ON STABLE EQUIVALENCES OF BG_{p}^{∧} FOR COMPACT LIE GROUPS, REVISITED

JOHN MARTINO AND STEWART PRIDDY

1. Introduction

Let G be a compact Lie group and p a prime. We continue our study of the relation between the p-local subgroup structure of G and the stable homotopy type of its classifying space BG_{p}^{∧} completed at p. A p-toral group Q is an extension of a torus T (possibly trivial) by a finite p-group π, i.e.,

1 → T → Q → π → 1.

For a p-toral group Q, let Inj_{fW} (Q, G) be the set of conjugacy classes of continuous monomorphisms α : Q → G with finite Weyl group N_{G}(Im(α))/Im(α).

Theorem 1. Let G and G′ be compact Lie groups such that for every p-toral group Q there is an isomorphism

𝔽_{p}Inj_{fW} (Q, G) ≈ 𝔽_{p}Inj_{fW} (Q, G′)

of 𝔽_{p}Out(Q) modules. Then BG_{p}^{∧} is stably homotopy equivalent to BG_{p}^{′∧}.

A key difference between this and the finite group case [MP1],[MP3] is the failure of the completed double Burnside ring to compute the stable homotopy classes of self-maps {BG_{p}^{∧}, BG_{p}^{∧}} for compact Lie groups. Only a weak version of the Segal Conjecture holds [Mi], [F]. Another crucial difference is the existence of infinitely many G-conjugacy classes of isomorphic p-toral subgroups with finite Weyl groups (see Example 8.2 of the Appendix). This led to the incorrect proof of this result given as Theorem 1.1 of [MP2]. It further invalidated the proofs of Theorems 1.2, 1.3. We plan to return to these questions at a later date. In the present paper we circumvent this difficulty by passing to homology which reduces the problem to finitely generated modules.

The proof follows by combining the main Propositions 3.5 and 3.6 below. The plan of the proof is to show the hypothesis of Theorem 1 implies that BG_{p}^{∧} and BG_{p}^{′∧} have the same indecomposable summands with the same multiplicities. The paper is organized as follows: In section 2 we show the modules associated with indecomposable summands are both finite and simple in homology. In section 3 we recall the functor P Hom and state Propositions 3.5 and 3.6 relating certain modules derived from 𝔽_{p}Inj_{fW} (Q, G) to the summands of BG_{p}^{∧}. Section 4 is devoted to the necessary Lie group theory. Various finiteness results necessary for our proofs are collected in Section 5. We use finiteness for an induction step key to the proof of the main propositions in Sections 6, 7.

Date: October 16, 2012.

To ease the notation all spaces and spectra are assumed to be completed at p. We reserve the letter Q and its variants Q′, Q_{i}, etc. for p-toral groups; G, G′ will always denote compact Lie groups.

2. Finiteness and simplicity in homology

Let {BQ, BG} be the set of stable homotopy classes of maps. If [f] ∈ {BQ, BG} then passing to homology f_{∗} : H_{∗}BQ → H_{∗}BG with simple 𝔽_{p} coefficients defines a homomorphism

ϕ : {BQ, BG} → Hom(H_{∗}BQ, H_{∗}BG)

ϕ([f]) = f_{∗}. Let 𝔼(Q, G) = Im(ϕ) which is finite by [H]. Define the 𝔽_{p} module J_{τ}(Q) ⊂ {BQ, BQ} to be generated by maps

BQ ⟶^{tr} BQ′ ⟶^{Bg} BQ

where Q′ ≨ Q, tr is the transfer, and g : Q′ → Q is a homomorphism. We denote by \overline{J}_{τ}(Q) the ideal of \{BQ, BQ\} generated by J_{τ}(Q). Set \overline{𝕁}_{τ}(Q) = ϕ(\overline{J}_{τ}(Q)). Then ϕ induces a surjective homomorphism of rings

ϕ : S(Q) → 𝕊(Q)

where S(Q) = 𝔽_{p}⊗\{BQ, BQ\}/\overline{J}_{τ}(Q), 𝕊(Q) = 𝔼(Q, Q)/\overline{𝕁}_{τ}(Q). Letting O(Q) = 𝔽_{p}Out(Q) and 𝕆(Q) = ϕ(O(Q)) ⊆ 𝔼(Q, Q), we note that O(Q) acts on 𝔼(Q, G) via the finite ring 𝕆(Q). This action will be used to study {BQ, BG}.

Suppose e ∈ S(Q) is a primitive idempotent such that e ∉ \overline{J}_{τ}(Q). Then X = eBQ is an indecomposable spectrum linked to a summand of BQ, equivalently any stable equivalence X → BQ ⟶^{tr} BQ′ → X has Q′ = Q in which case the transfer is the identity. For further discussion of linked summands see [MP2], 3.7.

ON STABLE EQUIVALENCES OF BG_{p}^{∧} 3

Lemma 2.1. For e ∈ S(Q), e_{∗} : H_{∗}BQ → H_{∗}BQ does not belong to \overline{𝕁}_{τ}(Q).

Proof Suppose e_{∗} ∈ \overline{𝕁}_{τ}(Q) then e_{∗} = f_{∗} for some f ∈ \overline{J}_{τ}(Q). Thus X → BQ ⟶^{f} BQ → X is an equivalence. By construction f_{∗} is a linear combination of maps of the form l_{∗}tr_{∗}k_{∗} where k : BQ → BQ, tr : BQ → BQ′, Q′ ≨ Q, and l : BQ′ → BQ. Since X is indecomposable {X, X} is a local ring. Hence some composite ltrk also yields an equivalence

X → BQ ⟶^{ltrk} BQ′ → X

However this implies X is not linked to a summand of BQ, a contradiction. □

Consider the simple S(Q) module M = S(Q)e/Rad(S(Q))e corresponding to X where Rad(·) is the Jacobson radical. Its corresponding 𝕊(Q) module in homology is 𝕄 = 𝕊(Q)e_{∗}/Rad(𝕊(Q))e_{∗} which, restricting via the ring homomorphism ϕ : S(Q) → 𝕊(Q), is also a S(Q) module. Since ϕ is surjective ϕ(Rad(S(Q)) ⊂ Rad(𝕊(Q)) and so ϕ induces a surjection of S(Q) modules

Φ : M → 𝕄.

Lemma 2.2. Φ : M → 𝕄 is an isomorphism of finite simple S(Q) modules.

