ON THE HOMOTOPY TYPE OF THE NON-COMPLETED CLASSIFYING SPACE OF A p-LOCAL FINITE GROUP

ASSAF LIBMAN AND ANTONIO VIRUEL

Abstract. We establish sufficient conditions of the nerve of the linking system of a p-local finite group (S, F, L) to have the homotopy type of an Eilenberg-MacLane space K(Γ, 1) for a group Γ which contains S. We prove that in this situation the entire p-local finite group can be reconstructed from Γ. Examples of p-local finite groups that our theorem applies to are the exotic ones which were constructed by the second author and Ruiz.

1. Introduction and the main results

1.1. This paper originated as an attempt to study the homotopy type of the nerve of the linking system of a p-local finite group (S, F, L). A brief summary of the definition and the basic properties of p-local finite groups can be found in §2. The canonical reference is [6].

Jesper Grodal was probably the first to observe that |L| needs not be an Eilenberg-MacLane space of type K(Γ, 1). Easy examples are given by Chevalley groups G of high enough Lie rank at the prime p. One can easily see that the nerve of their associated linking system, |L|, is the homotopy orbit space of the action of G/Z(G) on the Tits building of G. In particular |L| is the total space in a fibre bundle whose base space is K(G/Z(G), 1) and whose fibre is the Tits building. When the Lie rank of G is ≥ 2, the Tits building has the homotopy type of a wedge of high dimensional spheres and in particular |L| has infinitely many higher homotopy groups. On the other hand, in these cases, e.g. GL_{4}(p), it is also the case that π_{1}|L| = G/Z(G) so the fusion system of G is induced by π_{1}|L| because G and G/Z(G) are mod-p equivalent.

Another extreme was recently demonstrated by Chermak, Oliver and Shpectorov. They showed that π_{1}|L_{Sol(q)}| = 0 where L_{Sol(q)} is the centric linking system considered by Levi and Oliver in [19] known as “the Solomon group”.

In complete contrast, in this paper we study the implication of |L| having the homotopy type of K(Γ, 1) and study conditions for this to happen.

1.2. To begin with, let G be a discrete group containing a finite p-group S. There is a fusion system F_{S}(G) on S whose objects are the subgroups of S and whose morphisms are restrictions of inner automorphisms of G to the subgroups of S. More explicitly, the transporter of subgroups H, K ≤ G is

N_{G}(H, K) = {g ∈ G : gHg^{−1} ≤ K}.

Date: May 2, 2006.

Key words and phrases. Homology decompositions, p-local finite groups.

The first author was supported by the Nuffield Foundation Grant NAL/00735/G.

The second author was partially supported by MEC grant MTM2004-06262 and CEC-JA grant FQM213.

Then F_{S}(G)(P, Q) = N_{G}(P, Q)/C_{G}(P).

Let H be a family of subgroups of G. The transporter category T^{H}(G) has H as its object set. Morphism sets H → K are the elements of the transporters N_{G}(H, K). Composition is induced by the product in G. When H consists of subgroups of S we may consider F_{S}^{H}(G) which denotes the full subcategory of F_{S}(G) containing P ≤ S where P ∈ H as its object set. There is an obvious quotient functor T^{H}(G) → F_{S}^{H}(G). By convention T_{S}^{c}(G) denotes the full subcategory of T (G) on the F_{S}(G)-centric subgroups of S.

Next, consider a functor Θ : T_{S}^{c}(G)^{op} → {Groups} with the properties

(1) Θ(P) is a complement of Z(P) in C_{G}(P) and

(2) Θ(Q) ≤ gΘ(P)g^{−1} if g ∈ N_{G}(P, Q). These functors are known as signaliser functors. A slightly more explicit description is given in Section 3. One can then form a category L_{S,Θ}^{c} whose objects are the F-centric subgroups of S and whose morphism sets are given by the cosets N_{G}(P, Q)/Θ(P). There is an evident functor π : L_{S,Θ}^{c} → F_{S}(G). There are also evident monomorphisms δ_{P} : P → N_{G}(P)/Θ(P). It is not hard to show that L_{S,Θ}^{c} is an associated linking system for F_{S}(G). See Section 2.

Before stating the main results of this paper we recall that Bousfield and Kan constructed in [2] the p-completion functor X ↦ X_{p}^{∧} on spaces. It has the property that a map of spaces X → Y induces and isomorphism in H_{∗}(−; 𝔽_{p}) if and only if X_{p}^{∧} → Y_{p}^{∧} is a homotopy equivalence. Two spaces, X and Y, are called mod-p equivalent if X_{p}^{∧} ≃ Y_{p}^{∧}. A homomorphism of groups Γ → Γ′ is called a mod-p equivalence if it induces BΓ_{p}^{∧} ≃ BΓ′_{p}^{∧} where BΓ is the classifying space of Γ. Two groups Γ and Γ′ are called mod-p equivalent if BΓ_{p}^{∧} ≃ BΓ′_{p}^{∧}. Furthermore, the p-completion functor is coaugmented in the sense that it is equipped with a natural map X → X_{p}^{∧} which is frequently (but not always) a mod-p equivalence.

1.3. Theorem. Let A be a central subgroup of a p-local finite group (S, F, L). Assume that either

(a) |S/A| ≤ p^{5}.

(b) S/A has p-rank 2.

(c) S/A is resistant [9, Definition 3.1]

(d) (S/A, F/A, L/A) is one of the exotic p-local finite groups in [6, Examples

9.3 and 9.4]. Then

(1) |L| ≃ BΓ for some discrete group Γ and the natural map f : BS → |L| induces an inclusion S ≤ Γ by passing to fundamental groups.

(2) Γ is an amalgamated product of finite groups, all of which are subgroups of automorphism groups of objects of L.

(3) F = F_{S}(Γ) and

(4) There is a signaliser functor Θ on T_{S}^{c}(Γ) which induces L, that is L ≅ L_{S,Θ}^{c}. Moreover, if in addition (S, F, L) is induced by a finite group G where S ∈ Syl_{p}(G) then there exists a group Γ′ containing S together with mod-p equivalences

Γ ← Γ′ → G.

Furthermore, F = F_{S}(Γ′) and there is a signaliser functor Θ′ on T_{S}^{c}(Γ′) which induces L.

Proof. In general, we apply Theorem 5.5 to the p-local finite group (S/A, F/A, L/A), and then recover the information about (S, F, L) in view of [4, Theorem E] and Theorem 3.4. Hence we can assume that (S, F, L) is centerfree.

To prove the case (a) we apply Proposition 6.1 and Theorem 5.5 to the p-local finite group (S × V, F × F_{V} , L × L_{V}), where V is an elementary abelian p-group of size \frac{p^{5}.}{|S|} and (V, F_{V} , L_{V}) is the p-local groups induced by V. So we obtain that |L × L_{V} | = |L| × BV is aspherical, hence |L| is so and the result follows from Theorem 3.4.

Case (b) follows from Example 7.4 and case (c) from Example 7.5. Case (d) follows from Examples 7.6 and 7.7. When (S, F, L) is associated with a finite group, use Theorem 5.7. □

1.4. Notice that hypothesis (a) above is the weakest possible in terms of the size of the sylow of the p-local finite group. So for example, the 2-local group (S, F, L) induced by GL_{4}(2) is centerfree, has sylow of size 2^{6}, and as it was pointed in the introduction, |L| has infinitely many homotopy groups.

1.5. The exotic p-local finite groups of Ruiz and Viruel described in [23, Table 1.2] satisfy the conclusions (1)-(4) of the theorem. Indeed, they are defined over the extra special group P = p_{+}^{1+2} of order p^{3} and exponent p. In particular the nerves of their linking systems have the type of an Eilenberg-MacLane K(Γ, 1). Their fusion and centric linking systems are induced by Γ, i.e. F = F_{P} (Γ) and L is induced from a signaliser functor on the transporter category of Γ. A further study of these examples is given in [20].

1.6. Recall from Martino-Priddy [22] that the groups

G = SL_{2}(41)  and  K = SL_{2}(89)

are mod-2 equivalent, however there exists no zigzag of mod-2 equivalences of finite groups connecting G and K. One may ask what happens when infinite groups are allowed. Using the functoriality of the Kan-Thurston construction we get

KT(BG) ⟶ KT(BG_{p}^{∧}) ≃ KT(BK_{p}^{∧}) ⟵ KT(BK)
↓  ↓  ↓
BG ⟶  BG_{p}^{∧} ≃ BK_{p}^{∧}  ⟵ BK

where all the vertical maps are homology equivalences. Thus, we have connected G and K by means of mod-p equivalences between groups, albeit infinite. The problem is that the fundamental groups of the Kan-Thurston construction on these spaces are huge and there is little we can say on them. Fortunately, our theorem yields another zigzag of groups connecting G and K on which we have greater control.

Note that the 2-Sylow subgroups of G and K have order 16 = 2^{4}. Since they are mod-2 equivalent then L_{S}^{c}(G) = L_{S}^{c}(K). Using Theorem 1.3 there is a zigzag of mod-2 equivalences of groups

G ← Γ′_{G} → Γ_{G} = Γ_{K} ← Γ′_{K} → K

where BΓ_{G} ≃ |L_{S}(G)| = |L_{S}(K)| ≃ BΓ_{K}. The groups Γ := Γ_{G} = Γ_{K}, Γ′_{G} and Γ′_{K} are amalgamated products of finite groups. Moreover, S is contained in all these groups and

F_{S}(G) = F_{S}(Γ′_{G}) = F_{S}(Γ) = F_{S}(Γ′_{K}) = F_{S}(K).

Also there are signaliser functors on the transporter categories T_{S}^{c}(Γ), T_{S}^{c}(Γ′_{G}) and T_{S}^{c}(Γ′_{K}) which induce the linking systems of G and K.

The same argument applies to the mod-2 equivalent groups SL_{3}(3) and M_{11} which cannot be connected by a zigzag of mod-2 equivalent finite groups as well.

2. A quickie on p-local finite groups

Fix a group G. An element g ∈ G defines an inner automorphism c_{g}(x) = gxg^{−1}. If c_{g}(H) ≤ K where H, K ≤ G, we also use c_{g} to denote the restriction c_{g}|_{H}^{K}. The set Hom_{G}(H, K) denotes the set of all such monomorphisms.

2.1. Definition. Let S be a finite p-group. A fusion system on S is a subcategory F of the category of groups whose objects are the subgroups of S and whose morphisms are group monomorphisms satisfying the requirement that every morphism in F factors as an isomorphism in F followed by an inclusion of groups. In addition, Hom_{S}(P, Q) ⊆ F(P, Q) for every P, Q ∈ F.

Subgroups P and P′ of S are called F-conjugate if they are isomorphic as objects of F. Note that every endomorphism in F is an automorphism, and we therefore write Aut_{F}(P) for F(P, P). The group of outer F-automorphisms of P is the subgroup Out_{F}(P) = Aut_{F}(P)/Inn(P) of Out(P).

Suppose that F′ is a fusion system on S and that Δ_{1}, …, Δ_{m} are groups of automorphisms of subgroups Q_{1}, …, Q_{m} of S. We denote by

F := ⟨F′ ; Δ_{1}, …, Δ_{m}⟩

the smallest fusion system on S containing F′ and such that Δ_{i} ≤ Aut_{F}(Q_{i}). Explicitly, F has the same objects as F′. Morphisms in F are compositions of morphisms in F′ and restrictions of elements φ ∈ Δ_{i} to subgroups of Q_{i}.

2.2. Definition. Let F be a fusion system on S. A subgroup P ≤ S is called fully F-centralised if |C_{S}(P)| ≥ |C_{S}(P′)| for every P′ in the F-conjugacy class of P. A subgroup P ≤ S is called fully F-normalised if |N_{S}(P)| ≥ |N_{S}(P′)| for every P′ in the F-conjugacy class of P.

A subgroup P of S is called F-centric if C_{S}(P′) = Z(P′) for every P′ in the F-conjugacy class of P. The set of all the F-centric subgroups of S is denoted by F^{c}. We sometimes let F^{c} denote the full subcategory of F containing the F-centric subgroups of S.

