q. Then C_{S}(φ(Q)) ≤ ϕ^{−1}(Q) = φ(Q). □

Any constrained fusion system was proven to come from a finite group by Broto, Castellana, Grodal, Levi and Oliver.

Theorem 2.16. [BCG^{+}05, Theorem 4.3] Let F be a saturated constrained fusion system on a finite p-group S and set Q = O_{p}(F). Then there exists a unique up to isomorphism, finite group G having S as a Sylow p-subgroup and such that Q ⊴ G, C_{G}(Q) = Z(Q) and F = F_{S}(G). Furthermore G/Z(Q) ≃ Aut_{F}(Q).

A stronger uniqueness property was shown in [Asc08, 2.5]. It asserts that if G_{1} and G_{2}, two p′-reduced p-constrained finite groups, realize a saturated constrained fusion system F, then there is an isomorphism φ : G_{1} → G_{2} with φ_{|S} = Id_{S}. A result from elementary group theory which will be useful later.

Lemma 2.17. Suppose G is p′-reduced p-constrained finite group and let S be a Sylow psubgroup of G. If H is an overgroup of S in G then H is p′-reduced p-constrained.

Proof. The Lemma is a direct consequence of the fact that a group G is p′-reduced pconstrained if and only if C_{G}(O_{p}(G)) ≤ O_{p}(G). □

In our construction, we will also use the following result due to Aschbacher.

Proposition 2.18. [Asc, 1.1] Let F be a saturated constrained fusion system on a finite pgroup S and let G be a finite group that realizes F. Assume that E is a saturated constrained subsystem of F on S. Then there exists an overgroup H of S in G with E = F_{S}(H).

3. Chamber Systems

Chamber systems were introduced by Tits [Tit81]; the present treatment follows Scharlau [Sch95, Sections 1, 6] and Ronan [Ron89, Chapters 1, 4]. A comprehensive treatment of the theory of coverings of chamber systems can be found in [Ron80].

Basic notions. A chamber system over a set I is a nonempty set C (whose elements are called chambers) together with a family of equivalence relations (∼_{i}; i ∈ I) on C indexed by I. The equivalence classes with respect to ∼_{i} are called i-panels. Two distinct chambers c and d are called i-adjacent if they are contained in the same i-panel; we write c ∼_{i} d. A gallery of length n connecting two chambers c_{0} and c_{n} is a sequence γ : c_{0}, …, c_{n} of n + 1 chambers such that c_{i−1} and c_{i} are i_{j}-adjacent with i_{j} ∈ I, for all 1 ≤ j ≤ n. The sequence (i_{1}, …, i_{n}) is called the type of the gallery γ. If each i_{j} belongs to some subset J of I, then γ is called a J-gallery. The chamber system C is connected (or J-connected) if any two chambers can be joined by a gallery (or J-gallery). The J-connected components of C are called J-residues. Every Jresidue is a connected chamber system over the set J. The cardinality of the set I is the rank of the chamber system. The i-panels are rank 1 residues while the chambers are rank 0 residues.

3.1. We shall assume from now on that all chamber systems under consideration are connected and of finite rank.

A morphism φ : C → D between two chamber systems over I is a map defined on the chambers that preserves i-adjacency, i.e. if c, d ∈ C and c ∼_{i} d then φ(c) ∼_{i} φ(d) in D. We denote by Aut(C) the group of all automorphisms of C, where the term automorphism has the obvious meaning. If G is a group of automorphisms of C then C/G inherits a natural structure from C, and there is a chamber system C/G over I.

Coverings of chamber systems. A type preserving morphism of chamber systems φ : Ĉ → C is called an m-covering if it is surjective and if it maps each rank m residue of Ĉ isomorphically onto a rank m residue of C. Most of the properties discussed below can be formulated for m-coverings; however we shall restrict ourselves to the case when m = 2, which has greater relevance to the study of buildings as it transpares from [Tit81]. An elementary 2-homotopy of galleries is an alteration from a gallery of the form γωδ to a gallery γω′δ where ω and ω′ are galleries (with the same extremities) in a rank 2 residue. Two galleries are 2-homotopic if one can be transformed into the other by a sequence of elementary homotopies. The 2-homotopy relation is an equivalence relation on the set of galleries. If c is a chamber in a connected chamber system C, a closed gallery based at c will mean a gallery starting and ending at c. The fundamental group π^{2}(C, c) is the set of 2-homotopy classes [γ] of closed galleries γ based at c, together with the binary operation [γ] · [γ′] = [γγ′] where γγ′ means γ followed by γ′; using γ^{−1} to denote the reversal of γ, one has [γ]^{−1} = [γ^{−1}]. We call C simply 2-connected if it is connected and π^{2}(C, c) = 1. If b is another chamber in C, δ is a gallery from c to b and γ is a closed gallery based at c, the correspondence [γ] ↦ [δ^{−1}γδ] gives an isomorphism from π^{2}(C, c) to π^{2}(C, b). Given a morphism φ : (C, c) → (D, d) with φ(c) = d there is an induced map φ_{∗} : π^{2}(C, c) → π^{2}(D, d) via [γ] → [φ(γ)], which is a group homomorphism, and if φ is a 2-covering then φ_{∗} is injective; see [Ron89, Exercise 1, Chp. 4]. To put it briefly, 2-coverings of chamber systems have similar properties to the topological covers, with the appropriate adjustments on homotopy and simple connectivity; for details see [Ron80] and [Ron89, 4.2]. In particular we mention a few useful results, which are known from the covering theory of topological spaces.

Lemma 3.2. [Ron89, Lemma 4.4] Let φ : Ĉ → C be a 2-covering. Given a gallery γ in C starting at some chamber c, and given a chamber ĉ ∈ φ^{−1}(c), there is a unique gallery γ̂ in Ĉ starting at ĉ with φ(γ̂) = γ.

Proposition 3.3. [Ron80, Theorem 4.8] Let φ : (Ĉ, ĉ) → (C, c) be a 2-covering and let f : (D, d) → (C, c) be a morphism of chamber systems. The group f_{∗}(π^{2}(D, d)) is a subgroup of φ_{∗}(π^{2}(Ĉ, ĉ)) if and only if there is a unique chamber systems morphism f̂: (D, d) → (Ĉ, ĉ) such that φ∘f̂ = f.

Since the results of Proposition 3.3 will be used several times in the remainder of the section, we provide a proof.

Proof. If a morphism f̂: (D, d) → (Ĉ, ĉ) exists then for any [γ] ∈ π^{2}(D, d) we have

f_{∗}([γ]) = (φ∘f̂)_{∗}([γ]) = [(φ∘f̂)(γ)] = [φ(f̂(γ))] = φ_{∗}[f̂(γ)] ∈ φ_{∗}(π^{2}(Ĉ, ĉ))

and clearly f_{∗}(π^{2}(D, d)) ≤ φ_{∗}(π^{2}(Ĉ, ĉ)). Conversely, we may assume that f_{∗}(π^{2}(D, d)) ≤ φ_{∗}(π^{2}(Ĉ, ĉ)). Step 1: we construct a map f̂: (D, d) → (Ĉ, ĉ) with φ∘f̂ = f. Let d′ be a chamber in D, distinct from d. Let γ_{1} and γ_{2} be two galleries from d to d′ in D. By Lemma 3.2, the galleries f(γ_{1}) and f(γ_{2}) from c = f(d) to f(d′) have unique lifts to galleries γ̂_{1} and γ̂_{2} in Ĉ starting at ĉ, where φ(γ̂_{i}) = f(γ_{i}) for i = 1, 2. Next consider the closed gallery γ_{1}γ_{2}^{−1} based at d. Then f(γ_{1}γ_{2}^{−1}) is a closed gallery in C based at f(d) = c, whose 2-homotopy class lies in φ_{∗}(π^{2}(Ĉ, ĉ)) according to our assumption. Hence f(γ_{1}γ_{2}^{−1}) lifts uniquely to a closed gallery based at ĉ, and therefore γ̂_{1} and γ̂_{2} have the same endchamber, say ĉ′. Define f̂(d′) = ĉ′. The preceding argument shows that f̂: D → Ĉ is well-defined. Step 2: we show that f̂ is a morphism of chamber systems. Assume that d and d′ are i-adjacent in D. Then f(d) and f(d′) are i-adjacent and since φ is isomorphic on rank-1 residues it follows that φ^{−1}(f(d)) = f̂(d) and φ^{−1}(f(d′)) = f̂(d′) are i-adjacent, thus f̂ is indeed a chamber systems morphism. Step 3: we prove that the morphism f̂ is unique. Assume that f̂_{1} and f̂_{2} are two chamber systems morphisms, constructed as in Step 1. Let d′ be a chamber in D that is i-adjacent to d. Then f̂_{1}(d′) ∼_{i} f̂_{1}(d) = f̂_{2}(d) ∼_{i} f̂_{2}(d′) in Ĉ, but f(d) and f(d′) are i-adjacent in C and since φ is an isomorphism on rank 1 residues it follows that f̂_{1}(d′) = f̂_{2}(d′). Since D is connected (recall 3.1) the result follows. □

We record the following direct consequence of Proposition 3.3.

Corollary 3.4. [Sch95, Proposition 6.1.7] Let f be an automorphism of the chamber system C and let φ : (Ĉ, ĉ) → (C, c) be a 2-covering. Given ĉ′ ∈ φ^{−1}(f(c)) there exists a unique automorphism g of Ĉ satisfying g(ĉ) = ĉ′ with φ∘g = f ∘φ if and only if (f ∘φ)_{∗}(π^{2}(Ĉ, ĉ)) = φ_{∗}(π^{2}(Ĉ, ĉ′)).

Notation 3.5. We let Aut(φ) denote the group of deck transformation of the 2-covering φ : (Ĉ, ĉ) → (C, c); thus the automorphism g : Ĉ → Ĉ is an element of Aut(φ) if φ∘g = φ.

The following result characterizes 2-coverings of C which correspond to normal subgroups of the fundamental group π^{2}(C, c); such 2-coverings are called normal in topology.

Proposition 3.6. Let φ : (Ĉ, ĉ) → (C, c) be a 2-covering.

(i). φ_{∗}(π^{2}(Ĉ, ĉ)) ⊴ π^{2}(C, c) if and only if for each chamber ĉ′ ∈ φ^{−1}(c) there is a deck

transformation g ∈ Aut(φ) with the property g(ĉ) = ĉ′. (ii). Aut(φ) ≃ N_{π^{2}(C,c)}(φ_{∗}(π^{2}(Ĉ, ĉ)))/φ_{∗}(π^{2}(Ĉ, ĉ)). (iii). C ≃ Ĉ/Aut(φ) if and only if φ_{∗}(π^{2}(Ĉ, ĉ)) ⊴ π^{2}(C, c).