Proof: By Lemma 2.1, e_{∗} ≠ 0 in 𝕊(Q). Since 𝕊(Q) is finite, Rad(𝕊(Q)) is nilpotent hence e_{∗} ∉ Rad(𝕊(Q)) and 𝕄 ≠ 0. Finally M is simple hence Φ is an isomorphism and since 𝕄 is finite so is M. □

3. The functor PHom and the main propositions

We begin by recalling the definition and properties of the functor P Hom.

Definition 3.1. Let R be an algebra over 𝔽_{p} and suppose N is an Rmodule with a projective cover P_{R}(N) ⟶^{π} N. Then P Hom_{R}(M, N) ⊂ Hom_{R}(M, N) is the set of R-module homomorphisms ρ : M → N which factor through π : P_{R}(N) → N

P_{R}(N)
↗  ↘ ^{π}
M ⟶_{ρ}  N

Since P_{R}(N) is unique up to isomorphism, P Hom_{R}(−, N) is a functor from R-modules to 𝔽_{p}-vector spaces independent of the choice of P_{R}(N).

Lemma 3.2. Let ρ : R → S be a surjection of algebras over 𝔽_{p} satisfying the idempotent lifting property with S artinian. If N is a finitely generated S-module then viewed as an R-module via ρ, N has a projective cover P_{R}(N).

Proof This is Lemma 4.1 of [MP2].

Proposition 3.3. Let R → S be a surjection of 𝔽_{p}-algebras satisfying the idempotent lifting property with S artinian and quasi-frobenius. Let N be a finitely generated S-module. If x ∈ R there is an isomorphism of 𝔽_{p}-modules

F : P Hom_{R}(Rx, N) ⟶^{≈} xN = ρ(x)N

given by F (ϕ) = ϕ(x) for ϕ ∈ P Hom_{R}(Rx, N).

Proof [MP2] Prop. 4.3.

Definition 3.4. Let Cen(Q, G) ⊆ Inj_{fW} (Q, G) be the set of G-conjugacy classes of monomorphisms α : Q → G such that C_{G}(Imα)/Z(Imα) is a p′-group which is necessarily finite since N_{G}(Imα)/Imα is finite. Let nCen(Q, G) be the complement of Cen(Q, G) in Inj_{fW} (Q, G).

We can now state the main Propositions 3.5, 3.6 the proofs of which are given in Section 7.

Proposition 3.5. Under the hypotheses of Theorem 1, if 𝕄 is a finite 𝕊(Q) simple module considered as an O(Q) module then there are isomorphisms

P Hom_{O(Q)}(𝔽_{p}nCen(Q, G), 𝕄) ≈ P Hom_{O(Q)}(𝔽_{p}nCen(Q, G′), 𝕄)

P Hom_{O(Q)}(𝔽_{p}Cen(Q, G), 𝕄) ≈ P Hom_{O(Q)}(𝔽_{p}Cen(Q, G′), 𝕄).

Proposition 3.6. Suppose, under the hypothesis of Theorem 1, for every p-toral group Q and every finite simple 𝕊(Q) module 𝕄 associated to a stable summand linked in BQ_{p}^{∧} there is an isomorphism

P Hom_{O(Q)}(𝔽_{p}Cen(Q, G), 𝕄) ≈ P Hom_{O(Q)}(𝔽_{p}Cen(Q, G′), 𝕄)

as 𝔽_{p} modules. Then BG_{p}^{∧} is stably homotopy equivalent to BG_{p}^{′∧}.

4. Lie group-theoretic preliminaries

We establish some Lie group-theoretic properties needed in our proofs. Many of these results are very similar to those for finite groups but we include details for the convenience of the reader.

The groups Q̃ and their properties.

To study the set nCen(Q, G) we consider subgroups Q′ of G such that Q′ ≈ Q, C_{G}(Q′)/Z(Q′) is finite, and p divides |C_{G}(Q′)/Z(Q′)|. We let Q̃′ be a Sylow p-subgroup of subgroup of C_{G}(Q′) · Q′. That is, Q̃′ is a maximal p-toral subgroup of C_{G}(Q′)·Q′. Any two such are conjugate in C_{G}(Q′) · Q′ [JMO]. Thus Q ≤ Q̃ since Q is normal in C_{G}(Q′) · Q′. Under the hypothesis of Theorem 1, the maximal p-toral subgroups of G, G′ are isomorphic and so Q̃′ does not depend on G.

Definition 4.1. (i) Two such groups are said to be equivalent Q̃ ∼ Q̃′ if there is an isomorphism s : Q̃ ⟶^{≈} Q̃′ satisfying s(Q) = Q′.

(ii) Given a p-toral Q, let {Q̃_{j}}_{j∈J} be a set of equivalence classes of subgroups of G with Q ≈ Q_{j} and N(Q_{j})/Q_{j} finite.

If Q′, Q′′ are conjugate then Q̃′ ∼ Q̃′′, however the converse is false.

Definition 4.2.

Cen_{Q_{j}}(Q̃_{j}, Q̃_{k}, G) = \{β : Q̃_{j} → G | [β] ∈ Cen(Q̃_{j}, G), { \atop β(Q_{j})} ∼ Q̃_{k}\}/G.

In this definition β is required to be a monomorphism and it is important to note that Q_{j} ≈ Q_{k} but Q̃_{j} is isomorphic to a possibly proper subgroup of Q̃_{k}.

Define a set of conjugacy classes of monomorphisms

R(Q′, Q̃) = {α : Q′ → Q̃ | α(Q′) = Q}/Inn(Q̃).

and a group of outer automorphisms

Out(Q̃|Q) = {α : Q̃ → Q̃ | α(Q) = Q}/Inn(Q̃).

Observe that R(Q′, Q̃) is a left Out(Q′) set and a right Out(Q̃|Q) set since the image of Q is invariant. Similarly Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G) is a left Out(Q̃_{k}|Q_{k})-set. The following Lemma is a corrected Lie group version of [MP1] Lemma 4.6, see [MP3] Lemma 4.

Proposition 4.3. There is an isomorphism of Out(Q)-sets

∐_{k∈J} ψ : R(Q, Q̃_{k})×_{Out(Q̃_{k}|Q_{k})}Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G) → nCen(Q, G)

natural in G given by composition, i.e. ψ(γ × [δ]) = [δγ].