Fix a fusion system F on S and a morphism φ ∈ F(P, S). Set

N_{φ} = {g ∈ N_{S}(P) : φ ∘ c_{g} ∘ φ^{−1} ∈ Aut_{S}(φP)}.

2.3. Definition ([6, Definition 1.2]). A fusion system F on S is saturated if

I. Every fully F-normalised P ≤ S is fully F-centralised and Aut_{S}(P) ∈

Syl_{p}(Aut_{F}(P)). II. For every φ ∈ F(P, S) such that φ(P) is fully F-centralised, there exists a

morphism φ̃ ∈ F(N_{φ}, S) which extends φ.

2.4. Definition. A finite group G is called p-reduced if it contains no non-trivial normal p-subgroup.

2.5. Definition. A subgroup P ≤ S is called F-radical if Out_{F}(P) is p-reduced. The set of F-radical subgroups of S is denoted F^{r}. The set of the subgroups of S which are both F-centric and F-radical are denoted F^{cr}.

A striking consequence of these definitions is Alperin’s fusion theorem

2.6. Theorem ([6, Theorem A.10]). A saturated fusion system F is generated by the automorphism groups Aut_{F}(Q) where Q ∈ F^{cr}.

2.7. Definition. Let F be a saturated fusion system on S. An associated centric linking system consists of

1. A small category L whose object set is F^{c}.

2. A functor π : L → F which induces the inclusion F^{c} ⊆ Obj(F).

3. A collection of group monomorphisms {δ_{P} : P → Aut_{L}(P)} for all P ∈ F^{c}. This data is subject to

(A) For every P, Q ∈ F^{c}, the action of Z(P) on L(P, Q) via φ · z = φ ∘ δ_{P} (z) for every z ∈ Z(P), is free. Moreover the functor π sets up a bijection L(P, Q)/Z(P) ≈ F(P, Q).

(B) The composition P ⟶^{δ_{P}} Aut_{L}(P) ⟶^{π} Aut_{F}(P) coincides with the composi-

tion P → Inn(P) ⊆ Aut_{F}(P) for every P ∈ F^{c}. (C) For every φ ∈ L(P, Q) and every g ∈ P we have φ∘δ_{P} (g) = δ_{Q}(π(φ)(g))∘φ.

2.8. Definition. A p-local finite group is a triple (S, F, L) where F is a saturated fusion system on S and L is an associated centric linking system.

The p-completion of the nerve of L, i.e. the space |L|_{p}^{∧}, is called the classifying space of (S, F, L). The inclusion δ_{S} : S → Aut_{L}(S) gives rise to a canonical map f : BS → |L|. More generally the full subcategory of L on an object P gives rise to a natural map BAut_{L}(P) → |L|. More details are given in the remarks before the statement of Theorem 3.4.

2.9. The prime examples of p-local finite groups are those which are induced by finite groups. See Proposition 1.3 and the remarks after Definition 1.8 in [6]. More explicitly, fix a finite group G and S ∈ Syl_{p}(G). Consider the category F = F_{S}(G) whose objects are the subgroups of S and whose morphisms are restrictions of inner automorphisms of G. The linking system L = L_{S}^{c}(G) has F_{S}^{c}(G) as its object set. Morphism sets L(P, Q) are given by N_{G}(P, Q)/C′_{G}(P) where C′_{G}(P) is the subgroup of C_{G}(P) generated by the elements of order p′. In this case, when P is F-centric, Z(P) is a Sylow p-subgroup of C_{G}(P) and C′_{G}(P) is a complement. The maps δ_{P} are induced by the inclusion of P in N_{G}(P, Q). The projection π is induced by the obvious bijection F(P, Q) ≈ N_{G}(P, Q)/C_{G}(P).

A key feature of this construction is that |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}. Furthermore, two finite groups G and K are mod-p equivalent if and only if the have the same Sylow p-subgroups S and moreover L_{S}^{c}(G) ≅ L_{S}^{c}(K).

3. The fusion systems of aspherical spaces

Signaliser functors. Let S be a finite p-subgroup of a discrete group G, possibly infinite. In 1.2 we have defined the fusion system F_{S}(G) and the transporter category T_{S}^{c}(G). Also recall

3.1. Definition. A signaliser functor on T_{S}^{c}(G) is an assignment P ↦ Θ(P) for every F-centric subgroup of S such that (S1) Θ(P) is a complement of Z(P) in C_{G}(P). (S2) An inclusion P ≤ Q gives rise to an inclusion Θ(Q) ≤ Θ(P). (S3) For every F_{S}(G)-centric subgroup P ≤ S and every g ∈ G such that P^{g} ≤

S, hence P^{g} is F-centric, we have Θ(P^{g}) = Θ(P)^{g}.

We note that (S3) implies, in particular, that Θ(P) ◁ N_{G}(P). Given a signaliser functor on T^{c} = T_{S}^{c}(G) one can form the category L_{S,Θ} := T^{c}/Θ as follows. The objects of L_{S,Θ} are the F-centric subsgroups of S. Morphism sets are given by L_{S,Θ}(P, Q) = N_{G}(P, Q)/Θ(P), that is left cosets of Θ(P). Composition is obtained by multiplying representatives of cosets, which is well defined because of (S2) and (S3). Since P ∩ Θ(P) = 1, the inclusion P ⊆ N_{G}(P, Q) induces an injection

δ_{P} : P → N_{G}(P, Q)/Θ(P)

which renders L_{S,Θ} a free right Z(P)-set. Taking orbits of these Z(P)-actions one obtains a functor π : L_{S,Θ} → F^{c}.

One easily checks that L_{S,Θ} is a centric linking system associated to F_{S}(G). Indeed, with the notation of 2.7, (A) π : L_{S,Θ} → F_{S}^{c}(G) is the identity on objects and on morphism sets it is the quotient by the free action of Z(P) on N_{G}(P, Q)/Θ(P) via δ_{P}. Also (B) πδ_{P} (g) = c_{g} ∈ Aut_{F}(P) for all g ∈ P. In addition (C) for fΘ(P) in L_{S,Θ}(P, Q) and for g ∈ P we have by inspection δ_{Q}(π(f)(g)) ∘ f = f ∘ δ_{P} (g). Indeed using (S2), (S3)

fgf^{−1}Θ(Q) · fΘ(P) = fgΘ(Q)^{f}Θ(P) = fgΘ(P) = fΘ(P) · gΘ(P).

We deduce

3.2. Proposition. Let S be a finite p-subgroup of G and let Θ be a signaliser functor on T_{S}^{c}(G). Then (L_{S,Θ}, δ, π) is an associated linking system for F_{S}(G).

When S is a Sylow p-subgroup of a finite group G, if P ≤ S is F-centric then Z(P) is a Sylow p-subgroup of C_{G}(P) and there is a unique signaliser functor on T_{S}^{c}(G) where Θ(P) = C′_{G}(P) = O^{p}(C_{G}(P)). Here O^{p}(Γ) denotes the subgroup of Γ generated by the torsion elements of Γ of p′-order.

The main results of this section. Recall that any discrete group G is canonically associated with a category BG which contains a unique object ∘_{G} with G as its set of automorphisms. The geometric realisation of the nerve of BG is an Eilenberg-MacLane space of type K(G, 1) denoted BG. Throughout this paper whenever we write BG we refer to this particular model of K(G, 1). A group homomorphism φ : G → G′ gives rise to an obvious functor Bφ : BG → BG′. Conversely, every functor Φ : BG → BG′ has this form. Natural transformations Bφ → Bψ are in one to one correspondence with elements g′ ∈ G′ such that φ = c_{g′} ∘ ψ where c_{g′} is the inner automorphism of G′ induced by g′.

3.3. Fix a p-local finite group (S, F, L). The monomorphism δ_{P} : P → Aut_{L}(P) gives rise to a functor j_{P} : BP → L into the subcategory of L on the object P. The geometric realisation of j_{S} yields a canonical map

f : BS → |L|.

We now state the main results of this section

3.4. Theorem. Let (S, F, L) be a p-local finite group such that |L| is an Eilenberg-MacLane space of type K(G, 1) and such that f : BS → |L| induces the inclusion i : S → G. Then F = F_{S}(G). Furthermore, there exists a signaliser functor Θ on T_{S}^{c}(G) which induces L, namely there is an isomorphism of p-local finite groups L_{S,Θ} ≅ L.

An F-collection is a set C of subgroups of S which is closed to F-conjugacy, that is, if P ∈ C then so is any F-conjugate of P. An F-collection is called centric if it consists of F-centric subgroups of S. In this case L^{C} denotes the full subcategory of L whose object set is C.

Consider a p-local finite group (S, F, L) associated with a Sylow p-subgroup S of a finite group Γ and fix an F-collection C which contains S. The inclusion S ≤ N_{Γ}(S) gives a natural map

g : BS → |T_{S}^{C}(Γ)|.

If C is also F-centric we have a natural quotient functor T_{S}^{C}(Γ) → L_{S}^{C}(Γ) which induces a mod-p equivalence on nerves by [5, Lemma 1.3]. That is

(3.5)  |T_{S}^{C}(Γ)|_{p}^{∧} ≃ |L_{S}^{C}(Γ)|_{p}^{∧}.

It is shown in [3, Theorem B] that for any p-local finite group, if C is an F-centric collection which contains F^{cr} (Definition 2.5) then the inclusion induces a homotopy equivalence

(3.6)  |L^{C}| ⟶^{≃} |L|.

Our second result is

3.7. Theorem. Let (S, F, L) be a p-local finite group associated with a Sylow psubgroup S of a finite group Γ. Let C be an F-collection which contains all the F-centric F-radical subgroups of S. If |T_{S}^{C}(Γ)| has the homotopy type of K(G, 1) where S is a subgroup of G via g : BS → |T_{S}^{C}(Γ)|, then F = F_{S}(G) and moreover there exists a signaliser functor Θ on T_{S}^{c}(G) such that L_{S,Θ} ≅ L_{S}^{c}(Γ) = L.

Fusion systems associated to spaces. Fix topological spaces X and Y. The space of all the continuous functions X → Y is denoted map(X, Y). For f : X → Y, we let map(X, Y)_{f} denote the connected component of f with f as a basepoint. Now let H and G be discrete groups. The set Hom(H, G) modulo the action of Inn(G) is denoted Rep(H, G). It is well known that

[BH, BG] ≅ Rep(H, G) and map(BH, BG)_{Bρ} ≃ BC_{G}(ρ(H)).

The second equivalence is obtained by adjunction of the realisation of the functor

BC_{G}(ρ(H)) × BH ⟶^{(x,h)↦x·ρ(h)} BG.

Note that the first bijection implies, in particular, that for every subgroup H ≤ G and every g ∈ G the map Bc_{g} : BH → BG is homotopic to Bi_{H}^{G} where i_{H}^{G} is the inclusion H ≤ G.

We also recall that the space map_{∗}(BG, BH) of pointed maps BG → BH is homotopically discrete and is equivalent to the set Hom(G, H) of homomorphisms G → H.

Recall from [6, §7] that given a finite p-group S and a map of spaces f : BS → X, one constructs a fusion system F = F_{S,f}(X) as follows. The objects of F are all the subgroups of S. Morphism sets F(P, Q) consist of all the group monomorphisms φ : P → Q such that the following triangle is homotopy commutative

\xymatrix{BP \ar[rr]^{Bφ} \ar[dr]_{f∘Bi_{P}^{S}} &  & BQ \ar[dl]^{f∘Bi_{Q}^{S}} \\  & X.}

Theorem 7.3 in [6] asserts that for any p-local finite group one has F = F_{S,η∘f}(|L|_{p}^{∧}), where η is the coaugmentation of the completion.

3.8. Proposition. Consider a finite p-group S and a map f : BS → X. (1) If g : BS → X is homotopic to f then F_{S,f}(X) = F_{S,g}(X).

(2) A map g : X → Y induces an inclusion F_{S,f}(X) ⊆ F_{S,g∘f}(Y). Equality holds if g is a homotopy equivalence.