Proof. (i). Set N(H) = N_{π^{2}(C,c)}(φ_{∗}(π^{2}(Ĉ, ĉ))) with H = φ_{∗}(π^{2}(Ĉ, ĉ)). Observe that changing the base point ĉ ∈ φ^{−1}(c) to ĉ′ ∈ φ^{−1}(c) corresponds to conjugating H by an element [γ] ∈ π^{2}(C, c), where γ is a closed gallery based at c that lifts to a gallery γ̂ from ĉ to ĉ′. Thus [γ] is in N(H) if and only if φ_{∗}(π^{2}(Ĉ, ĉ)) = φ_{∗}(π^{2}(Ĉ, ĉ′)), which by Corollary 3.4 is equivalent to the existence of a (unique) deck transformation g ∈ Aut(φ) with g(ĉ) = ĉ′. (ii). Define the morphism Φ : N(H) → Aut(φ) given by Φ([γ]) = g where [γ] is sent to the deck transformation g which takes ĉ to ĉ′ (γ is as in part (i), which means that the unique lift γ̂ of γ at ĉ is a gallery from ĉ to ĉ′). There exists a unique such deck transformation g by Proposition 3.3 and Corollary 3.4. Observe that Φ is a group homomorphism, for if γ′ is another closed gallery based at c, with Φ([γ′]) = g′ and such that g′(ĉ) = ĉ′′ then γγ′ lifts to γ̂g(γ̂′) a gallery from ĉ to g(ĉ′′) = g(g′(ĉ)), given that γ̂′ is the unique lift of γ′ at ĉ. Hence g∘g′ is the deck transformation corresponding to [γ] · [γ′]. For g ∈ Aut(φ), let γ̂ be the gallery from ĉ to ĉ′ = g(ĉ). (Note here that Ĉ is assumed to be connected, see 3.1). Then for γ = φ(γ̂), [γ] lies in the preimage of g. So Φ is surjective. The kernel of Φ consists of those 2-homotopy classes [γ] which lift to closed galleries in Ĉ, that is the elements of H. (iii). Since φ_{∗}(π^{2}(Ĉ, ĉ)) ⊴ π^{2}(C, c), it follows by part (i) of this Proposition, that the group of deck transformations Aut(φ) acts chamber transitively on Ĉ. □

3.7. Let φ : (Ĉ, ĉ) → (C, c) be a 2-covering of chamber systems. A lift of f ∈ Aut(C) corresponding to φ, is a morphism f̂ ∈ Aut(Ĉ) with φ∘f̂ = f ∘φ.

Definition 3.8. A 2-covering φ : (Ĉ, ĉ) → (C, c) is called universal if whenever ψ : (C′, c′) → (C, c) is a 2-covering, there exists some 2-covering α : (Ĉ, ĉ) → (C′, c′) such that ψ∘α = φ. It was shown by Tits [Tit81, 5.1] that universal 2-coverings of chamber systems always exist; they are unique up to isomorphism. A 2-covering φ : (Ĉ, ĉ) → (C, c) is universal if and only if Ĉ is simply 2-connected; see [Ron89, Proposition 4.6].

In the special case when Ĉ is simply 2-connected, an application of Proposition 3.6 and Corollary 3.4 gives the following, cf. [Sch95, Proposition 6.1.8].

Proposition 3.9. Let φ : (Ĉ, ĉ) → (C, c) be a universal 2-covering.

(i). Aut(φ) ≃ π^{2}(C, c) and C ≃ Ĉ/Aut(φ).

(ii). For any subgroup G ≤ Aut(C) the set of liftings:

Ĝ := {ĝ ∈ Aut(Ĉ) : φ∘ĝ = g∘φ for some g ∈ G}

is a subgroup of Aut(Ĉ), and the map Ĝ → G, ĝ ↦ g, is a surjective group homomorphism with kernel Aut(φ). For any chamber ĉ ∈ Ĉ, the stabilizer StabĜ(ĉ) is mapped isomorphically onto Stab_{G}(φ(ĉ)).

Parabolic systems and chamber systems. Next, we shall discuss the relation between chamber systems and chamber transitive groups of automorphisms.

3.10. Let G be a group and let I be a finite index set of cardinality n. Let B, G_{i}, i ∈ I be a collection of subgroups of G satisfying the following properties:

(P_{1}). G = ⟨G_{i} ; i ∈ I⟩ ≠ ⟨G_{j} ; j ∈ J ⊊ I⟩;
(P_{2}). G_{i} ∩ G_{j} = B, for any distinct i, j ∈ I;
(P_{3}). B ≠ G_{i} for all i ∈ I;
(P_{4}). ∩_{g∈G}B^{g} = 1.

The family 𝒫(G) = (B, G_{i} ; i ∈ I) is called a parabolic system of rank n in G.

3.11. To 𝒫(G) we can associate a chamber system C(𝒫(G)) as follows. The left cosets of B in G correspond to the chambers. Two chambers gB and hB are i-adjacent if gG_{i} = hG_{i} where g, h ∈ G and i ∈ I, whence h^{−1}g ∈ G_{i}. Since G = ⟨G_{i} ; i ∈ I⟩ the chamber system C is connected. The group G acts on C via left multiplication, an element x ∈ G takes a chamber gB into a chamber xgB; if gB ∼_{i} hB is an i-panel, then gG_{i} = hG_{i}, which implies xgG_{i} = xhG_{i} and therefore xgB ∼_{i} xhB. The action of G on C is chamber transitive; it is also faithful because of (P_{4}). Henceforth the group G will be identified with the automorphisms of C induced by G.

Notation 3.12. Set G_{J} = ⟨G_{j} ; j ∈ J⟩ for all nonempty subsets J ⊆ I, with the convention that G_{∅} := B. In particular G_{I} = G, G_{i} = G_{{i}} and we write G_{ij} for G_{{i,j}}. It is a consequence of (P_{2}) that G_{J} is the stabilizer in G of the J-residue which contains the chamber B.

3.13. Given 𝒫(G) = (B, G_{i} ; i ∈ I) and 𝒫(G′) = (B′, G′_{i} ; i ∈ I), two parabolic systems of rank n over the same set of types; a group homomorphism φ : G → G′ is called a 2-morphism of parabolic systems, if φ(B) = B′, φ(G_{i}) = G′_{i} and φ(G_{ij}) = G′_{ij} for all i, j ∈ I with i ≠ j. In particular φ(G) = G′, thus φ is surjective.

Lemma 3.14. Let φ : 𝒫(G) → 𝒫(G′) be a 2-morphism of parabolic systems with the property that the restriction of φ to G_{ij} is a group isomorphism for every distinct i, j ∈ I. Then there is an induced 2-covering φ_{∗} : C(𝒫(G)) → C(𝒫(G′)) between the associated chamber systems. Furthermore Aut(φ_{∗}) ∩ G = Ker(φ) and C(𝒫(G′)) ≃ C(𝒫(G))/Ker(φ).

Proof. For gB a chamber in C(𝒫(G)), define φ_{∗}(gB) = φ(g)B′, and for an i-panel gG_{i} set φ_{∗}(gG_{i}) = φ(g)G′_{i}. It is straightforward to check that φ_{∗} is a morphism of chamber systems. Since φ is surjective it follows that φ_{∗} is also surjective, and a 2-covering of chamber systems. Clearly Ker(φ) ≤ Aut(φ_{∗}) ∩ G, so it remains to show the opposite inclusion. Let h ∈ Aut(φ_{∗}) ∩ G. Then, according to 3.5, φ(h)φ(g)B′ = φ(g)B′, for every g ∈ G. It follows that φ(h) ∈ xB′x^{−1} with x ∈ G′. Thus φ(h) ∈ ∩_{x∈G′}xB′x^{−1} = 1 by (P_{4}), and therefore Kerφ = Aut(φ_{∗}) ∩ G. Observe that, for all g, h ∈ G, φ_{∗}(gB) = φ_{∗}(hB) if and only if g^{−1}h ∈ φ^{−1}(B′) = Ker(φ)B. Hence the orbits of Ker(φ) in C(𝒫(G)) are just the fibres of φ_{∗} and C(𝒫(G′)) ≃ C(𝒫(G))/Ker(φ). □

Colimits and covers.

3.15. Let I be a finite index set. A diagram of (finite) groups is a pair A = (𝒢, Υ) which consists of a collection of finite groups 𝒢 = {B, G_{i}, G_{ij} ; i, j ∈ I, i ≠ j} and a family of injective group homomorphisms Υ = {τ_{B}^{G_{i}} : B → G_{i}, τ_{G_{i}}^{G_{ij}} : G_{i} → G_{ij} ; i, j ∈ I} such that τ_{G_{j}}^{G_{ij}} ∘τ_{B}^{G_{j}} = τ_{G_{i}}^{G_{ij}} ∘τ_{B}^{G_{i}}, for all i, j ∈ I. A completion of A is a pair (G̃, Φ) where G̃ is a group endowed with a family Φ of group homomorphisms ϕ : B → G̃, ϕ_{i} : G_{i} → G̃ and ϕ_{ij} : G_{ij} → G̃ which commute with the morphisms in Υ. The colimit (G̃, Φ) of A (also known as the universal completion of A) has the property that if (G̃′, Φ′) is another completion, there is exactly one group homomorphism f : G̃ → G̃′ such that f ∘ϕ = ϕ′, f ∘ϕ_{i} = ϕ′_{i} and f ∘ϕ_{ij} = ϕ′_{ij}, for all i, j ∈ I. The group G̃ always exists (although it might be the trivial group) and can be regarded as a group whose set of generating symbols is the disjoint union of elements of G_{i} and with relations given by the disjoint union of those relations holding in G_{ij} that involve only generators from G_{i} and G_{j}. Thus G̃ is the largest group realizing the diagram of groups, any other completion of A is a quotient of G̃. A completion for which the morphisms in Φ are injective is called faithful. The pair A = (𝒢, Υ) admits a faithful completion if and only if its universal completion is faithful.

3.16. If 𝒫(G) = (B, G_{i} ; i ∈ I) is a parabolic system in G there is a corresponding diagram of groups 𝒫 = {B, G_{i}, G_{ij} ; i, j ∈ I, i ≠ j} with G_{ij} := ⟨G_{i}, G_{j}⟩. Let G̃ be the universal completion of 𝒫. For every i ∈ I, the subgroup G_{i} of G lifts to a subgroup G̃_{i} of G̃, and similarly B lifts to B̃, thus 𝒫(G̃) = (B̃, G̃_{i} ; i ∈ I) is a parabolic system in G̃.

Theorem 3.17. [Sch95, Proposition 6.5.2] The chamber system C(𝒫(G̃)) is the universal 2covering of the chamber system C(𝒫(G)). In particular, the natural homomorphism G̃ → G is an isomorphism if and only if C(𝒫(G)) is simply 2-connected.

Proof. Set C̃ = C(𝒫(G̃)) and C = C(𝒫(G)). Let f : C̃ → C be the 2-covering defined as in Lemma 3.14. Let φ : Ĉ → C be the universal 2-covering of C, which exists by [Tit81, 5.1]. According to Proposition 3.9, the group G ≤ Aut(C) lifts through φ to a subgroup Ĝ of Aut(Ĉ). The action of Ĝ on Ĉ is chamber transitive; this is a direct consequence of Proposition 3.9(ii) and of Corollary 3.4 (or Proposition 3.6(i)). Let ĉ be a chamber of Ĉ with φ(ĉ) = c, where c is the chamber corresponding to B in 𝒫(G). Let 𝒫(Ĝ) = (B̂, Ĝ_{i} ; i ∈ I) be the parabolic system defined by ĉ in Ĝ, where B̂ = StabĜ(ĉ) and Ĝ_{i} are the stabilizers in Ĝ of the i-panels containing ĉ. Another application of Proposition 3.9 gives a chamber system isomorphism C ≃ Ĉ/Aut(φ) and a group isomorphism Aut(φ) ≃ π^{2}(C, c). The canonical projection ψ : Ĝ → G induces a 2-morphism of parabolic systems 𝒫(Ĝ) → 𝒫(G). Moreover, ψ is an isomorphism when restricted to B̂, Ĝ_{i}, Ĝ_{ij} with i, j ∈ I; see also [Ron89, Exercise 8, Chp. 4]. We can identify the subgroups B̂, Ĝ_{i} and Ĝ_{ij} of 𝒫(Ĝ) with their counterparts in 𝒫(G) and we can regard 𝒫(G) as a family of subgroups of Ĝ. As G̃ is the universal completion of the family 𝒫, there is a unique surjective homomorphism ζ : G̃ → Ĝ inducing the identity isomorphism on each of the subgroups B, G_{i}, G_{ij} with i, j ∈ I, i ≠ j. This group homomorphism induces a 2-covering ζ_{∗} : C̃ → Ĉ, and since Ĉ is a universal 2-covering, it follows that C̃ and Ĉ are isomorphic. □

4. Complexes of groups and saturated fusion systems

In this section, we consider discrete groups that contain finite Sylow p-subgroups. In particular, we investigate the case when G is obtained as a completion of a diagram of finite groups and using the properties of an associated chamber system we identify conditions under which the fusion system F_{S}(G) is saturated.