Proof By definition, ϕ(γ_{k} × [δ]) = [δγ_{k}]. Since δγ_{k}(Q) = Q_{k},

{ \atop δγ_{k}(Q)}/δγ_{k}(Q) = Q̃_{k}/Q_{k} ≤ C_{k}(Q_{k})/Z(Q_{k})

which contains an element of order p. Thus { \atop β(Q_{j})}/β(Q_{j}) ∈ nCen(Q, G). It is easily checked that ψ is well-defined. Injectivity:

Suppose ψ(γ_{k} × [δ_{k}]) = ψ(γ_{l} × [δ_{l}]). Then for some g ∈ G

δ_{l}(Q_{l}) = im(δ_{l}γ_{l}) = c_{g}im(δ_{k}γ_{k}) = c_{g}δ_{k}(Q_{k})

Hence c_{g}[C_{G}(δ_{k}(Q_{l})) · δ_{k}(Q_{k})] = C_{G}(δ_{l}(Q_{l})) · δ_{l}(Q_{l}) and similarly for their Sylow p-subgroups. Hence { \atop δ_{k}(Q_{k})} = { \atop δ_{l}(Q_{l})} up to conjugation sending δ_{k}(Q_{k}) to δ_{l}(Q_{l}). Thus Q̃_{k} ∼ Q̃_{l} which implies Q̃_{k} = Q̃_{l} and Q_{k} = Q_{l}. Therefore we have a commutative diagram

Q ⟶^{γ_{k}} Q̃_{k} ⟶^{δ_{k}} G
id↓  ^{c_{g}}↓
Q ⟶^{γ_{l}} Q̃_{k} ⟶^{δ_{l}} G

i.e., γ_{l} = δ_{l}^{−1}c_{g}δ_{k}γ_{k}. Thus

γ_{l} × [δ_{l}] = δ_{l}^{−1}c_{g}δ_{k}γ_{k} × [δ_{l}] = γ_{k} × [δ_{l}(δ_{l}^{−1}c_{g}δ_{k})] = γ_{k} × [δ_{k}]

since δ_{l}^{−1}c_{g}δ_{k} ∈ Aut(Q̃_{k}|Q_{k}). Surjectivity: There is the unique factorization

f : Q ⟶^{π} { \atop f(Q)} ↪^{i} G

where π is a monomorphism onto f(Q) and i is inclusion. By definition if [f] ∈ nCen(Q, G) then { \atop f(Q)} is equivalent to Q̃_{j} for some j. Thus we may choose an equivalence u : { \atop f(Q)} → Q̃_{j} such that u(f(Q)) = Q_{j}. Set γ = uπ, δ = iu^{−1}. By Lemma 4.6 (1), [δ] ∈ Cen_{Q_{j}}(Q̃_{j}, Q̃_{j}, G). Since γ ∈ R(Q, Q̃_{j}) we have ψ(γ, [δ]) = [δγ] = [f] as required. □

Definition 4.4. Define a set of conjugacy classes of monomorphisms

R_{Q_{i}}(Q̃_{i}, Q̃_{k}) = {α : Q̃_{i} → Q̃_{k} | α(Q_{i}) = Q_{k}}/Inn(Q̃_{k}).

Proposition 4.5. There is an isomorphism of Out(Q̃_{i}|Q_{i})-sets

ψ : R_{Q_{i}}(Q̃_{i}, Q̃_{k}) ×_{Out(Q̃_{k}|Q_{k})} Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G) → Cen_{Q_{i}}(Q̃_{i}, Q̃_{k}, G).

natural in G where ψ(γ, [δ]) = [δγ].

Proof Fix Q = Q_{i}. First we note that { \atop δγ(Q)} = { \atop δ(Q_{k})} ∼ Q̃_{k} Therefore [δγ] ∈ Cen_{Q}(Q̃, Q̃_{k}, G). It is now easy to check that ψ is well-defined. Injectivity: Fix Q = Q_{i} and suppose ψ(γ_{1} × [δ_{1}]) = ψ(γ_{2} × [δ_{2}]). Then

δ_{1}γ_{1} = c_{g}δ_{2}γ_{2}

for some g ∈ G. Hence γ_{2} = δ_{2}^{−1}c_{g}^{−1}δ_{1}γ_{1}. Thus γ_{2} × [δ_{2}] = δ_{2}^{−1}c_{g}^{−1}δ_{1}γ_{1} × [δ_{2}] = γ_{1} × [δ_{2}(δ_{2}^{−1}c_{g}^{−1}δ_{1})] = γ_{1} × [δ_{1}].

Surjectivity: Given [f] ∈ Cen_{Q}(Q̃, Q̃_{k}, G), there is a factorization

f : Q̃ ⟶^{π} f(Q̃) ↪^{i} G

and an equivalence u : { \atop f(Q)} → Q̃_{k} where π is a isomorphism onto f(Q̃), i is inclusion, and u(f(Q)) = Q_{k}. Since f(Q̃) ≤ C_{G}(f(Q)) · f(Q) we have c_{g}(f(Q̃)) ≤ { \atop f(Q)} for some g ∈ C_{G}(f(Q)) · f(Q). Let γ = uc_{g}π, δ = ic_{g}^{−1}u^{−1}. Then γ ∈ R_{Q}(Q̃, Q̃_{k}), [δ] ∈ Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G) and ψ(γ, [δ]) = [f]. □

The following lemma describes properties of Cen_{Q_{j}}(Q̃_{j}, Q̃_{k}, G) which allow us to more closely analyze Cen and nCen. For this purpose we define the height of Q̃_{j} by ht(Q̃_{j}) = |Q̃_{j}/Q_{j}|. Let rk(Q_{j}) = rank(Q_{j}) as a Lie group. We note ht(Q_{i}) is finite since rk(Q_{i}) = rk(Q̃_{i}) by the finiteness of the Weyl group N_{G}(Q_{i})/Q_{i}. By the definition of Q̃_{j}, ht(Q̃_{j}) > 1.

Lemma 4.6.

(1) \{β : Q̃_{j} → G | { \atop β(Q_{j})} ∼ Q̃_{j}\}/G ⊆ Cen_{Q_{j}}(Q̃_{j}, Q̃_{j}, G).

(2) If ht(Q̃_{m}) is maximal among {ht(Q̃_{j})}_{j∈J} then Cen(Q̃_{m}, G) = Inj_{fW} (Q̃_{m}, G) and Cen_{Q_{m}}(Q̃_{m}, Q̃_{m}, G) = Cen(Q̃_{m}, G).

(3) If Cen_{Q_{j}}(Q̃_{j}, Q̃_{k}, G) ≠ ϕ then ht(Q̃_{j}) ≤ ht(Q̃_{k}).