Proof. The first part of this proposition follows immediately from the definitions. For the second part note that F_{S,f}(X) and F_{S,g∘f}(Y) have the same object sets. The inclusion of categories follows from the fact that two homotopic maps BP → X remain homotopic after being composed with g. The equality F_{S,f}(X) = F_{S,g∘f}(Y) when g is a homotopy equivalence follows from the first part by choosing a homotopy inverse h for g yielding F_{S,f}(X) ⊆ F_{S,g∘f}(Y) ⊆ F_{S,h∘g∘f}(X) = F_{S,f}(X). □

3.9. Proposition. Let S be a finite p-subgroup of a discrete group G. Let X be an Eilenberg-MacLane space of type K(G, 1) and suppose that f : BS → X is a map which induces the inclusion i : S → G on fundamental groups. Then F_{S,f}(X) = F_{S}(G).

Proof. Fix a homotopy equivalence h : X → BG such that h ∘ f induces the inclusion i : S → G upon taking fundamental groups. Since [BS, BG] ≅ Rep(S, G), it follows that h ∘ f ≃ Bi. Now Proposition 3.8 implies that

F_{S,f}(X) = F_{S,h∘f}(BG) = F_{S,Bi}(BG).

It is left to show that F_{S,Bi}(BG) = F_{S}(G). Fix P, Q ≤ S and a monomorphism φ : P → Q. The bijection [BP, BG] ≅ Rep(P, G) shows that Bi_{Q}^{G} ∘ Bφ ≃ Bi_{P}^{G} if and only if [φ] = [i_{P}^{G}] in Rep(P, G), namely φ = c_{g}|_{P}^{Q} for some g ∈ G. Thus, φ ∈ F_{S,Bi}(BG) if and only if φ ∈ F_{S}(G). □

Linking systems of Eilenberg-Mac Lane type.

3.10. Proposition. Let (S, F, L) be a p-local finite group and assume that X is a space equipped with a map g : BS → X and a homotopy equivalence ℓ : X_{p}^{∧} ⟶^{≃} |L|_{p}^{∧}. Assume further that the compositions

BS ⟶^{g} X ⟶^{η} X_{p}^{∧} ⟶^{ℓ} |L|_{p}^{∧} and BS ⟶^{f} |L| ⟶^{η} |L|_{p}^{∧}

are homotopic. If F ⊆ F_{S,g}(X) then F = F_{S,g}(X).

Proof. The result follows from 3.8 and [6, 7.3] since F ⊆ F_{S,g}(X) ⊆ F_{S,η∘g}(X_{p}^{∧}) = F_{S,ℓ∘η∘g}(|L|_{p}^{∧}) = F_{S,η∘f}(|L|_{p}^{∧}) = F. □

3.11. Corollary. Let (S, F, L) be a p-local finite group. Then F_{S,f}(|L|) = F where f : BS → |L| is the natural map.

Proof. To apply Proposition 3.10 it only remains to check that F ⊆ F_{S,f}(|L|).

We first claim that F^{c} ⊆ F_{S,f}(|L|). Fix a morphism φ : P → S in F^{c}. We must show that f ∘ Bφ ≃ f ∘ Bi_{P}^{S} where i_{P}^{S} : P → S is the inclusion. Consider the following functors (see 3.3)

Φ : BP ⟶^{Bφ} BS ⟶^{j_{S}} L,

Γ : BP ⟶^{j_{P}} L,

Ψ : BP ⟶^{Bi_{P}^{S}} BS ⟶^{j_{S}} L.

Let ι_{P}^{S} be a lift in L for the inclusion i_{P}^{S} : P → S. Similarly let φ̃ be a lift in L for φ : P → S. Axiom (C) of the linking system is a restatement of the assertion that

P ⟶^{ι_{P}^{S}} S and P ⟶^{φ̃} S provide natural transformations Γ → Ψ and Γ → Φ. Indeed, j_{P} (g) = δ_{P} (g) for every g ∈ P. Consequently, f ∘ Bφ = |Φ| and f ∘ Bi_{P}^{S} = |Ψ| are homotopic because they are both homotopic to |Γ|.

By Proposition 3.8 and [6, Proposition 7.3], the map η : |L| → |L|_{p}^{∧} gives rise to inclusions

F^{c} ⊆ F_{S,f}(|L|) ⊆ F_{S,η∘f}(|L|_{p}^{∧}) = F.

Alperin’s fusion Theorem (Theorem 2.6) asserts that every morphism in F is obtained by composing restrictions of automorphisms of F-centric subgroups. Since F_{S,f}(|L|) contains F^{c} and all these compositions, it follows that F ⊆ F_{S,f}(|L|), hence equality. □

3.12. Corollary. Let (S, F, L) be a p-local finite group associated with a finite group Γ and some S ∈ Syl_{p}(Γ). Let C be a centric F-collection, that is C ⊆ F^{c}, which contains F^{cr}. Then F = F_{S,g}(|T_{S}^{C}(Γ)|) where g : BS → |T_{S}^{C}(Γ)| is the natural map.

Proof. The maps |T_{S}^{C}(Γ)|_{p}^{∧} → |L^{C}|_{p}^{∧} → |L|_{p}^{∧} are homotopy equivalences by (3.5) and (3.6). Furthermore, by inspection of 2.9, the composition

(3.13)  BS ⟶^{g} |T_{S}^{C}(Γ)| → |L_{S}^{C}(Γ)|

is equal to f : BS → |L_{S}^{C}(Γ)|. To apply Proposition 3.10 it only remains to prove that F ⊆ F_{S,g}(|T_{S}^{C}(Γ)|). The proof of the corresponding fact in Corollary 3.11 can be adapted easily. Lifts of φ ∈ F(P, S) are appropriate elements g ∈ N_{Γ}(P, S). □

Recall from [6, §7] that given a map f : BS → X one constructs a category L_{S,f}(X) whose objects are the subgroups P ≤ S and morphisms P → Q are pairs (φ, [H]) where φ : P → Q is a group monomorphism and [H] is a homotopy class of paths in map(BP, X) from f ∘ Bi_{Q}^{S} ∘ Bφ to f ∘ Bi_{P}^{S}. We let L_{S,f}^{c} denote the full subcategory of the F_{S,f}(X)-centric subgroups P ≤ S. There is an obvious projection functor π : L_{S,f}^{c}(X) → F_{S,f}(X) by forgetting the homotopy H. Clearly, if h : X → Y is a homotopy equivalence then L_{S,f}(X) ≅ L_{S,h∘f}(Y). Similarly, if f, f′ : BS → X are homotopic then L_{S,f}(X) ≅ L_{S,f′}(X).

Now let (S, F, L) be a p-local finite group. It is shown in [6, Proposition 7.3] that F = F_{S,η∘f}(|L|_{p}^{∧}) and that L ≅ L_{S,η∘f}^{c}(|L|_{p}^{∧}) where η : |L| → |L|_{p}^{∧} is the natural augmentation of the p-completion. In fact one obtains an isomorphism of p-local finite groups.

3.14. Proposition. Let (S, F, L) be a p-local finite group. Assume that X is an Eilenberg-MacLane space K(G, 1) where S ≤ G and that we are given a map k : BS → X which induces an inclusion i : S → G on fundamental groups. Assume further that F = F_{S,k}(X) and that

(1) There is a homotopy equivalence ℓ : X_{p}^{∧} ⟶^{≃} |L|_{p}^{∧}.

(2) The compositions BS ⟶^{k} X ⟶^{η} X_{p}^{∧} ⟶^{ℓ} |L|_{p}^{∧} and BS ⟶^{f} |L| ⟶^{η} |L|_{p}^{∧} are

homotopic.

(3) For any F-centric subgroup P ≤ S, the adjoint of BZ(P) × BP ⟶^{μ} BS ⟶^{k} X_{p}^{∧} gives rise to a homotopy equivalence BZ(P) ≃ map(BP, X_{p}^{∧})_{η∘k∘Bi_{P}^{S}}

Then there exists a signaliser functor Θ on T_{S}^{c}(G) such that L ≅ L_{S,Θ}. In fact, (S, F_{S}(G), L_{S,Θ}) is isomorphic to (S, F, L).

Proof. Let T denote the transporter category T_{S}^{c}(G). Choose a homotopy equivalence h : X → BG such that h ∘ k induces the inclusion i : S → G on fundamental groups. For an F-centric subgroup P ≤ S we have the commutative diagram

\xymatrix{BZ(P) \ar[r] \ar[dr]_{≃} & map(BP, X)_{k∘Bi_{P}^{S}} \ar[r]_{≃}^{h_{∗}} \ar[d]^{η_{∗}} & map(BP, BG)_{Bi_{P}^{G}} \ar[d]^{η_{∗}} \\  & map(BP, X_{p}^{∧})_{η∘k∘Bi_{P}^{S}} \ar[r]^{≃}_{_{h_{p}^{∧}_{∗}}} & map(BP, BG_{p}^{∧})_{η∘Bi_{P}^{G}}.}

The maps η are the the coaugmentation of the p-completion, and the diagonal arrow is a homotopy equivalence by hypothesis. It follows that the vertical arrow on the right induces a split surjection

π_{1}map(BP, BG)_{Bi_{P}^{G}} → π_{1}map(BP, BG_{p}^{∧})_{η∘Bi_{P}^{G}} ≅ Z(P).

By identifying π_{1}map(BP, BG)_{Bi_{P}^{G}} = C_{G}(P) we now define the signaliser functor Θ by

Θ(P) = Ker(C_{G}(P) → π_{1}map(BP, BG_{p}^{∧})_{η∘Bi_{P}^{G}}).

Clearly Θ(P) is a complement of Z(P) in C_{G}(P). If P ≤ Q, then Q is F-centric and we get a commutative diagram

BC_{G}(Q) ⟶^{≃} map(BQ, BG)_{Bi_{Q}^{G}} ⟶ map(BQ, BG_{p}^{∧})_{η∘Bi_{Q}^{G}} ≃ BZ(Q)
Bincl↓  ^{(Bi_{P}^{Q})^{∗}}↓  ↓^{(Bi_{P}^{Q})^{∗}}
BC_{G}(P) ⟶_{≃} map(BP, BG)_{Bi_{P}^{G}} ⟶ map(BP, BG_{p}^{∧})_{η∘Bi_{P}^{G}} ≃ BZ(P).

By applying π_{1}(−) to the diagram and taking kernels of the rows we find that Θ(Q) ≤ Θ(P). For g ∈ G such that P, P^{g} ≤ S we have P^{g} is F-centric because F = F_{S}(G) and we have a commutative diagram

BC_{G}(P) ⟶^{≃} map(BP, BG)_{Bi} ⟶ map(BP, BG_{p}^{∧})_{η∘Bi}
Bc_{g}^{−1}↓^{≃}  ^{≃}↓^{map(Bc_{g}^{−1},Bc_{g})}  ^{≃}↓^{map(Bc_{g}^{−1},Bc_{g} _{p}^{∧})}
BC_{G}(P^{g}) ⟶_{≃} map(BP^{g}, BG)_{Bi} ⟶ map(BP^{g}, BG_{p}^{∧})_{η∘Bi}.

This gives an isomorphism Θ(P) ⟶^{c_{g}} Θ(P^{g}) by taking kernels of the rows after passage to fundamental groups. It follows that Θ is a signalizer functor.

It remains to show that L_{S,Θ} ≅ L. Note that h ∘ k ≃ Bi_{S}^{G} because [BS, BG] = Rep(S, G). Since h : X → BG is a homotopy equivalence we have

L = L_{S,η∘f}^{c}(|L|_{p}^{∧}) = L_{S,ℓ∘η∘k}^{c}(|L|_{p}^{∧}) ≅ L_{S,η∘k}^{c}(X_{p}^{∧})

≅ L_{S,η∘h∘k}^{c}(BG_{p}^{∧}) = L_{S,η∘Bi_{S}^{G}}^{c}(BG_{p}^{∧}).

