Mathematische Zeitschrift manuscript No. (will be inserted by the editor)

Minami-Webb splittings and some exotic p-local finite groups

Assaf Libman ^{⋆}

Department of Mathematical Sciences, King’s College, University of Aberdeen, Aberdeen AB24 3UE, Scotland, U.K., e-mail: assaf@maths.abdn.ac.uk

Received: date / Revised version: date – \begin{matrix}c \\ ◯ \end{matrix} Springer-Verlag 2006

Abstract We give a new proof of the Minami-Webb formula for classifying spaces of finite groups by exploiting Symonds’s resolution of Webb’s conjecture. The methods are applicable to obtain a stable decomposition of Minami’s type for the classifying spaces of the three exotic p-local finite groups which were introduced by Ruiz and Viruel at the prime 7. We obtain a similar decomposition for the classifying spaces of a family of exotic p-local finite groups which were constructed by Broto, Levi and Oliver.

Key words Homology decompositions – p-local finite groups

Mathematics Subject Classification (2000): 55R35,55N91,20D15

1 Introduction

The goal of “homology decompositions” is to present, up to p-completion [7], a classifying space of a group by means of the classifying spaces of its proper subgroups. Milestones in this subject are Jackowski, McClure and Oliver’s work in [16] and [17,18] as well as Dwyer’s achievements in [12], [13].

The introduction of p-local finite groups by Broto, Levi and Oliver in [10], re-opened the subject. This time the goal is to decompose the classifying space of a p-local finite group by means of classifying spaces of finite groups. See e.g. [10, §2] and [19]. Every finite group G gives rise to a plocal finite group (S, F, L) where S is a Sylow p-subgroup of G. A p-local finite group is called “exotic” if its underlying fusion system F cannot be associated with a finite group. The classifying space of a p-local finite group (S, F, L) is by definition the p-completed nerve, |L|_{p}^{∧}, of the linking system L. In the non-exotic cases it is homotopy equivalent to BG_{p}^{∧}. A consequence of Minami’s theorem 6.5 in [22] and Webb’s observation [34, Proposition 2.7.1] is the theorem below.

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

Theorem 1.1. Let G be a finite group whose order is divisible by p and let Δ denote the simplicial complex associated to one of the following posets

P : All the non identity p-subgroups of G E : All the non identity elementary abelian p-subgroups of G R : All the non identity p-radical subgroups of G; See 5.3 or [4, p. 229].

Then in the homotopy category of spectra

(Σ_{+}^{∞}BG)_{p}^{∧} ≃ ∑_{σ∈Δ/G}(−1)^{dim σ}(Σ_{+}^{∞}BG_{σ})_{p}^{∧}

where G_{σ} is the stabilizer of a simplex σ ∈ Δ.

In this theorem Σ_{+}^{∞}X is the suspension spectrum of a space X to which one adds a disjoint base-point. The formal sum (wedge sum) of spectra on the right hand side can be transformed into a genuine equivalence of sums of spectra by transferring the negative terms in the right hand side to the left. We use (−)_{p}^{∧} for the p-completion of spectra. In the end of §4 we will give a new proof for theorem 1.1 and show that the formal sum of spectra is a consequence of a split exact sequence of spectra associated with Δ (see 4.3,4.4). This fact could not be deduced from Minami’s original proof because it depends on Webb’s result [33, Theorem A’].

A natural question to ask is how to extend Minami’s result to the classifying spaces of p-local finite groups. In this paper we prove

Theorem A. Let (S, F, L) be one of the following p-local finite groups

(1) The three exotic examples at p = 7 of Ruiz and Viruel; see table 1.2 in [25].

(2) The family of exotic examples described in the second row of the table in Example 9.3 of [10].

Then, in the homotopy category of spectra, the p-completed suspension spectrum Σ_{+}^{∞}|L|_{p}^{∧} is an alternating sum, that is, negative terms are allowed, of p-completed suspension spectra Σ_{+}^{∞}BG_{p}^{∧} of classifying spaces of finite groups.

We will prove Theorem A in §9. The p-completion in this theorem can be replaced with localization of spectra with respect to the Moore spectrum Sℤ_{(p)}, see the appendix §A. Moreover, we will show that the finite groups involved in the stable decomposition are subgroups of the automorphism groups of objects of the linking system under study. We shall also show that the alternating sum is a consequence of a split exact sequence of spectra.

Minami-Webb splittings and some exotic p-local finite groups  3

In [34] Webb proved an algebraic result closely related to theorem 1.1. He shows that when Δ is the simplicial complex associated to the poset P of all the non-trivial p-subgroups of a finite group G whose order is divisible by p, and when M is a suitable Mackey functor, there is a split exact sequence of ℤ_{(p)}-modules

0 → M(G) → ⨁_{σ∈Γ_{0}(Δ)/G} M(G_{σ}) →

→ ⨁_{σ∈Γ_{1}(Δ)/G} M(G_{σ}) → ⋯ → ⨁_{σ∈Γ_{d}(Δ)/G} M(G_{σ}) → 0. (1.2)

Here Γ_{n}(Δ) denotes the set of n-simplices of Δ and d denotes its dimension. One of the consequences of this result is that H̃^{∗}(P/G; ℤ_{(p)}) = 0. Suitable complexes and Mackey functors to which Webb’s result applies are listed in [34, Theorem 2.2.2].

In [13] Dwyer interpreted (1.2) in terms of vanishing of the Bredon cohomology groups H_{G}^{∗}(Δ; M^{i}) for the coefficient systems M^{i} = H^{i}(G; ℤ_{(p)}[−]) which assigns to an orbit G/K the i-th group cohomology of its associated permutation module ℤ_{(p)}[G/K]. The main observation of this paper is that Dwyer’s approach can be used to reverse Webb’s line of reasoning. For this procedure to have merit, one has to prove that Δ/G is an acyclic space and to show that Δ is G-acyclic for Bredon cohomology theories associated to a certain natural class of Mackey functors. This is in contrast with the adhoc collection of Mackey functors that Webb considers in [34] and the “one element class” of Mackey functors M(G/−) = H^{∗}(G; 𝔽_{p}[G/−]) which is sufficient for Dwyer’s arguments. The Mackey functors that are considered in this paper are M_{h^{j}} : G/H ↦ h^{j}(BH) where h^{∗} is a p-local generalised cohomology theory, that is, the values of h^{∗} are ℤ_{(p)}-modules.

The key lemma of the paper is 4.9 which we restate as Theorem B below. For subgroups H, K of a group G let G/H ⊠ G/K denote the G-subset of G/H × G/K consisting of the elements whose isotropy group is non-trivial. Clearly there is a G-equivariant projection map G/H ⊠ G/K ⟶^{π} G/K. The fibres of π are the preimages of one-point subsets of G/K. Since G acts transitively on G/K all the fibres of π have the same cardinality which we call by abuse of language, the fibre of π.

Recall that a G-poset 𝔛 gives rise to a G-simplicial complex Δ whose nsimplices are the subsets of 𝔛 which contain n + 1 elements and are linearly ordered in 𝔛. We say that Δ/G is finite if the set of simplices of Δ has a finite number of G-orbits. Equivalently, Δ is finite dimensional and there are finitely many orbits of n-simplices for every n ≥ 0.

Theorem B. Let S be a finite subgroup of a group G (possibly infinite) and let Δ be the simplicial complex associated to a G-poset 𝔛. Assume that Δ/G is finite and that

1. G acts on Δ with finite isotropy groups.

2. If H is an isotropy group of a simplex of Δ then the number of elements in the fibres of G/S ⊠ G/H → G/H is a unit in ℤ_{(p)}.

3. The fixed-point subspace Δ^{Q} is contractible for every non identity Q ≤ S and

4. H̃_{∗}(Δ/G; ℤ_{(p)}) = 0.

Then, in the homotopy category of spectra

(Σ_{+}^{∞}Δ_{hG})_{p}^{∧} = ∑_{σ∈Δ/G}(−1)^{dim σ}(Σ_{+}^{∞}BG_{σ})_{p}^{∧}

where Δ_{hG} denotes the Borel construction EG ×_{G} Δ. The alternating sum reflects a split exact sequence of spectra (4.5) and the p-completion can be replaced with localization with respect to the Moore spectrum Sℤ_{(p)}.

As an immediate corollary to Theorem B we quite effortlessly reprove Minami’s theorem 1.1 in §4. We need to make no reference to Webb’s results from [33].

The core of the proof of theorem A consists of showing that for each one of the exotic p-local finite groups (S, F, L) listed there, one can construct a discrete group G, albeit infinite, which is an amalgamated product of finite groups, together with a G-poset R such that

a. |L| ≃ Δ_{hG} where Δ is the G-simplicial complex associated to R, b. G contains S and F is the fusion system on S induced by conjugations

in G, c. Δ/G is finite and Δ is G-acyclic for the Bredon cohomology theories as-

sociated to the Mackey functors M_{h^{j}} for all p-local cohomology theories h.

This is established in §9 using theorem 8.5. We then apply the more general theorem B

2 Coefficient systems associated to cohomology theories

Fix a discrete group G, possibly infinite, and let S^{G} denote the category of G-sets. Recall that every G-set is a disjoint union of its orbits, namely its transitive G-subsets. Every orbit X is isomorphic to the coset space G/H for a subgroup H whose conjugacy class, denoted (H), is called the type of X. This is the set Iso_{G}(X) of all the isotropy groups of the points of X.

Recall that G-maps φ : G/H → G/K are in one-to-one correspondence with the set of all the left cosets gK for which g^{−1} is an element in the transporter set

N_{G}(H, K) = {g ∈ G : gHg^{−1} ≤ K}.  (2.1)

The G-map G/H → G/K which corresponds to gK is xH ↦ xgK.

Recall that a G-set X is called free if the isotropy groups G_{x} are trivial for all x ∈ X. Every G-set X is the disjoint union

X = X^{>e} ∐ X^{free}

where X^{free} is the union of all the free orbits of X. Thus, G_{x} ≠ 1 for every x ∈ X^{>e}. Note that if f : X → Y is a G-map then a non-free orbit of X must be mapped to a non-free orbit of Y. This shows that the assignment X ↦ X^{>e} is a functor on S^{G} which is equipped with a natural transformation to the identity functor on S^{G}.

We now choose once and for all a contractible G-CW-complex EG on which G acts freely. The Borel construction X_{hG} on a G-set, and more generally, a G-space X is

X_{hG} := (EG × X)/G the orbit space of the diagonal action.

Note that (G/H)_{hG} is homotopy equivalent to the classifying space BH of H. Furthermore (∐_{i} X_{i})_{hG} = ∐_{i}(X_{i})_{hG}. In particular, whenever X is a G-set then X_{hG} is a disjoint union of classifying spaces of subgroups of G.

Let f : X → Y be a map of G-sets. The fibre over y ∈ Y is f^{−1}(y). Clearly the cardinality of the fibres over the points in the orbit Gy is constant. It is denoted by χ(f, Gy).

Remark 2.2. Let f : X → Y be a map of G-sets with finite fibres. The fibre of f_{hG} : X_{hG} → Y_{hG} over a point in the component (Y_{i})_{hG} is finite and its cardinality is χ(f, Y_{i}).

Definition 2.3. Fix a commutative ring k. A coefficient system is a functor

M : (S^{G})^{op} → k - 𝔪𝔬𝔡

which satisfies M(∐_{i} X_{i}) = ∏_{i} M(X_{i}).

Note that the definition implies that M(∅) = 0. Consider now a generalised cohomology theory h^{∗}, cf. Adams [1] or Dyer [15]. For every integer q there is an associated coefficient functor M_{h^{q}} defined by

M_{h^{q}}(X) = h^{q}(X_{hG}).  (2.4)

There is another coefficient system M̃_{h^{q}} which we define below. Fix X ∈ S^{G} and choose a section X/G ⟶^{σ_{X}} X_{hG} to the projection X_{hG} ⟶^{π_{X}} X/G. Thus, σ_{X} chooses a point in every component of X_{hG}. Define

M̃_{h^{q}}(X) = ker (h^{q}(X_{hG}) ⟶^{σ_{X}^{∗}} h^{q}(X/G)).  (2.5)

Note that any two choices of σ_{X} are homotopic so M̃_{h^{q}} is well defined. There is a natural isomorphism

M̃_{h^{q}}(X) ≈ coker (h^{q}(X/G) ⟶^{π_{X}^{∗}} h^{q}(X_{hG}))  (2.6)

because π_{X} ∘ σ_{X} = id. It follows that M̃_{h^{q}} is a coefficient system because X_{hG} ⟶^{π_{X}} X/G is natural in X and because h^{q} transforms coproducts of spaces to products of abelian groups [1, p. 154].

Remark 2.7. Consider a G-set X and a set U together with a map U ⟶^{σ} X_{hG} which hits every connected component of X_{hG}. Note that there is a factorisation, unique up to homotopy, U ⟶^{σ′} X/G ⟶^{σ_{X}} X_{hG} where σ′ is surjective. Therefore,

M̃_{h^{q}}(X) = ker (h^{q}(X_{hG}) ⟶^{σ_{X}^{∗}} h^{q}(U))

because σ′^{∗} : h^{∗}(X/G) → h^{∗}(U) is injective.

Recall that by definition (or using Brown’s representability theorem) a cohomology theory h^{∗} is a representable functor on the homotopy category of spectra Ho^{s}. More explicitly, there exists a spectrum U such that, using the notation of Adams in [1], h^{∗}(X) = [X, U]_{−∗} for every spectrum X. Using Bousfield’s theory of localization of spectra, see the Appendix, we make the following

Definition 2.8. A cohomology theory h is called p-local if it is represented by a Sℤ_{(p)}-local spectrum E where Sℤ_{(p)} is the Moore spectrum of type (0, ℤ_{(p)}).

Using Proposition A.1 and the Atiyah-Hirzebruch spectral sequence [1, §III.7], one easily checks that h is p-local if and only if h^{∗}(X) are ℤ_{(p)}- modules for every space X or, indeed, for every spectrum X.