(4) As Out(Q̃_{j}|Q_{j}) sets

Cen(Q̃_{j}, G) = ∐_{k}Cen_{Q_{j}}(Q̃_{j}, Q̃_{k}, G)

taken over all Q̃_{k} such that ht(Q̃_{j}) ≤ ht(Q̃_{k}).

Proof (1) Let β ∈ \{β : Q̃_{j} → G | { \atop β(Q_{j})} ∼ Q̃_{j}\}. By definition we must show C_{G}(β(Q̃_{j}))/Z(β(Q̃_{j})) is a p′-group. As a p-toral subgroup of C_{G}(β(Q_{j})) · β(Q_{j}), β(Q̃_{j}) is conjugate to a subgroup of { \atop β(Q_{j})} by some element g ∈ C_{G}(β(Q_{j})) · β(Q_{j}). Consider the maps

Q̃_{j} ⟶^{β ≈} β(Q̃_{j}) ⟶^{c_{g}} { \atop β(Q_{j})} ⟶^{s ≈} Q̃_{j}

If c_{g} is not surjective then Q̃_{j} is isomorphic to a proper subgroup of itself which impossible since it is p-toral. Thus c_{g} : β(Q̃_{j}) ≈ { \atop β(Q_{j})} and the result follows from Lemma 5.3.

(2) Since Cen(Q̃_{m}, G) ⊆ Inj_{fW} (Q̃_{m}, G) we need only show the other inclusion. For [β] ∈ Inj_{fW} (Q̃_{m}, G) we must show C_{G}(β(Q̃_{m}))/Z(β(Q̃_{m})) is a finite p′-group. Using maximality this argument is similar to that of (1).

For the second part by definition Cen_{Q_{m}}(Q̃_{m}, Q̃_{m}, G) ⊆ Cen(Q̃_{m}, G). Let [β] ∈ Cen(Q̃_{m}, G) then [β] ∈ C_{Q_{m}}(Q̃_{m}, Q̃_{k}, G) for some k. Now consider the maps

Q̃_{m} ⟶^{β ≈} β(Q̃_{m}) ⟶^{c_{g}} { \atop β(Q_{m})} ⟶^{s ≈} Q̃_{k}

where g ∈ C_{G}(β(Q_{m})) · β(Q_{m}). Hence Q_{m} ≈ Q_{k}. If c_{g} is not surjective then Q̃_{m}/Q_{m} is isomorphic to a proper subgroup of Q̃_{k}/Q_{k} contradicting the maximality of ht(Q̃_{m}). Thus Q̃_{m} ∼ Q̃_{k}, i.e. Q̃_{m} = Q̃_{k}.

(3) If [β] ∈ Cen_{Q_{j}}(Q̃_{j}, Q̃_{k}, G) then as in (2) there is an monomorphism of Lie groups

Q̃_{j} ⟶^{β ≈} β(Q̃_{j}) ⟶^{c_{g}} { \atop β(Q_{j})} ⟶^{s ≈} Q̃_{k}

where g ∈ C_{G}(β(Q_{j})) · β(Q_{j}). Thus Q̃_{j}/Q_{j} injects into Q̃_{k}/Q_{k} by the weak four lemma.

(4) Let [β] ∈ Cen(Q̃_{j}, G). Then β(Q̃_{j})/β(Q_{j}) ≈ Q̃_{j}/Q_{j} contains an element of order p. Since β(Q̃_{j})/β(Q_{j}) is subconjugate to { \atop β(Q_{j})}/β(Q_{j}) it follows that { \atop β(Q_{j})}/β(Q_{j}) also contains an element of order p. Thus { \atop β(Q_{j})} ∼ Q̃_{k} for some unique k. Hence [β] ∈ Cen_{Q_{j}}(Q̃_{j}, Q̃_{k}, G) and ht(Q̃_{j}) ≤ ht(Q̃_{k}) as observed in (3). If ht(Q̃_{j}) = ht(Q̃_{k}) then it easily follows that Q̃_{j} = Q̃_{k}. □

An adjoint isomorphism for P Hom.

If B, C are left S modules we have defined

P Hom_{S}(B, C) ⊂ Hom_{S}(B, C)

to be the set of homomorphisms which factor through the projective cover π : P_{C} → C.

Given two rings R, S and modules _{R}A, _{S}B_{R}, _{S}C then there is a natural adjoint isomorphism [M] Ex. V.3.4

η : Hom_{S}(B ⊗_{R} A, C) ⟶^{≈} Hom_{R}(A, Hom_{S}(B, C))

given by η(f)(a)(b) = f(b ⊗ a).

Lemma 4.7. Suppose R → S is a homomorphism of rings and _{R}A, SB_{R} = S, _{S}C are modules . Then there is a natural isomorphism of abelian groups

η : P Hom_{S}(S ⊗_{R} A, C) → P Hom_{R}(A, C).

Proof If f ∈ P Hom(B ⊗_{R} A, C) then we have a factorization f :

B ⊗_{R} A ⟶^{f̂} P_{C} ⟶^{π} C for some f̂. From the adjoint formula

η(f)(a)(b) = f(b ⊗ a) = πf̂(b ⊗ a) = π_{∗}(η(f̂)(a))(b)

therefore η(f) factors

η(f) : A ⟶^{η(f̂)} Hom_{S}(B, P_{C}) ⟶^{π_{∗}} Hom_{S}(B, C).

= A ⟶^{η(f̂)} P_{C} ⟶^{π_{∗}} C.

Thus η(f) ∈ P Hom_{R}(A, C)).

□

This gives an alternate form for the domain of ψ in Proposition 4.3 after applying P Hom. Let R_{k} = 𝔽_{p}Out(Q̃_{k}|Q_{k}) → O(Q_{k}) = 𝔽_{p}Out(Q_{k}) be the obvious ring homomorphism.

Corollary 4.8. Suppose 𝕄 is an 𝕊(Q_{k}) module considered as an O(Q_{k}) module via O(Q_{k}) → 𝕊(Q_{k}) then there is a natural isomorphism

η : P Hom_{O(Q_{k})}(𝔽_{p}R(Q, Q_{k}) ⊗_{R_{k}} 𝔽_{p}Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G), 𝕄)

⟶^{≈} P Hom_{R_{k}}(𝔽_{p}Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G), 𝕄).

Remark: In Prop. 4.3, Q ≈ Q_{k}, hence O(Q_{k}) ≈ 𝔽_{p}R(Q, Q̃_{k}) as a left O(Q_{k}) and a right R_{k} module.

5. Some finiteness results.

In this section we assume Q is a fixed p-toral subgroup of G.