Hypothesis 4.1. Let A = (𝒢, Υ) be a diagram of groups with 𝒢 = {B, G_{i}, G_{ij} ; i, j ∈ I, i ≠ j}, a collection of finite groups, Υ = {τ_{B}^{G_{i}} : B → G_{i}, τ_{G_{i}}^{G_{ij}} : G_{i} → G_{ij} ; i, j ∈ I}, a family of inclusion maps, and I a finite index set. Let G denote a completion of A, as defined in 3.15. We shall assume that this completion is faithful, so B, G_{i}, G_{ij}, with i, j ∈ I can be regarded as subgroups of G. To 𝒢 and G we associate a chamber system C in the standard way. The chambers of C are the left cosets of B in G, two chambers gB and hB are i-adjacent if gG_{i} = hG_{i}. The group G acts chamber transitively, by left multiplication on C. Further, we assume that C^{P} is connected for every finite p-subgroup of G. Since the empty set is disconnected, it follows that C^{P} ≠ ∅.

4.2. For any pair of groups H, G let Rep(H, G) := Inn(G)\Inj(H, G) and let [α] ∈ Rep(H, G) denote the class of α ∈ Inj(H, G).

4.3. If H is a finite group, we denote by Rep(H, C) the chamber system whose chambers are the elements of Rep(H, B). The i-panels are simply represented by the elements of

Rep(H, B, G_{i}) := {[γ] ∈ Rep(H, G_{i}) : γ ∈ Inj(H, G_{i}) with γ(H) ≤ B}.

Two chambers [α] and [β] are i-adjacent if [τ_{B}^{G_{i}}∘α] = [τ_{B}^{G_{i}}∘β] in Rep(H, G_{i}) which means that τ_{B}^{G_{i}}∘α = c_{g}∘τ_{B}^{G_{i}}∘β for some element g ∈ G_{i}. In particular, the i-panel [γ] contains the chamber [α] if [τ_{B}^{G_{i}}∘α] = [γ] in Rep(H, G_{i}). Observe that Aut(H) acts on Rep(H, C) via φ · [α] = [α∘φ^{−1}], and if H ≤ G, then N_{G}(H) acts on Rep(H, C) via g · [α] = [α∘c_{g^{−1}}].

The next result is Lemma 4.1 from [BLO06] written in terms of chamber systems. We shall use the customary notation π_{0}(X) from topology to denote the set of connected components of a space X.

Lemma 4.4. Let G and C be as in 4.1 with P a p-subgroup of B. The following hold.

(a). There is an N_{G}(P)-equivariant isomorphism of chamber systems between C^{P} /C_{G}(P) and Rep(P, C)_{0}, the connected component of Rep(P, C) which contains [τ_{P}^{B}], where τ_{P}^{B} denotes the inclusion map of P into B.

(b). The natural map Φ_{P} : π_{0}(Rep(P, C)) → Rep(P, B, G) is a bijection. In particular, [α] lies in the connected component of [τ_{P}^{B}] if and only if α ∈ Hom_{G}(P, B).

Proof. Note that if K ≤ G and gK ∈ (G/K)^{P} then P ≤ gKg^{−1}. Hence the class [c_{g^{−1}}] in Rep(P, K) of c_{g^{−1}} : P → K given by x ↦ g^{−1}xg, depends on the coset gK only. Define the following map

f_{P} : C^{P} → Rep(P, C)

given by f_{P} (gB) = [c_{g^{−1}}] where [c_{g^{−1}}] ∈ Rep(P, B) and g ∈ G. Step 1: f_{P} is a morphism of chamber systems. We have to show that if gB and hB are i-adjacent chambers in C^{P}, then f_{P} (gB) = [c_{g^{−1}}] and f_{P} (hB) = [c_{h^{−1}}] are i-adjacent in Rep(P, C). Since h^{−1}g ∈ G_{i} it follows that h^{−1} = pg^{−1} for some p ∈ G_{i} and that gG_{i}g^{−1} = hG_{i}h^{−1}. Hence τ_{B}^{G_{i}}∘c_{h^{−1}} = τ_{B}^{G_{i}}∘c_{p}∘c_{g^{−1}} which says that [c_{h^{−1}}] = [c_{g^{−1}}] in Rep(P, B, G_{i}) and therefore f_{P} (gB) and f_{P} (hB) are i-adjacent in Rep(P, C). Step 2: Im(f_{P}) is Rep(P, C)_{0}, the connected component of [τ_{P}^{B}] in Rep(P, C). Let [α] ∈ Rep(P, B) be i-adjacent to some chamber [c_{g^{−1}}] ∈ Im(f_{P}). Then there exists h ∈ G_{i} such that α = c_{h^{−1}}∘c_{g^{−1}} = c_{(gh)^{−1}} and so [α] ∈ Im(f_{P}). This shows that an i-panel lies in Im(f_{P}) if one of its chambers lies in Im(f_{P}) and therefore Im(f_{P}) is a union of connected components of Rep(P, C). But since C^{P} is assumed to be connected, we obtain that Im(f_{P}) is connected and hence a connected component of Rep(P, C). Step 3: f_{P} is an N_{G}(P)-equivariant map. The group N_{G}(P) acts on Rep(P, C) via g · [α] = [α∘c_{g^{−1}}] and it is easy to see that this action preserves i-adjacency. Since G acts on C, N_{G}(P) acts on the chamber subsystem C^{P} fixed by the action of P. If n ∈ N_{G}(P) and gB is a chamber in C^{P} then

f_{P} (n(gB)) = f_{P} (ngB) = [c_{(ng)^{−1}}] = [c_{g^{−1}}∘c_{n^{−1}}] = n · [c_{g^{−1}}] = n · f_{P} (gB)

Hence the chamber systems morphism f_{P} is N_{G}(P)-equivariant. It also follows that f_{P} is C_{G}(P)-equivariant and there is an induced chamber systems morphism between the quotients C^{P} /C_{G}(P) ⟶ Rep(P, C)/C_{G}(P). But C_{G}(P)-acts trivially on Rep(P, C) by definition and therefore there is a morphism C^{P} /C_{G}(P) ⟶ Rep(P, C). Step 4: f_{P} induces an isomorphism of chamber systems C^{P} /C_{G}(P) ≃ Im(f_{P}). Two chambers gB and hB of C^{P} have the same image f_{P} (gB) = f_{P} (hB) if and only if [c_{g^{−1}}] = [c_{h^{−1}}] in Rep(P, B). Then c_{h^{−1}} = c_{y^{−1}}∘c_{g^{−1}} with y ∈ B and h^{−1} = (gy)^{−1}z^{−1} where z ∈ C_{G}(P). Thus h = zgy which shows that h ∈ C_{G}(P)gB and therefore gB and hB are in the same C_{G}(P)-orbit of C. Part (a) of the Lemma follows from Steps 1-4. It remains to show that the map Φ_{P} is bijective. Surjectivity is clear. Let α, β ∈ Inj(P, B) be such that Φ_{P} ([α]) = Φ_{P} ([β]). Then β∘α^{−1} = c_{g} for some g ∈ G. So [β∘α^{−1}] ∈ Im(f_{α(P)}) and thus by Step 2, [β∘α^{−1}] lies in the same connected component as [τ_{α(P)}^{B}] in Rep(α(P), C). Therefore [α] and [β] are in the same connected component in Rep(P, C). It follows that Φ_{P} is injective. □

Lemma 4.5. Let G and C be as in 4.1. If S is a Sylow p-subgroup of B then S is a Sylow p-subgroup of G.

Proof. Let P be a finite p-subgroup of G. Since C^{P} is nonempty and G is chamber transitive, there is an element g ∈ G such that gB ∈ C^{P}. Hence P ≤ gBg^{−1} and since gSg^{−1} ∈ Syl_{p}(gBg^{−1}) this shows that P is G-conjugate to a subgroup of S. □

Remark 4.6. Henceforth it makes sense to consider the fusion system F_{S}(G) over S realized by G, as defined in 2.13.

Lemma 4.7. Let G and C be as in 4.1. Assume that S is a Sylow p-subgroup of B and set F = F_{S}(G). Then every morphism in F is the composite of morphisms φ_{1}, …, φ_{n} with φ_{i} ∈ F_{S}(G_{j_{i}}) for some j_{i} ∈ I.

Proof. Let φ = c_{g} ∈ Hom_{F}(P, Q) for P, Q ≤ S and g ∈ G. Then P, ^{g}P ≤ S ≤ B and thus P fixes the chambers B and g^{−1}B in C. So since C^{P} is connected, there is a gallery γ in C^{P} from B to g^{−1}B. Recall that two chambers xB and yB are i-adjacent if and only if x = yg_{i} for some g_{i} ∈ G_{i}. Hence we can write γ = (g_{0}B, g_{0}g_{1}B, …, g_{0}g_{1} ⋯ g_{n}B) with g_{0} = 1, g_{0}g_{1} ⋯ g_{n} = g^{−1} and g_{i} ∈ G_{j_{i}} for some j_{i} ∈ I. Set g̃_{i} := g_{0} ⋯ g_{i}. As P stabilizes the gallery γ, P g̃_{i} ≤ B. Since S ∈ Syl_{p}(B) we can choose b_{i} ∈ B such that P^{g̃_{i}b_{i}} ≤ S for every 1 ≤ i ≤ n. As P g̃_{0} = P ≤ S and P g̃_{n} = P^{g^{−1}} ≤ S we may take b_{0} = 1 = b_{n}. Set h_{i} := b_{i−1}^{−1}g_{i}b_{i} ∈ G_{j_{i}} for i = 1, … n. Then:

h_{1} ⋯ h_{i} = (g_{1}b_{1}) · (b_{1}^{−1}g_{2}b_{2}) ⋯ (b_{i−1}^{−1}g_{i}b_{i}) = g_{1} ⋯ g_{i}b_{i} = g̃_{i}b_{i}

Hence P^{h_{1}···h_{i}} ≤ S for all i ≤ n, and g^{−1} = g̃_{n} = h_{1} … h_{n}. Thus c_{g} factors as c_{g} = c_{h_{n}^{−1}}∘ … ∘c_{h_{1}^{−1}} with c_{h_{i}^{−1}} : P^{h_{1}···h_{i−1}} → P^{h_{1}···h_{i}} a morphism in F_{S}(G_{j_{i}}). This completes the proof. □

Lemma 4.8. Let G and C be as in 4.1. Let S be a Sylow p-subgroup of B. Set F = F_{S}(G). Assume that every subgroup P of S that is essential in F_{S}(G_{i}), for i ∈ I, is F-centric. Then every morphism in F is a composite of restrictions of morphisms between F-centric subgroups.