2.9. Recall that Σ_{+}^{∞} is a functor from spaces to Ho^{s} which assigns to every space X the suspension spectrum of the space X with a disjoint base-point added to it. It transforms coproducts (disjoint unions of spaces) into sums (“wedge”) of spectra.

When f : X → B is a (Serre) fibration with a compact fibre F and a connected base B, Becker and Gottlieb [3] and Dwyer [14] constructed a stable transfer map

τ(f) : Σ_{+}^{∞}B → Σ_{+}^{∞}X.

The stable transfer map enjoys the following properties

(T1) Naturality. If g : B′ → B is a map of connected spaces and f′ : X′ → B′ is the pullback of f along g then the following square of spectra commutes

Σ_{+}^{∞}X′ ⟶^{Σ_{+}^{∞}ḡ} Σ_{+}^{∞}X
τ(f′)↑  ↑^{τ(f)}
Σ_{+}^{∞}B′ ⟶_{Σ_{+}^{∞}g} Σ_{+}^{∞}B

(T2) Normalisation. The composition Σ_{+}^{∞}B ⟶^{τ(f)} Σ_{+}^{∞}X ⟶^{Σ_{+}^{∞}f} Σ_{+}^{∞}B induces multiplication by χ(F) in H_{∗}(−; ℤ).

The construction of τ(f) can be extended to fibrations f with an arbitrary base spaces B provided f has compact fibres over each connected component of B. This is done by defining τ(f) = ∨_{i}τ(f_{i}) where f_{i} is the restriction of f to f^{−1}(B_{i}) for every component B_{i} of B. Property (T2) is immediate. Property (T1) is verified by first reducing to a connected base space B and then examining each component of B′ separately. We leave the details to the reader.

The main result of this section is

Proposition 2.10. Fix a group G and a p-local cohomology theory h and an integer q. We have remarked that the coefficient systems M = M_{h^{q}} and M̃ = M̃_{h^{q}} take values in ℤ_{(p)} - 𝔪𝔬𝔡. Then

(1) For every G-set X there is a naturally split short exact sequence of ℤ_{(p)}-

modules

0 → h^{q}(X/G) → M(X) → M̃(X) → 0.

(2) If X is a free G-set then M̃(X) = 0. More generally, if X is a G-set

then

M̃(X) = M̃(X^{>e}).

Moreover, there is a map tr(f) : M̃(X) → M̃(Y) associated with every map f : X → Y of G-sets with finite fibres which enjoys the following properties.

(3) Consider a pullback diagram of G-sets in the left hand square below where f has finite fibres. Then f̄ has finite fibres and the square of ℤ_{(p)}-modules on the right is commutative.

\begin{matrix}X′ & ⟶^{ḡ} & X \\ f̄↓ & & ↓^{f} \\ Y′ & ⟶_{g} & Y. \end{matrix}  ⇝  \begin{matrix}M̃(X′) & ⟵^{ḡ^{∗}} & M̃(X) \\ ^{tr(f̄)}↓ & & ↓^{tr(f)} \\ M̃(Y′) & ⟶_{g^{∗}} & M̃(Y). \end{matrix}

(4) Suppose that f : X → Y is a map in S^{G} such that the cardinalities of all its fibres are finite and units in ℤ_{(p)}. Then the composition below is an isomorphism

M̃(Y) ⟶^{f^{∗}} M̃(X) ⟶^{tr(f)} M̃(Y)

Proof. (1) Using the notation of (2.6), σ_{X}^{∗} is a left inverse of π_{X}^{∗} and hence

splits 0 → h^{q}(X/G) ⟶^{π_{X}^{∗}} M(X). The splitting is natural because σ_{X} is unique up to homotopy.

(2) If X is a free G-set then X_{hG} → X/G is a homotopy equivalence, hence M̃(X) = 0. More generally, for any G-set X

M̃(X) = M̃(X^{>e}) × M̃(X^{free}) = M̃(X^{>e}).

We now construct the transfer map tr(f) associated with f : X → Y with finite fibres. Note that f_{hG} has finite fibres (2.2) hence (2.9) there is a stable transfer map τ(f_{hG}) : Σ_{+}^{∞}Y_{hG} → Σ_{+}^{∞}X_{hG}. Chose a set U_{Y} and a map σ : U_{Y} → Y_{hG} which surjects on connected components. We form the pullback of f_{hG} along σ and obtain the left hand square in the display below. In this square U_{X} is a discrete set because f_{hG} is a covering map. Also, σ̄ surjects on components because σ does. Since the fibres of f_{hG}, hence of f̄, are finite, there are associated stable transfer maps τ(f_{hG}) and τ(f̄) which give rise to the right-hand commutative square of ℤ_{(p)}-modules below, by applying h^{q}(−) to (T1).

\begin{matrix}U_{X} & ⟶^{σ̄} & X_{hG} \\ f̄↓ & & ↓^{f_{hG}} \\ U_{Y} & ⟶^{σ} & Y_{hG} \end{matrix}  ⇝  \begin{matrix}h^{q}(X_{hG}) & ⟶^{h^{q}(σ̄)} & h^{q}(U_{X}) \\ ↓^{h^{q}(τ(f_{hG}))} & ↓^{h^{q}(τ(f̄))} \\ h^{q}(Y_{hG}) & ⟶^{h^{q}(σ)} & h^{q}(U_{Y}) \end{matrix}

The transfer map tr(f) : M̃(X) → M̃(Y) is obtained by taking the kernels of the rows in the right hand square above and using remark 2.7.

(3) Set U_{Y} = Y/G and choose a section U_{Y} ⟶^{σ_{Y}} Y_{hG}. Consider the commutative diagram below where both squares are pullback squares. Thus, all the horizontal maps are covering projections and U_{X} and U_{Y′} are discrete sets.

\xymatrix{U_{Y′} \ar@{->>}[r]^{σ_{g}} \ar[d]_{σ_{Y ′}} & U_{Y} \ar[d]^{σ_{Y}} & U_{X} \ar@{->>}[l]_{σ_{f}} \ar[d]^{σ_{X}} \\ Y′_{hG} \ar@{->>}[r]_{g_{hG}} & Y_{hG} & X_{hG} \ar@{->>}[l]^{f_{hG}}}  (2.11)

Furthermore, σ_{X} and σ_{Y′} hit every component of X_{hG} and Y′_{hG} because σ_{Y} has the same property. We observe that (−)_{hG} preserves pullback squares of G-sets. By taking the pullback of each row of (2.11) we obtain a morphism of pullback squares

\xymatrix{U_{X′} \ar[dr]^{σ_{X′}} \ar@{->>}[rrr]^{σ̄_{g}} \ar[ddd]_{σ̄_{f}} &  &  & U_{X} \ar[dl]^{σ_{X}} \ar[ddd]^{σ_{f}} \\  & _{X′_{hG}} \ar@{->>}[r]^{ḡ} \ar@{->>}[d]_{f̄} & X_{hG} \ar@{->>}[d]^{f_{hG}} \\  & Y′_{hG} \ar@{->>}[r]_{g_{hG}} & Y_{hG} \\ U_{Y′} \ar[ur]^{σ_{Y ′}} \ar@{->>}[rrr]_{σ_{g}} &  &  & U_{Y} \ar[ul]_{σ_{Y}}}

Clearly U_{X′} is a discrete set and σ_{X′} hits every component of X′_{hG} because σ_{X}, σ_{Y′} have the same property and σ_{Y} is bijective on components. Moreover, all the squares and trapezoids are pullback diagrams. Clearly f̄ has finite fibres because f_{hG} has the same property. Consequently σ_{f} and σ̄_{f} have finite fibres too. We now apply 2.9(T1) to obtain a commutative diagram in Ho^{s}

\xymatrix{Σ_{+}^{∞}U_{X′} \ar[dr]^{σ_{X′}} \ar[rrr]^{σ̄_{g}} &  &  & Σ_{+}^{∞}U_{X} \ar[dl]^{σ_{X}} \\  & Σ_{+}^{∞}X′_{hG} \ar[r]^{ḡ} & Σ_{+}^{∞}X_{hG} \\  & Σ_{+}^{∞}Y′_{hG} \ar[r]_{g} \ar[u]_{τ(f̄)} & Σ_{+}^{∞}Y_{hG} \ar[u]_{τ(f)} \\ Σ_{+}^{∞}U_{Y′} \ar[uuu]^{τ(σ̄_{f})} \ar[ur]^{σ_{Y ′}} \ar[rrr]_{σ_{g}} &  &  & Σ_{+}^{∞}U_{Y} \ar[uuu]_{τ(σ_{f})} \ar[ul]_{σ_{Y}}}

The right hand commutative diagram in point (3) of this proposition is obtained by applying h^{q}(−) to this diagram and using remark 2.7.

(4). Evidently, we only need to consider the case when Y_{hG} is connected, namely Y is an orbit. Note that X must have finitely many orbits because χ(f, Y) is finite.

Choose a point ∗ in Y_{hG} and let F denote the fibre of f_{hG} over this point. Let f̄ denote f_{hG}|_{F}. Clearly, the inclusion F ⟶^{σ} X_{hG} hits every component of X_{hG}. We now obtain using 2.9(T1) a commutative diagram in Ho^{s}

Σ_{+}^{∞}∗ ⟶^{τ(f̄)} Σ_{+}^{∞}F ⟶^{f̄} Σ_{+}^{∞}∗
↓  ↓^{σ}  ↓
Σ_{+}^{∞}Y_{hG} ⟶_{τ(f_{hG})} Σ_{+}^{∞}X_{hG} ⟶_{f_{hG}} Σ_{+}^{∞}Y_{hG}

Using (T2), the composition f_{hG} ∘ τ(f_{hG}) in the bottom row induces multiplication by χ, hence an isomorphism, in H_{∗}(−; ℤ_{(p)}). Since Σ_{+}^{∞}X_{hG} is a connective spectrum, then this implies by Theorem A.2 that L_{Sℤ_{(p)}}(f_{hG} ∘ τ(f_{hG})) is an equivalence. Since h^{∗} is p-local it follows from Proposition A.3 that h^{∗}(f_{hG} ∘ τ(f_{hG})) is an isomorphism. Clearly h^{q}(f̄∘ τ(f̄)) is multiplication by χ hence an isomorphism because F is a finite set. The result follows by applying h^{q} to the diagram above and using remark 2.7

3 Bredon homology and G-acyclic spaces for proper actions

Fix a ring k. The homotopy groups π_{∗}U of a simplicial k-module U are by definition the homology of the chain complex CU

⋯ → U_{n} \underrightarrow{∑_{j=0}^{n} −1^{j}d_{j}} U_{n−1} → ⋯ → U_{1} ⟶^{d_{1}−d_{0}} U_{0}.

Similarly, the cohomotopy groups π^{∗}V of a cosimplicial k-module V are the cohomology groups of the analogous associated cochain complex CV, cf. [5, §2]. A short exact sequence of cosimplicial modules gives rise to a long exact sequence of cohomotopy groups. See Weibel [35, §8.2] for more details.

Throughout this section a space means a simplicial set, see e.g. Bousfield-Kan [7] or May [21]. A G-spaces is a simplicial objects in S^{G}.

Definition 3.1. Let X be a G-space and M a coefficient system for G. The Bredon cohomology groups of X with coefficients in M are

H_{G}^{∗}(X; M) = π^{∗}M(X)

where M(X) is the cosimplicial group n ↦ M(X_{n}). The space X is called acyclic for M if H_{G}^{i}(X; M) = 0 for all i > 0.

Fix a discrete group G, possibly infinite.

Definition 3.2. Let X, Y be G-sets. Define

Y ⊠ X = (Y × X)^{>e}.

That is, Y ⊠ X is the union of all the non-free orbits of Y × X. There is a natural composite maps

Y ⊠ X ⟶^{incl} Y × X ⟶^{π} Y.

Recall that a set H of subgroups of a group G is called a collection if it is a union of conjugacy classes. A G-space is said to have type H if Iso_{G}(X) ⊆ H.

3.3. Fix a subgroup S of a discrete group G and a collection H of subgroups of G. Throughout we shall assume

(H1) The collection H consists of finite subgroups of G.

(H2) For every H ∈ H the G-set G/S ⊠ G/H has only finitely many orbits.

Proposition 3.4. With the hypotheses above, if X is a G-set of type H then G/S ⊠ X → X has finite fibres.

Proof. It suffices to assume that X is an orbit G/H for some H ∈ H. The result is immediate from (H2) by observing that the isotropy groups of G/S ⊠ G/H are finite because of (H1).

Fix a cohomology theory h^{∗} and consider the coefficient system M̃_{h^{q}} in (2.4).

Proposition 3.5. With the hypotheses 3.3, every G-map f : X → Y, where X, Y are G-sets of type H, gives rise to a commutative square in which tr(π_{X}) and tr(π_{Y}) are defined in 2.10

M̃(G/S ⊠ Y) ⟵^{(G/S⊠f)^{∗}} M̃(G/S ⊠ X)
tr(π_{X})↓  ↓^{tr(π_{Y} )}
M̃(X)  ⟵_{f^{∗}}  M̃(Y)

Proof. Consider the following pullback square

W ⟶ G/S ⊠ Y
π_{X} ↓  ↓^{π_{Y}}
X ⟶_{f}  Y

and observe that W is the union of G/S ⊠ X with several free G-orbits in G/S × X. Now, π_{Y} has finite fibres by 3.4. Consequently W has finite fibres and the result follows from points (2) and (3) of Proposition 2.10.

The main result of this section is the following lemma.

Lemma 3.6. Fix a prime p. Let H be a collection of finite subgroups of G and let S be a non-trivial finite subgroup of G. Assume that for every H ∈ H

(1) G/S ⊠ G/H has finitely many orbits and

(2) The cardinality of the fibre of G/S ⊠ G/H → G/H is a unit in ℤ_{(p)}.

Let X be a G-space such that Iso_{G}(X) ⊆ H and assume that

(3) H̃^{∗}(X/G; ℤ_{(p)}) = 0 and

(4) X^{Q} is weakly contractible for every non identity Q ≤ S.

Then X is acyclic for all coefficient systems M = M_{h^{q}} associated with p-local cohomology theories h^{∗} defined in (2.4).