Lemma 5.1. {Q̃_{j}}_{j∈J} is a finite set.

Proof By Prop. IV 6.3 [TD] there are finitely many conjugacy classes of subgroups which are centralizers of closed subgroups of G. By abuse of notation we list a complete set of representatives of the non-conjugate C_{G}(Q_{i}) as

N = {C_{G}(Q_{i}) : i = 1, 2, ..., N}.

Fix C_{G}(Q_{i}) ∈ N and suppose C_{G}(Q_{j}) ∉ N is conjugate to C_{G}(Q_{i}) via c_{g} : C_{G}(Q_{i}) → C_{G}(Q_{j}) for some g ∈ G. Thus their maximal tori T_{i}, T_{j} are also conjugate. By the finite Weyl group hypothesis C_{G}(Q_{i})/ZQ_{i}, C_{G}(Q_{j})/ZQ_{j} are finite so C_{G}(Q_{i})_{0} = T_{i}, C_{G}(Q_{j})_{0} = T_{j}. Hence C_{G}(Q_{j})/T_{j} is also finite, we have ZQ_{j} = A_{j} × T_{j} where A_{j} ≤ C_{G}(Q_{j})/T_{j} ≈ C_{G}(Q_{i})/T_{i} is a finite abelian group. Further, C_{G}(Q_{j})/Z(Q_{j}) is a quotient of the finite group C_{G}(Q_{j})/T_{j}. Thus as C_{G}(Q_{j}) varies over the conjugacy class of C_{G}(Q_{i}) there are, up to isomorphism, finitely many possibilities for C_{G}(Q_{j})/Z(Q_{j}).

Now C_{G}(Q_{j}) · Q_{j} is an extension of the form

Q → E → C_{G}(Q_{j})/ZQ_{j}

with Q ≈ Q_{j} and trivial characteristic function χ_{E} : C_{G}(Q_{j})/ZQ_{j} → Out(Q). Such extensions E are classified up to congruence by

H^{2}(C_{G}(Q_{j})/ZQ_{j}; ZQ) = H^{2}(C_{G}(Q_{j})/ZQ_{j}; A_{j} × T_{j}).

This group is finite by the finiteness of the group cohomology of a finite group with finitely generated coefficients. Two such congruent extensions

Q ⟶^{≈} Q_{j} ⟶ C_{G}(Q_{j}) · Q_{j} ⟶ C_{G}(Q_{j})/Z(Q_{j}) ≈ C_{G}(Q_{k})/Z(Q_{k})
id↓  ↓  ^{id}↓
Q ⟶^{≈} Q_{k} ⟶ C_{G}(Q_{k}) · Q_{k} ⟶  C_{G}(Q_{k})/Z(Q_{k})

give rise to an equivalence Q̃_{j} ∼ Q̃_{k} in our sense (Def. 4.1). Therefore as C_{G}(Q_{j}) varies over the conjugacy class of C_{G}(Q_{i}) there are, up to equivalence, finitely many possibilities for C_{G}(Q_{j}) · Q_{j} and therefore for Q̃_{j} as well. □

We recall N̄_{G}(Q)_{∗} = ∑_{x} c_{x}_{∗} : H_{∗}(BQ) → H_{∗}(BQ), x ∈ N_{G}(Q)/Q.

Lemma 5.2. In 𝕊(Q_{i}) = 𝔼(Q_{i}, Q_{i})/\overline{𝕁}_{τ}(Q_{i}), N̄_{G}(Q_{i})_{∗} = 0 except for a finite number of i, where Q_{i} ranges over a set of conjugacy class representatives of subgroups of G isomorphic to Q.

Proof Suppose N̄_{G}(Q_{j})_{∗} ≠ 0 modulo \overline{𝕁}_{τ}(Q_{j})_{∗} for j = 1, 2, .... Each element of Inj(Q, G) is represented, up to conjugation, by an injection

Q → G which factors as Q ⟶^{α_{i}} Q_{i} ⟶^{inc_{i}} G where α_{i} : Q ⟶^{≈} Q_{i}

is an isomorphism and inc_{i} is inclusion. Since H^{∗}(BG) is a finitely generated algebra there are only finitely many homomorphisms f_{∗} : H_{∗}(BQ) → H_{∗}(BG) induced by (unstable) topological maps f : BQ → BG. Thus the collection of distinct images {Im(inc_{j}_{∗}α_{j}_{∗})} is finite so Im(inc_{i}_{∗}α_{i}_{∗}) = Im(inc_{j}_{∗}α_{j}_{∗}) for some i ≠ j.

We have induced maps in homology

H_{∗}(BQ) ⟶^{α_{i}_{∗}≈} H_{∗}(BQ_{i}) ⟶^{inc_{i}_{∗}} H_{∗}(BG) ⟶^{tr_{j}_{∗}} H_{∗}(BQ_{j})

H_{∗}(BQ) ⟶^{α_{j}_{∗}≈} H_{∗}(Q_{j}) ⟶^{inc_{j}_{∗}} H_{∗}(BG) ⟶^{tr_{j}_{∗}} H_{∗}(BQ_{j}).

By the double coset formula [F] the first composite is zero mod \overline{𝕁}_{τ}(Q_{i}, Q_{j})_{∗}α_{i}_{∗} since Q_{i} and Q_{j} are not conjugate. Hence

Im(inc_{j}_{∗}α_{j}_{∗}) = Im(inc_{i}_{∗}α_{i}_{∗}∗) ⊆ ker(tr_{j}_{∗}) mod \overline{𝕁}_{τ}(Q_{i}, Q_{j})α_{i}_{∗}. Thus the second composite is zero mod \overline{𝕁}_{τ}(Q_{i}, Q_{j})_{∗}α_{i}_{∗}. On the other hand by the double coset formula this composite is N̄_{G}(Q_{j})α_{j}_{∗} mod \overline{𝕁}_{τ}(Q_{j}, Q_{j})_{∗}α_{j}_{∗}. Finally by naturality of the transfer, \overline{𝕁}_{τ}(Q_{i}, Q_{j})_{∗}α_{i}_{∗} = \overline{𝕁}_{τ}(Q_{j}, Q_{j})α_{j}_{∗} and so N̄_{G}(Q_{j})_{∗}α_{j}_{∗} = 0 mod \overline{𝕁}_{τ}(Q_{j}, Q_{j})_{∗}α_{j}_{∗}. Hence N̄_{G}(Q_{j})_{∗} = 0 mod \overline{𝕁}_{τ}(Q_{j}, Q_{j})_{∗}, a contradiction. □

Lemma 5.3. Let Q̃ be a maximal p-toral subgroup of C_{G}(Q) · Q. Then C_{G}(Q̃)/Z(Q̃) is finite p′-group.