Proof. The F-morphism φ = c_{g} ∈ Hom_{F}(P, Q) is a composite of morphisms φ_{1}, …, φ_{n} with φ_{i} ∈ F_{S}(G_{j_{i}}), for j_{i} ∈ I; see Lemma 4.7. Each group G_{j_{i}} is finite, S is a Sylow p-subgroup of G_{j_{i}} and therefore the fusion systems F_{i} = F_{S}(G_{j_{i}}) are all saturated. By an application of Alperin-Goldschmidt theorem for fusion systems, see 2.10, we obtain that each φ_{i} can be written as a composite of restrictions of automorphisms of S and of automorphisms of fully F_{S}(G_{j_{i}})-normalized F_{S}(G_{j_{i}})-essential subgroups of S. But S itself is F-centric while F_{S}(G_{j_{i}})-essential subgroups are F-centric for all j_{i} ∈ I, by assumption. Therefore the Lemma is proved. □

Proposition 4.9. Let G and C be as in 4.1. Assume that S is a Sylow p-subgroup of B and set F = F_{S}(G). Let P be any subgroup of S that is F-centric and fully F-normalized. Suppose that:

(i). the group Aut_{G}(P) acts chamber transitively on C^{P} /C_{G}(P); (ii). given any p-subgroup R of Aut_{G}(P), the chamber subsystem of C^{P} /C_{G}(P) fixed by the

action of R is connected.

Then Aut_{S}(P) is a Sylow p-subgroup of Aut_{F}(P).

Proof. Let P be a F-centric fully F-normalized subgroup of S. The group Aut_{G}(P) acts on C^{P} /C_{G}(P) and according to Steps 3 and 4 from the proof of Lemma 4.4, it also acts on Rep(P, C)_{0}. Given φ ∈ Aut_{G}(P), observe that φ · [τ_{P}^{B}] = [τ_{P}^{B}∘φ^{−1}] = [τ_{P}^{B}] if and only if τ_{P}^{B}∘φ = c_{b}∘τ_{P}^{B} for some b ∈ B. Hence φ = c_{b |P} and the stabilizer of the chamber [τ_{P}^{B}] is Aut_{B}(P). As P is fully F-normalized, it is fully F_{B}(S)-normalized and Aut_{S}(P) is a Sylow p-subgroup of Aut_{F_{S}(B)}(P) = Aut_{B}(P). To obtain the conclusion, apply the argument from the proof of Lemma 4.5, with C^{P} /C_{G}(P) in place of C and with Aut_{G}(P) in place of G. □

Proposition 4.10. Let G and C be as in 4.1. Assume that S is a Sylow p-subgroup of B and set F = F_{S}(G). Let P be a F-centric subgroup of S. Assume that if R is a p-subgroup of Aut_{G}(P) then the chamber subsystem fixed by the action of R on C^{P} /C_{G}(P) is connected. Then for any φ ∈ Hom_{F}(P, S) there is a morphism \overline{φ} ∈ Hom_{F}(N_{φ}, S) with the property that \overline{φ}_{|P} = φ.

Proof. The proof of the present Proposition will be achieved in two steps. We start with some necessary notation. Let φ ∈ Hom_{F}(P, S) where P is F-centric and let N_{φ} be as in 2.3. Set K = Aut_{N_{φ}}(P) = N_{φ}/Z(P) and observe that since P ≤ N_{φ}, the subgroup N_{φ} is also F-centric. Consider the map

Γ : Rep(N_{φ}, C)_{0} ⟶ Rep(P, C)_{0}

induced by the restriction N_{φ} → P, between the connected components of the inclusion maps τ_{N_{φ}}^{B} and τ_{P}^{B}. The map is well-defined by Lemma 4.4(b). Let Rep(P, C)_{0}^{K} denote the chamber subsystem of Rep(P, C)_{0} fixed by K. Step 1: Im(Γ) = Rep(P, C)_{0}^{K}. Let [ψ] ∈ Rep(N_{φ}, C)_{0} and let h ∈ N_{φ} and consider the following commutative diagram

P ⟶^{τ_{P}^{Nφ}} N_{φ} ⟶^{ψ} B
c_{h}↓  ^{c_{h}}↓  ^{c_{ψ(h)}}↓
P ⟶^{τ_{P}^{Nφ}} N_{φ} ⟶^{ψ} B

from which we see that c_{ψ(h)} = ψ_{|P} ∘c_{h}∘ψ_{|P}^{−1}. Thus the Γ-image of [ψ] lies in Rep(P, C)_{0}^{K}.

We will show that given any chamber [α] in Rep(P, C)_{0}^{K} and any i-panel [β] in Rep(N_{φ}, C)_{0} with the property that [β_{|P}] is a panel of [α], the chamber [α] lies in Im(Γ). This will finish the proof of this Step since (C^{P} /C_{G}(P))^{K} which is isomorphic as a chamber system to Rep(P, C)_{0}^{K}, by Lemma 4.4, is assumed to be connected. Denote K′ = αKα^{−1} ≤ Aut(P′) with P′ = α(P). Set N′ = {a ∈ N_{B}(P′) | c_{a} ∈ K′} and observe that Aut_{N′}(P′) ≤ K′. We shall prove that Aut_{N′}(P′) = K′. Since [α] is fixed by K, for any h ∈ N_{φ}, [α∘c_{h}] = [α] in Rep(P, B), thus there is an element b ∈ B such that c_{b} = α∘c_{h}∘α^{−1} : P′ → P′, using that c_{h} ∈ K. Therefore K′ ≤ Aut_{B}(P′) and in fact K′ ≤ Aut_{N_{B}(P′)}(P′). Let f ∈ K′ so there exists a ∈ N_{B}(P′) with f = c_{a}. This implies a ∈ N′ and c_{a} ∈ Aut_{N′}(P′) and therefore K′ ≤ Aut_{N′}(P′). Next we note that K′ = Aut_{Q}(P′), where Q ∈ Syl_{p}(N′) which is true since K′ is a p-group.

Recall that β : N_{φ} → G_{i}, and since [α] lies in the i-panel [β_{|P}], it follows that β_{|P} = c_{g}∘α, for some g ∈ G_{i}. Given any z ∈ Q, the morphism c_{g}∘c_{z}∘c_{g^{−1}} : β(P) → β(P) corresponds to conjugation by an element in gQg^{−1}. It follows from the definition of K′ that for any z ∈ Q there is an element y ∈ N_{φ} such that c_{z} = α∘c_{y}∘α^{−1}. Therefore c_{g}∘c_{z}∘c_{g^{−1}} = c_{g}∘α∘c_{y}∘α^{−1}∘c_{g^{−1}} = β∘c_{y}∘β^{−1} = c_{β(y)} with β(y) ∈ β(N_{φ}) and it follows that gQg^{−1} ≤ β(N_{φ}) · C_{G_{i}}(β(P)) := H. Observe that if P is F-centric then P is F_{S}(G_{i})-centric and therefore P is p-centric in G_{i}. Therefore β(P) is p-centric in G_{i}. We claim that both gQg^{−1} and β(N_{φ}) are Sylow psubgroups of H. Notice that Z(β(P)) = β(Z(P)) ∈ Syl_{p}(C_{G_{i}}(β(P))) so the unique Sylow p-subgroup of C_{G_{i}}(β(P)) is already in β(N_{φ}). Next note that K = Aut_{N_{φ}}(P) = N_{φ}/Z(P) ≃ K′ = Aut_{Q}(P′) = Q/Z(P′) so |Q| = |N_{φ}| = |β(N_{φ})|. The claim is proved; gQg^{−1} and β(N_{φ}) are Sylow p-subgroups of H. It follows that there is an element h ∈ C_{G_{i}}(β(P)) with the property that h(gQg^{−1})h^{−1} = β(N_{φ}) so Q = c_{(hg)^{−1}}(β(N_{φ})). Set \overline{α} = c_{(hg)^{−1}}∘β : N_{φ} → Q with hg ∈ G_{i}. Finally observe that \overline{α}_{|P} = c_{(hg)^{−1}}∘β_{|P} = c_{g^{−1}}∘β_{|P} = α since h centralizes β(P). Thus [\overline{α}] = [β] in Rep(N_{φ}, B, G_{i}) showing that [α] is in the image of the restriction map. Step 2: Given φ ∈ Hom_{F}(P, S) with P is F-centric, then there is a morphism \overline{φ} ∈ Hom_{F}(N_{φ}, S) with \overline{φ}_{|P} = φ. Consider the chamber [φ] which lies in Rep(P, C)_{0}^{K} since the automorphisms in K are induced by elements of N_{φ}. By Step 1 there is a morphism ψ : N_{φ} → B with [ψ] ∈ Rep(N_{φ}, C)_{0} and φ = c_{g^{−1}}∘ψ_{|P} for some g ∈ B. We will use the properties of the saturated fusion system F_{S}(B) and the fact that c_{g^{−1}} is a morphism in this fusion system. We may assume that ψ(N_{φ}) ≤ S; if this is not the case choose an element h ∈ B with hψ(N_{φ})h^{−1} ≤ S and replace ψ by ψ′ = c_{h}∘ψ. This is possible since [ψ] = [ψ′] in Rep(N_{φ}, B). Next we prove that ψ(N_{φ}) ≤ N_{φ∘ψ^{−1}} = N_{c_{g^{−1}}} with g ∈ B. Let z ∈ ψ(N_{φ}) so z = ψ(y) for y ∈ N_{φ} and consider φ∘ψ^{−1}∘c_{z}∘ψ∘φ^{−1} = φ∘c_{y}∘φ^{−1} ∈ Aut_{S}(φ(P)) since y ∈ N_{φ}. Hence z ∈ N_{φ∘ψ^{−1}}. Thus φ∘ψ^{−1} = c_{g^{−1}} extends to a map χ : ψ(N_{φ}) → S in F_{S}(B) and therefore \overline{φ} := χ∘ψ : N_{φ} → S is such that χ∘ψ(y) = φ(y) for all y ∈ P. This ends the proof of the second saturation condition for F_{S}(G). □

After assembling these results we obtain the following generalization to chamber systems of [BLO06, Theorem 4.2]. Broto, Levi and Oliver considered in their construction the case when C was a finite tree of finite groups.

Theorem 4.11. Let G and C be as in 4.1. Assume that S is a Sylow p-subgroup of B and set F = F_{S}(G). Suppose the following hold.

(a). If P is a subgroup of S that is F-centric and fully F-normalized then Aut_{G}(P) acts

chamber transitively on C^{P} /C_{G}(P). (b). If P is a subgroup of S that is F-centric and if R is a p-subgroup of Aut_{G}(P) then

(C^{P} /C_{G}(P))^{R} is connected. (c). If P ≤ S is an essential p-subgroup of F_{S}(G_{i}), for i ∈ I, then P is F-centric.

Then F is a saturated fusion system over S.

Proof. Our proof is a compilation of the last two Propositions and three Lemmas. After we obtain that S is a Sylow p-subgroup of G, see Lemma 4.5, we show that every morphism in F can be written as a composition of restrictions of morphisms between F-centric subgroups, see Lemmas 4.7 and 4.8. According to a result of [BCG^{+}05, Theorem 2.2], it then suffices to verify the two saturation axioms in 2.4 for the collection of F-centric subgroups only. The first saturation condition is proved in Proposition 4.9, while the second one is proved in Proposition 4.10. □

5. Parabolic families for fusion systems

In this section we discuss fusion systems F which contain families of subsystems {F_{i} ; i ∈ I} with certain properties, denoted below (F1 - F4). To such a fusion system we associate a discrete group G and a chamber system C, on which the group acts. We give some sufficient conditions C has to fulfill in order to ensure saturation of F.