Proof. Note that H satisfies the hypotheses (H1) and (H2) of 3.3. Fix a p-local cohomology theory h^{∗} and set M = M_{h^{q}} and M̃ = M̃_{h^{q}} as in (2.4) and (2.5).

Throughout the proof we consider X as a functor X : Δ^{op} → S^{G}. To avoid confusion, whenever we wish to refer to X as a space we shall denote it |X|. Thus, X_{hG} denotes the simplicial space n ↦ (X_{n})_{hG}, whereas |X|_{hG} denotes the Borel construction on X. Similarly h^{q}(X/G) is the cosimplicial ℤ_{(p)}-module n ↦ h^{q}(X_{n}/G), that is, h^{q}(X/G) = Hom(ℤ[X/G], h^{q}(pt)).

The map of simplicial spaces X_{hG} ⟶^{π_{X}} X/G gives rise, by Proposition 2.10(1), to a short exact sequence of cosimplicial ℤ_{(p)}-modules

0 → h^{q}(X/G) ⟶^{h^{q}(π_{X})} M(X) ⟶ M̃(X) → 0.

Since h is p-local, hypothesis (3) of this lemma implies that

π^{i}h^{q}(X/G) = H^{i}(|X/G|, h^{q}(pt)) = 0 for all i > 0.

The long exact sequence in Bredon cohomology shows that X is acyclic for M if and only if it is acyclic for M̃.

Consider the coefficient functor M̃ ↓_{S}^{G} on S-sets, defined by (see e.g. [34, §3.3])

M̃ ↓_{S}^{G} (Y) = M̃(G ×_{S} Y), for every S-set Y.

If U is a G-set then there is a natural isomorphism G ×_{S} U ≈ G/S × U. Functoriality of the assignment U ↦ G/S ⊠ U shows that G/S ⊠ X is a sub-simplicial G-set of G/S × X. We obtain from Proposition 2.10(2)

M̃ ↓_{S}^{G} (X) ≈ M̃(G/S × X) = M̃(G/S ⊠ X).  (3.7)

The simplicial G-map G/S ⊠ X ⟶^{π_{X}} X yields a map of cosimplicial ℤ_{(p)}- modules

M̃(X) ⟶^{π_{X}^{∗}} M̃(G/S ⊠ X).

Proposition 3.5 shows that the simplicial map G/S ⊠ X → X gives rise to a a morphism of cosimplicial ℤ_{(p)}-modules

M̃(G/S ⊠ X) ⟶^{tr(π_{X})} M̃(X).

Note that the cardinality of the fibres of G/S ⊠ X_{n} → X_{n} are units in ℤ_{(p)} by hypothesis (2). Proposition 2.10(4) now shows that the composition

M̃(X) ⟶^{M̃(π_{X})} M̃(G/S ⊠ X) ⟶^{tr(π_{X})} M̃(X)

is an isomorphism. To show that X is acyclic for M̃ it is therefore sufficient to prove that G/S ⊠ X is acyclic for M̃. Equivalently (3.7), we must show that X is acyclic for M̃ ↓_{S}^{G}. Note that M̃ vanishes on free G-sets, hence M̃ ↓_{S}^{G} vanishes on free S-sets, so

M̃ ↓_{S}^{G} (X) = M̃ ↓_{S}^{G} ((X ↓_{S}^{G})^{>e})

where (X ↓_{S}^{G})^{>e} denotes the S-subspace of X of all the non-free S-orbits in X. It remains to show that (X ↓_{S}^{G})^{>e} is acyclic for M̃ ↓_{S}^{G}. Finally, S is a finite non-trivial group and (X ↓_{S}^{G})^{>e} is S-equivalent to a point by hypothesis (4) and Proposition [12, Proposition 4.1]. Therefore it is acyclic for all coefficient functors by [13, Proposition 4.8].

4 A Webb formula for simplicial G-complexes

4.1. Let 𝔛 be a G-poset. The G-simplicial complex associated to it is denoted by sd 𝔛. The set sd_{n} 𝔛 of its n simplices consists of the sequences x_{0} ⪇ … ⪇ x_{n} in 𝔛. The geometric realisation of sd 𝔛 is denoted | sd 𝔛|.

The poset 𝔛 may also be considered as a small category and its nerve X is a G-space. The set of non-degenerate n-simplices of X is exactly sd_{n} 𝔛 and sd 𝔛 becomes a restricted simplicial G-sub-set of X by discarding the degeneracy maps. Also note that |X| and | sd 𝔛| are canonically homeomorphic.

4.2. Consider a cosimplicial k-module V. The cochain complex CV contains a sub-complex NCV called the normalised cochain complex associated to V where

NC^{n}V = ⋂_{j=0}^{n−1} ker s^{j}, (s^{j} : V^{n} → V^{n−1}).

The inclusion of NCV in CV is a cochain homotopy equivalence. See Weibel [35, §8.3] and Bousfield [5, pp. 306, 307].

4.3. Recall that the homotopy category of spectra is additive. A chain complex of spectra is a sequence A of spectra

⋯ ⟶^{∂_{2}} A_{1} ⟶^{∂_{1}} A_{0} ⟶^{∂_{0}} A_{−1} ⟶^{∂_{−1}} ⋯

such that ∂_{n+1} ∘ ∂_{n} = 0. A chain complex A is called split exact if it is equivalent to ∨_{i∈ℤ}A_{i} where A_{i} has the form

⋯ → 0 → 0 → A_{i} ⟶^{≃} A_{i} → 0 → 0 → ⋯

Lemma 4.4. Fix a G-poset 𝔛 such that m = dim sd 𝔛 is finite. Assume that the nerve of 𝔛 is G-acyclic for every coefficient functor M associated to a p-local cohomology theory h^{∗} (cf. (2.4)). Then there is a chain complex of spectra

0 → ⋁_{σ∈sd_{m} 𝔛/G} Σ_{+}^{∞}BG_{σ} → ⋁_{σ∈sd_{m−1} 𝔛/G} Σ_{+}^{∞}BG_{σ} → ⋯

⋯ → ⋁_{σ∈sd_{1} 𝔛/G} Σ_{+}^{∞}BG_{σ} → ⋁_{σ∈sd_{0} 𝔛/G} Σ_{+}^{∞}BG_{σ} → Σ_{+}^{∞}| sd 𝔛|_{hG} → 0 (4.5)

which becomes split exact after localization at the Moore spectrum Sℤ_{(p)} (see the appendix.) The arrows are induced by alternating sums of face maps in sd 𝔛 and σ runs through representatives of orbits of simplices. If sd 𝔛/G is finite, then

(Σ_{+}^{∞}| sd 𝔛|_{hG})_{Sℤ_{(p)}} = ∑_{σ∈sd 𝔛/G}(−1)^{dim σ}(Σ_{+}^{∞}BG_{σ})_{Sℤ_{(p)}}  (4.6)

Proof. Let X denote the nerve of 𝔛. As in the proof of Lemma 3.6, we regard X as a simplicial object of G-sets and use the notation |X| when we want to consider it as a G-space. Thus, X_{hG} denotes the simplicial space n ↦ (X_{n})_{hG} and |X|_{hG} denotes the Borel construction on |X|. Similarly sd 𝔛_{hG} denotes the restricted simplicial space n ↦ (sd_{n} 𝔛)_{hG}.

Note that there is a natural G-equivariant weak equivalence |X| ≃ hocolim_{Δ^{op}} X, cf. [7, p. 338]. Since homotopy colimits commute with each other,

and the Borel construction is one, we can identify |X|_{hG} with hocolim_{Δ^{op}} (X_{hG}) and obtain an augmented simplicial object in the homotopy category of spaces

X_{hG} ⟶^{d_{0}} |X|_{hG}.

By restriction, we also obtain an augmented restricted simplicial object in the homotopy category of spaces

sd 𝔛_{hG} → |X|_{hG}.

It gives rise to an augmented chain complex of spectra

0 → Σ_{+}^{∞} sd_{m} 𝔛_{hG} \underrightarrow{∑_{j=0}^{m}(−1)^{j}d_{j}} Σ_{+}^{∞} sd_{m−1} 𝔛_{hG} → ⋯

⋯ ⟶^{d_{0}−d_{1}} Σ_{+}^{∞} sd_{0} 𝔛_{hG} ⟶^{d_{0}} Σ_{+}^{∞}|X|_{hG}.

We shall denote its Sℤ_{(p)}-localization by T ⟶^{d_{0}} T_{−1}. That is

T_{n} ≃ ⋁_{σ∈sd_{n} 𝔛/G}(Σ_{+}^{∞}BG_{σ})_{Sℤ_{(p)}}  and  T_{−1} = (Σ_{+}^{∞}|X|_{hG})_{Sℤ_{(p)}}.

It is the chain complex T → T_{−1} that we need to show that is split exact.

Fix a p-local cohomology theory h^{∗} and a coefficient system M = M_{h^{q}} (2.4). Consider the cochain complex CM(X). Clearly C^{n}M(X) = ∏_{σ∈X_{n}/G} M(G · σ). Since degeneracy maps of X map orbits of simplices of X_{n} isomorphically onto orbits of X_{n+1} it follows by inspection that NC^{n}M(X) = M(sd_{n} 𝔛). Proposition A.3 implies the following equality of augmented cosimplicial ℤ_{(p)}-modules

{h^{q}(|X|_{hG} ⟶^{d^{0}} NCM(X))} = {h^{q}( sd 𝔛_{hG} → |X|_{hG})} = {h^{q}(T → T_{−1})}

In particular

H^{∗}h^{q}(T) = H^{∗}NCM(X) = H^{∗}CM(X) = π^{∗}M(X) = H_{G}^{∗}(|X|; M).

(4.7)

We are now ready to prove that the chain complex T → T_{−1} is split exact.

Assume that m > 0 and consider the cohomology theory h represented by the spectrum T_{m}. Since by hypothesis H_{G}^{m}(X, M_{h^{0}}) = 0, we see that the cochain complex h^{0}(T) is exact at m namely

[T_{m−1}, T_{m}] \underrightarrow{∑_{j=0}^{m} −1^{j}[d_{j},T_{m}]} [T_{m}, T_{m}]

is surjective. In particular T_{m} → T_{m−1} has a left inverse, so T takes the form

0 → T_{m} → T_{m} ∨ T′_{m−1} → T_{m−2} → ⋯ T_{1} → T_{0}.

If m − 1 > 0 we let h be the cohomology theory represented by T′_{m−1}. Note that T′_{m−1} is Sℤ_{(p)}-local because it is a summand of T_{m−1} which is Sℤ_{(p)}-local. Therefore h is p-local and by hypothesis H_{G}^{m−1}(|X|; M_{h^{0}}) = 0. In other words, [T_{m−2}, T′_{m−1}] → [T′_{m−1}, T′_{m−1}] is surjective. We deduce that T′_{m−1} → T_{m−2} has a left inverse, i.e T_{m−2} ≃ T′_{m−1} ∨ T′_{m−2}. Continuing in this way we see that T has the form

0 → \underbrace{T′_{m}}_{T_{m}} → \underbrace{T′_{m} ∨ T′_{m−1}}_{T_{m−1}} → \underbrace{T′_{m−1} ∨ T′_{m−2}}_{T_{m−2}} → ⋯

⋯ → \underbrace{T′_{2} ∨ T′_{1}}_{T_{1}} → \underbrace{T′_{1} ∨ T′_{0}}_{T_{0}} → T_{−1} (4.8)

It remains to prove that T′_{0} → T_{−1} is an equivalence of spectra. There is a first quadrant (cohomological) Bousfield-Kan spectral sequence

E_{2}^{i,j} = \underleftarrow{lim}^{i}_{n∈Δ^{op}} H_{j}((X_{n})_{hG}; ℤ_{(p)}) ⇒ H^{i+j}(|X|_{hG}; ℤ_{(p)}).

Bousfield and Kan describe E_{2} in [7, pp. 284-285] and together with (4.7)

E_{2}^{i,j} = π^{i}H^{j}((X)_{hG}; ℤ_{(p)}) = H^{i}H^{j}(T; ℤ_{(p)}).

The splitting (4.8) shows that the spectral sequence collapses at its E^{2}-page, hence

H^{j}(T′_{0}; ℤ_{(p)}) = E_{2}^{0,j} ≈ H^{j}(|X|_{hG}; ℤ_{(p)}).

That is T′_{0} → T_{−1} is an H^{∗}(−; ℤ_{(p)})-isomorphism. Furthermore T′_{0}, which is a retract of T_{0}, and T_{−1} are Sℤ_{(p)}-local. We deduce that T′_{0} → T_{−1} is an equivalence by proposition A.4.

Theorem 4.9. Fix a group G and a non-trivial finite subgroup S and let H be a collection of finite subgroups of G. Let 𝔛 be a G-poset such that

(1) sd 𝔛 is finite dimensional and Iso(sd 𝔛) ⊆ H.

(2) For every H ∈ H, the cardinality of the fibre of G/S ⊠ G/H → G/H is a unit in ℤ_{(p)}.

(3) For every 1 ≠ Q ≤ S the space | sd 𝔛|^{Q} is weakly equivalent to a point.

(4) H̃_{∗}(| sd 𝔛|/G; ℤ_{(p)}) = 0.

Then

(a) | sd 𝔛| is G-acyclic for any p-local coefficient functor M_{h^{q}} (2.4) and (b) If sd 𝔛/G is finite then (4.5) is a split exact sequence of Sℤ_{(p)}-local

spectra.

Proof. Let X denote the nerve of 𝔛. Recall that |X| and | sd 𝔛| are Gequivariantly homeomorphic. We can now apply Lemma 3.6 to obtain (a). Point (b) follows from (a) and Lemma 4.4.

As an application we can now recover Minami’s Theorem 1.1

Proof (of Theorem 1.1). Let 𝔛 denote one of the collections in the hypotheses of the theorem and let H = Iso_{G}(sd 𝔛). We verify the conditions of Theorem 4.9. Obviously sd 𝔛/G is finite and H consists of finite groups whence (1). Lemmas 5.7 and 5.8 below imply (2) . Proposition 5.2 below and the fact that the inclusions E → P ← R induce G-equivalences on the associated simplicial complexes [4, Theorems 6.6.1, 6.6.6] imply (3). Symonds’s theorem [29], known as Webb’s conjecture, asserts that | sd P|/G is contractible. Since sd E → sd P and sd R → sd P are G-equivalences, it follows (e.g. from 6.7 or from tom Dieck [30, Proposition 2.7, Chapter II]) that | sd E|/G and | sd R|/G are contractible whence condition (4).