Proof If H = C_{G}(Q̃)/Z(Q̃) is not finite then it has a non-trivial maximal torus T. Hence some maximal torus T′ of C_{G}(Q̃) is not a subset of Q̃. Let T′′ be a maximal torus of Q̃. Since C_{G}(Q̃) ⊆ C_{G}(Q), T′ · T′′ is a compact, connected, abelian subgroup. Hence T′ · T′′ is a torus of C_{G}(Q) · Q strictly larger than T′′. This contradicts the fact that T′′ is a maximal torus of C_{G}(Q) · Q.

Suppose x ∈ C_{G}(Q̃) projects to an non-trivial element \overline{x} ∈ C_{G}(Q̃)/Z(Q̃) of order p. Let K =< Q̃, x >. Since π = Q̃/T′′ is a finite p-group and x ∉ Q̃, K/T′′ = π × ℤ/p < x̄ > is also finite p-group. Thus K is p-toral subgroup of C_{G}(Q) · Q. However Q̃ ⊊ K which contradicts the maximality of Q̃. □

Proposition 5.4. If 𝕄 is a finite 𝕊(Q) module then

P Hom_{O(Q)}(𝔽_{p}Cen(Q, G), 𝕄)

is finite.

Proof Let {Q_{i} : i ∈ I} be a complete set of conjugacy class representatives of subgroups isomorphic to Q such that C_{G}(Q_{i})/Z(Q_{i}) is a finite p′-group. As an O(Q) module, 𝔽_{p}Cen(Q, G) decomposes as a sum of permutation modules

𝔽_{p}Cen(Q, G) ≈ ⨁_{i∈I} O(Q)/O(Q)𝔽_{p}W_{G}(Q_{i}) ≈ ⨁_{i∈I} O(Q)\overline{W}_{G}(Q_{i}).

where \overline{W}_{G}(Q_{i}) = ∑ c_{x}, x ∈ W_{G}(Q_{i}) := N_{G}(Q_{i})/Q_{i} · C_{G}(Q_{i}). Via the surjection of rings O(Q) → 𝕆(Q), 𝕄 is an O(Q) module. This surjection satisfies the idempotent lifting property and 𝕆(Q) is finite, hence Prop. 4.3 [MP2] yields

P Hom_{O(Q)}(O(Q)\overline{W}_{G}(H_{i}), 𝕄) ≈ \overline{W}_{G}(Q_{i})𝕄.

Therefore

P Hom_{O(Q)}(𝔽_{p}Cen(Q, G), 𝕄) = ∏_{i} \overline{W}_{G}(Q_{i})𝕄.

Let \overline{N}_{G}(Q_{i}) := ∑_{x} c_{x}, x ∈ N_{G}(Q_{i})/Q_{i}. Since C_{G}(Q_{i})/Q_{i} is a p′-group \overline{N}_{G}(Q)𝕄 is a p′ multiple of \overline{W}_{G}(Q_{i})𝕄 and by Lemma 5.2, \overline{N}_{G}(Q_{i})_{∗} = 0 mod \overline{𝕁}_{τ}(Q_{i})_{∗} except for a finite number of i. □

6. Proof of Proposition 6.1

To ease notation let R_{j} = 𝔽_{p}Out(Q̃_{j}|Q_{j}), T_{j} = 𝔽_{p}Out(Q̃_{j}), S(Q̃_{j}) = 𝔽_{p} ⊗ {BQ̃_{j}, BQ̃_{j}}/J̄_{τ}(Q̃_{j}). We note the obvious homomorphisms R_{j} → T_{j} → S(Q̃_{j}) → 𝕊(Q̃_{j}). Finally we recall that the hypothesis of Theorem 1 implies G and G′ have isomorphic maximal p-toral subgroups which we denote jointly by P.

Proposition 6.1. If, under the hypotheses of Theorem 1, 𝕄_{j} is a finite 𝕊(Q̃_{j}) module then there are isomorphisms of finite 𝔽_{p} modules

(A) P Hom_{T_{j}}(𝔽_{p}Cen(Q̃_{j}, G), 𝕄_{j}) ≈ P Hom_{T_{j}}(𝔽_{p}Cen(Q̃_{j}, G′), 𝕄_{j})

(B) P Hom_{R_{j}}(𝔽_{p}Cen_{Q_{j}}(Q̃_{j}, Q̃_{j}, G), 𝕄_{j}) ≈ P Hom_{R_{j}}(𝔽_{p}Cen_{Q_{j}}(Q̃_{j}, Q̃_{j}, G′), 𝕄_{j}).

Proof (A) We use induction on height ht(·). Starting with subgroups of maximal height and applying Lemma 4.6(2) we may assume by hypothesis and downward induction

P Hom_{T_{i}}(𝔽_{p}Cen(Q̃_{i}, G), 𝕄_{i}) ≈ P Hom_{T_{i}}(𝔽_{p}Cen(Q̃_{i}, G′), 𝕄_{i}) (1)

for all Q̃_{i} such that ht(Q̃_{i}) > t for some t such that 1 ≤ t < m = max{ht(Q̃_{k}) | k ∈ J}.

\underline{Claim:} There are isomorphisms of finite 𝔽_{p} modules

P Hom_{R_{i}}(𝔽_{p}Cen_{Q_{i}}(Q̃_{i}, Q̃_{i}, G), 𝕄_{i}) ≈ P Hom_{R_{i}}(𝔽_{p}Cen_{Q_{i}}(Q̃_{i}, Q̃_{i}, G′), 𝕄_{i})

for all Q̃_{i} such that ht(Q̃_{i}) > t.