Definition 5.1. Let F be a fusion system over a finite p-group S and set B = N_{F}(S). We say that F has a family of parabolic subsystems if F contains a collection {F_{i}; i ∈ I} of saturated, constrained fusion subsystems, each of essential rank one^{2} with the following properties: (F1). B is a proper subsystem of F_{i} for all i ∈ I; (F2). F = ⟨F_{i}; i ∈ I⟩ and no proper subset {F_{j}; j ∈ J ⊂ I} generates F; (F3). F_{i} ∩ F_{j} = B for any pair of distinct elements F_{i} and F_{j}; (F4). F_{ij} := ⟨F_{i}, F_{j}⟩ is saturated constrained subsystem of F for all i, j ∈ I.

Proposition 5.2. Let F be a fusion system over a finite p-group S. If F contains a family of parabolic subsystems then there are p′-reduced p-constrained finite groups B, G_{i}, G_{ij} that realize B, F_{i}, F_{ij} respectively (for i, j ∈ I), and injective group homomorphisms ψ_{i} : B → G_{i}, ψ_{ij} : G_{i} → G_{ij} such that A = {(B, G_{i}, G_{ij}), (ψ_{i}, ψ_{ij}); i, j ∈ I} is a diagram of groups.

Proof. First notice that B is a saturated, constrained fusion system. Next, recall that, according to [BCG^{+}05, Theorem 4.3], (also see [AKO11, Theorem I.4.9]), for every saturated constrained fusion system over a finite p-group S, there exists a p′-reduced p-constrained finite group, unique up to isomorphism, which realizes the fusion system. Thus we can find such finite groups B, G_{i}, G_{ij}, i, j ∈ I with the property that B = F_{S}(B), F_{i} = F_{S}(G_{i}) and F_{ij} = F_{S}(G_{ij}). Set U_{i} = O_{p}(F_{i}) and let U_{ij} = O_{p}(F_{ij}) for all i, j ∈ I. The fusion system B is also realized by N_{G_{i}}(S) and by N_{G_{ij}}(S), see [Lib08, Proposition 3.8]. Then Lemma 2.17 gives that the groups N_{G_{i}}(S) and N_{G_{ij}}(S) are p′-reduced p-constrained. Thus [AKO11, Theorem I.4.9(b)] tells us that there exist isomorphisms B ≃ N_{G_{i}}(S) ≃ N_{G_{ij}}(S) which are the identity on the Sylow p-subgroup S. Set B_{i} = N_{G_{i}}(S) and B_{ij} = N_{G_{ij}}(S). Denote these isomorphisms by α_{i} : B → B_{i} and by α_{ij} : B_{i} → B_{ij} with α_{i|S} = Id_{S} = α_{ij|S}. Let τ_{i} : B_{i} → G_{i} be the inclusion map, and set ψ_{i} = τ_{i}∘α_{i}. Let q_{ij} : N_{G_{ij}}(U_{i}) ↠ Aut_{F_{ij}}(U_{i}) denote the canonical quotient map. We use the argument in the proof of [Asc, 1.1] to construct a subgroup of G_{ij} that is isomorphic to G_{i}. Observe that Aut_{F_{i}}(U_{i}) ⊆ Aut_{F_{ij}}(U_{i}) and introduce the notation G_{i}^{(j)} = q_{ij}^{−1}(Aut_{F_{i}}(U_{i})). Since F_{i} ⊆ F_{ij}, the group G_{i}^{(j)} is an overgroup of S in G_{ij} with the property that F_{i} = F_{S}(G_{i}^{(j)}). Let τ_{i}^{(j)} : G_{i}^{(j)} → G_{ij} be the inclusion map. The group G_{i}^{(j)} is p′-reduced p-constrained as follows from Lemma 2.17. Using [Asc97, 21.7], we construct isomorphisms φ_{i}^{(j)} : G_{i} → G_{i}^{(j)}. We need the bottom two rows of the following diagram:

2This means that there is one F_{i}-conjugacy class of F_{i}-essential subgroups.

1 ⟶ Z(U_{ij}) ⟶ G_{ij} ⟶ Aut_{F_{ij}}(U_{ij}) ⟶ 1
↑
1 ⟶ Z(U_{i}) ⟶ N_{G_{ij}}(U_{i}) ⟶^{q_{ij}} Aut_{F_{ij}}(U_{i}) ⟶ 1
∥  ↑  ↑
1 ⟶ Z(U_{i}) ⟶ G_{i}^{(j)} ⟶ Aut_{F_{i}}(U_{i}) ⟶ 1
∥  ∥
1 ⟶ Z(U_{i}) ⟶ G_{i} ⟶ Aut_{F_{i}}(U_{i}) ⟶ 1

where the vertical arrows correspond to inclusions. We also need the restrictions of these bottom two rows to the subgroups B_{ij} and B_{i}:

1 ⟶ Z(U_{i}) ⟶ B_{ij} ⟶ Res_{U_{i}}^{S}(Aut_{F_{i}}(S)) ⟶ 1
∥  ↑^{α_{ij}}  ∥
1 ⟶ Z(U_{i}) ⟶ B_{i} ⟶ Res_{U_{i}}^{S}(Aut_{F_{i}}(S)) ⟶ 1

For every pair i < j in I there exist isomorphisms φ_{i}^{(j)} : G_{i} → G_{i}^{(j)} which extend the isomorphisms α_{ij} : B_{i} → B_{ij}.

For pairs i > j, let φ_{i}^{(j)} : G_{i} → G_{i}^{(j)} be isomorphisms extending α_{ji}∘α_{j}∘α_{i}^{−1} : B_{i} → B_{ij}, yielding the commutativity of the following diagram:

B ⟶^{α_{j}} B_{j} ⟶^{τ_{j}} G_{j} ⟶^{φ_{j}^{(i)}} G_{j}^{(i)} ⟶^{τ_{j}^{(i)}} G_{ij}
∥  ∥
B ⟶^{α_{i}} B_{i} ⟶^{τ_{i}} G_{i} ⟶^{φ_{i}^{(j)}} G_{i}^{(j)} ⟶^{τ_{i}^{(j)}} G_{ij}

Let ψ_{ij} : G_{i} → G_{ij} be ψ_{ij} = τ_{i}^{(j)}∘φ_{i}^{(j)}. Consequently, we obtain the following diagram of groups A = {(B, G_{i}, G_{ij}), (ψ_{i}, ψ_{ij}); i, j ∈ I}. □

We record the following useful fact for further reference:

5.3. Let F be a saturated constrained fusion system over a finite p-group S with the property that F = ⟨F_{1}, F_{2}⟩, where F_{1} and F_{2} are saturated constrained subsystems over S. Let G, G_{1} and G_{2} be p′-reduced p-constrained finite groups that realize F, F_{1} and F_{2} respectively, chosen so that G_{1}, G_{2} ≤ G. We claim that G = ⟨G_{1}, G_{2}⟩. Let H := ⟨G_{1}, G_{2}⟩ and observe that ⟨F_{1}, F_{2}⟩ ⊆ F_{S}(H) ⊆ F. Hence F_{S}(H) = F_{S}(G). But F_{S}(H) is saturated and constrained, also H is p′-reduced p-constrained (see Lemma 2.17). Finally, combine the fact that H ≤ G with Theorem 2.16 to conclude that H = G. In particular, we remark that G_{ij} = ⟨G_{i}, G_{j}⟩.

Lemma 5.4. Let F be a fusion system over a finite p-group S. Assume O_{p}(F) = 1 and that F contains a family of parabolic subsystems. Let G be a faithful completion of the diagram of groups A from 5.2. Then the collection of groups {B, G_{i} ; i ∈ I} is a parabolic system in G.

Proof. If G is a faithful completion of A, then we can identify the groups B, G_{i}, i ∈ I with subgroups of G. We need to check that conditions (P1)-(P4) from 3.10 are fulfilled. Properties (P1) and (P3) are easy consequences of the properties of F and the way G was constructed. Next we show that given any distinct i, j ∈ I, G_{i} ∩ G_{j} = B. It is clear that B ⊆ G_{i} ∩ G_{j}. It remains to show the opposite inclusion. Observe that H := G_{i} ∩ G_{j} is an overgroup of S in G_{i} and also in G_{j}; and by 2.17 the group H is p′-reduced p-constrained. Thus F_{S}(H) is a saturated constrained fusion system on S and B ⊆ F_{S}(H) ⊆ F_{i} ∩ F_{j}. Then, using (F 3) we obtain that F_{S}(H) = B. But since B ≤ H and both H and B are p′-reduced p-constrained finite groups, realizing the same saturated constrained fusion system it follows that B ≃ H = G_{i} ∩ G_{j}, proving (P2). Set B_{G} = ∩_{g∈G}B^{g} and observe that O_{p}(B_{G}) is a Sylow p-subgroup of B_{G}. If O_{p}(B_{G}) = 1 then B_{G} is a p′-group. Since B_{G} is a subgroup of B that is normal in G, it follows that B_{G} ⊴ G_{i}, for every i ∈ I. But this implies that O_{p′}(G_{i}) ≠ 1, a contradiction with the fact that G_{i} is assumed to be p′-reduced. So we must have O_{p}(B_{G}) ≠ 1. In this case O_{p}(B_{G}) is a p-subgroup of S which is normal in every G_{i}, i ∈ I. It follows that O_{p}(B_{G}) ⊴ F_{i}, for all i ∈ I, and since F = ⟨F_{i} ; i ∈ I⟩, the p-subgroup O_{p}(B_{G}) is normal in F. Hence O_{p}(B_{G}) ≤ O_{p}(F) = 1 and property (P4) holds. □

Definition 5.5. A fusion-chamber system pair, denoted by (F, C), consists of: (i). a fusion system F, with O_{p}(F) = 1, which contains a family of parabolic subsystems

(as in 5.1); (ii). a chamber system C = C(G; B, G_{i}, i ∈ I), with G a faithful completion of the diagram

of groups A from 5.2.

Proposition 5.6. Let (F, C) be a fusion-chamber system pair, and let S denote a Sylow p-subgroup of B. Suppose that, for any finite p-subgroup P of G, C^{P} is connected. Then:

(i). S is a Sylow p-subgroup of G; (ii). F is the fusion system of G over S, denoted F_{S}(G); (iii). Every morphism in F is a composition of restrictions of morphisms between F-centric

subgroups.

Proof. (i). The first part follows from Lemma 4.5. (ii). It is clear that F ⊆ F_{S}(G). The opposite inclusion follows from Lemmas 4.7 and 5.4. (iii). Recall that F_{i} has essential rank one and the only F_{i}-essential subgroup E_{i} of F_{i} must contain U_{i} = O_{p}(F_{i}). But F_{i} is saturated and constrained, and according to Proposition 2.15, the group U_{i} is F-centric, for all i ∈ I. It then follows that E_{i} is F-centric, and the result is obtained by an application of Lemma 4.8. □

The next Proposition is a slight variation of a technical result due to Linckelmann [Lin06, Proposition 1.6] and Stancu [Sta04, Proposition 4.3], which applies to any fusion system.