Dwyer shows in [12] that |𝔛|_{hG} → BG is an H_{∗}(−; 𝔽_{p})-isomorphism, hence an H_{∗}(−; ℤ_{(p)})-isomorphism because H_{∗}(| sd 𝔛|; ℤ) is finitely generated in every dimension. The result follows from Theorem 4.9 and from Theorem A.2 which imply (Σ_{+}^{∞}|𝔛|_{hG})_{Sℤ_{(p)}} ≃ (Σ_{+}^{∞}BG)_{Sℤ_{(p)}}.

5 Sylow p-subgroups and collections of finite p-subgroups

Let G be a group. Recall that a G-collection is a set C of subgroups of G closed under conjugation. Note that G acts on the simplicial complex sd C by conjugation, cf. Brown [11] and Quillen [24].

Throughout P_{G} denotes the collection of all the non identity finite psubgroups of G. Whenever there is no danger of confusion we shall simply write P for P_{G}.

Definition 5.1. A finite p-subgroup S of G is called a Sylow p-subgroup if every finite p-subgroup of G is conjugate to a subgroup of S.

Not every group admits a Sylow subgroup. However if G has one then the simplicial complex sd P is finite dimensional, in fact, sd P/G is finite. Indeed, every simplex of sd P is conjugate to a simplex consisting of subgroups of S.

Also note that when G has a Sylow p-subgroup, then for every subgroup H the product of all its normal finite p-subgroups is a characteristic finite p-subgroup of H denoted O_{p}(H). Indeed, every ascending chain of normal finite p-subgroups of H must become constant.

Proposition 5.2. Fix a group G with a Sylow p-subgroup and consider the collection P. If H is a subgroup of G such that O_{p}(H) ≠ 1 then |P|^{H} is contractible.

Proof. For every Q ∈ P^{H} we have a zigzag of inclusions of subgroups in P^{H}

Q ≤ Q · O_{p}(H) ≥ O_{p}(H)

which connects the identity functor on P^{H} to the constant functor on the object O_{p}(H). This shows that |P|^{H} = |P^{H}| is contractible, cf. Dwyer [13, 7.2].

Definition 5.3. A finite p-subgroup P of G is called p-radical if N_{G}(P) is finite and O_{p}(N_{G}(P)/P) = 1. Let R_{p}(G) denote the collection of all the non-identity p-radical subgroups of G.

Remark 5.4. If P is p-radical in G then it contains every normal finite p-subgroup of G. See [17, Lemma 1.5(i)] and its proof.

Proposition 5.5. Let G be a group with a non-trivial Sylow p-subgroup and assume that N_{G}(P) is finite for every non-trivial p-subgroup P of G. Then the inclusion R_{p}(G) → P_{G} induces a G-equivalence upon taking nerves, that is |R_{p}(G)|^{H} → |P|^{H} is a weak equivalence for every subgroup H ≤ G.

Proof. This result is due to Thévenaz-Webb [31] in the case where G is finite. See e.g. [4, Theorem 6.6.6]. The proof only uses the fact that P/G is finite and the fact that normalisers of elements of P are finite subgroups of G.

If X, Y are subsets of a group G we let X · Y denote the set of elements of G of the form xy where x ∈ X and y ∈ Y. The following is a variant of the second isomorphism theorem. Its proof is elementary.

Proposition 5.6. Let G be a group and X a subset of G. Assume that L is a subgroup of G such that L ⊆ X and XL = X. Let H be a subgroup of G containing L and assume that for every x, y ∈ X, if xH = yH then y^{−1}x ∈ L. Then the obvious map X/L → XH/H is a bijection. If K ≤ X satisfies K · X = X then this bijection is equivariant with respect to the action of K on X by left translations.

For example, if K, H are subgroups of G then KH/H ≅ K/K ∩ H.

Lemma 5.7. Let S be a Sylow p-subgroup of a group G. Fix a finite subgroup H of G such that G/S ⊠ G/H has finitely many orbits. Let F denote the fibre of the projection map G/S ⊠ G/H → G/H and let T be a Sylow p-subgroup of H. Then

|F | = |S\N_{G}(T, S)H/H| · |H : T | mod p.

Proof. Set X = N_{G}(T, S)H, see (2.1). We claim that X is the set of elements g ∈ G such that S^{g} contains a Sylow p-subgroup of H. Indeed if g ∈ N_{G}(T, S) and h ∈ H then S^{gh} contains T^{h} ∈ Syl_{p}(H). Conversely, if S^{x} contains a Sylow p-subgroup of H, then S^{x} ≥ T^{h} for some h ∈ H. Consequently xh^{−1} ∈ N_{G}(T, S).

Consider the G-set G/S × G/H ≅ ∐_{g∈S\G/H} G/(S^{g} ∩ H). We have

|F | = ∑_{g∈S\G/H} {[H : S^{g} ∩ H] : S^{g} ∩ H ≠ 1}

= ∑_{g∈S\G/H} {[H : S^{g} ∩ H] : S^{g} ∩ H ∈ Syl_{p}(H)} mod p

which yields |F | = |S\N_{G}(T, S)H/H| · |H : T | mod p.

Lemma 5.8. Let S be a Sylow p-subgroup of a finite group G where p||G|. Let U be a subgroup of G whose order is divisible by p. Then

|S\N_{G}(T, S)U/U| = \frac{[G : S]}{[U : T]} mod p.

Proof. Let F denote the fibre of G/S ⊠G/U → G/U. Since G/S is the fibre of G/S × G/U → G/U and |U| ≠ 0 mod p, it follows that |F | = |G/S| mod p. The result follows from Lemma 5.7.

6 Collections of p-centric subgroups

The material in this section is contained in §§2,4 of Dwyer’s work in [12]. Dwyer states his results for finite groups and we only observe that his approach is applicable unchanged to arbitrary infinite groups. Throughout we fix a group G with a finite Sylow p-subgroup S (5.1).

Definition 6.1. A finite p-subgroup P of G is called p-centric if Z(P) is a Sylow p-subgroup of C_{G}(P) in the sense of 5.1.

Note that in this case Z(P) is a unique Sylow p-subgroup of C_{G}(P) because it is central there. A map f : X → Y of spaces is called a mod-p equivalence if it induces an H_{∗}(−; 𝔽_{p})-isomorphism. Our aim is to recover Dwyer’s result [12, Theorem 8.3].

Theorem 6.2. Let S be a Sylow p-subgroup of G and assume that N_{G}(P) is a finite group for every non identity finite p-subgroup P of G. Let C denote the collection of the non-identity p-centric subgroups of G. Then the inclusion C ⊆ P induces a mod-p equivalence |C|_{hG} → |P|_{hG}.

Dwyer’s proof of of [12, Theorem 8.3], in the case where G is finite, can almost be read verbatim. His argument uses only the finiteness of normalisers of non-trivial p-subgroups of G and the finiteness of the number of conjugacy classes in P. We shall briefly outline Dwyer’s argument.

6.3. Recall from [12, Lemma 8.4] that if K is a subgroup of G and X is an K-space then (G ×_{K} X)_{hG} = X_{hK}. Here G ×_{K} X = (G × X)/K where K acts via k · (g, x) = (gk^{−1}, kx). Also recall from [12, Lemma 8.5] that when X is a G-space and N is a normal subgroup of G which acts trivially on X, then EG/N is a model for BN and X_{hG} = (X × BN)_{hW} where W = G/N acts diagonally on X × BN.

Let C be a collection of subgroups of G. Let O_{C}^{G} be the full subcategory of G-sets whose objects are the orbits G/H for all H ∈ C. Let J : O_{C}^{G} → S^{G} denote the inclusion functor and define

EC = hocolim_{O_{C}^{G}} J

The following two results are contained in [12, §2]. The arguments there do not involve the finiteness of G and carry over verbatim to collections of subgroups of arbitrary discrete groups G.

For a subgroup K of G let C(K) denote the sub-poset of C which consists of all the elements H ∈ C which contain K. Evidently N_{G}(K) acts on C(K) by conjugation.

Proposition 6.4. For every K ≤ G there is a N_{G}(K)-equivariant map EC^{K} → |C(K)| which is a weak homotopy equivalence. In particular EC → |C| is a weak homotopy equivalence and if K ∈ C then EC^{K} is contractible.

Proof. See Dwyer, [12, §2.12, 2.14 and 2.15].

Recall that a map f : X → Y of G-spaces is called a G-equivalence if the maps f^{H} : X^{H} → Y^{H} on fixed points are weak homotopy equivalence for all subgroups H of G.

The next two results are contained in §4 of [12]. Though Dwyer states them for finite groups, his method of proof is general enough to apply it with no changes to infinite groups G.

Lemma 6.5 ([12, Proposition 4.1]). Consider a map f : X → Y of G-spaces. If f^{H} : X^{H} → Y^{H} is a weak homotopy equivalence for every H ∈ Iso(X) ∪ Iso(Y) then f is a G-equivalence.

Proposition 6.6 ([12, Theorem 4.2]). Let X be a G-space and C a collection of subgroup of G which contains Iso_{G}(X). Then there is a natural zigzag of G-maps

X ←^{f} X′ ⟶^{h} EC

where f is a G-equivalence. If X^{H} is contractible (resp. 𝔽_{p}-acyclic) for all H ∈ C then h is a G-equivalence (resp. G-𝔽_{p}-equivalence.)

The following result is well known for finite groups but less so for arbitrary discrete ones Proposition 6.7. Let f : X → Y be a G-equivalence of G-spaces. Then the induced map X/G → Y/G is a weak homotopy equivalence.

Proof. We follow Dwyer’s idea in the proof of [12, Theorem 4.8]. Fix C = Iso(X) ∪ Iso(Y). Let O_{C} denote the full subcategory of G-Sets consisting of the orbits G/H where H ∈ C. For a G-space Z let Z_{C} denote hocoend(∗, Map_{G}(−, Z)) where Map_{G}(−, Z) is the fixed point functor of Z over O_{C} defined by G/H ↦ Z^{H}. There is a natural map Z_{C} → Z.

When Z = U × G/H, where U is a space with a trivial G-action, one sees that Z_{C} ≅ |(O_{C} ↓ G/H)|×U ×G/H where |(O_{C} ↓ G/H)| is contractible because H ∈ C. It follows that Z_{C}/G → Z/G is a weak equivalence in this case.

We now observe that homotopy coends preserve weak equivalences, coproducts, pushouts in which one of the arrows is a monomorphism of simplicial sets and sequential colimits. It follows, using an inductive argument on the skeleta of Z that Z_{C}/G → Z/G is a weak equivalence for arbitrary Z.

Finally, the naturality of Z ↦ Z_{C} gives (see also the commutative square in the proof of [12, 4.1])

X/G ≃ X_{C}/G ≃ hocoend(∗, Map_{G}(−, X)) ≃ ≃ hocoend(∗, Map_{G}(−, Y)) ≃ Y_{C}/G ≃ Y/G.

Finally we recall one of the key results in [12] of which we state a special case.

Proposition 6.8 (Dwyer [12, Proposition 6.7]). Let W be a finite group and let Y be a W -space. Let P_{W} denote the collection of all the non-trivial p-subgroups of W and consider the diagonal action of W on the space Y × |P_{W} |. If there exists an element of order p in W which acts trivially on H_{∗}(Y; 𝔽_{p}) then the map (Y × |P_{W} |)_{hW} → Y_{hW} is a mod-p equivalence.

Proof (proof of Theorem 6.2). Following Słominska in [27] we call a collection D of p-subgroups of G concave if whenever Q is a p-subgroup of G which contains an element of D then Q ∈ D. Clearly the collections P and C are concave. The theorem follows from the stronger statement that the inclusion C ⊆ D induces a mod-p equivalence |C|_{hG} → |D|_{hG} for every concave collection D of p-subgroups which contains C.

Note that D consists of finitely many conjugacy classes because G has a finite Sylow p-subgroup S. We use induction on |D/G|, the case D = C is trivial. Let P be a minimal element of D\C and put

D′ = D\{the conjugacy class of P }.

Clearly D′ is concave and by induction hypothesis |C|_{hG} → |D′|_{hG} is a mod-p equivalence. It remains to show that |D′|_{hG} → |D|_{hG} is a mod-p equivalence. In light of Proposition 6.4 it suffices to show that D′ ⊆ D induces a mod-p equivalence

(ED′)_{hG} → (ED)_{hG}.  (6.9)

By inspection, Iso_{G}(ED)\ Iso_{G}(ED′) is the conjugacy class of P (cf. [12, Lemma 4.4]). We now have a pushout square in which N = N_{G}(P) and the vertical arrows are inclusions (cofibrations) of spaces (cf. [12, Lemma 8.6])

\begin{matrix}G ×_{N} (ED′)^{P} & ⟶ & ED′ \\ ↓ & & ↓ \\ G ×_{N} ED^{P} & ⟶ & ED \end{matrix}  (6.10)

To prove that (6.9) is a mod-p equivalence, it suffices to prove that the vertical left arrow in (6.10) induces a mod-p equivalence after applying (−)_{hG}. That is, we need to prove that (ED′^{P})_{hN} → (ED^{P})_{hN} is a mod-p equivalence (see 6.3). Note that ED^{P} ≃ ∗ by 6.4, so it remains to prove that (BP × ED′^{P})_{hW} → BP_{hW} is a mod-p equivalence where W = N/P. Now, P is not p-centric hence \frac{C_{G}(P)P}{P} contains an element of order p which acts trivially on H_{∗}(BP). Using Proposition 6.8, it remains to prove that (ED′)^{P} and P_{W} can be connected by a zigzag of W -equivariant weak equivalences.