It remains to find an isomorphism L_{S,Θ} ≅ L_{S,η∘Bi_{S}^{G}}^{c}(BG_{p}^{∧}). Fix an F-centric subgroup P ≤ S and an element g ∈ G such that ^{g}P ≤ S. Conjugation induces a functor c_{g} : BP → BG. There is a canonical path from Bi_{P}^{G} to Bc_{g}|_{P}^{G} in map(BP, BG) which is given by the realisation of the functor H_{g} : {0 ⟶^{ι_{0}} 1} × BP → BG defined by

H_{g}(0 × o_{P} ⟶^{ι_{0}} 1 × o_{P}) = o_{G} ⟶^{g} o_{G},

H_{g}(0 × o_{P} ⟶^{x∈P} 0 × o_{P}) = o_{G} ⟶^{x} o_{G},

H_{g}(1 × o_{P} ⟶^{x∈P} 1 × o_{P}) = o_{G} ⟶^{c_{g}(x)} o_{G}.

The choice of the path H_{g} is natural in the following sense. Elements g, g′ ∈ G which define P ⟶^{c_{g}} Q ⟶^{c_{g′}} S give rise to a path from Bi_{P}^{G} to Bc_{g′·g} in map(BP, BG) by concatenation of H_{g′} ∘ Bc_{g} after H_{g′} where H_{g′} ∘ Bc_{g} is the image of H_{g′} under

map(BQ, BG)_{Bi_{Q}^{G}} ⟶^{(Bc_{g})^{∗}} map(BP, BG)_{Bi_{P}^{G}}.

This path is homotopic to the path H_{g′·g}; An explicit homotopy is given by means of the realisation of the following functor K : {0 ⟶^{ι_{0}} 1 ⟶^{ι_{1}} 2} × BP → BG

K(n × o_{P} ⟶^{x∈P} n × o_{P} ) = \{ \begin{matrix}o_{G} ⟶^{x} o_{G} & , & n = 0 \\ o_{G} ⟶^{c_{g}(x)} o_{G} & , & n = 1 \\ o_{G} ⟶^{c_{g′g}(x)} o_{G} & , & n = 2 \end{matrix}

K(n × o_{P} ⟶^{ι_{n}} (n + 1) × o_{P} ) = \{ \begin{matrix}o_{G} ⟶^{g} o_{G} & , & n = 0 \\ o_{G} ⟶^{g′} o_{G} & , & n = 1. \end{matrix}

Indeed, the realisation of K is a 2-simplex in map(BP, BG) whose restriction to the 0 − 1, 1 − 2 and 0 − 2 faces are respectively H_{g}, H_{g′} ∘ Bc_{g} and H_{g′g}.

Note that g ∈ Θ(P) induces the inclusion i_{P}^{G} : P → G. Therefore, H_{g} is a loop based at Bi_{P}^{G} which corresponds to g ∈ π_{1}BC_{G}(P). Consider the maps

BC_{G}(P) ≃ map(BP, BG)_{Bi_{P}^{S}} ⟶^{η_{∗}} map(BP, BG_{p}^{∧})_{η∘Bi_{P}^{S}} ≃ BZ(P).

By definition, g is in the kernel of the map induced on fundamental groups, namely its image in map(BP, BG_{p}^{∧})_{η∘Bi_{P}^{S}} is a null homotopic loop. We deduce that the assignment Λ : N_{G}(P, Q)/Θ(P) → L_{S,η∘Bi}^{c}(BG_{p}^{∧})(P, Q) given by

Λ : gΘ(P) ↦ (c_{g}, [η ∘ H_{g}])

is well defined. It is also clearly bijective because η ∘ H_{g} is null homotopic for g ∈ C_{g}(P) if and only if g ∈ Θ(P). Functoriality of Λ follows from the homotopy H_{g′g} ≃ H_{g} ∗H_{g′} ∘Bc_{g}. Thus, Λ : L_{S,Θ} → L_{S,η∘Bi_{S}^{G}}(BG_{p}^{∧}) is a functor. Examination of the construction of the linking system L_{S,f}(|L|_{p}^{∧}) in [6, pp. 826, 827] shows that Λ commutes with the maps δ_{P}. In particular, both morphism sets have a free Z(P)-action whose orbit set is F(P, Q). It follows that Λ is an isomorphism of categories and therefore an isomorphism of p-local finite groups. □

Proof of Theorem 3.4. First, F = F_{S,f}(|L|) = F_{S}(G) by Corollaries 3.11 and 3.9. Conditions (1) and (2) of Proposition 3.14 hold trivially. The equivalence

Z(P) ≃ map(BP, |L|_{p}^{∧})_{η∘f∘Bi_{P}^{S}}

for F-centric P ≤ S is proved in [6, Theorem 4.4(c)]. □

Proof of Theorem 3.7. Put X = |T_{S}^{c}(G)|. Then F = F_{S,g}(X) = F_{S}(G) by Corollaries 3.12 and 3.9. Condition (1) of Proposition 3.14 is verified using (3.5) and (3.6). Condition (2) is explained in (3.13). Finally, condition (3)

BZ(P) ≃ map(BP, |L|_{p}^{∧})_{η∘f∘Bi_{P}^{S}} ≃ map(BP, |L^{C}|_{p}^{∧})_{η∘f∘Bi_{P}^{S}}

≃ map(BP, |T_{S}^{C}(G)|_{p}^{∧})_{η∘g∘Bi_{P}^{S}}

follows by using [6, Theorem 4.4(c)] and (3.5) and (3.6). □

4. Trees of spaces

Recall that a graph G consists of a finite set V of vertices and a finite set E of edges together with a function ι : E → P(V) where for every e ∈ E one has ι(e) = {v_{1}, v_{2}} for two distinct vertices of G. One easily sees that G is in fact an 1-dimensional simplicial complex [27, Ch. 3.1]. A tree T is a graph with no cycles. Equivalently, it is a graph whose underlying simplicial complex is a contractible space.

A graph G = (V, E) gives rise to a subdivision category sdG whose object set is the disjoint union V ∪ E and apart from identity morphisms there exists a unique morphism e → v whenever e is an edge of G incident with the vertex v.

Let T be a tree. A functor X : sdT → Spaces is called a tree of spaces and is denoted by (X, T ). It can be thought of as an assignment of spaces X(v), X(e) for every vertex v and edge e of T , together with maps X(e) → X(v) whenever e is and edge incident with a vertex v. Following Serre in [26], by a tree of groups (G, T ) we mean a functor G : sdT → Groups such that G(e) → G(v) are group monomorphisms. A tree of Eilenberg-MacLane spaces is a tree of spaces (X, T ) such that all the spaces X(v) and X(e) are Eilenberg-MacLane spaces and such that the maps X(e) → X(v) induce group monomorphisms π_{1}X(e) → π_{1}X(v); The choice of basepoints is immaterial.

Two trees of spaces (X, T ) and (Y, T ) are called weakly equivalent if they are weakly equivalent as functors X, Y : sdT → Spaces, see A.1. That is, there are sdT -diagrams of spaces X = X_{0}, …, X_{r} = Y and natural transformations X_{0} ← X_{1} → ⋯ ← X_{r−1} → X_{r} all of which are weak homotopy equivalences of diagrams.

A diagram X : D → Spaces is called pointed if it is equipped with a consistent choice of a basepoint for every space X(d) namely a choice of points x_{d} ∈ X(d) for every d ∈ D such that X(φ)(x_{d}) = x_{d′} for every morphism φ ∈ D(d, d′).

4.1. Proposition. Let (X, T ) be a tree of non-empty connected spaces. Then X is weakly homotopy equivalent to a pointed tree of spaces (X′, T ).

Proof. Let F denote the geometric realisation of the functor (sdT /−) which assigns to every object σ of sdT the nerve of the comma category (sdT ↓ σ). See [2, p. 292]. More explicitly, let I denote the unit interval [0, 1] with 1 as a basepoint. Then for every edge e of T we have F (e) = {0} as a subspace of I. For a vertex v of T we define

F (v) = ⋁_{e→v} I

where the wedge sum is over all the edges e of T which are incident with the vertex v. For e → v in sdT the map F (e) → F (v) is the inclusion {0} → I into the e → v summand of F (v).

For every vertex v and every edge e in T choose points x_{v} ∈ X(v) and x_{e} ∈ X(e) arbitrarily. For every morphism ι_{v}^{e} : e → v in sdT choose a path α_{v}^{e} in X(v) from X(ι_{v}^{e})(x_{e}) to x_{v}. Define a natural transformation Φ : F → X by

Φ(e) : F (e) ⟶^{x_{e}} X(e),  and  Φ(v) : F (v) = ⋁_{e→v} \underrightarrow{∑ α_{v}^{e}} X(v).

It is straightforward to check that Φ is a natural transformation. Let X′ denote the mapping cone (cofibre) of Φ. That is at every object σ of sdT we let X′ be the mapping cone of Φ(σ). The basepoints of the mapping cones render X′ pointed. The natural map X → X′ is a weak homotopy equivalence because F is objectwise contractible. □

4.2. Remark. Proposition 4.1 shows that every tree (X, T ) of Eilenberg-MacLane spaces can be replaced with a pointed weakly equivalent one. By applying the functor π_{1}(−) to the latter we obtain a tree of groups which by abuse of notation we denote (π_{1}X, T ).

Following Serre [26], the direct limit of a tree of groups (G, T ) is denoted \underrightarrow{lim}(G, T ). That is \underrightarrow{lim}(G, T ) = colim_{sdT}G. Serre shows in [26] that the natural

maps G(v), G(e) → \underrightarrow{lim}(G, T ) are injective.

Recall that a homomorphism of groups φ : K → G induces a map Bφ : BK → BG on their classifying spaces. If φ is injective then Bφ is up to homotopy a covering projection whose fibre is the coset space G/K.

4.3. Proposition. Let (X, T ) be a tree of Eilenberg-MacLane spaces and consider Y = hocolim_{sdT} X. Then Y is an Eilenberg-MacLane space K(Γ, 1) where Γ =

\underrightarrow{lim}(π_{1}X, T ). The natural maps X(v) → Y induce group monomorphisms π_{1}X(v) → π_{1}(Y) ≈ Γ.

Proof. We may assume that X is pointed by Proposition 4.1. Note that |sdT | is contractible because T is a tree. Since X is pointed, we can apply a generalised version of the Seifert-Van Kampen theorem, see [11, Theorem 5.2 and Proposition 5.3], to conclude that

(4.4)  π_{1}(Y) ≈ colim_{sdT} π_{1}X = \underrightarrow{lim}(π_{1}X, T ) ≡ Γ.

Using Serre’s results, the homomorphisms π_{1}X(v) → π_{1}Y are injective. It is only left to prove that Y is an Eilenberg-MacLane space.

We now replace X with an equivalent cofibrant diagram, namely one such that the natural map hocolim X → colim X is a (weak) homotopy equivalence (see e.g. [12]). Let Y = colim X. Since a tree is connected and the spaces of X are not empty and connected it follows that Y is also connected. Let Y → P_{1}Y denote the first Postnikov-Moore section of Y. Thus, P_{1}Y ≃ K(Γ, 1) where Γ is defined in (4.4). By choosing a basepoint in P_{1}Y we can form a tree (F, T ) of spaces where for every object t ∈ sdT the space F (t) is the homotopy fibre of the composition

X(t) → Y → P_{1}Y.

We obtain a natural transformation (F, T ) → (X, T ) of trees of spaces over P_{1}Y. By Puppe’s theorem A.6 we obtain a fibre sequence

(4.5)  hocolim_{sdT} F → hocolim_{sdT} X → P_{1}Y.

For every object t ∈ sdT we have seen that the map X(t) → P_{1}Y is a map between Eilenberg-MacLane spaces inducing a monomorphism on fundamental groups. Hence, up to homotopy, it is a covering projection. Therefore, F (t) is homotopically discrete. Since sdT has an 1-dimensional nerve it follows that hocolim_{sdT}F is an 1-dimensional complex, hence aspherical. Using the homotopy long exact sequence for the fibre sequence (4.5) we conclude that Y = hocolim X is aspherical, hence an Eilenberg-MacLane space of type K(Γ, 1). □

5. The normaliser decomposition and aspherical linking systems

The main results. Fix a p-local finite group (S, F, L). A centric F-collection C consists of a set C of F-centric subgroups of S which is closed to F-conjugacy. An n-simplex P in C is a chain of proper inclusions P_{0} < P_{1} < ⋯ < P_{n} of elements of C. We say that two n-simplices P and P′ are F-conjugate if there exists an isomorphism φ : P_{n} → P′_{n} in F such that φ(P) = P′, that is φ(P_{i}) = P′_{i} for all i = 0, …, n. The set of F-automorphisms of P is the set of all the F-automorphisms of P_{n} which leave P invariant. We say that a k-simplex Q is a face of P if it is a sub-simplex of P.