Proposition 5.7. Let F be a fusion system over a finite p-group S. Assume that:

(i). Aut_{S}(S) is a Sylow p-subgroup of Aut_{F}(S); (ii). given an F-centric subgroup P of S and φ ∈ Hom_{F}(P, S), there is a morphism \overline{φ} ∈

Hom_{F}(N_{φ}, S) with the property that \overline{φ}_{|P} = φ.

If P is F-centric and fully F-normalized then Aut_{S}(P) is a Sylow p-subgroup of Aut_{F}(P).

Proof. Let Q be an F-centric fully F-normalized subgroup of maximal order such that Aut_{S}(Q) is not a Sylow p-subgroup of Aut_{F}(Q). Then Q is a proper subgroup of S, as it follows from part (i) of the hypothesis. Choose a p-subgroup R of Aut_{F}(Q) such that Aut_{S}(Q) is a proper normal subgroup of R. Let ϕ ∈ R \ Aut_{S}(Q). Since ϕ normalizes Aut_{S}(Q), for every y ∈ N_{S}(Q) there is an element z ∈ N_{S}(Q) such that ϕ(yuy^{−1}) = zϕ(u)z^{−1}, for all u ∈ Q. Thus N_{ϕ} = N_{S}(Q). Since Q is F-centric, it follows from part (ii) of the hypothesis that ϕ extends to \overline{ϕ} : N_{ϕ} → N_{S}(Q), so that \overline{ϕ} ∈ Aut_{F}(N_{S}(Q)). Since ϕ has p-power order, by decomposing \overline{ϕ} into its p-part and its p′-part we may assume that \overline{ϕ} has p-power order. Let ψ : N_{S}(Q) → S be a morphism in F such that ψ(N_{S}(Q)) = N′ is fully F-normalized. As the order of N′ is greater that the order of Q (also observe that N′ is F-centric), we have that Aut_{S}(N′) ∈ Syl_{p}(Aut_{F}(N′)). Now ψ∘\overline{ϕ}∘ψ^{−1} is a p-element of Aut_{F}(N′), thus conjugated to an element in Aut_{S}(N′). Therefore we may choose ψ in such a way that there is y ∈ N_{S}(N′) satisfying ψ∘\overline{ϕ}∘ψ^{−1}(v) = c_{y}(v) for all v ∈ N′. Since \overline{ϕ}_{|Q} = ϕ, it follows that ψ∘\overline{ϕ}∘ψ^{−1}(ψ(Q)) = ψ(Q) and y ∈ N_{S}(ψ(Q)). But Q is fully F-normalized and since ψ(N_{S}(Q)) ⊆ N_{S}(ψ(Q)) we have that ψ(N_{S}(Q)) = N_{S}(ψ(Q)). Hence \overline{ϕ}(u) = ψ^{−1}∘c_{y}∘ψ(u) for all u ∈ N_{S}(Q). In particular ϕ ∈ Aut_{S}(Q) contradicting our choice of ϕ. □

To this end we can combine the results of this section in the following:

Theorem 5.8. Let (F, C) be a fusion-chamber system pair. Assume the following hold.

(a). C^{P} is connected for all p-subgroups P of G. (b). If P is an F-centric subgroup of S and if R is a p-subgroup of Aut_{G}(P), then (C^{P} /C_{G}(P))^{R}

is connected.

Then F = F_{S}(G) is a saturated fusion system over S.

Proof. Assume that F is a fusion system over a finite p-group S, which contains a family of parabolic subsystems which fulfill the properties (F1)-(F4) from 5.1. Let B, G_{i}, G_{ij}, with i, j ∈ I, be p′-reduced p-constrained finite groups that realize B, F_{i}, F_{ij}, i, j ∈ I. Under our assumption that G is a faithful completion, the groups B, G_{i}, G_{ij}, i, j ∈ I can be regarded as subgroups of G, and according to Lemma 5.4, they form a parabolic system in G, in the sense of 3.10. Let C = C(G; B, G_{i}, i ∈ I) be the associated chamber system described in 3.11. Next, assuming that C^{P} is connected, for every P ≤ G and using Proposition 5.6, it is obtained that F is realized by G, in other words F = F_{S}(G). It remains to show that the fusion system F is saturated. First, it is shown that every morphism in F can be written as a composition of restrictions of morphisms between Fcentric subgroups; this is the result of Proposition 5.6(iii). It follows from [BCG^{+}05, Theorem 2.2], that it suffices to verify the two saturation axioms in 2.4 for the collection of F-centric subgroups only. The second saturation condition (II) is obtained from Proposition 4.10. Further Aut_{F}(S) = Aut_{B}(S) and S ∈ Syl_{p}(B), hence S/Z(S) = Aut_{S}(S) ∈ Syl_{p}(Aut_{B}(S)). Therefore the conditions from the hypothesis of Proposition 5.7 are in place and the first saturation axiom (I) follows. This concludes the proof of the fact that F is a saturated fusion system over S. □

6. A subsystem with a parabolic family

We show that a fusion system F over S with a family of parabolic subsystems contains a certain saturated subsystem F̂ over S which also has an associated parabolic family. Before proceeding with our construction we shall review some standard facts and properties of certain normal subsystems.

Subsystems of index prime to p. The material included in this overview appeared elsewhere in the literature, we refer the reader to [Pui09, Chapter 12] and to [BCG^{+}07, Section 5] for earlier sources. We will follow the more recent approach from [AKO11, Section I.7].

Proposition 6.1. Let F and G be fusion systems over S with G ⊆ F and F saturated. Assume that O^{p′}(Aut_{F}(Q)) ≤ Aut_{G}(Q) for every subgroup Q of S. Then:

(i). Hom_{F}(Q, S) = Aut_{F}(S)∘Hom_{G}(Q, S); (ii). Q is fully F-normalized (F-centralized) iff it is fully G-normalized (G-centralized); (iii). Q is F-centric if and only if it is G-centric; (iv). Q is F-essential if and only if it is G-essential.

Proof. (i). Let Q be a subgroup of S. Clearly we have Aut_{F}(S)∘Hom_{G}(Q, S) ⊆ Hom_{F}(Q, S). It remains to prove that for any morphism φ ∈ Hom_{F}(Q, S) there exist α ∈ Aut_{F}(S) and ζ ∈ Hom_{G}(Q, S) with φ = α∘ζ. We argue by induction on the index |S : Q|. If S = Q then φ ∈ Aut_{F}(S) and we are done. So we may suppose that |S : Q| > 1, which means that Q < S. By a standard argument (see the proof of Theorem A.10 in [BLO03]) we can show that it suffices to find the sought decomposition for an automorphism φ ∈ Aut_{F}(Q) of a fully F-normalized subgroup Q of S. By a general Frattini argument, using the fact that F is saturated and hence Aut_{S}(Q) ∈ Syl_{p}(Aut_{F}(Q)) we obtain:

Aut_{F}(Q) = N_{Aut_{F} (Q)}(Aut_{S}(Q)) · O^{p′}(Aut_{F}(Q)).

Thus φ = ϕ∘η with ϕ ∈ N_{Aut_{F} (Q)}(Aut_{S}(Q)) and η ∈ O^{p′}(Aut_{F}(Q)). Next observe that

N_{Aut_{F} (Q)}(Aut_{S}(Q)) = {ρ ∈ Aut_{F}(Q) | N_{ρ} = N_{S}(Q)}.

Since Q is fully F-normalized, it is fully F-centralized, and the saturation axiom (II) implies that ϕ extends to a map ϕ̂: N_{S}(Q) → S which has the property that ϕ̂(Q) = ϕ(Q) = Q. Since Q < N_{S}(Q) ≤ S, the induction hypothesis gives that ϕ̂ = α_{|χ̂(N_{S}(Q))}∘χ̂ where α ∈ Aut_{F}(S) and χ̂ ∈ Hom_{G}(N_{S}(Q), S). Consequently ϕ = α_{|χ̂(Q)}∘χ̂_{|Q} has the desired form. Therefore φ = ϕ∘η = α_{|χ̂(Q)}∘χ̂_{|Q}∘η with χ̂_{|Q}∘η ∈ Hom_{G}(Q, S) and φ has the required form also.

(ii). If Q is fully F-normalized then Q is also fully G-normalized, given G ⊆ F. Conversely, if Q is fully G-normalized and ψ : N_{S}(Q) → S is an F-morphism, we obtain that ψ = α∘ζ with α ∈ Aut_{F}(S) and ζ ∈ Hom_{G}(N_{S}(Q), S). Also |N_{S}(ζ(Q))| ≤ |N_{S}(Q)|. On the other side, we always have ζ(N_{S}(Q)) ≤ N_{S}(ζ(Q)) and since ζ is injective, N_{S}(ζ(Q)) = ζ(N_{S}(Q)). Therefore

ψ(N_{S}(Q)) = (α∘ζ)(N_{S}(Q)) = α(N_{S}(ζ(Q))) = N_{S}(α∘ζ(Q)) = N_{S}(ψ(Q))

which shows that Q is fully F-normalized. The other statement from (ii) can be proved in a similar way. (iii). Clearly, each F-centric subgroup P is also G-centric. Conversely, if P is G-centric then, by part (i), each F-conjugate of P has the form α(Q) with α ∈ Aut_{F}(S) and Q a G-conjugate of P. Then C_{S}(α(Q)) ≤ α(Q) since C_{S}(Q) ≤ Q. (iv). Observe O^{p′}(Aut_{F}(Q)) = O^{p′}(Aut_{G}(Q)) and thus O^{p′}(Out_{F}(Q)) = O^{p′}(Out_{G}(Q)). By the Frattini argument, a finite group G has a strongly p-embedded subgroup if and only if O^{p′}(G) has one. Hence Out_{F}(Q) has a strongly p-embedded subgroup if and only if Out_{G}(Q) has one. Now the assertion follows from part (iii). □

We use the result of [AKO11, Theorem I.7.7] to formulate the following:

Definition 6.2. Given a saturated fusion system F on a finite p-group S, we let O^{p′}(F) denote the smallest saturated fusion subsystem of F which has the property that Aut_{O^{p′}(F)}(P) ≥ O^{p′}(Aut_{F}(P)) for every subgroup P of S.

It follows from Proposition 6.1 that the Frattini argument holds for saturated fusion systems.

Corollary 6.3. If F is a saturated fusion system over a finite p-group S then there is a decomposition F = ⟨O^{p′}(F), N_{F}(S)⟩.

If F = F_{S}(G) then O^{p′}(F) does not necessarily correspond to O^{p′}(G). However, there are particular cases in which the correspondence is attained.

Proposition 6.4. Let F = F_{S}(G) be a saturated constrained fusion system over a finite p-group S with G a p′-reduced p-constrained finite group. Then O^{p′}(F) = F_{S}(O^{p′}(G)).

Proof. Let F and G be as in the hypothesis. Since O^{p′}(G) is a normal subgroup of G, it follows that F_{S}(O^{p′}(G)) is a normal saturated fusion subsystem of F. On the other side, F is constrained and Aschbacher’s Theorem [Asc08, Theorem 1] asserts the existence of a unique normal subgroup O_{G} of G with O^{p′}(F) = F_{S}(O_{G}). Set F′ = F_{S}(O^{p′}(G)) and F̂ = O^{p′}(F). The inclusion F′ ⊆ F̂ follows from O^{p′}(G) ≤ O_{G}. To prove that F̂ ⊆ F′, let φ ∈ Aut_{F}(Q), Q ≤ S be a p-element. Then φ = c_{g} for some g ∈ N_{G}(Q). Then we may choose g to be a p-element as well and so g ∈ O^{p′}(G). Hence φ = c_{g} ∈ Aut_{F′}(Q). This proves O^{p′}(Aut_{F}(Q)) ≤ Aut_{F′}(Q) and thus F̂ ⊆ F′. □

A reduction result. For the rest of this section we shall assume that (F, C) is a fusionchamber system pair as defined in 5.5. According to Lemma 5.4, the family (B, G_{i} ; i ∈ I) is a parabolic system in G, a faithful completion of the diagram of groups A from Proposition 5.2. Thus B ≤ G_{i} for all i ∈ I, and by 5.3 we also have G_{ij} = ⟨G_{i}, G_{j}⟩.