To see this, observe that if x ∈ (ED′)^{P} then Iso_{G}(x) = R for some R ∈ D′. Thus, R is a p-subgroup of G which properly contain P. Therefore

Iso_{N}(x) = R ∩ N = N_{R}(P) ⪈ P.

This shows that Iso_{W} ((ED′)^{P}) ⊆ P_{W}. Furthermore, if Q/P ∈ P_{W} then Q ∈ D′ hence (ED′^{P})^{Q/P} = (ED′)^{Q} ≃ ∗. We obtain a zigzag of W -equivariant weak equivalences (ED′)^{P} ← X′ → EP_{W} → |P_{W} | as required using 6.6 and 6.4.

7 Fusion systems and their associated linking systems

Definition 7.1. 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 such that every morphism in F factors as an isomorphism followed by an inclusion of groups. 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).

If S is a finite p-subgroup of a group G, we let F_{S}(G) denote the fusion system on S whose morphisms are restrictions of inner automorphisms of G to the subgroups of S. More explicitly, the morphisms set F(P, Q) of P, Q ≤ S are the homomorphisms c_{g} : P → Q where c_{g}(x) = gxg^{−1} for g ∈ N_{G}(P, Q). When P ≤ S and H ≤ G we write Aut_{H}(P) for the set of automorphisms of P induced by conjugation by elements of N_{H}(P).

Of particular importance are fusion systems F_{S}(G) associated to a Sylow p-subgroup S of a finite group G. Lluis Puig [23] was the first to isolate the essential properties of these fusion systems which he then took as axioms. He called fusion systems satisfying this set of axioms “full” (saturé in French). Broto, Levi and Oliver gave a somewhat different, yet equivalent, set of axions and called the fusion systems satisfying them “saturated” [10, §1]. We briefly recall their notation and terminology.

We fix a fusion system F on S. A subgroup P ≤ S is fully F-centralised (resp. fully F-normalised) if |C_{S}(P)| ≥ |C_{S}(P′)| (resp. |N_{S}(P)| ≥ |N_{S}(P′)|) for every P′ in the F-conjugacy class of P. A subgroup P of S is called Fcentric if C_{S}(P′) = Z(P′) for every P′ in the F-conjugacy class of P. We let F^{c} denote the full subcategory of F containing the F-centric subgroups of S.

For a morphism φ ∈ F(P, S) set

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

Definition 7.2 ([10, 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) is a

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

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

Let P_{1}, …, P_{k} be subgroups of S and let Δ_{i} be subgroups of Aut(P_{i}). We let ⟨F; F_{P_{1}}(Δ_{1}), …, F_{P_{k}}(Δ_{k})⟩ denote the minimal fusion system on S which contains F and all the fusion systems F_{P_{i}}(Δ_{i}). See e.g. [10, §9]. The morphisms of this fusion system are compositions of morphisms in F and restrictions of the automorphisms of Δ_{i} to subgroups of P_{i}. This fusion system need not be saturated.

Recall that P ≤ S is called F-radical if Inn(P) = O_{p}(Aut_{F}(P)). A remarkable result on saturated fusion systems is known as Alperin’s fusion theorem

Theorem 7.3 ([10, Theorem A.10]). Let F be a saturated fusion system on S and let P_{1}, …, P_{k} denote the F-centric F-radical subgroups of S. Then F is generated by {F_{P_{i}}(Δ_{i})}_{i=1}^{k}.

Remark 7.4. Fix an F-centric subgroup P of S. Assume that S is the only F-radical subgroup which properly contains any F-conjugate P′ of P. Alperin’s theorem implies that every isomorphism φ ∈ F(P, P′) where P and P′ are distinct, is a restriction of an F-automorphism of S.

Definition 7.5. 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 the set of F-centric subgroups of S.

2. A functor π : L → F which induces the inclusion Obj(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 the axioms

(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 com-

position 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.

Definition 7.6. 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.

Every finite group G with a Sylow p-subgroup S gives rise to a p-local finite group whose fusion system is F_{S}(G), see [10].

Broto, Levi and Oliver prove in [10, Proposition 1.11] the existence of a choice of morphisms ι_{P}^{Q} ∈ L(P, Q) for every F-centric P ≤ Q which enjoy the properties

(a) π(ι_{P}^{Q}) is the inclusion incl_{P}^{Q} ∈ F(P, Q) and ι_{P}^{P} is the identity on P in L.

(b) ι_{Q}^{R} ∘ ι_{P}^{Q} = ι_{P}^{R} for every P ≤ Q ≤ R.

Lemma 1.10(a) of [10] implies that every morphism φ : P → Q in L can be factored as φ = ι_{P′}^{Q} ∘ ψ where P′ = π(φ)(P) and ψ : P → P′ is an isomorphism in L.

Definition 7.7. Fix F-centric subgroups P < Q in S. Let Aut_{L}(P < Q) be the subgroup of Aut_{L}(P) consisting of the elements φ̃ for which there exists ψ̃ ∈ Aut_{L}(Q) such that ι_{P}^{Q} ∘ φ̃ = ψ̃ ∘ ι_{P}^{Q}.

Proposition 7.8. Fix F-centric subgroups P < Q and consider automorphisms φ ∈ Aut_{F}(P) and λ ∈ Aut_{F}(Q) such that λ|_{P} = φ. Then

(1) For every lift λ̃ ∈ Aut_{L}(Q) of λ there exists a unique lift φ̃ ∈ Aut_{L}(P) of φ such that ι_{P}^{Q} ∘ φ̃ = λ̃ ∘ ι_{P}^{Q}.

(2) For every lift φ̃ ∈ Aut_{L}(P) of φ there exists a unique ψ̃ ∈ Aut_{L}(Q) such that ι_{P}^{Q} ∘ φ̃ = ψ̃ ∘ ι_{P}^{Q}. (Note: π(ψ̃) ≠ λ.)

Proof. (1) follows immediately from [10, Lemma 1.10(a)]. For (2), choose an arbitrary lift λ̃ for λ in L. By axiom 7.5(A) there exists g ∈ Z(P) such that ι_{P}^{Q} ∘ φ̃ = λ̃ ∘ ι_{P}^{Q} ∘ δ_{P} (g). Since P is F-centric, Z(P) ≤ Q and we define ψ̃ = λ̃ ∘ δ_{Q}(g). By 7.5(C) we have ψ̃ ∘ ι_{P}^{Q} = λ̃ ∘ δ_{Q}(g) ∘ ι_{P}^{Q} = λ̃ ∘ ι_{P}^{Q} ∘ δ_{P} (g) = ι_{P}^{Q} ∘ φ̃.

The uniqueness in the statements of the proposition shows that φ̃ ↦ ψ̃ identifies Aut_{L}(P < Q) as a subgroup of Aut_{L}(Q). It also allow us to extend δ_{P} to a group monomorphism δ_{P} : Q → Aut_{L}(P) provided P ◁Q. Indeed, for g ∈ Q we let δ_{P} (g) be the unique morphism such that ι_{P}^{Q}∘δ_{P} (g) = δ_{Q}(g)∘ι_{P}^{Q}. In this way we identify a canonical copy of Q in Aut_{L}(P).

For the proof of the next result it will also be convenient to observe that P is identified as a subgroup of Aut_{L}(P) via δ_{P}. Every φ̃ ∈ Aut_{L}(P) induces by axion 7.5(C) by conjugation a homomorphism g ↦ φ̃∘δ_{P} (g)∘φ̃^{−1} which coincides on P with φ = π(φ̃).

Proposition 7.9. Fix an F-centric P ⪇ S and consider Q = N_{S}(P). Then

(1) N_{Aut_{L}(P)}(Q) = Aut_{L}(P < Q).

(2) N_{Aut_{L}(Q)}(P) = Aut_{L}(P < Q).

Proof. We shall exploit the fact proved in [8, Corollary 3.10] that every morphism in L is both an epimorphism and a monomorphism. (1). Consider φ̃ ∈ N_{Aut_{L}(P)}(Q). Thus, for every g ∈ Q there exists g′ ∈ Q such that φ̃ ∘ δ_{P} (g) ∘ φ̃^{−1} = δ_{P} (g′). Let φ denote π(φ̃) We see that g ∈ N_{φ} (cf. 7.2), that is Q = N_{φ}. Let λ ∈ F(Q, S) be an extension of φ. We clearly have λ(Q) ≤ N_{S}(φP) = N_{S}(P) = Q, hence λ ∈ Aut_{F}(Q). Proposition 7.8(2) implies that φ̃ ∈ Aut_{L}(P < Q).

Conversely, if φ̃ ∈ Aut_{L}(P < Q), choose ψ̃ ∈ Aut_{L}(Q) such that ι_{P}^{Q}∘φ̃ = ψ̃ ∘ ι_{P}^{Q}. Fix g ∈ Q. We have to show that φ̃ ∘ δ_{P} (g) ∘ φ̃^{−1} = δ_{P} (g′) for some g′ ∈ Q. Now,

ι_{P}^{Q} ∘ (φ̃ ∘ δ_{P} (g) ∘ φ̃^{−1}) ∘ φ̃ = ι_{P}^{Q} ∘ φ̃ ∘ δ_{P} (g) = ψ̃ ∘ ι_{P}^{Q} ∘ δ_{P} (g)

= ψ̃ ∘ δ_{Q}(g) ∘ ι_{P}^{Q} = δ_{Q}(ψg) ∘ ψ̃ ∘ ι_{P}^{Q} = δ_{Q}(ψg) ∘ ι_{P}^{Q} ∘ φ̃ = ι_{P}^{Q} ∘ δ_{P} (ψg) ∘ φ̃.

We can cancel φ̃ from both sides because it is an isomorphism and deduce that φ̃ ∘ δ_{P} (g) ∘ φ̃^{−1} = δ_{P} (ψg) because ι_{P}^{Q} is an epimorphism in L.

(2) Consider φ̃ ∈ Aut_{L}(P < Q) and let ψ̃ denote its image in Aut_{L}(Q), that is ι_{P}^{Q} ∘ φ̃ = ψ̃∘ ι_{P}^{Q}. Fix g ∈ P. We have to show that ψ̃∘ δ_{Q}(g) ∘ ψ̃^{−1} = δ_{Q}(g′) for some g′ ∈ P. Using axiom 7.5(C) we have

(ψ̃ ∘ δ_{Q}(g) ∘ ψ̃^{−1}) ∘ ι_{P}^{Q} ∘ φ̃ = ψ̃ ∘ δ_{Q}(g) ∘ ι_{P}^{Q} = ψ̃ ∘ ι_{P}^{Q} ∘ δ_{P} (g)

= ι_{P}^{Q} ∘ φ̃ ∘ δ_{P} (g) = ι_{P}^{Q} ∘ δ_{P} (φg) ∘ φ̃ = δ_{Q}(φg) ∘ ι_{P}^{Q} ∘ φ̃.

Now, φ̃ can be cancelled from both sides because it is an isomorphism and ι_{P}^{Q} is an epimorphism in L so ψ̃ ∘ δ_{Q}(g) ∘ ψ̃^{−1} = δ_{Q}(φg) as needed.

Conversely, choose ψ̃ ∈ N_{Aut_{L}(Q)}(P). Since conjugation by ψ̃ in Aut_{L}(Q) induces ψ = π(ψ̃) on Q, it follows that ψ|_{P} ∈ Aut_{F}(P). Proposition 7.8(1) shows that there exists φ̃ ∈ Aut_{L}(P < Q) whose image in Aut_{L}(Q) is ψ̃.

7.10. Let S be a non identity finite p-subgroup of a group G, possibly infinite. Throughout we assume

(F1) S is a non identity Sylow p-subgroup of G and (F2) If P is a non identity finite p-subgroup of G then N_{G}(P) is finite.

Under these assumptions, if P is a p-centric subgroup (6.1) of G then C_{G}(P) is finite and contains Z(P) as a central Sylow hence C_{G}(P) = Z(P)× C′_{G}(P) where C′_{G}(P) is the set of elements of order prime to p in C_{G}(P).

Definition 7.11. Let L_{p}^{c}(G) denote the category whose objects are the p-centric subgroups of G and whose morphism sets L(P, Q) are given by N_{G}(P, Q)/C′_{G}(P).

Let L_{S}^{c}(G) denote the full subcategory of L_{p}^{c}(G) whose objects are the pcentric subgroups of G contained in S. Let π : L_{S}^{c}(G) → F_{S}(G) be the functor which is the inclusion on objects and the quotient N_{G}(P, Q)/C′_{G}(P) → N_{G}(P, Q)/C_{G}(P) on morphisms. For P ∈ F^{c} let δ_{P} : P → Aut_{L}(P) be the monomorphism g ↦ gC′_{G}(P).

When S is a Sylow p-subgroup of a finite group G, then L_{S}^{c}(G) is an associated centric linking system of F_{S}(G) in the sense of Broto, Levi and Oliver in [10, §1]. When G is infinite we have

Proposition 7.12. Under the assumptions (F1)and (F2) of 7.10

(a) F_{S}(G) is a saturated fusion system on S.

(b) A subgroup P of S is p-centric if and only if it is F_{S}(G)-centric.

(c) L_{S}^{c}(G) is an associated centric linking system to F_{S}(G).

Proof. (a) The proof of [10, Proposition 1.3] works verbatim; It only uses the fact that normalisers of non identity p-subgroups of G are finite and the fact that for every finite subgroup H of G there exists a conjugate of S which contains a Sylow of H. This is readily verified to follow from the definition 5.1. See also [2].

(b) Set F = F_{S}(G). If P ≤ S is p-centric then Z(P) is a normal Sylow p-subgroup of C_{G}(P). This must be the case for any subgroup P′ = ^{g}P, thus P is F-centric. Conversely, if P ≤ S is F-centric then S contains a Sylow p-subgroup of ^{g}(P C_{G}(P)) for some g ∈ G, hence C_{S}(^{g}P) = Z(^{g}P) is a Sylow p-subgroup of C_{G}(^{g}P). Therefore this is the case for the inclusion Z(^{h}P) ≤ C_{G}(^{h}P) for any h ∈ G.

(c) Use (b) to ensure that L = L_{S}^{c}(G) is defined on the correct object set. The verification of the axioms of an associated centric linking system is straightforward.