For P ∈ C denote

[P, −)_{C} = {Q ∈ C : P ≤ Q},

(P, −)_{C} = {Q ∈ C : P < Q}, (5.1)

[P, P′]_{C} = {Q ∈ C : P ≤ Q ≤ P′}, etc.

5.2. Definition. We shall say that a k-simplex P = {P_{0} < … < P_{k}} in C contains gaps if there exists some i = 0, …, k − 1 and Q ∈ C such that P_{i} < Q < P_{i+1}. We say that P contains no gaps otherwise.

5.3. Definition. A collection of F-centric subgroups of a p-local finite group (S, F, L) is called compressible if

(C1) C contains all the F-centric F-radical subgroups of S.

(C2) If P ∈ C and P ≠ S then there exists a unique Q ∈ C containing P such

that the 1-simplex P < Q contains no gaps. We denote Q by suc(P). (C3) If P < Q and P′ < Q′ are 1-simplices in C containing no gaps, and if P is

F-conjugate to P′ then {P < Q} is F-conjugate to {P′ < Q′}.

The following is evident in light of the finiteness of C

5.4. Proposition. Hypothesis (C2) is equivalent to the condition that the subposet [P, −)_{C} is linearly ordered by inclusion for every P ∈ C.

5.5. Theorem. If a p-local finite group (S, F, L) admits a compressible centric F-collection C then

(a) |L| is an Eilenberg-MacLane space K(Γ, 1). Moreover Γ is a limit of a tree of finite groups.

(b) The natural map BS → |L| induces an inclusion S ≤ Γ. More generally the natural maps BAut_{L}(P) → |L| induce an injection on fundamental groups for all P ∈ C.

(c) F = F_{S}(Γ). Moreover there exists a signaliser functor Θ on T_{S}^{c}(Γ) which induces L, namely L ≅ L_{S,Θ}.

5.6. Addendum. We will obtain an explicit description of the tree of groups whose limit is Γ in (5.25).

5.7. Theorem. Let (S, F, L) be a p-local finite group associated with a Sylow psubgroup S of a finite group G. If it admits a compressible centric F-collection C then

(a) |T_{S}^{C}(G)| is an Eilenberg-MacLane space K(Γ′, 1). Moreover Γ′ is a limit of a tree of finite groups.

(b) The natural map BS → |T_{S}^{C}(G)| induces an inclusion S ≤ Γ′. More precisely, for every P ∈ C the natural maps BN_{G}(P) → |T_{S}^{C}(G)| induce an injection on fundamental groups.

(c) F = F_{S}(Γ′). Moreover, there exists a signaliser functor Θ′ on T_{S}^{c}(Γ′) such that L ≅ L_{S,Θ′}.

(d) There is a zigzag of mod-p equivalences BΓ ← BΓ′ → BG.

5.8. Addendum. The tree of groups whose limit is Γ′ is described in (5.26).

To apply these theorems we need a criterion to identify compressible collections.

5.9. Lemma. Let (S, F, L) be a p-local finite group and let C be an F-collection of F-centric subgroups containing all the F-centric F-radical subgroups of S. Assume furthermore that for every P ∈ C, if P ≠ S then N_{S}P ∈ C and |N_{S}P : P | = p. Then C is compressible.

Proof. Condition (C1) is included in the hypotheses of the lemma. We now verify (C2). Consider an 1-simplex P < Q containing no gaps. Note that P < N_{Q}P. By assumption |N_{S}P : P | = p and since N_{Q}P ≤ N_{S}P we conclude that N_{Q}P = N_{S}P. It follows that P < N_{S}P ≤ Q. By assumption N_{S}P ∈ C and P < Q contains no gaps so Q = N_{S}(P).

We now verify (C3). Let P < Q and P′ < Q′ be 1-simplices in C. Assume that they contain no gaps and that there exists an isomorphism φ : P → P′ in F. By Alperin’s fusion Theorem, Theorem 2.6, there are F-conjugate subgroups P = P_{1}, P_{2}, …, P_{n} = P′ of S and F-centric F-radical subgroups R_{1}, …, R_{n−1} of S and automorphisms ψ_{1}, …, ψ_{n−1} of R_{1}, …, R_{n−1} in F such that P_{i}, P_{i+1} ≤ R_{i}, ψ_{i}(P_{i}) = P_{i+1} and φ = ψ_{n−1} ∘ ⋯ ∘ ψ_{1}. This is depicted by

[figure: P = P_{1}; incl; P_{2}; incl; incl; P_{3}; incl; P_{n} = P′; incl; incl; R_{1}; ψ_{1}; R_{2}; ψ_{2}; R_{n−2}; ψ_{n−2}; R_{n−1}.; ψ_{n−1}]

We may assume that P_{i} ≠ P_{i+1}, otherwise by omitting ψ_{i} from the sequence we obtain another F-isomorphism P → P′ that φ can be replaced with. Thus, we may assume that R_{i} properly contains P_{i}, P_{i+1} for all i = 1, …, n − 2. Note that N_{S}P_{i}, N_{S}P_{i+1} ∈ C by assumption and R_{i} ∈ C because F^{cr} ⊆ C. Also note that N_{S}P_{i} = N_{R_{i}}P_{i} because of the index assumption:

1 < |N_{R_{i}}(P_{i}) : P_{i}| ≤ |N_{S}P_{i} : P_{i}| = p.

Similarly N_{S}P_{i+1} = N_{R_{i}}P_{i+1}. Since ψ_{i} is an automorphism of R_{i} carrying P_{i} to P_{i+1}, it carries N_{S}P_{i} to N_{S}P_{i+1}. By composing all the appropriate restrictions of the ψ_{i}’s we see that φ extends to Q = N_{S}(P_{i}) and φ(Q) = φ(N_{S}P) = N_{S}P′ = Q′ so {P < Q} is F-conjugate to {P′ < Q′}. □

The normaliser decomposition. Fix a p-local finite group (S, F, L). Using Propositions 1.10 and 1.11 in [6] we can fix once and for all morphisms ι_{P}^{Q} in L for every inclusion P ≤ Q of F-centric subgroups of S which satisfy the following conditions. First, ι_{P}^{Q} are lifts of the inclusions i : P → Q. Second, for inclusions P ≤ Q ≤ R of F-centric subgroups we have ι_{Q}^{R} ∘ ι_{P}^{Q} = ι_{P}^{R}.

Fix a centric F-collection C. For an n-simplex P define the group of L-automorphisms Aut_{L}(P) as the set of (n + 1)-tuples (φ_{i}) in ∏_{i=0}^{n} Aut_{L}(P_{i}) which render the following commutative ladder in L

(5.10)  \begin{matrix}P_{0} & ⟶^{ι_{P_{0}}^{P_{1}}} & P_{1} & ⟶^{ι_{P_{1}}^{P_{2}}} & ⋯ & ⟶ & P_{n−1} & ⟶^{ι_{P_{n−1}}^{Pn}} & P_{n} \\ ^{φ_{0}}↓ & ^{φ_{1}}↓ & & & & ↓^{φ_{n−1}} & ↓^{φ_{n}} \\ P_{0} & ⟶^{ι_{P_{0}}^{P_{1}}} & P_{1} & ⟶^{ι_{P_{1}}^{P_{2}}} & ⋯ & ⟶ & P_{n−1} & ⟶^{ι_{P_{n−1}}^{Pn}} & P_{n}. \end{matrix}

Note that the projection induces a monomorphism Aut_{L}(P) → Aut_{L}(P_{n}) because every morphism in L is both a monomorphism and an epimorphism by [3, Corollary 3.10]. In fact, using Proposition 1.10 in [6], the image of this homomorphism is the subgroup of Aut_{L}(P_{n}) of those automorphisms that are lifts of Aut_{F}(P).

We recall from [21] the construction of the normaliser decomposition associated to the collection C. The category s̄dC is the poset whose objects are the F-conjugacy classes of the simplices in C denoted [P] or [n, P] if we want to specify the dimension of the simplex. A unique morphism [P] → [Q] exists in s̄dC if and only if Q is Fconjugate to a face of P. Recall that L^{C} denotes the full subcategory of L on the object set C. It is shown in [3, Theorem B] that if C contains F^{cr} then the inclusion L^{C} ⊆ L induces a homotopy equivalence on nerves.

5.11. Theorem (Libman [21]). There exists a functor δ_{C} : s̄dC → Spaces with the following properties

(1) There is a natural weak equivalence hocolim_{s̄dC} δ_{C} → |L^{C}|.

(2) There is a natural weak equivalence δ_{C}([P]) ≃ BAut_{L}(P) for every nsimplex P. The natural map δ_{C}([P]) → |L^{C}| is induced by the realisation of the inclusion BAut_{L}(P_{n}) ⊆ L^{C}.

(3) If Q is a face of P then δ_{C}([P]) → δ_{C}([Q]) can be identified with Bincl for the inclusion Aut_{L}(P) ≤ Aut_{L}(Q).

5.12. Suppose now that (S, F, L) is associated with a finite group G and a Sylow psubgroup S as in [6, p. 787]. Then δ_{C}([P]) is homotopy equivalent to the classifying space of

(N_{G}(P_{0}) ∩ … ∩ N_{G}(P_{n})) / C′_{G}(P_{0})

where C′_{G}(P_{0}) is the subgroup of C_{G}(P_{0}) of the elements of p′-order.

By taking all the conjugates of the elements of C we obtain a collection H of p-subgroups of G, that is H is a union of conjugacy classes in G of p-subgroups. One can form a poset s̄dH whose objects are the G-conjugacy classes of simplices in H and morphisms are induced by taking faces. It is easy to deduce from Sylow’s theorems and F = F_{S}(G) that s̄dC = s̄dH. Dwyer constructed in [13] a functor δ_{H} : s̄dH → Spaces such that δ_{H}([P]) is homotopy equivalent to the classifying space of the group N_{G}(P_{0}) ∩ … ∩ N_{G}(P_{n}). An inclusion of simplices Q ⊂ P gives rise to δ_{H}([P]) → δ_{H}([Q]) which is induced by the inclusion

⋂_{P ∈P} N_{G}(P) ≤ ⋂_{P ∈Q} N_{G}(P).

Dwyer shows in [13, §3.3] that hocolim δ_{H} ≃ |T^{H}(G)|. We further note that the F-centric F-radical subgroups of S are the principal p-radical subgroups of G contained in S. Hence H contains the principal p-radical in the sense of Grodal. He proves in [16, Theorem 1.5] that hocolim δ_{H} is mod-p equivalent to BG. Furthermore we show in [21] that the functors δ_{C} and δ_{H} are mod-p equivalent via a zigzag of natural transformations δ_{H} ≃ δ_{C} which at the object [P] is given by the mod-p equivalence

B(N_{G}(P_{0}) ∩ … ∩ N_{G}(P_{n})) → B(\frac{N_{G}(P_{0}) ∩ … ∩ N_{G}(P_{n})}{C′_{G}(P_{0})}).

Proof of 5.5 and 5.7. Using the notation of (5.1), Proposition 5.4 justifies

5.13. Definition. Let P = {P_{0} < … < P_{k}} be a k-simplex in a compressible F-centric collection C. Its closure \overline{P} is the simplex [P_{0}, P_{k}]_{C}.

5.14. Remark. Note that \overline{P} contains P as a subsimplex. It contains no gaps and moreover, if P ⊆ P′ and P′ contains no gaps then \overline{P} ⊆ P′. Also observe that if P is F-conjugate to P′ via some F-isomorphism φ : P_{k} → P′_{k} then \overline{P} is conjugate to \overline{P′} via φ.

