CORRECTION: STABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧}

JOHN MARTINO AND STEWART PRIDDY

In this note we correct the proof of Theorem 1.1 of our paper [MP], given as Theorem 1 below. Let G, G′ be finite groups and p be a prime number. The goal is to give necessary and sufficient algebraic conditions on the p-subgroups of G and G′ which determine if their p-completions BG_{p}^{∧} and BG_{p}^{′∧} are stably homotopy equivalent.

Theorem 1. For finite groups G, G′ the following are equivalent: (1) BG_{p}^{∧} and BG_{p}^{′∧} are stably homotopy equivalent.

(2) For every p–group Q,

F_{p}Rep(Q, G) ≈ F_{p}Rep(Q, G′)

as Out(Q) modules. Rep(Q, G) = Hom(Q, G)/G with G acting by conjugation.

(3) For every p–group Q,

F_{p}Inj(Q, G) ≈ F_{p}Inj(Q, G′)

as Out(Q) modules. Inj(Q, G) ⊆ Rep(Q, G) consists of conjugacy classes of injective homomorphisms.

Our notation follows that of [MP]. Throughout all groups are finite and hereafter all group maps are monomorphisms.

The key step in the proof is the next Proposition. For Q a p-group we let Cen(Q, G) ⊆ Inj(Q, G) be the equivalence classes of injections α : Q ⟶ G such that C_{G}(Imα)/Z(Imα) is a p′-group.

Proposition 2. If F_{p}Inj(Q, G) ≈ F_{p}Inj(Q, G′) as Out(Q) modules for all p-groups Q then F_{p}Cen(Q, G) ≈ F_{p}Cen(Q, G′) as Out(Q) modules for all Q.

Let nCen(Q, G) be the complement of Cen(Q, G) in Inj(Q, G). To study these sets we shall consider subgroups H ≤ G such that H ≈ Q and p divides |C_{G}(H)/Z(H)|. Let H̃ be a Sylow p-subgroup of H · C_{G}(H). By definition |H| < |H̃|. An equivalence s : H̃_{1} ⟶^{≈} H̃_{2} between such groups will be an isomorphism such that s(H_{1}) = H_{2}. Let {Q̃_{j}} be a set of representatives for equivalence classes of such subgroups.

Date: March 3, 2010.

The proof of Proposition 2 depends on a close analysis of Cen(Q, G). Define

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

The following lemma describes some elementary properties of Cen_{Q_{j}}(Q̃_{j}, Q̃_{k}, G). Proofs are given at the end of the note.

Lemma 3. Let Q̃_{m} denote a subgroup of maximum order in {Q̃_{j}}.

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

(2) Cen(Q̃_{m}, G) = Inj(Q̃_{m}, G).

(3) Cen_{Q_{m}}(Q̃_{m}, Q̃_{m}, G) = Cen(Q̃_{m}, G).

(4) If |Q̃_{j}| = |Q̃_{k}| and Cen_{Q_{j}}(Q̃_{j}, Q̃_{k}, G) ≠ ϕ then Q̃_{j} ∼ Q̃_{k}.

(5) Cen(Q̃_{j}, G) = Cen_{Q_{j}}(Q̃_{j}, Q̃_{j}, G) ∐_{k} Cen_{Q_{j}}(Q̃_{j}, Q̃_{k}, G) where k ranges over {k : |Q̃_{k}| > |Q̃_{j}|}.

Define a set of monomorphisms

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

and a group of automorphisms

Aut(Q̃|Q) = {α : Q̃ → Q̃ | α(Q) = Q}.

Observe that R(Q′, Q̃) and Cen_{Q}(Q̃, Q̃_{k}, G) are Aut(Q̃|Q) sets since the image of Q is invariant. The following Lemma is a corrected version of Lemma 4.6 [MP] which contained an error pointed out to us by Kari Ragnarsson [R].

Lemma 4. There is an isomorphism of Out(Q) sets

∐_{k} ψ : R(Q, Q̃_{k})×_{Aut(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].

Define a set of monomorphisms

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

Arguing as in Lemma 4 we have Lemma 5. There is an isomorphism of Aut(Q̃_{i}|Q_{i}) sets

ψ : R_{Q_{i}}(Q̃_{i}, Q̃_{k}) ×_{Aut(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].

Proposition 6. If F_{p}Inj(Q, G) ≈ F_{p}Inj(Q, G′) as Out(Q) modules for all p-groups Q then F_{p}Cen(Q̃_{j}, G) ≈ F_{p}Cen(Q̃_{j}, G′) as Out(Q̃_{j}) modules for all j and

F_{p}Cen_{Q_{j}}(Q̃_{j}, Q̃_{j}, G) ≈ F_{p}Cen_{Q_{i}}(Q̃_{j}, Q̃_{j}, G′)

as Aut(Q̃_{j}|Q_{j}) modules for all j.

Proofs.