Proposition 7.13. With the hypotheses (F1) and (F2) of 7.10 on S ≤ G we have

(a) The inclusion L_{S}^{c}(G) ⊆ L_{p}^{c}(G) induces a weak homotopy equivalence on nerves.

(b) There is a natural mod-p equivalence |C|_{hG} → |L_{p}^{c}(G)| where C denotes the poset of p-centric subgroups of G.

Proof. (a) Clearly L_{S}^{c}(G) contains an object from every isomorphism class of objects in L^{c}(G) because S is a Sylow.

(b) The argument is essentially present in Broto, Levi and Oliver [9, Lemma 1.2]. Let \underline{G} denote the category with one object and G as its automorphism set. We regard the poset C as a small category. As such, the action of G on the poset C gives rise to a functor \underline{G} → Cat, which by abuse of notation we also denote by C. The functor C sends the unique object of \underline{G} to the poset C and a morphism g ∈ Mor(\underline{G}) to the endofunctor it induces on C via the action of G on C.

Note that |C|_{hG} ≃ hocolim_{\underline{G}} |C|. By inspection, Thomason’s construc-

tion T = \underline{G} ∫ C, see [32] is the small category whose objects are the elements (subgroups) P of C and whose morphism sets T (P, Q) are the transporters N_{G}(P, Q). Thomason proves that there is a natural homotopy equivalence |\underline{G} ∫ C| ≃ hocolim_{\underline{G}} |C|, whence |T | ≃ |C|_{hG}. There is an obvi-

ous functor T → L_{p}^{c}(G) which is the identity on objects and the quotient N_{G}(P, Q) → N_{G}(P, Q)/C′_{G}(P) on morphism sets. The result follows from [9, Lemma 1.3] which guarantees that |T | → |L^{c}(G)| induces an isomorphism on H_{∗}(−; ℤ_{(p)}).

Fix a saturated fusion system F on S. Recall that a right ideal is a full subcategory C of F such that for any two objects P ∈ C and Q ∈ F, if F(P, Q) ≠ ∅ then Q ∈ C. For example the collection of all the non-identity subgroups of S forms a right ideal. Also the collection of the F-centric subgroups of S forms a right ideal.

A right ideal C clearly forms a poset by inclusion of groups. We let S(C) denote the simplicial complex associated to it. Define an equivalence relation ∼ on the simplices of S(C) as follows. Two k-simplices P_{0} < … < P_{k} and P′_{0} < … < P′_{k} are declared to be equivalent (or “conjugate”) if there exists f ∈ Iso_{F}(P_{k}, P′_{k}) such that f(P_{i}) = P′_{i} for all i. The quotient space S(C)/ ∼ is denoted [S(C)].

Theorem 7.14 (Linckelmann [20, Theorem 1.1]). Let F be a saturated fusion system on a finite p-group S and let C be a right ideal in F. Then [S(C)] is contractible.

Corollary 7.15. Fix a group G and a finite p-subgroup S which satisfy the hypotheses of 7.10. Then | sd P|/G is contractible where P is either the collection of all the non-identity subgroups of G or the collection of the non-identity p-centric p-subgroups of G.

Proof. The fusion system F = F_{S}(G) is saturated according to 7.12. Let C denote the elements P ∈ P which are subgroups of S. Clearly C is a right ideal in F. Also note that since S is a Sylow in G it follows that | sd P|/G = [S(C)]. The result follows from Linckelmann’s theorem.

8 An amalgam of Ruiz-Viruel type

We first recall several standard facts from Serre’s seminal work [26] on trees. A graph Γ consists of a set X = V (Γ) of vertices and a set Y = E(Γ) of (oriented) edges together with functions

Y → X × X,  y ↦ (o(y), t(y))
Y → Y,  y ↦ ȳ

which are subject to the conditions (i) ¯ ȳ = y, (ii) ȳ ≠ y and (iii) o(y) = t(ȳ) for all y ∈ Y. The vertices o(y) and t(y) are called the origin and terminus of the oriented edge y.

A group action on Γ is a homomorphism G → Aut(Γ). Of particular importance is the case when G acts on Γ without inversions (see [26, §3.1]). That is, for every edge y of Γ and every element g ∈ G we have gy ≠ ȳ.

A tree is a connected non-empty graph T without circuits. That is, there exists no path without backtracks of positive length in T from a vertex to itself. Every two vertices x, y in T are connected by a unique path without backtracks called the geodesic from x to y and denoted x − y (See [26, §I.2.2]). Its length is called the distance between x and y and is denoted by ℓ(x, y) which is a G-equivariant metric on T.

A tree T gives rise to a small category sd T whose objects are the vertices P of T an the pairs {y, ȳ} of edges in T. There is a morphism {y, ȳ} → P if P is incident with y. It is the category of the simplices of a contractible 1-dimensional simplicial complex. A tree of groups (G, T) is a functor G : sd T → Gps which carries the morphisms in sd T to group monomorphisms. The limit of (G, T) is the direct limit in the category of groups

G_{T} = \underrightarrow{lim} (G, T) = \underrightarrow{lim}_{sd T} G.

The group G_{T} is called the amalgam of the groups {G_{P} }_{P ∈V (T)} along the G_{y}’s. It is shown in [26] that the natural maps G_{P} → G_{T} are injective and we identify each of the groups G_{P} and G_{y} as a subgroup of G_{T} in this way. In this embedding G_{y} = G_{P} ∩ G_{Q} if P and Q are the extremities of y. Associated with (G, T) is the following tree X

vert(X) = ∐_{P ∈V (T)} G/G_{P} , edge(X) = ∐_{y∈E(T)} G/G_{y}

where o(gG_{y}) = gG_{o(y)} and t(gG_{y}) = gG_{t(y)}. Also \overline{gG_{y}} = gGȳ. Clearly G acts without inversions on X by left translations. The Tree T is embedded in X by means of the cosets eG_{P}. Also, the isotropy subgroups of every vertex P and every edge y of T satisfy Stab_{G}(P) = G_{P} and Stab_{G}(y) = G_{y}. A path from eG_{P} to gG_{Q} in X has the form

G_{P_{0}}  g_{1}G_{P_{1}}  g_{1}g_{2}G_{P_{2}}  ⋯  (g_{1}g_{2} ⋯ g_{n})G_{P_{n}}

where P = P_{0} and Q = P_{n} and

– P_{0}, …, P_{n} are vertices of T such that P_{i} is adjacent to P_{i+1}.

– g_{i} ∈ G_{P_{i−1}} for all i = 1, …, n – g = g_{1}g_{2} ⋯ g_{n} mod G_{Q}.

We also note that if the groups G_{P} are finite then the tree X is locally finite because T is finite. The following is a standard result.

Proposition 8.1. Let G be the amalgam of a tree of groups (G, T). If H is a finite subgroup of G then H is conjugate to a subgroup of G_{P} for some vertex P ∈ T.

Proof. This follows from Corollary 2 and Proposition 27 in [26, §I.6.5] applied to the tree X associated to (G, T).

Proposition 8.2. Let (G, T) be a tree and let G = G_{T}. Assume that the groups G_{P} are finite and that |G_{y}| ⪇ |G_{P} | for every vertex P incident with an edge y.

Then G_{P} is not a subgroup of (G_{Q})^{g} for any Q ∈ V (T) and any g ∈ G unless Q = P and g ∈ G_{P}. In particular, if K is a finite subgroup of G containing G_{P} for some P, then K = G_{P}.

Proof. Consider the tree X associated to (G, T) and let H denote G_{P}. Clearly eH ∈ X^{H}. We have to show that eH is the only element of X^{H}. Fix an element gG_{Q} of X^{H} and consider the geodesic γ from eH to gG_{Q}. Clearly γ is fixed by H. Thus, if eH ≠ gG_{Q} then γ is not trivial hence H must fix the edge hG_{y} in γ incident with eH. This is impossible because |G_{y}| < |H|.

If K ≤ G is finite and contains G_{P} then by 8.1 some conjugate of K, hence of G_{P} is contained in G_{Q} for some Q ∈ V (T). We have just proved that necessarily Q = P and G_{P} ≤ K^{g} ≤ G_{P} for some g ∈ G_{P}.

Under the assumption of 8.2, by associating to every vertex x of X its isotropy group, we obtain a natural bijection between the vertices of X and the conjugacy classes of the groups G_{P}.

8.3. In the remainder of this section we will consider a tree of groups (G, T) of the form

[figure: K; L_{1}; L_{2}; L_{n}; H_{1}; H_{2}; H_{n}]

where all the groups are finite and the groups L_{i} are proper subgroups of H_{i} and K. We also assume that there exists a p-subgroup S ≤ K and subgroups

P_{i} < R_{i} ≤ S, i = 1, …, n

such that

(G1) S is a Sylow p-subgroup of K. Put Z = Z(S). Then |Z| = p and Z ◁ K. (G2) R_{i} ◁ L_{i} is a Sylow p-subgroup. Moreover |R_{i}| = p^{3} and it is not abelian. (G3) R_{i} is a Sylow p-subgroup of H_{i} and L_{i} = N_{H_{i}}(R_{i})

(G4) P_{i} ≅ C_{p}×C_{p} contain Z(S). Furthermore P_{i}◁H_{i} and H_{i} acts transitively

on the lines of P_{i}. In addition L_{i} = N_{K}(P_{i}). (G5) L_{i} = N_{H_{i}}(Z) where Z = Z(S). (G6) No P_{i} is K-conjugate to P_{j} for i ≠ j.

The main result of this section is

Proposition 8.4. Fix a tree of groups as in 8.3 and let G = \underrightarrow{lim}(G, T). Then

(a) S is a Sylow p-subgroup of G.

(b) If P is a non-trivial p-subgroup of G then N_{G}(P) is finite.

(c) (i) N_{G}(S) = N_{K}(S), (ii) N_{G}(P_{i}) = H_{i} and (iii) N_{G}(Z) = K.

(d) Let U be a finite subgroup of G. Then G/S ⊠ G/U has finitely many orbits.

(e) Fix a chain Q_{0} < ⋯ < Q_{m} of p-subgroups of G where |Q_{0}| > p. Put U = N_{G}(Q_{0}) ∩ … ∩ N_{G}(Q_{m}). Then U is finite and if F denote the fibre of the projection G/S ⊠ G/U → G/U then |F | is a unit in ℤ_{(p)}.

As a corollary we obtain

Theorem 8.5. Let G be an amalgamated product of a tree of groups which satisfies the hypotheses of 8.3. Then

(a) G satisfies (F1) and (F2) of 7.10, hence F_{S}(G) is a saturated fusion system with an associated centric linking system L_{S}^{c}(G).

(b) F_{S}(G) = ⟨F_{S}(K); F_{P_{i}}(H_{i}) : i = 1, …, n⟩.

(c) Let R denote the collection of the non-identity p-radical subgroups of G and assume that R contains no subgroups of order p. Then after localizing at the Moore spectrum Sℤ_{(p)} the chain complex of spectra below is split exact where m is the dimension of sd R.

0 → ⋁_{σ∈sd_{m} R/G} Σ_{+}^{∞}BG_{σ} → ⋯ → ⋁_{σ∈sd_{1} R/G} Σ_{+}^{∞}BG_{σ} →

→ ⋁_{σ∈sd_{0} R/G} Σ_{+}^{∞}BG_{σ} → Σ_{+}^{∞}|L_{S}^{c}(G)| → 0

Proof. (a) First, (F1) and (F2) of 7.10 are (a) and (b) of Proposition 8.4 hence Proposition 7.12 applies.

(b) Put F = ⟨F_{S}(K); F_{P_{i}}(H_{i}) : i = 1, …, n⟩. Clearly F ⊆ F_{S}(G) because K, H_{i} ≤ G. To show equality, it suffices, by Alperin’s fusion theorem, to show that Aut_{F}(Q) = Aut_{F_{S}(G)}(Q) for every Q ≤ S which is p-centric in G. Note that we may assume that |Q| > p because |S| ≥ p^{3}.

If Q is not conjugate to P_{i} then Proposition 8.11 implies that N_{G}(Q) = N_{K}(Q) hence Aut_{F_{S}(G)}(Q) = Aut_{K}(Q). If Q = P_{i} then (c)(ii) of 8.4 shows that Aut_{F_{S}(G)}(Q) = Aut_{H_{i}}(Q).

(c) Let H = Iso_{G}(sd R). Note that H is a collection of finite groups by 8.4(b). Also note that R contains no subgroups of order p by hypothesis.

Now, 8.4(e) shows that whenever U = G_{σ} for some simplex σ ∈ sd R, then the cardinality of the fibre of G/S ⊠ G/U → G/U is a unit in ℤ_{(p)}.

If 1 ≠ Q ≤ S then | sd R|^{Q} is contractible by Propositions 5.2 and 5.5. Also, | sd R|/G ≃ | sd P_{G}|/G ≃ ∗ by Propositions 5.5, 6.7 and 7.15. The split exact sequence of spectra is now a consequence of Theorem 4.9(b) and the observation that Theorem A.2 together with Proposition 7.13(a),(b), Theorem 6.2 and Proposition 5.5, imply

(Σ_{+}^{∞}|L_{S}^{c}(G)|)_{Sℤ_{(p)}} ≃ (Σ_{+}^{∞}|C|_{hG})_{Sℤ_{(p)}} ≃ (Σ_{+}^{∞}|P|_{hG})_{Sℤ_{(p)}} ≃ (Σ_{+}^{∞}|R|_{hG})_{Sℤ_{(p)}}

where C denote the collection of p-centric subgroups of G.

The proof of Proposition 8.4 depends on the following two results.

Proposition 8.6. Assume the hypotheses of 8.3 and notation of Proposition 8.4. Let X denote the tree associated to (G, T) and consider an element g ∈ G.

(a) If ℓ(K, gK) = 2 then K∩^{g}K contains ^{g_{1}}P_{i} as a normal Sylow p-subgroup for some g_{1} ∈ K and some i = 1, …, n.

(b) If ℓ(K, gK) ≥ 6 then K ∩^{g}K is a finite group of order prime to p.