Proof of Claim: We first observe that by Lemma 4.6(2),(3)

𝔽_{p}Cen_{Q_{m}}(Q̃_{m}, Q̃_{m}, G) ≈ 𝔽_{p}Cen_{Q_{m}}(Q̃_{m}, Q̃_{m}, G′).

as R_{m} modules. Setting R(i, k) := 𝔽_{p}R_{Q_{i}}(Q̃_{i}, Q̃_{k}) (Def 4.4) which is independent of G we have

P Hom_{R_{i}}(R(i, m) ⊗_{R_{m}} 𝔽_{p}Cen_{Q_{m}}(Q̃_{m}, Q̃_{m}, G), 𝕄_{i}) ≈
P Hom_{R_{i}}(R(i, m) ⊗_{R_{m}} 𝔽_{p}Cen_{Q_{m}}(Q̃_{m}, Q̃_{m}, G′), 𝕄_{i})

which starts the downward induction. Now we assume

P Hom_{R_{i}}(R(i, k) ⊗_{R_{k}} 𝔽_{p}Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G), 𝕄_{i}) ≈
P Hom_{R_{i}}(R(i, k) ⊗_{R_{k}} 𝔽_{p}Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G′), 𝕄_{i})

for all Q_{k} such that ht(Q̃_{k}) > v for some v such that t < v < m = max{ht(Q̃_{k})}. Therefore by Prop. 4.5

∏_{k} P Hom_{R_{i}}(𝔽_{p}Cen_{Q_{i}}(Q̃_{i}, Q̃_{k}, G), 𝕄_{i}) ≈

∏_{k} P Hom_{R_{i}}(𝔽_{p}Cen_{Q_{i}}(Q̃_{i}, Q̃_{k}, G′), 𝕄_{i})  (2)

taken over all k such that ht(Q̃_{k}) > v. Via the map R_{i} → T_{i} equation

(1) is true with T_{j} replaced by R_{i}. Hence by Lemma 4.6(4) we obtain

(2) taken over all k such that ht(Q̃_{k}) ≥ v. These modules are finite by Prop. 5.4 and Lemma 4.6 (4). The index k runs over a finite set by Lemma 5.1 hence the Krull-Schmidt Theorem applies and we have cancellation. Therefore

P Hom_{R_{i}}(𝔽_{p}Cen_{Q_{i}}(Q̃_{i}, Q̃_{i}, G), 𝕄_{i}) ≈ P Hom_{R_{i}}(𝔽_{p}Cen_{Q_{i}}(Q̃_{i}, Q̃_{i}, G′), 𝕄_{i})

for ht(Q̃_{i}) = v. This proves the Claim by induction. □

Returning to the proof of the Proposition and applying the adjoint isomorphism of Cor. 4.8 to the isomorphisms of the Claim yields

P Hom_{R_{i}}(𝔽_{p}R(Q_{i}, Q̃_{k}) ⊗_{R_{k}} 𝔽_{p}Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G), 𝕄_{i}) ≈
P Hom_{R_{i}}(𝔽_{p}R(Q_{i}, Q̃_{k}) ⊗_{R_{k}} 𝔽_{p}Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G′), 𝕄_{i})

for all k ∈ J. Hence by Proposition 4.3 we obtain an isomorphism

P Hom_{R_{i}}(𝔽_{p}nCen(Q̃_{i}, G), 𝕄_{i}) ≈ P Hom_{R_{i}}(𝔽_{p}nCen(Q̃_{i}, G′), 𝕄_{i}).

of finite modules. By definition

𝔽_{p}Inj_{fW} (Q̃_{i}, G) = 𝔽_{p}Cen(Q̃_{i}, G) ⊕ 𝔽_{p}nCen(Q̃_{i}, G).

Applying P Hom_{R_{i}}(·, 𝕄_{i}) to this yields (A), again by Krull-Schmidt and cancellation.

Repeating the argument of the Claim and induction we also have proved (B). □

7. Proofs of Propositions 3.5, 3.6

Proof of Proposition 3.5 Let R_{k} = 𝔽_{p}Out(Q̃_{k}|Q_{k}), By Proposition 4.3 there is an isomorphism

⨁_{k∈J} P Hom_{O(Q)}(𝔽_{p}R(Q, Q_{k}) ⊗_{R_{k}} 𝔽_{p}Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G)), 𝕄)

⟶^{≈} P Hom_{O(Q)}(𝔽_{p}nCen(Q, G), 𝕄).  (1)

By Corollary 4.8, there is an isomorphism

P Hom_{O(Q)}(𝔽_{p}R(Q, Q_{k}) ⊗_{R_{k}} 𝔽_{p}Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G), 𝕄)
⟶^{≈} P Hom_{R_{k}}(𝔽_{p}Cen_{Q_{k}}(Q̃_{k}, Q̃_{k}, G), 𝕄).  (2)

Hence for G, G′, the targets of (2) are isomorphic by Propositon 6.1(B). Thus (2) together with (1) implies the first isomorphism of the Proposition:

P Hom_{O(Q)}(𝔽_{p}nCen(Q, G), 𝕄) ⟶^{≈} P Hom_{O(Q)}(𝔽_{p}nCen(Q, G′), 𝕄).

Applying P Hom_{O(Q)}(·, M) to the decomposition

𝔽_{p}Inj_{fW} (Q, G) = 𝔽_{p}Cen(Q, G) ⊕ 𝔽_{p}nCen(Q, G)

Proposition 5.4, the Krull-Schmidt theorem, and cancellation yield

P Hom_{O(Q)}(𝔽_{p}Cen(Q, G), 𝕄) ≈ P Hom_{O(Q)}(𝔽_{p}Cen(Q, G′), 𝕄).

□

Proof of Proposition 3.6

Suppose X is an indecomposable spectrum. The idea of the proof is to show that X has the same multiplicity as a stable summand in BG_{p}^{∧} and BG_{p}^{′∧}. Let L(X) be a complete set of representatives of isomorphism classes of p-toral groups in which X is a linked summand of BQ for Q ≤ P where P is the common maximal p-toral subgroup of G,G′. For a fixed Q ∈ 𝕃(X), let M be the corresponding simple S(Q) module and let {H_{i}} (resp. {H′_{j}}) be a complete set of representatives of G- (resp. G′-) conjugacy classes of subgroups isomorphic to Q satisfying the additional requirement that C_{G}(H_{i})/Z(H_{i}) (resp. C_{G′}(H′_{j})/Z(H′_{j})) is a finite p′-group.

As an O(Q) module, 𝔽_{p}Cen(Q, G) decomposes as a sum of permutation modules

𝔽_{p}Cen(Q, G) ≈ ⨁_{i} O(Q)/O(Q)𝔽_{p}W_{G}(H_{i}) ≈ ⨁_{i} O(Q)\overline{W}_{G}(H_{i}).

where \overline{W}_{G}(H_{i}) = ∑ c_{x}, x ∈ W_{G}(H_{i}) := N_{G}(H_{i})/H_{i} ·C_{G}(H_{i}). Similarly for G′

𝔽_{p}Cen(Q, G′) ≈ ⨁_{j} O(Q)/O(Q)𝔽_{p}W_{G′}(H′_{j}) ≈ ⨁_{j} O(Q)\overline{W}_{G′}(H′_{j}).