We shall now consider a functor F : s̄dC → Spaces and assume that it satisfies (5.15) Hypothesis. For every n-simplex P the map F ([\overline{P}]) → F ([P]) is a homo-

topy equivalence. Clearly the constant functor satisfies this assumption. More interesting is

5.16. Proposition. Let (S, F, L) be a p-local finite group. Consider the normaliser decomposition δ_{C} associated with a centric F-collection C which satisfies hypothesis (C2) of Definition 5.3. Then δ_{C} satisfies (5.15), namely, δ_{C}([\overline{P}]) → δ([P]) is a homotopy equivalence for every k-simplex P = {P_{0} < … < P_{k}}.

Proof. In light of Theorem 5.11(3) it suffices to prove that the images of Aut_{L}(\overline{P}) and Aut_{L}(P) in Aut_{L}(P_{k}) are equal. Clearly Aut_{L}(\overline{P}) ≤ Aut_{L}(P). Fix φ in the image of Aut_{L}(P) → Aut_{L}(P_{k}). Consider Q ∈ \overline{P} = [P_{0}, P_{k}]_{C}. Since φ ∈ Aut_{L}(P) then φ(P_{k}) = P_{k} and φ(P_{0}) = P_{0}, hence P_{0} ≤ φ(Q) ≤ P_{k}. Here we abuse notation and write φ(Q) for the image of Q under π(φ). Since |φ(Q)| = |Q| and [P_{0}, P_{k}]_{C} is ordered linearly by inclusion, we see that φ(Q) = Q. Proposition 1.10 in [6] implies that there exists a unique morphism ψ ∈ Aut_{L}(Q) such that ι_{Q}^{P_{k}} ∘ ψ = φ ∘ ι_{Q}^{P_{k}}. Denote φ|Q := ψ. This can be done for every Q ∈ \overline{P} and the resulting sequence (φ|_{Q}) renders (5.10) commutative because the morphisms ι_{P}^{P′} are both monomorphisms and epimorphisms in L ([6, p. 788]). It follows that φ is in the image of Aut_{L}(\overline{P}). □

A similar argument working with elements of a finite group G instead of automorphisms of objects in L can be used to prove the next result.

5.17. Proposition. Let (S, F, L) be a p-local finite group associated with a finite group G and let C be a centric F-collection which satisfies (C2) of Definition 5.3. Let H denote the collection of the p-subgroups of G that are conjugate to the elements of C and let δ_{H} denote Dwyer’s normaliser decomposition 5.12. Then δ_{H} satisfies hypothesis (5.15), namely, δ_{H}([\overline{P}]) → δ_{H}([P]) is a homotopy equivalence for every k-simplex P in C.

Proof. Let P_{0} and P_{k} be the extremities of P. If g ∈ G normalises P_{0} and P_{k} then c_{g} ∈ F normalises \overline{P} because for any Q ∈ P we have P_{0} = P_{0}^{g} ≤ Q^{g} ≤ P_{k}^{g} = P_{k} so Q = Q^{g} because |Q| = |Q^{g}| and [P_{0}, P_{k}]_{C} is linearly ordered by inclusion. Therefore

⋂_{Q∈P} N_{G}(Q) = ⋂_{Q∈\overline{P}} N_{G}(Q)

and the result follows. □

Clearly if P is a k-simplex containing no gaps, so is any k-simplex which is F-conjugate to it. This justifies the following

5.18. Definition. Let s̄dC^{∗} denote the full subcategory of s̄dC containing all the objects of the form [P] where P is a k-simplex in C which contains no gaps.

5.19. Proposition. Let (S, F, L) be a p-local finite group and let C be a centric F-collection satisfying (C2) and (C3) of Definition 5.3. Let F : s̄dC → Spaces be a functor satisfying hypothesis (5.15). Then the inclusion j : s̄dC^{∗} → s̄dC induces a homotopy equivalence

hocolim_{s̄dC^{∗}} j^{∗}F ⟶^{≃} hocolim_{s̄dC} F.

Proof. Using the pushdown theorem, A.5, it suffices to show that the natural map

j_{∗}j^{∗}F → F

induces a weak equivalence of diagrams where j_{∗} stands for the homotopy left Kan extension.

Fix a k-simplex P = {P_{0} < … < P_{k}} in C and consider (j ↓ [P]). This category can be identified with the full subcategory of s̄dC of objects of the form [Q] where Q contains no gaps, and P is F-conjugate to a subsimplex of Q. Clearly [\overline{P}] is an object of (j ↓ [P]) and by Remark 5.14 it is in fact a terminal object. Assumption (5.15) implies that (see A.5)

(j_{∗}j^{∗}F)([P]) ≃ F ([\overline{P}]) ⟶^{≃} F ([P])

is an equivalence. □

5.20. Definition. Let s̄d_{1}C^{∗} denote the full subcategory of s̄dC^{∗} containing all the objects of the form [P] where P is either a 0-simplex or an 1-simplex in C with no gaps.

Let T_{C}^{∗,1} be the graph whose vertex set is the set of all the 0-simplices of s̄dC and whose edge set consists of all the 1-simplices in s̄dC that contain no gaps. That is

V = {[P] : P ∈ C},
E = {[P < Q] : P < Q contains no gaps}.

An edge [P < Q] is incident with the vertices [P] and [Q]. It is evident from the definitions that

(5.21)  s̄d_{1}C^{∗} ≈ sdT_{C}^{∗,1}.

5.22. Proposition. If C satisfies hypotheses (C2) and (C3) of Definition 5.3 then T_{C}^{∗,1} is a tree and the inclusion k : s̄d_{1}C^{∗} → s̄dC^{∗} is terminal (A.3).

Proof. For every P ∈ C there is a chain of inclusions

(5.23)  P < suc(P) < suc^{(2)}(P) < … < suc^{(n)}(P) = S

which gives rise to a path from the vertex [P] to the vertex [S] in T_{C}^{∗,1}. This shows that T_{C}^{∗,1} is connected. Hypothesis (C3) and Remark 5.14 imply that if [P] is a vertex of T_{C}^{∗,1} then [P < suc(P)] is independent of the choice of the representative P of [P]. Hence, the assignment [P] ↦ [P < suc(P)] establishes a bijection between V (T_{C}^{∗,1})\{[S]} and E(T_{C}^{∗,1}). It follows that |E(T_{C}^{∗,1})| = |V (T_{C}^{∗,1})|−1 and since T_{C}^{∗,1} is connected, it is a tree by [26, Ch. I, Proposition 12].

We now prove that k is terminal. Fix an object [P] of s̄dC^{∗}. Recall that P = {P_{0} < … < P_{n}} is an n-simplex containing no gaps. Therefore the comma category ([P] ↓ k) can be depicted as follows

[figure: [P_{0}]; [P_{1}]; [P_{2}]; [P_{n−1}]; [P_{n}]; [P_{0} < P_{1}]; [P_{1} < P_{2}]; [P_{n−1} < P_{n}].]

It clearly has a contractible nerve. □

We can now prove the first key lemma of this section

5.24. Lemma. Assume that C satisfies (C2) and (C3) of Definition 5.3 and that F : s̄dC → Spaces satisfies (5.15). Then the inclusion J : s̄d_{1}C^{∗} → s̄dC induces a homotopy equivalence

hocolim_{s̄d_{1}C^{∗}} J^{∗}F ⟶^{≃} hocolim_{s̄dC} F.

Proof. Decompose the inclusion J as s̄d_{1}C^{∗} ⟶^{k} s̄dC^{∗} ⟶^{j} s̄dC. By Proposition 5.19 the natural map hocolim j^{∗}F → hocolim F is an equivalence. Now, k is terminal so hocolim k^{∗}j^{∗}F → hocolim j^{∗}F is also an equivalence. □

We are now ready to prove the main results of this section

Proof of Theorem 5.5. Consider the normaliser decomposition δ_{C} : s̄dC → Spaces (Theorem 5.11). It satisfies (5.15) by Proposition 5.16. Lemma 5.24, the fact that C ⊇ F^{cr} and (3.6) yield

|L| ≃ |L^{C}| ≃ hocolim_{s̄dC} δ_{C} ≃ hocolim_{s̄d_{1}C^{∗}}J^{∗}δ_{C}.

Proposition 5.22 and (5.21) show that s̄d_{1}C^{∗} = sdT_{C}^{∗,1} where T_{C}^{∗,1} is a tree. It follows from Theorem 5.11, that J^{∗}δ_{C} is a tree of Eilenberg-MacLane spaces. We deduce from Proposition 4.3 that |L| is an Eilenberg-MacLane space K(Γ, 1) where, using Remark 4.2,

(5.25)  Γ = π_{1}|L^{C}| = \underrightarrow{lim}(π_{1}J^{∗}δ_{C} , T_{C}^{∗,1}).

Furthermore, for every P ∈ C there are group monomorphisms

Aut_{L}(P) = π_{1}δ([P]) → π_{1}|L| = Γ.

We identify S as a subgroup of Γ via the inclusion S ≤ Aut_{L}(S) → Γ. Part (c) follows from Theorem 3.4. □

Proof of Theorem 5.7. Consider the collection H of subgroups of G that are conjugate to the elements of C. We have remarked earlier that Dwyer and Grodal have shown that

|T^{H}(G)| ≃ hocolim δ_{H} → BG

is a mod-p equivalence. In Proposition 5.17 we show that δ_{H} satisfies (5.15). Furthermore, J^{∗}δ_{H}, where J is the inclusion of s̄d_{1}C^{∗} in s̄dC, is a Tree of Eilenberg-MacLane spaces by (5.21) and Proposition 5.22. Now Proposition 4.3 implies that |T^{H}(G)| has the homotopy type of a K(Γ′, 1) where

(5.26)  Γ′ = \underrightarrow{lim}(π_{1}J^{∗}δ_{H} , T_{C}^{∗,1}).

Also, the maps N_{G}(P) = π_{1}δ_{H}([P]) → Γ′ are injective. This completes the proof of (a) and (b).

Part (c) follows from Theorem 3.7. For (d) we use the mod-p equivalence δ_{H} → δ_{C} (5.12) which gives rise to mod-p equivalences

hocolim δ_{C} ← hocolim δ_{H} → BG

where the space on the left is BΓ by Theorem 5.5. □

6. Small Sylow subgrous

Recall that F^{cr} denotes the collection of the F-centric F-radical subgroups of S. In this section we prove

6.1. Proposition. A p-local finite group (S, F, L) admits a compressible F-centric collection C (Definition 5.3) if |S| = p^{5}.

Proof. Let C be the following F-collection of subgroups of S (1) All the F-centric F-radical subgroups P of order p^{2}.

(2) All the F-centric subgroups Q of order p^{3} that are:

• isomorphic to the extraspecial group of exponent p denoted p_{+}^{1+2} when p > 2.

• isomorphic to the dihedral or quaternion 8-groups if p = 2.

(3) all the F-centric F-radical subgroups Q of order p^{3} isomorphic to p×p×p.

(4) All the F-centric subgroups R of order p^{4}.

(5) The group S.

We now verify the hypotheses of Lemma 5.9. First we need several facts on

6.2. The automorphisms of groups of order p^{3}.

The groups of order p^{3} are classified, see e.g. Burnside [7, §73–74]. The nonabelian ones are:

• the extraspecial group p_{+}^{1+2} of exponent p and the metacyclic group P (p, 2) = ⟨x, y | x^{p^{2}} = y^{p} = 1, yxy^{−1} = x^{1+p}⟩, when p > 2.

• the dihedral and quaternion groups of size 8, denoted by D_{8} and Q_{8} re-

spectively, when p = 2. In the case of p > 2, in [15] it is shown that Out(p_{+}^{1+2}) = GL_{2}(p). The group

Out(P (p, 2)) contains a normal Sylow p-subgroup. This is shown, for instance, in [24]. Also, Burnside shows in [7, point 109] that P (p, 2) contains exactly p distinct cyclic groups of order p^{2}. This leaves room for p^{2} elements of order p which belong to a unique elementary abelian p-subgroup E = ⟨x^{p}, y⟩. The inclusion Z(P (p, 2)) < E can also be used to prove that Aut(P (p, 2)) contains a normal Sylow p-subgroup.

When p = 2, it is well known that Out(D_{8}) ≅ 2 and Out(Q_{8}) ≅ Σ_{3}.