Proof. Let γ denote the geodesic K − gK in X. Clearly G_{γ} = K ∩^{g}K. We have seen that γ has the form

K − g_{1}H_{i_{1}} − (g_{1}g_{2})K − (g_{1}g_{2}g_{3})H_{i_{2}} − (g_{1}g_{2}g_{3}g_{4})K − · · · · · · − (g_{1} ⋯ g_{k})K.

(8.7)

In particular (g_{1} ⋯ g_{k})K = gK and k = ℓ(K, gK).

(a) In the notation of (8.7) k = 2 and g_{1} ∈ K. We shall prove that ^{g_{1}}P_{i_{1}} is a normal Sylow of G_{γ}. Note that g_{2} ∈ H_{i_{1}}\L_{i_{1}} because g_{1}g_{2}K = gK ≠ K. Clearly

G_{γ} = K ∩ (^{g_{1}}H_{i_{1}}) ∩ (^{g_{1}g_{2}}K) = (^{g_{1}}L_{i_{1}}) ∩ (^{g_{1}g_{2}}L_{i_{1}}) = ^{g_{1}}(L_{i_{1}} ∩^{g_{2}}L_{i_{1}}).

Now, R_{i_{1}} ∩^{g_{2}}R_{i_{1}} is a normal Sylow p-subgroup of L_{i_{1}} ∩^{g_{2}}L_{i_{1}} by (G2). Also, (G3) implies that R_{i_{1}} ≠ ^{g_{2}}R_{i_{1}} because g_{2} ∈ H_{i_{1}}\L_{i_{1}}. Finally (G4) implies that g_{2} normalises P_{i_{1}} hence P_{i_{1}} ≤ R_{i_{1}} ∩ ^{g_{2}}R_{i_{1}}. Equality holds because |P_{i_{1}}| = p^{2}.

(b) Now assume that ℓ(K, gK) ≥ 6. Present the geodesic γ as a concatenation of paths of length 2. That is, γ = γ_{1} · γ_{2} · γ_{3} ⋯ where

\begin{aligned}γ_{1} & : & K − g_{1}H_{i_{1}} − g_{1}g_{2}K \\ γ_{2} & : & g_{1}g_{2}(K − g_{3}H_{i_{2}} − g_{3}g_{4}K) \\ γ_{3} & : & g_{1}g_{2}g_{3}g_{4}(K − g_{5}H_{i_{3}} − g_{5}g_{6}K) \end{aligned}

In fact, as we shall explain below

g_{1} ∈ K , g_{2} ∈ H_{i_{1}}\L_{i_{1}}
g_{3} ∈ \{ \begin{matrix}K & if i_{2} ≠ i_{1} \\ K\L_{i_{1}} & if i_{2} = i_{1} \end{matrix} , g_{4} ∈ H_{i_{2}}\L_{i_{2}}
g_{5} ∈ \{ \begin{matrix}K & if i_{3} ≠ i_{2} \\ K\L_{i_{2}} & if i_{3} = i_{2} \end{matrix} , g_{6} ∈ H_{i_{3}}\L_{i_{3}}

Clearly g_{2} ∉ L_{i_{1}} because g_{1}g_{2}K ≠ K and γ_{1} is a geodesic. For the same reason g_{4} ∉ L_{i_{2}} and g_{6} ∉ L_{i_{3}}. Note that if i_{1} = i_{2} then g_{3} cannot belong to L_{i_{1}} because in this case g_{1}H_{i_{1}} = g_{1}g_{2}g_{3}H_{i_{1}} which is impossible because γ is a geodesic. Similarly, if i_{2} = i_{3} then g_{5} ∉ L_{i_{2}}.

We showed in (a) that ^{g_{1}}P_{i_{1}} is a normal Sylow p-subgroup of G_{γ_{1}}. Similarly ^{g_{1}g_{2}g_{3}}P_{i_{2}} is a normal Sylow of G_{γ_{2}}. Now, using (G4)

g_{1}P_{i_{1}} ∩^{g_{1}g_{2}g_{3}}P_{i_{2}} = ^{g_{1}g_{2}}(P_{i_{1}} ∩^{g_{3}}P_{i_{2}})  (8.8)

is a normal Sylow p-subgroup of G_{γ_{1}} ∩ G_{γ_{2}}. Also note that Z ≤ P_{i_{1}} ∩^{g_{3}}P_{i_{2}} by (G1) and (G4). Now we claim that P_{i_{1}} ≠ ^{g_{3}}P_{i_{2}}. Indeed, when i_{1} ≠ i_{2} this follows from (G6). When i_{1} = i_{2} then g_{3} ∈ K\L_{i_{1}} and we use (G4). It follows that

g_{1}g_{2}(P_{i_{1}} ∩^{g_{3}}P_{i_{2}}) = ^{g_{1}g_{2}}Z.  (8.9)

Similarly, the following is a normal Sylow in G_{γ_{2}} ∩ G_{γ_{3}}

g_{1}g_{2}g_{3}P_{i_{2}} ∩^{g_{1}g_{2}g_{3}g_{4}g_{5}}P_{i_{3}} = ^{g_{1}g_{2}g_{3}g_{4}}(P_{i_{2}} ∩^{g_{5}}P_{i_{5}}) = ^{g_{1}g_{2}g_{3}g_{4}}Z.  (8.10)

Now, G_{γ} ≤ G_{γ_{1}} ∩ G_{γ_{2}} ∩ G_{γ_{3}}. Using (8.9) and (8.10) we see that the latter group has a normal Sylow p-subgroup equal to

g_{1}g_{2}Z ∩^{g_{1}g_{2}g_{3}g_{4}}Z = ^{g_{1}g_{2}}(Z ∩^{g_{3}g_{4}}Z) = ^{g_{1}g_{2}g_{3}}(Z ∩^{g_{4}}Z).

Now (G5) implies that Z ≠ ^{g_{4}}Z because g_{4} ∉ L_{i_{2}}, hence their intersection must be trivial because |Z| = p. This shows that G_{γ} is a p′-group.

Proposition 8.11. Keeping the hypotheses and notation of 8.3 and 8.4, if T is a subgroup of S which is not conjugate in K to a subgroup of P_{i} for some i, then

N_{G}(T, S) = N_{K}(T, S).

In particular N_{G}(T) = N_{K}(T).

Proof. Choose g ∈ N_{G}(T, S). Clearly T ⊆ S ∩ S^{g} ≤ K ∩ K^{g}. Proposition 8.6(a) implies that ℓ(K, g^{−1}K) < 2 because T is not conjugate in K to any subgroup of P_{i}. Now, ℓ(K, gK) is an even integer, therefore ℓ(K, g^{−1}K) = 0, namely g ∈ K.

Proof (proof of Proposition 8.4). (a) Consider a finite p-subgroup P of G. Then by 8.1 P is conjugate to a subgroup of K or a subgroup of H_{i} for some i. They contain S and R_{i} ≤ S as Sylow p-subgroups. Therefore P is conjugate to a subgroup of S.

(b) Let X denote the tree associated to (G, T). If P is a non trivial subgroup of S then P ≤ K ∩ ^{g}K for every g ∈ N_{G}(P). Proposition 8.6 shows that ℓ(K, gK) < 6. The tree X is locally finite because the groups of G are finite. Therefore

{gK : g ∈ N_{G}(P)} = N_{G}(P)K/K

is a finite set because it is contained in a ball of radius 6 in X. It follows that N_{G}(P) is finite because K is finite.

(c)(i) Clearly S is not conjugate to a subgroup of P_{i} because |S| ≥ p^{3}. Proposition 8.11 implies that N_{G}(S) = N_{G}(S, S) = N_{K}(S).

To prove (ii) and (iii) note that N_{G}(P_{i}) and N_{G}(Z) are finite by (b). Furthermore H_{i} ≤ N_{G}(P_{i}) by (G4) and K ≤ N_{G}(Z) by (G1). Equality holds by Proposition 8.2.

(d) It suffices to show that U ∩ S^{g} ≠ 1 for finitely many g ∈ G. Let T_{1}, …, T_{m} denote the Sylow p-subgroups of U. Then T_{i} ≤ ^{g_{i}}S for some g_{i} ∈ G because S is a Sylow of G. In the tree X associated with (G, T) set

d = max{ℓ(K, g_{i}K)}_{i=1}^{m}.

Consider g ∈ G and assume that ℓ(K, gK) > d+6. By the triangle inequality ℓ(gK, g_{i}K) > 6, hence ℓ(K, g^{−1}g_{i}K) > 6. Proposition 8.6(b) shows that (^{g}K)∩(^{g_{i}}K) = ^{g}(K ∩^{g^{−1}g_{i}}K) is a finite group of order prime to p, therefore gS ∩^{g_{i}}S = 1. In particular ^{g}S ∩ T_{i} = 1 for all i, hence U ∩^{g}S = 1. We see that U ∩^{g}S = 1 for all g ∈ G such that ℓ(K, gK) > d + 6. The result follows because X is locally finite.

(e) Consider first the case when either (i) m > 0 or (ii) Q_{0} is not conjugate to a subgroup of P_{i} for some i. Without loss of generality, we may assume that Q_{m} ≤ S. Therefore U ≤ K because either (i) U ≤ N_{G}(Q_{1}) = N_{K}(Q_{1}) or (ii) U = N_{G}(Q_{0}) = N_{K}(Q_{0}) by Proposition 8.11. Clearly U is finite by (d) and we choose T ∈ Syl_{p}(U). Then T is not conjugate to a subgroup of P_{j} because either (i) N_{Q_{1}}(Q_{0}) ≤ U has order > p^{2} or (ii) Q_{0} ≤ T. We deduce from Proposition 8.11 that N_{G}(T, S) ⊆ K. Furthermore, we may assume that U is a subgroup of K because Since K is finite and S ∈ Syl_{p}(K), Lemma 5.8 implies that

|S \N_{G}(T, S)U/ U| = |S \N_{K}(T, S)U/ U| ≠ 0 mod p.

It follows from Lemma 5.7 that |F | ∈ ℤ_{(p)}^{×}.

It remains to consider m = 0 and Q_{0} = P_{i} for some i. In this case U = H_{i} by (c)(ii) and we may choose T = R_{i}. Consider x, y ∈ N_{G}(R_{i}, S) such that y = xu for some u ∈ H_{i}. Note that N_{G}(R_{i}, S) ≤ K by Proposition 8.11. Note that ^{xu}R_{i} = ^{y}R_{i} ≤ S so ^{u}R_{i} ≤ K. Furthermore R_{i} ≤ H_{i} so uR_{i} ≤ H_{i}. It follows that ^{u}R_{i} ≤ L_{i} and (G3) implies that ^{u}R_{i} = R_{i} namely u ∈ N_{H_{i}}(R_{i}) = L_{i}. We can now use 5.6 and 5.8 to see that

|S \N_{G}(R_{i}, S)H_{i}/ H_{i}| = |S \N_{K}(R_{i}, S)H_{i}/ H_{i}| = |S \N_{K}(R_{i}, S)/ L_{i}| ≠ 0 mod p.

Finally Lemma 5.7 shows that |F | ∈ ℤ_{(p)}^{×}.

9 The exotic examples of Ruiz-Viruel and Broto-Levi-Oliver

Let S denote the extraspecial group p_{+}^{1+2}, that is S is the group of upper triangular unipotent matrices in GL_{3}(p). Recall that every element of S has order p and that Z = Z(S) is a group of order p. Every subgroup of S of order p^{2} is elementary abelian and contains Z. Also there are exactly p + 1 such subgroups of S because S/Z is elementary abelian of rank 2.

Ruiz and Viruel classified in [25] all the p-local finite groups over S = p_{+}^{1+2}. They showed that there are only 3 exotic ones, denoted in this paper RV_{1}, RV_{2} and RV_{3}, all of which occur at the prime p = 7. In these exotic p-local finite groups, all the subgroups of S of order p^{2} are F-centric and F-radical. The table below summarizes the information about the automorphism groups of these subgroups in each of the 3 examples, cf. [25, Table 1.2]

Out_{F}(S) #{V ∈ F^{c} : |V | = p^{2}}  Aut_{F}(V)
RV 1  6^{2} : 2  6 + 2  SL_{2}(p) : 2, GL_{2}(p)
RV2 D_{16} × 3  4 + 4  SL_{2}(p) : 2, SL_{2}(p) : 2
RV3 SD_{32} × 3  8  SL_{2}(p) : 2

Table 9.1

The second column in the table gives the isomorphism type of Out_{F}(S) = Aut_{F}(S)/Inn(S). The third column gives the sizes of the conjugacy classes of subgroups V of S of order p^{2}.

Proof (of Theorem A(1)). Let (S, F, L) denote one of the p-local finite groups RV_{1}, RV_{2} or RV_{3}. Choose an elementary abelian group V_{i} of rank 2 from each F-conjugacy class (i = 1, …, n). Thus,n = 2 for RV_{1}, RV_{2} and n = 1 for RV_{3}. Define (7.7)

K = Aut_{L}(S),
H_{i} = Aut_{L}(V_{i})
L_{i} = Aut_{L}(V_{i} < S)  (i = 1, …, n).

Let G denote the amalgam of the tree of groups

H_{1} ^{L_{1}} K ^{L_{2}} H_{2}  for RV_{1}, RV_{2}  (9.2)
H_{1} ^{L_{1}} K  for RV_{3}  (9.3)

We will show below that (9.2) and (9.3) satisfy (G1)-(G6) of 8.3. Theorem 8.5(a) guarantees that F_{S}(G) is a saturated fusion system with an associated centric linking system L_{S}^{c}(G). Proposition 8.4(c)(i) shows that

Aut_{F_{S}(G)}(S) = K/Z(S) = Aut_{F}(S).

Proposition 8.4(c)(ii) shows that for every V_{i}

Aut_{F_{S}(G)}(V_{i}) = H_{i}/C_{H_{i}}(V_{i}) = H_{i}/V_{i} = Aut_{F}(V_{i}).