Proof of Lemma 3: (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. Let x ∈ C_{G}(β(Q̃_{j})) ⊆ C_{G}(β(Q_{j})) have order a power of p. Since { \atop β(Q_{j})} ∼ Q̃_{j},

|Q̃_{j}| = |β(Q̃_{j})| ≤ |{ \atop β(Q_{j})}| = |Q̃_{j}|. Therefore β(Q̃_{j}) = { \atop β(Q_{j})} and so

β(Q̃_{j}) is a Sylow p-subgroup of C_{G}(β(Q_{j})) · β(Q_{j}). However ⟨x, β(Q̃_{j})⟩ is a p-subgroup of C_{G}(β(Q_{i}))·Q_{i} which means x ∈ β(Q̃_{j}) and therefore x ∈ Z(β(Q̃_{j})).

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

(3) By definition Cen_{Q_{m}}(Q̃_{m}, Q̃_{m}, G) ⊆ Cen(Q̃_{m}, G). Let [β] ∈ Cen(Q̃_{m}, G). By maximality we have equality |Q̃_{m}| = |β(Q̃_{m})| = |{ \atop β(Q_{m})}| = |Q̃_{m}|. Since { \atop β(Q_{m})} is a Sylow p-subgroup of C_{G}(β(Q_{m})) · β(Q_{m}) there is an isomorphism

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

where g ∈ C_{G}(β(Q_{m})) · β(Q_{m}). Hence Q̃_{m} ∼ { \atop β(Q_{m})} and so [β] ∈ Cen_{Q_{m}}(Q̃_{m}, Q̃_{m}, G).

(4) If [β] ∈ Cen_{Q_{j}}(Q̃_{j}, Q̃_{k}, G) then |Q̃_{j}| = β(Q̃_{j})| ≤ |{ \atop β(Q_{j})}| = |Q̃_{k}|.

Thus β : Q̃_{j} ⟶^{≈} { \atop β(Q_{j})} and it follows that Q̃_{j} ∼ Q̃_{k}.

(5) [β] ∈ Cen(Q̃_{j}, G) iff β : Q̃_{j} → G is a monomorphism and C_{G}(β(Q̃_{j}))/Z(β(Q̃_{j})) is a p′-group. Now { \atop β(Q_{j})} ∼ Q̃_{k} for some unique k. Hence [β] ∈ Cen_{Q_{j}}(Q̃_{j}, Q̃_{k}, G). The restriction on the range of k follows from (4). The other inclusion is clear. □

Proof of Lemma 4: If R(Q, Q̃_{k}) ≠ ϕ then |Q| = |Q_{k}| < |Q̃_{k}|. Therefore ϕ(γ_{k} × [δ_{l}]) ∈ 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: If [f] ∈ nCen(Q, G) then 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. Choose an equivalence u : { \atop f(Q)} → Q̃_{j} for some j such that u(f(Q)) = Q_{j} and set γ = uπ, δ = iu^{−1}. By Lemma 3 (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. □

Proof of Lemma 5: 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]. □

Proof of Proposition 6: By Lemma 3 (2) and downward induction on the order of Q̃_{i} we may assume

F_{p}Cen(Q̃_{i}, G) ≈ F_{p}Cen(Q̃_{i}, G′)  (1)

as Out(Q̃_{i}) modules for all Q̃_{i} such that |Q̃_{i}| > |Q̃_{j}| for some j. Claim:

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

as Aut(Q̃_{i}|Q_{i}) modules for all Q̃_{i} such that |Q̃_{i}| > |Q̃_{j}|. By Lemma 3 (3) we may assume by downward induction that

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

as Aut(Q̃_{k}|Q_{k}) modules for all Q̃_{k} such that |Q̃_{k}| > |Q̃_{i}| for some i such that |Q̃_{i}| > |Q̃_{j}| . Therefore by Lemma 5,

⊕_{k,k≠i}F_{p}Cen_{Q_{i}}(Q̃_{i}, Q̃_{k}, G) ≈ ⊕_{k,k≠i}F_{p}Cen_{Q_{i}}(Q̃_{i}, Q̃_{k}, G′)

as Aut(Q̃_{i}|Q_{i}))-modules. By equation (1), Lemma 3 (5),

⊕_{k}F_{p}Cen_{Q_{i}}(Q̃_{i}, Q̃_{k}, G) ≈ ⊕_{k}F_{p}Cen_{Q_{i}}(Q̃_{i}, Q̃_{k}, G′)

as Aut(Q̃_{i}|Q_{i})-modules. By the Krull-Schmidt Theorem we have cancellation, hence

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

which proves the claim by induction.

Setting Q = Q̃_{j} in Lemma 4 we conclude

F_{p}nCen(Q̃_{j}, G) ≈ F_{p}nCen(Q̃_{j}, G′).

as Out(Q̃_{j}) modules. By definition

F_{p}Inj(Q̃_{j}, G) = F_{p}Cen(Q̃_{j}, G) ⊕ F_{p}nCen(Q̃_{j}, G)

and we have cancellation. It follows that F_{p}Cen(Q̃_{j}, G) ≈ F_{p}Cen(Q̃_{j}, G′) for all j. Repeating the argument of the claim we also have

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

for all j. □

Proof of Proposition 2: By Prop 6 and Lemma 4,

F_{p}nCen(Q, G) ≈ F_{p}nCen(Q, G′).

The result follows by the decomposition

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

and cancellation. □

Proof of Theorem 1: With the aid of Proposition 2, the proof of this result now follows as in [MP].

References

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

[R] K. Ragnarsson Private Correspondence.

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