Now let p be any prime. We now consider the abelian groups of order p^{3}. The cyclic group p^{3} has an abelian automorphism group of order p^{2}(p − 1).

The group p^{2} × p contains a Frattini subgroup Φ of order p. It also contains the subgroup Ω_{1} of all the elements of order p as a characteristic subgroup of order p^{2}. Thus the image of Aut(p^{2} × p) → Aut(Ω_{1}) is contained in the subgroup of upper triangular matrices in GL_{2}(p) and its kernel is easily seen to be a group of order p^{2}. It follows that Aut(p^{2} × p) contains a normal Sylow p-subgroup.

Finally, p×p×p has GL_{3}(p) as its automorphism group and we will make use of the lemma below.

6.3. Lemma. A p-reduced subgroup H of GL_{3}(p) cannot have a Sylow p-subgroup of order p^{2}.

Proof. Let V denote (𝔽_{p})^{3} with the usual action of GL_{3}(p) and with the standard basis {e_{1}, e_{2}, e_{3}}. Let B_{1,2} and B_{2,1} denote the stabilisers of the flags

0 < ⟨e_{1}⟩ < V  and  0 < ⟨e_{1}, e_{2}⟩ < V.

Explicitly

B_{1,2} = \{(\begin{matrix}∗ & ∗ & ∗ \\ 0 & ∗ & ∗ \\ 0 & ∗ & ∗ \end{matrix})\} B_{2,1} = \{(\begin{matrix}∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ \\ 0 & 0 & ∗ \end{matrix})\} .

Projections to the flag quotients give rise to epimorphisms

π_{1,2} : B_{1,2} → GL_{2}(p) × 𝔽_{p}^{∗}  and  π_{2,1} : B_{2,1} → GL_{2}(p) × 𝔽_{p}^{∗}

whose kernels are elementary abelian groups p × p on which the quotients GL_{2}(p) act in the natural way.

Sublemma. Let P and Q be subgroups of B_{1,2} (resp. B_{2,1}) such that Q ≰ P and |P | = p^{2}. Then the subgroup generated by P and Q contains a subgroup of order p^{3}. Proof. Set π = π_{1,2}. If P = Ker π then |QP | = p^{3} and we therefore assume that P ≠ Ker π. Then, |π(P)| = p and the order of R = P ∩ Ker π is p. If Q ≤ Ker π then R · Q = Ker π because Q ⊈ R, hence ⟨P, Q⟩ = P · Ker π whose order is p^{3}. We therefore assume that Q ⊈ Ker π. If π(P) = π(Q) then P ≤ Q · Ker π and therefore P < ⟨P, Q⟩ ≤ Q · Ker π whose order is p^{3}, thus |⟨P, Q⟩| = p^{3}. If, on the other hand π(P) ≠ π(Q) then they generate SL_{2}(p) which acts transitively on the lines of Ker π. Since R is such a line, it follows that ⟨P, Q⟩ contains all the lines in Ker π and as a consequence it contains P · Ker π whose order is p^{3}. Q.E.D.

We now fix a p-reduced subgroup H ≤ GL_{3}(p) and assume that the order of its Sylow p-subgroups is p^{2}. We will obtain a contradiction.

Claim 1. If P and Q are distinct Sylow p-subgroups of H then P ∩ Q = 1. Proof. Denote R = P ∩ Q and assume that |R| = p. Then U := V^{R} is a nontrivial subspace of V. It is stable under P and Q because [P, R] = [Q, R] = 1. It follows that P and Q are contained in the stabiliser of the flag 0 < U < V and the sublemma implies that the subgroup they generate, and consequently H, contain a subgroup of order p^{3}. This is impossible. Q.E.D.

Claim 2. If P and Q are distinct Sylow p-subgroups of H then V^{P} ∩ V^{Q} = 0. Proof. If this is not the case then P and Q stabilize a non-trivial subspace U = V^{P} ∩ V^{Q}. In particular, P and Q stabilize the flag 0 < U < V and the sublemma implies that the subgroup that P and Q generate, hence H, contains a subgroup of order p^{3}. This is impossible. Q.E.D

We now deduce that V^{P} is a line in V for every P ∈ Syl_{p}(H). To see this, note that if V^{P} is a plane, then this is the case for all Q ∈ Syl_{p}(H) because they are conjugate to P. Since H must have two distinct Sylow p-subgroups P and Q, and since any two planes in V must intersect non-trivially for dimensional reasons, it follows that V^{Q} ∩ V^{P} ≠ 0, which is impossible by Claim 2.

Let ℙV denote the projective space of V, i.e. the set of all the lines in V. It has p^{2} + p + 1 elements which shall be denoted by [v] for some 0 ≠ v ∈ V. The orbit of [v] under H is denoted H.[v].

Claim 3. For every [v] ∈ ℙV, the elements of the orbit H.[v] generate V as a vector space. Proof. Clearly the subspace U generated by H.[v] is invariant under H. If it is not all of V, then H is a subgroup of the stabiliser of the flag 0 < U < V. Since it is p-reduced it must contain at least two distinct Sylow p-subgroups. Therefore H contains a subgroup of order p^{3} by the sublemma. This is impossible. Q.E.D.

Claim 4. H acts transitively on ℙV. Proof. Choose P ∈ Syl_{p}(H). We have seen that V^{P} is a line in V. Since H is p-reduced, there exists Q ∈ Syl_{p}(H) distinct from P. Fix 1 ≠ g ∈ Q. It generates a subgroup Q′ of order p which is not contained in P by Claim 1. The sublemma implies that Q′ cannot stabilise [V^{P}] because Q′ and P cannot be subgroups of the stabiliser of the flag 0 < V^{P} < V. Thus, no non-trivial element of Q stabilises [V^{P}], therefore the orbit Q.[V^{P}] contains p^{2} lines. Clearly [V^{Q}] is an orbit of Q. This leaves room for p more lines in ℙV on which Q must act transitively because [V^{Q}] is the only line fixed by Q. Indeed, if Q fixes another line ℓ, it must fix it pointwise and therefore Q must fix the plane generated by ℓ and V^{Q}, which is impossible.

Up to conjugacy, we may assume that Q is contained in the group of upper tiangular matrices in GL_{3}(p) which is contained, of course, in B_{2,1}. Since Q does not fix a plane, π_{2,1}(Q) is the subgroup of unipotent upper triangular matrices in GL_{2}(p). Thus,

Q.[e_{1}] = {[e_{1}]}  and  Q.[e_{2}] = {[(a, 1, 0)] : a ∈ 𝔽_{p}}

are the orbits of size 1 and p in the action of Q on ℙV. The third orbit Q.[e_{3}] contains p^{2} lines. Now, P.[e_{1}] ∪ P.[e_{2}] does not generate V. It follows from Claim 3 that H.[e_{1}] and H.[e_{2}] must intersect P.[e_{3}] ⊆ H.[e_{3}]. Q.E.D.

We have seen that V^{P} are lines for P ∈ Syl_{p}(H). Claim 2 and Claim 4 show that P ↦ [V^{P}] establishes a bijection Syl_{p}(H) ≈ ℙV, in particular |Syl_{p}(H)| = p^{2}+p+1.

Fix some P ∈ Syl_{p}(H) and let it act on Syl_{p}(H) by conjugation. Clearly {P } is an orbit. The other orbits have order p^{2} by the following argument. Every Q ∈ Syl_{p}(H) distinct from P is a normal Sylow p-subgroup in its H-normaliser. Claim 1 implies that P ∩ N_{H}(Q) = P ∩ Q = 1 so the P -orbit of Q has |P | = p^{2} elements. It follows that |Syl_{p}(H)| = 1 mod p^{2} which conflicts with |Syl_{p}(H)| = p^{2} + p + 1. □

To begin the verification of the conditions in Lemma 5.9 we first show that

6.4. C contains F^{cr}. Consider a subgroup Q ∈ F^{cr}. In particular, C_{S}(Q) ≤ Q and Out_{F}(Q) is p-reduced (Definition 2.4). To begin with, Q cannot have order p because in this case C_{S}Q = N_{S}Q ⪈ Q which is impossible because Q is F-centric. Also if either |Q| = p^{2} or |Q| = p^{4} or Q = S then Q ∈ C. It remains to examine the case |Q| = p^{3}.

Note that since Q is F-centric then N_{S}Q/QC_{S}Q = N_{S}Q/Q is a non-trivial psubgroup of Out_{F}(Q). Thus Out_{F}(Q) is a p-reduced subgroup of Out(Q) whose order is divisible by p. This rules out the possibility that Q is isomorphic to:

• p^{3} and p^{2} × p for any p, or

• P (p, 2) for p > 2, or

• D_{8} if p = 2 (although centric subgroups isomorphic to this one are in C), because their outer automorphism groups contain a normal Sylow p-subgroup by 6.2. It follows that Q ≅ p×p×p for any p, or Q ≅ p_{+}^{1+2} if p > 2, or Q ≅ Q_{8} if p = 2, hence Q ∈ C.

We now verify that for every H ∈ C different from S we have |N_{S}H : H| = p and N_{S}H ∈ C. We do so by examining the possibilities (1)-(4) in the definition of C given in the beginning of the proof.

6.5. Consider P ∈ C of order p^{2}. Thus, P ∈ F^{cr}. We now prove that P ≅ p×p, that |N_{S}P : P | = p, that N_{S}P = p_{+}^{1+2} if p > 2, or N_{S}P = D_{8} if p = 2, and therefore N_{S}P ∈ C.

First, N_{S}P properly contains P. Since P is F-centric and abelian, N_{S}P/P is a non trivial p-subgroup of Out_{F}(P). Since P if F-radical, P cannot be cyclic because Aut(p^{2}) is abelian. Hence P ≅ p×p and therefore |N_{S}P : P | = p because N_{S}P/P is a subgroup of GL_{2}(p). Finally, choose g ∈ N_{S}N_{S}P \N_{S}P. Then P^{g} is an elementary abelian p-subgroup of rank 2 in N_{S}P distinct from P. Therefore N_{S}P cannot be isomorphic to

• P (p, 2), if p > 2, which has a unique such subgroup, see 6.2.

• Q_{8}, if p = 2, which has no such subgroups. Furthermore, N_{S}P is not abelian because P is F-centric. It follows that

• N_{S}P ≅ p_{+}^{1+2}, if p > 2.

• N_{S}P ≅ D_{8}, if p = 2.

It is F-centric because it contains P and therefore N_{S}P ∈ C.

6.6. Consider Q ≅ p_{+}^{1+2} in C, p > 2. Since Q is F-centric, N_{S}Q/Q = N_{S}Q/Q·C_{S}Q is a non trivial p-subgroup of Out_{F}(Q) ≤ Out(p_{+}^{1+2}) = GL_{2}(p). Therefore |N_{S}Q : Q| = p. Finally |N_{S}Q| = p^{4} and it is F-centric because it contains Q, hence N_{S}Q ∈ C.

6.7. Consider Q ≅ D_{8} or Q_{8} in C, p = 2. Since Q is F-centric, N_{S}Q/Q = N_{S}Q/Q · C_{S}Q is a non trivial 2-subgroup of Out_{F}(Q) ≤ Out(p_{+}^{1+2}) = 2 or Σ_{3}. Thus |N_{S}Q : Q| = 2, i.e. N_{S}Q has order 2^{4} and it is F-centric because it contains Q, hence N_{S}Q ∈ C.

6.8. Consider Q ≅ p×p×p in C. Thus Q ∈ F^{cr}. Choose a fully F-normalised Q′ which is F-conjugate to Q. Note that N_{S}Q′/Q′ is a non-trivial p-subgroup of Out_{F}(Q′) because Q′ is F-centric. It is also a Sylow p-subgroup of Out_{F}(Q′) by Definition 2.3 and its order is at most |S : Q| = p^{2}. Since Q′ is F-radical, Out_{F}(Q′) = Aut_{F}(Q′) is a p-reduced subgroup of GL_{3}(p). Lemma 6.3 implies that |N_{S}Q′ : Q′| = p and consequently, since Q′ is fully F-normalised, |N_{S}Q : Q| = p. In particular the order of N_{S}Q is p^{4} and it is F-centric because it contains Q so N_{S}Q ∈ C.