Now, F and F_{S}(G) are both saturated, they have the same centric objects with the same automorphism groups. Alperin’s fusion theorem implies that F = F_{S}(G). Moreover, since F admits a unique centric linking system [25, Theorem 1.1] we conclude that (S, F_{S}(G), L_{S}^{c}(G)) is the p-local finite group RV_{i}. It remains to prove that (G1)-(G6) hold. Put P_{i} = V_{i} and R_{i} = S as in the notation of 8.3. Also let Z = Z(S). Note that P_{i} ◁ S because they have index p.

(G1). Clearly S is a Sylow subgroup of K because K = Aut_{L}(S). In fact S ◁ K hence Z ◁ K. Also Z has order p.

(G2). Clearly R_{i} = S ◁ L_{i} because S ◁ K. Also |S| = p^{3} and it is not abelian.

(G3). R_{i} is a Sylow subgroup of H_{i} because P_{i} ◁ S. Also L_{i} = N_{H_{i}}(S) by proposition 7.9(1).

(G4). P_{i} are elementary abelian of rank 2 and contain Z. Also P_{i}◁H_{i} = Aut_{L}(P_{i}). Furthermore Aut_{F}(P_{i}) = H_{i}/P_{i} acts transitively on the lines of P_{i} because it contains SL_{2}(p) by Table 9.1. Also L_{i} = N_{K}(P_{i}) by proposition 7.9(2).

(G5). Clearly L_{i} = N_{H_{i}}(S) ≤ N_{H_{i}}(Z) because Z is characteristic in S. It suffices to show that [H_{i} : L_{i}] = [H_{i} : N_{H_{i}}(Z)]. Note that H_{i} acts transitively on the lines of P_{i} so

[H_{i} : N_{H_{i}}(Z)] = p + 1 = 8.

Also note that L_{i} = N_{K}(P_{i}) so

[K : L_{i}] = |{conjugacy class of P_{i} in K}|.

Remark 7.4 shows that Aut_{F}(S), and consequently K, act transitively on each F-conjugacy class of the V_{i}’s. This shows that

[K : L_{i}] = |{F − conjugacy class of P_{i}}|.

Thus, the proof that L_{i} = N_{H_{i}}(Z) will be complete if we show that

[H_{i} : L_{i}] = \frac{|H_{i}|}{|K|} · [K : L_{i}] = \frac{|Aut_{L}(V_{i})|}{|Aut_{L}(S)|} · |\{F − conjugacy class of P_{i}\}|

is equal to 8. This is verified from Table 9.1 using |Aut_{L}(S)| = 7^{3}·| Out_{F}(S)| and |Aut_{L}(V)| = 7^{2} · |Aut_{F}(V)|.

(G6). The choice of the P_{i}’s is such that they are not F-conjugate, hence not K-conjugate.

Note that P_{i} and S form a complete set of representatives for the conjugacy classes of the p-radical subgroups of G. Also note that P_{i} < S form a complete set of representatives for the G-conjugacy classes of the 1-simplices of sd R. Indeed, if V < S is a 1-simplex and V is conjugate to P_{i} in G then by remark 7.4 and 8.4(c)(i) V < S is conjugate to P_{i} < S. It follows that sd R/G has the form

[figure: \begin{matrix}sd^{1}R/G \\ sd^{0}R/G \end{matrix}; (P_{1} < S); ∂_{1}; ∂_{0}; (P_{2} < S); ∂_{1}; ∂_{0}; (P_{1}); (S); (P_{2}); (P_{1} < S); ∂_{0}; ∂_{1}; (P_{1}); (S)]

RV_{1} and RV_{2}  RV_{3}

Finally, we observe that no subgroup of order p of S is p-radical in G because (G4) implies that all these subgroups are conjugate to Z and S ◁ N_{G}(Z) = K by 8.4(c)(iii).

The split exact sequences of Sℤ_{(p)}-localized spectra which we have to obtain is a consequence of Theorem 8.5 and Proposition 7.13 upon observing that R coincides with the collection of p-centric subgroups of G.

Proof (proof of Theorem A(2).). We take a somewhat different approach to the construction of fusion systems underlying the exotic p-local finite groups in [10, Example 9.3] and to the proof of their saturation. We also obtain an explicit construction of their associated centric linking systems.

Fix a prime p ≥ 5 and consider Σ_{p} as the permutation group on the elements of ℤ/p = {0, 1, …, p − 1}. Clearly ℤ/p embeds in Σ_{p} using the regular presentation whose image is generated by the element τ = (0 · 1 · 2 · ⋯ p − 1).

Fix ℓ ≥ 1. Let μ be a generator of the cyclic subgroup C_{p−1} in the automorphism group of ℤ/p^{ℓ}. Clearly μ acts on ℤ/p ≤ ℤ/p^{ℓ} by means of multiplication by a generator of ℤ/p^{×} which by abuse of notation we also denote by μ.

Put M = (ℤ/p^{ℓ})^{p} and let A be the subgroup of the p-tuples whose sum of their entries vanish. Let Γ denote C_{p−1} × Σ_{p} and consider M and A as Γ -modules where C_{p−1} acts diagonally and Σ_{p} by permuting the entries of the p-tuples in M.

Define K = A ⋉ Γ. Let D be the diagonal subgroup of M and let Z = D∩A. Clearly Z ≅ ℤ/p and we fix z = (1, …, 1) a generator. Moreover Z ◁ K.

Put S = ⟨A, τ⟩. Clearly S ∈ Syl_{p}(K). Also observe that Z = Z(S). Consider the set X = S\A. It consists of elements of order p in K. We now claim that the elements of X belong to exactly two conjugacy classes in K.

To see this we first observe that N_{Σ_{p}}(⟨τ⟩) = ⟨τ⟩ ⋉ ℤ/p^{×}. We fix a generator σ ∈ ℤ/p^{×}. It has the form σ = (0 · k · 2k · · · · · · · (p − 1)k) for some generator k ∈ ℤ/p^{×}. The elements of the form a · τ for a ∈ A exhaust the conjugacy classes of the elements of X because σ acts on X. Define a group homomorphism φ : M → A by φ(x) = x − τ(x). It is not difficult to prove that there is a bijection

\{ \begin{matrix}Conjugacy classes of \\ the elements of X \end{matrix} \} ↔ \{ \begin{matrix}A/φ(A) modulu \\ the action of C_{p−1} \end{matrix} \}

Consider the set B = {(t, 0, …, 0) : t ∈ ℤ/p} in M. It is easy to check that φ(B) forms a complete set of representatives for the cosets of φ(A) in A. Also B has exactly two orbits under the action of C_{p−1} = ⟨μ⟩.

It follows that X consists of two K-conjugacy classes and we choose representatives x_{1} = τ and x_{2} = b_{1}τ. Put

P_{1} = ⟨z, τ⟩

Clearly P_{1} is elementary abelian of rank 2 and contains Z. By inspection P_{1} is self centralising in K. Also N_{K}(P_{1})/P_{1} must be contained in the group of upper triangular matrices in GL(P_{1}), which we denote by U, because Z ◁ K.

Let w denote the element (0, 1, 2, …, p − 1) in A ∩ (ℤ/p)^{p} (it should not be confused with τ). Put

R_{1} = ⟨P_{1}, w⟩.

By inspection [w, τ] ∈ Z and Z is central in R_{1}, therefore R_{1} ≅ p_{+}^{1+2}. Also note that [μ, σ] = 1 and that [μ, w], [σ, w] ∈ ⟨w⟩. It follows that ⟨w, σ, μ⟩ can be identified with U endowed with its canonical action on P_{1}. Define

L_{1} = ⟨P_{1}, w, σ, μ⟩ = P_{1} ⋉ U

and note that L_{1} = N_{K}(P_{1}) because N_{K}(P_{1})/P_{1} ≤ U = L_{1}/P_{1}. Define

H_{1} = P_{1} ⋉ GL(P_{1}).

Consider now the group K̃ = M ⋉ Γ. Note that all the elements of X are conjugate in K̃ because φ : M → A is surjective. One can find g ∈ K which fixes z and τ^{g} = b_{1}τ. Put

P_{2} = P_{1}^{g} = ⟨Z, b_{1}τ⟩,  R_{2} = R_{1}^{g},  L_{2} = L_{1}^{g},  H_{2} = P_{2} ⋉ GL(P_{2}).

Note that L_{2} ≤ K because K ◁ K̃.

Let (G, T) denote the tree of groups

H_{1} ^{L_{1}} K ^{L_{2}} H_{2}

and let G denote its amalgamation. The hypotheses (G1)-(G6) of 8.3 were mostly verified during the construction of the tree. One only notices that U is the normaliser of a Sylow p-subgroup of GL_{2}(p) therefore L_{i} = N_{H_{i}}(R_{i}) as required in (G3). Also U is the stabilizer of Z in the action of GL(P_{1}) hence (G5). Finally, by construction P_{1} and P_{2} are not conjugate in K hence (G6).

Theorem 8.5 implies that F_{S}(G) is saturated and has L_{S}^{c}(G) as an associated centric linking system. In fact F_{S}(G) = ⟨F_{S}(K); F_{P_{i}}(H_{i}), i = 1, 2⟩ so F_{S}(G) is the fusion system described in the second row of the table in Example 9.3 of [10] where it is also shown to be exotic. It is also shown that it admits a unique associated linking system. Hence our construction gives rise to the same family of p-local finite groups, but is more explicit.

Finally, note that A ◁ K. It follows using 5.4 and 8.11 that that the p-radical subgroups of G are the conjugates of S, A and P_{1}, P_{2} because only A and S are p-radical in K. Remark 7.4 now implies that sd R/G has the form

sd_{0} R/G = {(S), (A), (P_{1}), (P_{2})},
sd_{1} R/G = {(A < S), (P_{1} < S), (P_{2} < S)}.

Also note that N_{G}(S) = N_{K}(S) = N_{G}(A < S) by 8.5(c)(i) and 8.11. Also N_{G}(A) = K by 8.11 and N_{G}(P_{i}) = H_{i} by 8.5(c)(ii). Theorem 8.5(c) now shows that after localization at Sℤ_{(p)}, we obtain the following equality in the homotopy category of spectra

|L| = BK + BH_{1} + BH_{2} − BL_{1} − BK_{2}.

A Appendix - Homology localization of spectra

We let Ho^{s} denote a stable category of spectra, e.g. Boardman’s category. Recall that Ho^{s} is equipped with smash products A∧B and internal function spectra [A, B] with the usual adjunction properties. The category Ho^{s} is triangulated. A spectrum E gives rise to a homology theory E_{∗}X = [S^{0}, E ∧ X]_{∗} and a cohomology theory E^{∗}(X) = [X, E]_{−∗}. In fact, every generalized cohomology theory is given as a representable functor in this way.

A space X is associated with a spectrum Σ_{+}^{∞}X which is the suspension spectrum of the spaces X_{+}, that is X with a disjoint base-point.

Fix a spectrum E. A spectrum X is called E-acyclic if E_{∗}X = 0. A map f : A → B of spectra is called an E-equivalence if E_{∗}(f) is an isomorphism. A spectrum T is called E-local if [f, T]_{∗} is an isomorphism for every Eequivalence f : A → B. Equivalently T is E-local if [A, T]_{∗} = 0 for every E-acyclic spectrum A.

Bousfield proves in [6] that every spectrum X is associated with a canonical triangle

EX → X ⟶^{η} L_{E}(X) ⟶^{Σ} _{E}X

where _{E}X is E-acyclic and L_{E}X is E-local. Bousfield uses the notation X_{E} for L_{E}X. It is clear that η is an E-equivalence and L_{E}X is called the E-localization of X. It is shown that

(i) There is a natural equivalence L_{E}(X ∨ Y) ≃ L_{E}X ∨ L_{E}Y (ii) There is a natural equivalence L_{E}ΣX ≃ ΣL_{E}X (iii) L_{E} preserves triangles.

Let G be an abelian group. Let SG denote the Moore spectrum of type (G, 0), that is

(i) π_{i}SG = 0 if i < 0 (ii) H_{0}SG ≈ G (iii) H_{i}SG = 0 if i > 0.

Let J be a set of primes and ℤ_{(J)} be the integers localized at J.

Proposition A.1. (Bousfield [6, Proposition 2.4]) There is a natural equivalence L_{Sℤ_{(J)}}X ≃ Sℤ_{(J)} ∧X for all X ∈ Ho^{s} and π_{∗}L_{Sℤ_{(J)}}X = ℤ_{(J)} ⊗π_{∗}X. Moreover X is Sℤ_{(J)}-local if and only if the groups π_{∗}X are J-local, namely uniquely p ∉ J divisible.

Recall that if h is a (co)homology theory represented by E, then h(−; Λ), where Λ is any abelian group, is represented by the spectrum E ∧ SΛ. In particular H^{∗}(−; ℤ_{(p)}) is represented by H ∧Sℤ_{(p)} where H is the Eilenberg-Mac Lane spectrum representing ordinary homology theory.

Theorem A.2. ([6, Thoerem 3.1]). If X is a connective spectrum then

L_{Hℤ_{(p)}}X ≃ L_{Sℤ_{(p)}}X ≃ X ∧ Sℤ_{(p)}.

Proposition A.3. For every Sℤ_{(p)}-local spectrum T and every spectrum X, T^{∗}(X_{Sℤ_{(p)}}) ≅ T^{∗}(X).

Proof. We use Proposition A.1

T^{∗}(X_{Sℤ_{(p)}}) = [X ∧ Sℤ_{(p)}, T]_{−∗} = [X, F (Sℤ_{(p)}, T)]_{−∗} =

= [X, F (S^{0} ∧ Sℤ_{(p)}, T)]_{−∗} ≈ [X, F (S^{0}, T)]_{−∗} = T^{∗}(X)

Proposition A.4. Every H^{∗}(−; ℤ_{(p)})-isomorphism of Sℤ_{(p)}-local spectra is an equivalence.

Proof. Let Z be the cofibre of f. Then H^{∗}(Z; ℤ_{(p)}) = 0 and we need to prove that Z = 0. Let h denote the cohomology theory represented by Z. The Atiyah-Hirzebruch spectral sequence H^{i}(Z; π_{j}(Z)) ⇒ h^{i+j}(Z) = π_{i+j}F (Z, Z) shows that F (Z, Z) = 0.

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