Let 𝕄 be the corresponding simple 𝕊(Q) module described in Section 2. Via the ring homomorphism 𝕆(Q) → 𝕊(Q), 𝕄 is an 𝕆(Q) module. It is also an O(Q) module via O(Q) → 𝕆(Q). This surjection satisfies the idempotent lifting property by [Mit] Cor. 1.4, and 𝕆(Q) is finite hence Prop. 4.3 [MP2] yields

P Hom_{O(Q)}(O(Q)\overline{W}_{G}(H_{i}), 𝕄) ≈ \overline{W}_{G}(H_{i})𝕄.

(resp. for G′

P Hom_{O(Q)}(O(Q)\overline{W}_{G′}(H′_{j}), 𝕄) ≈ \overline{W}_{G′}(H′_{j})𝕄.)

By hypothesis, Proposition 3.5 holds, hence

∏_{i} \overline{W}_{G}(H_{i})𝕄 ≈ ∏_{j} \overline{W}_{G′}(H′_{j})𝕄

and so by Lemma 2.2 replacing 𝕄 by M we have

∏_{i} \overline{W}_{G}(H_{i})M ≈ ∏_{j} \overline{W}_{G′}(H′_{j})M. (1)

The next step is to repeat this process for each Q ∈ L(X) and its associated module M. We note that these modules depend only on Q and are independent of G, G′. Taking the isomorphisms (1) together we have an isomorphism

∏_{l} \overline{W}_{G}(Q_{l})M_{l} ≈ ∏_{m} \overline{W}_{G′}(Q′_{m})M′_{m} (2)

where Q_{l} ranges over a complete set of G-conjugacy classes of p-toral subgroups in which X is linked(resp. Q′_{m} ranges over a complete set of G′-conjugacy classes of p-toral subgroups in which X is linked and M′_{m} is the associated simple module).

Now we wish to relate all of this to the multiplicities of X in BG and BG′. Setting

N̄_{G}(Q_{l}) = ∑_{x} c_{x}, x ∈ N_{G}(Q_{l})/Q_{l}

we have N̄_{G}(Q_{l})M_{l} = q\overline{W}_{G}(Q_{l})M_{l} where q is prime to p since C_{G}(Q_{l})/ZQ_{l} is a finite p′ group. Thus there is an isomorphism

∏_{l} N̄_{G}(Q_{l})M_{l} ≈ ∏_{l} \overline{W}_{G}(Q_{l})M_{l}

and similarly

∏_{m} N̄_{G′}(Q′_{m})M′_{m} ≈ ∏_{m} \overline{W}_{G′}(Q_{m})M′_{m}.

Thus from (2)

∏_{l} N̄_{G}(Q_{l})M_{l} ≈ ∏_{m} N̄_{G′}(Q′_{m})M′_{m}. (3)

By Theorem 3.12, 3.13 of [MP2] the multiplicities of the summand X are given by

m(X, BG) = ∑_{l} dim_{k}N̄(Q_{l})M_{l}.

(resp.

m(X, BG′) = ∑_{m} dim_{k}N̄(Q′_{m})M′_{m}.)

All but a finite number of these summands are zero by Lemma 5.2; this also follows since otherwise the multiplicities would be infinite contradicting [H]. Thus all the direct products above are finite direct sums. Combining this with the isomorphism of (3) we have

m(X, BG) = m(X, BG′). □

8. Addendum

Although {BG, BG} is not computed by a completed two sided Burnside ring as in the finite group case it retains some nice properties (e.g. [L] Prop. 3.5). We recall that a ring R is called semiperfect if R/Rad(R) is semisimple and idempotents lift modulo Rad(R), the Jacobson radical. Equivalently R has a complete set of primitive orthogonal idempotents e_{1}, e_{2}, ..., e_{n} with each e_{i}Re_{i} a local ring.

Proposition 8.1. If G is a compact Lie group then {BG, BG} is semiperfect.

Proof By [Mit] 𝔼(G, G) has a complete set of primitive, orthogonal idempotents f_{1}, f_{2}, ..., f_{n} which lift via

R = {BG, BG} → 𝔼(G, G)

to a complete set of primitive orthogonal idempotents e_{1}, e_{2}, ..., e_{n}. Also e_{i}Re_{i} = {e_{i}BG, e_{i}BG} is local since e_{i}BG is indecomposable. □

We end with an example of a compact Lie group with infinitely many conjugacy classes of isomorphic p-toral subgroups having finite Weyl groups.

Example 8.2. Consider S^{1} = ℝ/ℤ and let G = (S^{1} × S^{1}) ⋊ π where π = ⟨t⟩ has order two and acts by t(x, y)t^{−1} = (−x, −y) The subgroups S_{m} = {(x, mx)t^{e} | m ∈ ℚ, e = 0, 1} are non-conjugate, isomorphic to O(2), with finite Weyl groups N_{G}(S_{m})/S_{m} = ℤ/2.

References

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[H] H.-W. Henn. Finiteness properties of injective resolutions of certain unstable modules over the Steenrod algebra, Math. Ann. 291 (1991), 191–203.

[JMO] S. Jackowski, J. McClure, B. Oliver. Homotopy classifications of self-maps of BG via G-actions, Annals of Mathematics, (1992), 35, 183–224.

[L] C.-N. Lee On stable splittings of classifying spaces of Lie groups,Topology 32, (1993),55–59.

[M] S. MacLane. Homology, Die Gundlehren 114, Springer-Verlag, Berlin,(1963).

[MP1] J. Martino and S. Priddy. Stable homotopy classification of BG_{p}^{∧}, Topology 34 (1995), 633–649.

[MP2] J. Martino and S. Priddy. A classification of the stable type of BG_{p}^{∧} for compact Lie groups, Topology 38 (1999), 1–21.

[MP3] J. Martino and S. Priddy. Corrigendum to ”Stable homotopy classification of BG_{p}^{∧}”, Topology 48 Supplement 1 (2011) eS1–eS4.

[Mi] N. Minami. The relative Burnside module and the stable self maps between classifying spaces of compact Lie groups, Trans. Amer. Math. Soc. 347 (1995), 461–498.

[Mit] S. Mitchell. Splitting B(Z/p)^{n} and BT^{n} via modular representation theory, Math. Zeit. 189 (1985), 1–9.

Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008

  E-mail address: john.martino@wmich.edu

Department of Mathematics, Northwestern University, Evanston, IL 60208

  E-mail address: priddy@math.northwestern.edu