6.9. Consider R ∈ C of order p^{4}. Clearly N_{S}R = S ∈ C.

Combining 6.5-6.9 and 6.4, the conditions of Lemma 5.9 are verified and the proof is complete.

We summarise the structure of C in the following tables in which every row describes a chain of successive S-normalisers of elements of C.

Order (p > 2)  p^{2}  p^{3}  p^{4}  p^{5}
Description  p × p ∈ F^{cr} p_{+}^{1+2} ∈ F^{c} R ∈ F^{c} S
p_{+}^{1+2} ∈ F^{c} R ∈ F^{c} S
p×p×p ∈ F^{cr} R ∈ F^{c} S
R ∈ F^{c} S
Order (p = 2)  2^{2}  2^{3}  2^{4}  2^{5}
Description  2 × 2 ∈ F^{cr}  D_{8} ∈ F^{c}  R ∈ F^{c} S
Q_{8} ∈ F^{c}  R ∈ F^{c} S
2 × 2 × 2 ∈ F^{cr} R ∈ F^{c} S
R ∈ F^{c} S
□

7. Depth 2

7.1. Definition. A collection C of F-centric subgroups of S is said to have depth 2 if the the longest chain of proper inclusions in C has length 2.

In this case (C1) and (C2) of Definition 5.3 are automatic and (C3) is a direct consequence of Alperin’s fusion Theorem 2.6. Indeed, any F-isomorphism φ : P_{1} → P_{2} of elements of C is a composition of an F-automorphism of P_{1} followed by a restriction of an F-automorphism of S. Also note that in this case s̄dC = s̄d_{1}C^{∗}. Thus

7.2. Proposition. If the collection F^{cr} (Definition 2.5) in a p-local finite group (S, F, L) has depth 2 then it is compressible (Definition 5.3).

7.3. Example. Let S be a group of order at most p^{3} and let F be a saturated fusion system on S. A subgroup P of order p cannot be F-centric. If follows immediately that F^{cr} has depth 2 and it is therefore compressible.

7.4. Example. Sylow of p-rank two. The centerfree p-local finite groups (S, F, L) where S has p-rank 2 were classified in [9] (when p > 2) and [18] (when p = 2). In fact, the collections F^{cr} are determined there and it is shown that the depth of F^{cr} is always at most 2. In particular it is compressible.

7.5. Example. Recall that a p-groups S is called resistant [9, Definition 3.1] if whenever F is a saturated fusion system over S then N_{F}(S) = F. In other words, if S is resistant and F is a saturated fusion over S, then F^{cr} = {S} and it has depth 1.

7.6. Example. The exotic p-local finite groups in [6, Example 9.3]. Fix a prime p ≥ 5 and let Σ_{p} act on (ℤ/p^{ℓ})^{p} in the natural way. We let A = (ℤ/p^{ℓ})^{p−1} denote the submodule of the p-tuples whose entries’s sums vanish. We let Γ = C_{p−1} × Σ_{p} act on (ℤ/p^{ℓ})^{p−1} in the natural way. Let τ = (12 ⋯ p) be a generator of a Sylow p-subgroup S̄ in Σ_{p}. Then S = A ⋊ S̄ is a Sylow subgroup of G = A ⋊ Γ. Let X denote a set of representatives for the G-conjugacy classes of elements of order p in S − A. As pointed out in [6, pp. 843] there are 2 elements in X. For x ∈ X let Q_{x} denote the subgroup generated by x and Z = Z(S). It is easy to see that Z(S) is the set of diagonal elements in A and that Q_{x} is elementary Abelian of rank 2. Put Δ_{x} = Aut(Q_{x}) = GL_{2}(p) and

F := ⟨F_{S}(G); F_{Q_{x}}(Δ_{x}) | x ∈ X⟩.

That is F is the smallest fusion system on S which contains F_{S}(G) and F_{Q_{x}}(Δ_{x}). It is shown in [6, Example 9.3] that F is a saturated fusion system of an exotic p-local finite group (S, F, L).

Note that F_{S}(G)^{cr} = {A, S} because A is normal in F_{S}(G) and therefore A is contained in every F_{S}(G)-radical subgroup. Consider a subgroup P ≤ S which is not F-subconjugate to Q_{x}. Then clearly the F-conjugacy class of P is equal to the F_{S}(G)-conjugacy class of P and moreover Aut_{F}(P) = Aut_{F_{S}(G)}(P). This shows that if P ∈ F^{cr} then P = A or P = S. If P is a subgroup of S which is F-subconjugate to Q_{x} then either [P] = [Q_{x}] in which case it is F-radical because GL_{2}(p) is p-reduced, or it has order p hence cannot be F-centric.

This shows that the only elements of F^{cr} are S, A and the conjugates of Q_{x}. Finally we note that none of the Q_{x}’s is F-subconjugate to A because this must imply that this is the case in F_{S}(G) using the way morphisms in F are generated, however A ◁ G. Conclusion. The collection C = F^{cr} has depth 2 and it is therefore compressible. In fact T_{C} has the form

[figure: [A]; [S]; [Q_{x_{1}}]; [Q_{x_{2}}].]

A similar argument can be applied to

7.7. Example. The exotic examples in [6, Example 9.4]. In this case we set A = (C_{p^{ℓ}})^{p} with the natural Σ_{p} action. Choose subgroups μ + t ≤ μ_{s} ≤ (ℤ/p^{ℓ})^{∗} and let Γ = Γ(s, t; p) denote the subgroup of GL_{p}(ℤ/p^{ℓ}) of the permutation matrices whose non-zero entries are in μ_{s} and whose determinant is in μ_{t}. Clearly Γ(s, s; p) = C_{s}≀Σ_{p} and Γ(s, s; p) : Γ(s, t; p)| = s/t. Let S̄ = ⟨(12 ⋯ p)⟩ be a Sylow p-subgroup of Γ. Let G = A ⋊ Γ and let S = A ⋊ S̄ be a Sylow p-subgroup. Observe that Z = Z(S) = {(x, …, x)|x ∈ ℤ/p^{ℓ}}. Put

Q = ⟨Z, (1, z, z^{2}, …, z^{p−1}), τ⟩

for some z ∈ C_{p^{ℓ}} of order p and some generator τ ∈ S̄. Let Δ ≅ SL_{2}(p) be the group of automorphisms of Q which induce the identity on Z. Note that Z = Z(Q) and set

F(s, t; p) := ⟨F_{S}(G); F_{Q}(Δ)⟩.

It is shown in [6] that if t < \frac{1}{2}s then F is an exotic fusion system with a unique associated linking system.

Note that A ◁ G so if P ≤ S is in F_{S}(G)^{cr} then A ≤ P. Hence F_{S}(G)^{cr} = {A, S}. Consider a subgroup P ≤ S in F^{cr}. If it is not F-subconjugate to Q then P ∈ F_{S}(G)^{cr} so P = A or P = S. Clearly Q and any of its F-conjugates are F-centric because the F-conjugacy class of Q is equal to its F_{S}(G)-conjugacy class. Hence, P is F-radical.

Assume now that P is subconjugate to a proper subgroup of Q. Without loss of generality we may assume that P ≤ Q. Since Z = Z(Q) it follows that Z ≤ P. Obviously P ≠ Z because then C_{S}(P) ≥ A. Thus |Q : P | = p and we can find some y ∈ P − Z such that y and Z generate P. Using an automorphism φ ∈ Aut_{F}(Q) such that φ(y) = (1, z, …, z^{p−1}) mod Z we see that φ(P) ≤ A which is impossible because A is abelian.

We conclude that the only objects in F^{cr} are S, A and the conjucagy class of Q. Also note that Q cannot be F-subconjugate to A because that would have been the case in F_{S}(G), but A ◁ G. Conclusion. The collection C = F^{cr} has depth 2 and it is therefore compressible. In fact T_{C} has the form

[A]  [S]  [Q].

Appendix A: Homotopy colimits and Kan extensions

A.1. Let D be a small category. A diagram of spaces is a functor F : D → Spaces. Throughout, we work simplicially so a “space” means a simplicial set. There is a functor

hocolim Spaces^{D} → Spaces

called the homotopy colimit functor, see [2, Ch. XII]. Two D-diagrams F, F′ of (cofibrant) spaces are called weakly equivalent if there exists a natural transformation φ : F → F′ such that φ : F (d) → F′(d) is a weak homotopy equivalence for every object d ∈ D. In this case the induced natural map

hocolim F → hocolim F′

is a weak homotopy equivalence. We say that F and F′ are weakly equivalent if there is a zigzag of weak homotopy equivalences of D-diagrams connecting F and F′. In this case hocolim F and hocolim F′ are weakly equivalent spaces. For more details see [14].

A.2. Given a small functor f : C → D and an object d ∈ D one defines the over category (f ↓ d) as follows. Objects are pairs (c, f(c) ⟶^{φ} d) where c ∈ C and φ ∈ D.

Morphisms (c, φ) → (c′, φ′) are morphisms ψ ∈ C(c, c′) rendering the triangle below commutative (in D)

\xymatrix{f(c) \ar[rr]^{f(ψ)} \ar[dr]_{φ} &  & f(c′) \ar[dl]^{φ′} \\  & d.}

Similarly, the under category (d ↓ f) is defined to have objects the pairs (c, φ) where c ∈ C and φ ∈ D(d, f(c)) while morphisms are determined by ψ ∈ C(c, c′) rendering the triangle

\xymatrix{ & d \ar[dl]^{φ} \ar[dr]^{φ′} \\ f(c) \ar[rr]^{ψ} &  & f(c′)}

commutative (in D). There are obvious projection functors (f ↓ d) → C and (d ↓ f) → C.

Note that when j : C → D is a full embedding of posets then (f ↓ d) can be identified with the subposet of C consisting of the objects c which admits a morphism to d.

A.3. Given f : C → D above and a functor X : D → Spaces (of cofibrant space), there is an induced map

hocolim f^{∗}X → hocolim X.

It is a homotopy equivalence if f is terminal namely the categories (d ↓ f) are contractible for all objects d ∈ D [2].

A.4. Given f as above and X : C → Spaces then there is an induced functor f_{∗}X : D → Spaces called the homotopy left Kan extension of X. One can express f_{∗}X using a certain homotopy coend formula from which is becomes clear that for every d ∈ D

(f_{∗}X)(d) = hocolim [(f ↓ d) ⟶^{π} C ⟶^{f} Spaces].

If X has cofibrant values then the pushdown theorem, originally due to Segal [25], says that there is a natural homotopy equivalence (see [17, 5.5])

hocolim f_{∗}X → hocolim X.

A.5. There is a natural map f_{∗}f^{∗}X → X given objectwise using the cone

(f_{∗}f^{∗}X)(d) = hocolim [(f ↓ d) ⟶^{π} C ⟶^{f} Spaces] ⟶^{{f(c)→d}} X(d).

Similarly there is X → f^{∗}f_{∗}X which on c is induced by the inclusion of {(c, id_{f(c)})} ⊂ (f ↓ f(c)).

It is not hard to check that the following triangle commutes

\xymatrix{hocolim_{C} f ∗ X \ar[r] \ar[d]_{≃} & hocolim_{D} X \\ hocolim_{D} f_{∗}f^{∗}X \ar[ur]}

A.6 (Puppe’s Theorem). Let E be a D-diagram of spaces and consider a space B. A cone E ⟶^{f_{•}} B consists of maps E(d) ⟶^{f_{d}} B for every object d ∈ D such that E(φ) ∘ f_{d} = f_{d′} for every φ ∈ D(d′, d).

Assume that B is connected and that F is a D-diagram together with a natural transformation F → E such that for every d the maps

F (d) → E(d) → B

form, up to homotopy, a fibre sequence. Puppe’s theorem in its general form, see Dror-Farjoun [10, pp. 180] or Chachólski [8, Appendix], that

hocolim_{D} F → hocolim_{D} E → B

is up to homotopy a fibre sequence.

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Department of Mathematical Sciences, King’s College, University of Aberdeen, Aberdeen AB24 3UE , Scotland, U.K.

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

Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Apdo correos 59, 29080 Málaga, Spain.

   E-mail address: viruel@agt.cie.uma.es