Notation 6.5. The fusion systems F_{i} = F_{S}(G_{i}) and F_{ij} = F_{S}(G_{ij}), i, j ∈ I, are saturated and constrained with G_{i}, G_{ij} respectively, p′-reduced p-constrained finite groups. Denote Ĝ_{i} = O^{p′}(G_{i}) and set F̂_{i} = O^{p′}(F_{i}), for i ∈ I. According to Proposition 6.4, F̂_{i} = F_{S}(Ĝ_{i}) and the fusion system F̂_{i} is saturated and constrained. Define F̂ = ⟨F̂_{i}; i ∈ I⟩ and let B̂ = ⟨NF̂_{i}(S); i ∈ I⟩.

Lemma 6.6. The fusion subsystem B̂ is saturated and constrained, and B̂ = NF̂(S). Further B̂ = F_{S}(B̂) with B̂ = ⟨NĜ_{i}(S); i ∈ I⟩.

Proof. Because NF̂_{i}(S) ⊆ N_{F}(S) it follows that B̂ ⊆ B. Thus every morphism in B̂ extends to an F-automorphism of S, and this implies that B̂ is a saturated fusion system. It is obviously constrained since S is a normal B̂-centric subgroup. Next observe that since NF̂_{i}(S) ⊆ NF̂(S) for all i ∈ I, it follows that B̂ ⊆ NF̂(S). Let now φ ∈ Hom_{NF̂(S)}(P, Q) for P, Q ≤ S. There exists a morphism φ̂ ∈ AutF̂(S) which extends φ. Since F̂ = ⟨F̂_{i}; i ∈ I⟩, there are morphisms ψ̂_{j} ∈ AutF̂_{i}(S), i ∈ I and j = 1, … n, such that φ̂ = ψ̂_{n}∘ … ∘ψ̂_{1}. It follows now that there are subgroups P = P_{0}, P_{1}, … P_{n} = Q of S and morphisms ψ_{j} : P_{j−1} → P_{j} with ψ_{j} = ψ̂_{j}|_{P_{j−1}} and φ = ψ_{n}∘ … ∘ψ_{1}. Because ψ_{j} is the restriction of an F̂_{i}-automorphism of S, it follows that ψ_{j} is a morphism in NF̂_{i}(S); this proves that φ ∈ HomB̂(P, Q).

Since B̂ ⊆ B, according to Proposition 2.18 and Lemma 2.17, there exists a p′-reduced p-constrained subgroup B̂ of B which realizes B̂. Using [Lib08, Proposition 3.8], we can identify B with N_{G_{i}}(S), for all i ∈ I. Set T := ⟨NĜ_{i}(S); i ∈ I⟩ ≤ B and notice that F_{S}(T) is a saturated constrained fusion system. We will show that B̂ = T. Since each F̂_{i} = F_{S}(Ĝ_{i}) is saturated, another application of [Lib08, Proposition 3.8] gives that NF̂_{i}(S) = F_{S}(NĜ_{i}(S)). Thus B̂ ⊆ F_{S}(T) and, using Proposition 2.18, we can choose B̂ ≤ T. Conversely, let g ∈ T and P ≤ S be such that ^{g}P ≤ S. But g = g_{m} ⋯ g_{1} for g_{i} ∈ NĜ_{j_{i}}(S) and j_{i} ∈ I. Thus c_{g} : P → ^{g}P can be decomposed as P → ^{g_{1}}P → … → ^{g_{m}···g_{1}}P = ^{g}P with ^{g_{i}···g_{1}}P ≤ S since each g_{i} normalizes S. Therefore c_{g_{i}} : ^{g_{i−1}···g_{1}}P → ^{g_{i}···g_{1}}P is in fact a morphism in NF̂_{j_{i}}(S) ⊆ B̂. Thus B̂ = F_{S}(T) and Theorem 2.16 together with the fact that B̂ ≤ T give that B̂ = T. □

Lemma 6.7. The fusion systems G_{i} := ⟨F̂_{i}, B̂⟩ and G_{ij} := ⟨G_{i}, G_{j}⟩ are saturated and constrained for all i, j ∈ I.

Proof. Since G_{i} ⊆ F_{i}, it follows that O_{p}(F_{i}) is normal in G_{i}, it is G_{i}-centric because it is F_{i}-centric. Thus G_{i} is a constrained fusion system on S; similarly G_{ij} is also constrained.

Let B̂ ⊆ B ⊆ F_{i} and the corresponding p′-reduced p-constrained finite groups B̂ ≤ B ≤ G_{i}, where B̂ is as in Lemma 6.6. We will show that G_{i} = F_{S}(H_{i}) with H_{i} := Ĝ_{i}B̂, a p′-reduced p-constrained subgroup of G_{i}. This will suffice to prove the saturation of G_{i} (according to 2.14). Clearly F̂_{i}, B̂ ⊆ F_{S}(H_{i}). Conversely, let h ∈ H_{i} and P ≤ S be such that ^{h}P ≤ S so c_{h} ∈ Hom_{F_{S}(H_{i})}(P, S). But h = gb ∈ Ĝ_{i} with g ∈ Ĝ_{i} and b ∈ B̂. Hence c_{h} : P → ^{b}P → gbP = ^{h}P, where P, ^{b}P, ^{h}P are subgroups of S, showing that c_{h} is indeed a morphism in G_{i}.

To show that G_{ij} is saturated, we will prove that G_{ij} = F_{S}(H_{ij}) where H_{ij} := ⟨Ĝ_{i}, Ĝ_{j}⟩B̂. Set G̃_{ij} := ⟨Ĝ_{i}, Ĝ_{j}⟩ and observe that since B ≤ G_{i} ∩ G_{j}, the subgroup G̃_{ij} is normal in G_{ij}. Thus H_{ij} is indeed a subgroup of G_{ij}. Next, notice that H_{ij} = ⟨Ĝ_{i}, Ĝ_{j}⟩B̂ = ⟨Ĝ_{i}B̂, Ĝ_{j}B̂⟩ = ⟨H_{i}, H_{j}⟩. Hence G_{ij} ⊆ F_{S}(H_{ij}). To prove the opposite inclusion, let h ∈ H_{ij} and P ≤ S with ^{h}P ≤ S. Write h = gb with g ∈ G̃_{ij} and b ∈ B̂ and argue as before. □

Lemma 6.8. Maintain the notations from above. The fusion systems (G_{i}; i ∈ I) form a parabolic family in F̂.

Proof. It follows from Lemma 6.7 that G_{i} and G_{ij}, for all i, j ∈ I, are saturated, constrained subsystems of F̂, and contain the subsystem B̂ = NF̂(S). Step 1: Each G_{i} has essential rank one. According to Proposition 6.1(iv), Q is G_{i}-essential if and only if Q is F̂_{i}-essential if and only if Q is F_{i}-essential. Since F_{i} has essential rank one, G_{i} also has essential rank one. Step 2: B̂ is a proper subsystem of G_{i}. Assume by contradiction that F̂_{i} ⊆ B̂, for some i ∈ I and recall that B ⊊ F_{i}. Then by Corollary 6.3, F_{i} = ⟨F̂_{i}, B⟩ = B, a contradiction. Step 3: The collection G_{i}, i ∈ I is a minimal generating set for F̂. Assume by contradiction that F̂ is generated by G_{j}, j ∈ J with J a proper subset of I. It follows from Corollary 6.3 that F_{i} = ⟨F̂_{i}, B⟩. Thus we have the following equalities:

F =⟨F_{i}, i ∈ I⟩ = ⟨⟨F̂_{i}, B⟩, i ∈ I⟩ = ⟨F̂_{i}, i ∈ I; B⟩ = ⟨F̂, B⟩
=⟨F̂_{j}, j ∈ J; B⟩ = ⟨⟨F̂_{j}, B⟩, j ∈ J⟩ = ⟨F_{j}, j ∈ J⟩

which contradicts the fact that F_{i}, i ∈ I is a minimal generating set for F. Step 4: The following inclusions hold:

B̂ ⊆ G_{i} ∩ G_{j} ⊆ (F_{i} ∩ F_{j}) ∩ F̂ = B ∩ F̂ ⊆ NF̂(S) = B̂

which verify (F3). Hence the set (G_{i}, i ∈ I) forms a parabolic family in the fusion system F̂. □

6.9. Let B̂, H_{i} and H_{ij}, i, j ∈ I be the groups constructed in 6.7 and 6.8. Set Ĝ := ⟨H_{i}; i ∈ I⟩ ≤ G and observe that

Ĝ =⟨H_{i}; i ∈ I⟩ = ⟨Ĝ_{i}B̂; i ∈ I⟩ = ⟨Ĝ_{i}, B̂; i ∈ I⟩ =
⟨Ĝ_{i}, ⟨NĜ_{j}(S); j ∈ I⟩; i ∈ I⟩ = ⟨Ĝ_{i}, NĜ_{i}(S); i ∈ I⟩ = ⟨Ĝ_{i}; i ∈ I⟩

Recall that C = C(G; B, G_{i}, i ∈ I) denotes the chamber system associated to F and let Ĉ = C(Ĝ, B̂, H_{i}, i ∈ I) be the chamber system associated to F̂.

We can now formulate the main result of this section:

Theorem 6.10. Let (F, C) be a fusion-chamber system pair. Let F̂ be defined as in 6.5 and let Ĝ and Ĉ be as in 6.9. If O_{p}(F̂) = 1 then (F̂, Ĉ) is a fusion-chamber system pair. Further, assume that the following two conditions hold:

(a). Ĉ^{P} is connected for all subgroups P of Ĝ; (b). If P is F̂-centric and if R is a p-subgroup of AutĜ(P) then (Ĉ^{P} /CĜ(P))^{R} is connected.

Then:

(i). F̂ = F_{S}(Ĝ) is a saturated fusion system that is normal in F; (ii). The map φ : Ĉ → C given by φ(gB̂) = gB, for g ∈ Ĝ, is a 2-covering of chamber

systems.

Proof. By Lemma 6.8 the set {G_{i} ; i ∈ I} forms a family of parabolic subsystems in F; this gives rise to the collection of groups {B̂, H_{i} ; i ∈ I} which is a parabolic system in Ĝ, according to Lemma 5.4. Recall that, by construction, the groups B and H_{i}, for i ∈ I, are subgroups of Ĝ. Hence (F, Ĉ) is a fusion-chamber system pair. (i). The saturation of F̂ as well as the fact that F̂ is realized by Ĝ follow from an application of Theorem 5.8. To prove that F̂ ⊴ F, recall that F = ⟨F_{i}, i ∈ I⟩ = ⟨F̂_{i}, B, i ∈ I⟩ = ⟨F̂, B⟩. We have to show that B normalizes F̂. But F̂_{i} ⊴ F_{i} ⊇ B and since B normalizes each F̂_{i}, it follows that B indeed normalizes F̂. (ii). The second part of the Theorem follows from [Tim87, Lemma 2.5], for completeness we provide the proof^{3}. We have that (recall ??):

G = ⟨G_{i}; i ∈ I⟩ = ⟨Ĝ_{i}B; i ∈ I⟩ = ⟨Ĝ_{i}; i ∈ I⟩B = ĜB

thus the map φ : gB̂ ↦ gB, for g ∈ Ĝ, is well-defined and surjective. Let gB̂ and hB̂ be two i-adjacent chambers in Ĉ. Then h^{−1}g ∈ H_{i}. Since H_{i} ≤ G_{i}, it follows that gG_{i} = hG_{i} and the chambers gB and hB are i-adjacent in C. Thus φ is a morphism of chamber systems. To show that φ is a 2-covering, we must prove that φ_{ij}, the restriction of φ to a rank two {i, j}-residue is bijective, for each i, j ∈ I. Clearly φ_{ij} is surjective. As B = N_{F}(S) = N_{F_{ij}}(S), and similarly B̂ = NF̂(S) = N_{G_{ij}}(S), we may use [Lib08, Proposition 3.8], Lemma 2.17 and Theorem 2.16, to write B = N_{G_{ij}}(S) and B̂ = N_{H_{ij}}(S). Thus B̂ = B ∩H_{ij}. An application of the Frattini argument, together with the fact that H_{ij} = G̃_{ij}B̂ give that G_{ij} = G̃_{ij}B = H_{ij}B. The product formula gives:

|H_{ij}B| · |H_{ij} ∩ B| = |H_{ij}B| · |B̂| = |G_{ij}| · |B̂| = |H_{ij}| · |B|

which shows that the (ij)-residues in Ĉ and C have the same number of chambers. Hence φ_{ij} is injective. This concludes our proof that φ is a 2-covering. □

7. An application: classical parabolic families in fusion systems

In this section we investigate fusion systems which contain parabolic families with specific properties. In finite group theory, the parabolic systems of type ℳ, defined below, are at the core of so called amalgam method and emphasize the deep connections between local analysis, on one side, and group geometries on the other side. For a comprehensive overview of the early results on parabolic systems of type ℳ, see Meixner [Mei90]. For an up to date succinct overview on higher rank amalgams see Parker and Rowley [PR02, Chapter 24].

3See Section 3 for chamber systems and their covers.

The terminology is inspired by the structure of the groups of Lie type, in which a parabolic system (B, G_{i}; i ∈ I) consists of the Borel subgroup B, together with the minimal parabolic subgroups G_{i} containing it. We will show that these notions have natural generalizations to the context of fusion systems. We start with reviewing some of the standard terminology.

Chamber systems and classical parabolic systems. A chamber system C of rank two is a generalized digon if and only if C is the chamber system associated to a parabolic system of the form P(G) = (G_{i}∩G_{j}, G_{i}, G_{j}) with G = G_{i}G_{j} = G_{j}G_{j} and G_{i} ≠ G_{j}. A rank two chamber system C is a classical generalized m_{ij}-gon, for m_{ij} ≥ 3, if C is isomorphic to C(G; B, G_{i}, G_{j}) where G is an (essentially^{4}) simple rank two group of Lie type in characteristic p, B = G_{i} ∩G_{j} is a Borel subgroup of G and G_{i} and G_{j} are the two maximal parabolic subgroups of G containing B; here m_{ij} denotes the integer that defines the Weyl group of G.

7.1. A connected chamber system C over I is a classical Tits chamber system if all rank two residues are either generalized digons or classical generalized m_{ij}-gons, for i, j ∈ I with i ≠ j. Here m_{ij} is fixed for each type. C is called locally finite, if all rank two residues are finite. The diagram (or the type of C) is denoted ℳ = ℳ(I) and it is a graph whose vertices are labeled by the elements of I, the nodes i and j are connected by a bond of strength m_{ij} − 2.

7.2. A collection of finite subgroups (B, G_{i}; i ∈ I) in a group G is a classical parabolic system^{5} if it is a parabolic system in the sense of 3.10 and in addition, fulfills the following conditions.

(i). S = O_{p}(B) ∈ Syl_{p}(G_{ij}) with G_{ij} = ⟨G_{i}, G_{j}⟩, for all i, j ∈ I. (ii). For each i ∈ I, G_{i}/O_{p}(G_{i}) is a rank one group of Lie type in characteristic p. (iii). For each pair i, j ∈ I, either G_{ij} = G_{i}G_{j} = G_{j}G_{i} or G_{ij}/O_{p}(G_{ij}) is a rank two Lie type

group in characteristic p.

The associated chamber system C = C(G; B, G_{i}, i ∈ I) is a locally finite Tits chamber system of classical type; see [Tim83, Lemma 4.2] for a proof. The diagram ℳ is constructed as in 7.1; in particular, m_{ij} is 2 if G_{ij} = G_{i}G_{j} and it is determined by the Weyl group of G_{ij} otherwise.

We end this review with the following standard result:

Proposition 7.3. [Tim85, 3.1] Let C be a locally finite classical Tits chamber system over I, with |I| ≥ 3, with spherical diagram ℳ and chamber transitive automorphism group G. Then one of the following holds:

(i). C is a finite spherical building of type ℳ and G is an extension of a simple group of

Lie type ℳ by diagonal and field automorphisms or G ≃ A_{7} and ℳ = A_{3}. (ii). C is the Neumeier chamber system obtained from A_{7}, ℳ = C_{3} and G ≃ A_{7}.

The diagram is spherical if the associated Weyl group is finite. Buildings are special cases of chamber systems of type ℳ, for a detailed treatment see Ronan [Ron89] or the monumental paper of Tits [Tit81]. For a description of the Neumeier chamber system of type C_{3} in the alternating group A_{7} see [Ron89, Example 2, pg. 50].

4The following groups are also allowed: A_{6}, S_{6}, G_{2}(2)′ ≃ U_{3}(3), G_{2}(2), ^{2}F_{4}(2)′ and ^{2}F_{4}(2).

5Conform to [FSW00].

Classical parabolic families in fusion systems. In this section we apply Theorem 5.8 to a fusion system F which contains a classical family of parabolic systems of type ℳ, as defined below. We shall use the notation introduced at the beginning of Section 5. In particular, recall that F contains a collection of saturated constrained fusion subsystems {F_{i} ; i ∈ I}, and for each pair i, j ∈ I, the subsystems F_{ij} = ⟨F_{i}, F_{j}⟩ are also saturated and constrained. Also B = N_{F}(S). We set U_{i} = O_{p}(F_{i}) and U_{ij} = O_{p}(F_{ij}) for all i, j ∈ I.

Definition 7.4. Let F be a fusion system over a finite p-group S. We say that F contains a classical family of parabolic systems of type ℳ if the following hold.

(i). F contains a family of parabolic systems, in the sense of 5.1. (ii). For each i ∈ I, Out_{F_{i}}(U_{i}) is a rank one finite group of Lie type in characteristic p. (iii). For each pair i, j ∈ I, Out_{F_{ij}}(U_{ij}) is either a rank two finite group of Lie type in

characteristic p or it is a (central) product of two rank one finite groups of Lie type in characteristic p.

To such a fusion system we can associate a diagram ℳ on I in the following way: if Out_{F_{ij}}(U_{ij}) is a product of two rank one groups of Lie type then i and j are not connected, if Out_{F_{ij}}(U_{ij}) is a rank two group of Lie type we take the corresponding Coxeter diagram for the edge between i and j.

We arrive at the main result of this Section, a generalization of Proposition 7.3 to fusion systems.

Proposition 7.5. Let (F, C) be a fusion-chamber system pair with |I| ≥ 3. Assume that:

(i). F contains a classical family of parabolic systems of type ℳ; (ii). ℳ is a spherical diagram.

Then F is the fusion system of a finite simple group of Lie type in characteristic p extended by diagonal and field automorphisms.

Proof. Let (F, C) be a fusion-chamber system pair as defined in 5.5. First recall that F contains a family of parabolic subsystems (see 5.1). There is a collection of p′-reduced, p-constrained finite groups B, G_{i} and G_{ij}, that realize B, F_{i} and F_{ij} respectively. By Proposition 5.2, these groups (together with appropriate group homomorphisms) form a diagram of groups (as defined in 3.15). If G is a faithful completion of this diagram of groups then {B, G_{i}; i ∈ I} is a parabolic system in G, by Lemma 5.4. For each i ∈ I, the fusion system F_{i} = F_{S}(G_{i}) is saturated and constrained, thus U_{i} is F_{i}-centric and U_{i} = O_{p}(G_{i}). An application of [AKO11, Proposition III.5.8] gives that Out_{F_{i}}(U_{i}) ≃ G_{i}/U_{i}. Similarly, we obtain Out_{F_{ij}}(U_{ij}) = G_{ij}/U_{ij} and U_{ij} = O_{p}(G_{ij}). Also recall that G_{ij} = ⟨G_{i}, G_{j}⟩, conform to 5.3. Let now C = C(G; B, G_{i}, i ∈ I). Since G = ⟨G_{i}, i ∈ I⟩, the chamber system C is connected. By transitivity each {i, j}-residue of C is isomorphic to {gB| gB ⊆ G_{ij}}. Since the parabolic family in F is of classical type, (ii) and (iii) in Definition 7.4 imply (ii) and (iii) from 7.2. As 7.2(i) clearly holds, it follows that {B, G_{i} ; i ∈ I} is a classical parabolic system in G. This implies that C is a locally finite Tits chamber system of classical type. Since ℳ is spherical, according to Timmesfeld results [Tim83, 3.14, 4.3] and [Tim85, 3.1], see also Proposition 7.3, C is a finite spherical building of type ℳ and G is an extension of a simple group of Lie type ℳ by diagonal and field automorphisms, or G ≃ A_{7} and ℳ is of type A_{3}, or C is the Neumeier chamber system obtained from A_{7}, ℳ = C_{3} and G ≃ A_{7}.

In the former case with C a building, it is well known that C^{P} are contractible for all subgroups P ≤ S; an elegant proof of this fact can be found in [Qui78, proof of Theorem 3.1]. The claim follows by an application of Proposition 5.6.

Let now G ≃ A_{7} and let B denote some Sylow 2-subgroup of G. Then G has exactly four subgroups X_{1}, …, X_{4} containing B and which are isomorphic to S_{4} (the symmetric group on four letters). For suitable labeling, see [Mei90, Example 1.2], we get:

a. C_{i} = C(G; B, X_{1}, X_{2}, X_{i}), for i = 3, 4, two isomorphic Neumeier chamber systems of

type C_{3}; b. C′ = C(G; B, X_{3}, X_{2}, X_{4}), the chamber system of type A_{3} over the field with two ele-

ments.

Hence, in either case B = F_{D_{8}}(D_{8}) and F_{i} = F_{D_{8}}(S_{4}) for i = 1, 2, 3. The three fusion subsystems F_{i}, do not form a minimal generating set for F_{D_{8}}(A_{7}); only two subsystems are sufficient to generate F (see [AKO11, Example I.2.7]). Thus, we do not obtain families of parabolic subsystems for F_{D_{8}}(A_{7}) (in the sense of 5.1). □

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Department of Mathematics, The Ohio State University, 100 Mathematics Tower, 231 West 18^{th} Avenue, Columbus, Ohio 43210, USA

E-mail address: onofrei@math.ohio-state.edu