revised March 29, 1994

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

John Martino and Stewart Priddy*

1. Introduction and Statement of Main Results.

If G is a finite group then its classifying space BG is an Eilenberg-MacLane space K(G, 1) so G determines BG up to homotopy equivalence and BG determines G up to isomorphism class. However, after Z–completion this entirely satisfactory state of affairs ceases to hold. For example, Z_{∞}(BD_{2p} ×Z/4) ≅ Z_{∞}(BQ_{4p} ×Z/2), where p is an odd prime, Q_{4p} is the generalized quaternion group [CE] of order 4p, and D_{2p} is the dihedral group of order 2p. This example is due to N. Minami. The situation is even worse upon stabilization; there are examples where Z_{∞}BG and Z_{∞}BG′ are stably homotopy equivalent but not unstably (see Example 5.2).

Since

BG ≃ ⋁_{p||G|} BG_{p}^{∧}

stably, where BG_{p}^{∧} is p–completion of BG, it is sufficient to work at a fixed prime p. It is the purpose of this paper to provide a necessary and sufficient condition for BG_{p}^{∧} and BG′_{p}^{∧} to be stably homotopy equivalent. Since p-completion and plocalization agree for BG we could equally well state our results using localization.

Theorem 1.1. (Stable classification) For two finite groups G and 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.

As an immediate corollary of Theorem 1.1 (3), we have a theorem of G. Nishida:

*the first author is partially supported by NSF Grant DMS-9007361, the second by NSF Grant DMS-880067 and the Alexander von Humboldt Foundation.

Theorem 1.2 [N]. If BG_{p}^{∧} and BG′_{p}^{∧} are stably equivalent then G and G′ have isomorphic Sylow p-subgroups.

More generally Theorem 1.1 (3) implies

Theorem 1.3. If BG_{p}^{∧} and BG′_{p}^{∧} are stably equivalent then G and G′ have the same number of conjugacy classes of subgroups isomorphic to Q, for every finite p-group Q.

Let W_{G}(H) denote the Weyl group of H ≤ G, i.e.

W_{G}(H) = N_{G}(H)/H · C_{G}(H)

where N_{G}(H) is the normalizer and C_{G}(H) is the centralizer of H in G. Then W_{G}(H) ≤ Out(H).

Now let P denote the common (up to isomorphism) Sylow p-subgroup. Then it is easy to see that condition (3) of Theorem 1.1 can be written more explicitly as

⨁ 1_{W_{G}(Q_{1})}^{Out(Q)} ≈ ⨁ 1_{W_{G′} (Q_{1})}^{Out(Q)}

as Out(Q) modules for all Q ≤ P, where the sum runs over conjugacy classes of Q_{1} ≤ P such that Q_{1} ≈ Q. Tensoring this isomorphism with F_{p} over F_{p}Out(Q) implies Theorem 1.3. In Section 4 we shall show that the sum can be restricted to those Q_{1} such that C_{G}(Q_{1})/Z(Q_{1}) is a p′-group.

An important application of Theorem 1.1 is to the case of a normal Sylow p– subgroup, in particular the cyclic mod p groups.

Definition 1.4. Two subgroups H, K ≤ G are called pointwise conjugate in G if there is a bijection of sets H ⟶^{α} K such that α(h) = g_{h}^{−1}hg_{h} for g_{h} ∈ G depending on h ∈ H.

Alternately it is easy to see that an equivalent condition is

|H ∩ (g)| = |K ∩ (g)|

for all g ∈ G, where (g) denotes the conjugacy class of g.

Theorem 1.5. (Normal Sylow p–subgroups) Suppose G, G′ are finite groups with normal Sylow p–subgroups P, P′. Then BG_{p}^{∧} and BG′_{p}^{∧} have the same stable homotopy type if and only if there is an isomorphism ϕ : P → P′ and W_{G}(P) is pointwise conjugate to ϕ^{−1}(W_{G′}(P′))ϕ in Out(P).

The choice of isomorphism ϕ : P → P′ is the choice of a representative from the conjugacy class of ϕ^{−1}(W_{G′}(P′))ϕ, and thus pointwise conjugacy is independent of ϕ.

STABLE HOMOTOPY CLASSIFICATION OF BG_{p}^{∧} 3

Definition 1.6. G is called cyclic mod p (or p–hypoelementary) if a Sylow p– subgroup P is normal and C = G/P is a cyclic p′–group, i.e. has order prime to p. We say G is reduced if O_{p′}(G) = 1.

For a cyclic mod p group G, being reduced is equivalent to W_{G}(P) = C. It follows that P ⋊ W_{G}(P) is itself reduced. Therefore up to stable equivalence at p, G can be replaced by the reduced group P ⋊W_{G}(P) since the projection BG → B(P ⋊W_{G}(P)) induces an isomorphism in mod-p cohomology.

Theorem 1.7. (Cyclic mod p groups) Suppose G, G′ are reduced cyclic mod p groups. Then BG_{p}^{∧} ≃ BG′_{p}^{∧} stably at p if and only if G ≈ G′.

Cyclic mod p groups are important for several reasons. Their mod p cohomology is computed as the ring of invariants H^{∗}(G) = H^{∗}(P)^{C}. On the level of stable homotopy one also has the Minami–Webb Formula [Mi]: Let C_{p}(G) be the set of cyclic mod p subgroups of G. Then

BG_{p}^{∧} ≃ ⋁_{(H)} \frac{f(H)}{[N_{G}(H) : H]} BH_{p}^{∧}

stably, where H runs over the conjugacy classes of C_{p}(G) and f : C_{p}(G) ⟶ Z is the Möbius function given by

∑_{J≤K} f(K) = 1

for J, K ∈ C_{p}(G).

This expression is to be interpreted in a Grothendieck group of spectra, that is, multiplying through by a common denominator and moving the negative terms to the left side results in a valid formula in the category of p–complete spectra.

The next result shows the uniqueness of the Minami–Webb Formula.

Theorem 1.8. (Linear independence of cyclic mod p groups) Suppose

⋁_{i=1}^{m} α_{i}BH_{i}_{p}^{∧} ≃ ⋁_{j=1}^{n} α′_{j}BH′_{j}_{p}^{∧}

stably, for H_{i}, H′_{j} reduced cyclic mod p groups with α_{i}, α′_{j} ∈ Q. Then m = n and up to a permutation of indices α_{i} = α′_{j}, H_{i} ≈ H′_{j}.

The paper is organized as follows. In Section 2 we classify the stable summands in the splitting of BG_{p}^{∧} and develop the concept of linkage. In Section 3 we define a matrix which will allow us to determine the stable homotopy type of BG_{p}^{∧}. Section 4 contains the proof of Theorem 1.1. Section 5 deals with the case of a normal Sylow p–subgroup. Section 6 is concerned with cyclic mod p groups.

Unless stated specifically otherwise all groups in this paper are finite.

We thank David Benson, Bill Dwyer, Nick Kuhn, Guido Mislin, and especially Hans–Werner Henn for several useful conversations on the topics in this paper. We are also grateful to the University of Heidelberg for support during part of the time this research was done. Finally we thank the referee whose many comments significantly improved the exposition of this paper.

Many of the results of this paper were announced in [MP2].

2. Linkage and a classification of stable summands.

In this section we develop the theory which will allow us to use representation theory in describing the stable summands of the p–localization of BG. We begin with a quick review of Nishida’s theory of stable summands [N]. For further preliminaries see [MP1]. Let BG_{+} denote BG with a disjoint basepoint. Stably BG_{+} = BG ∨ S^{0}, thus the stable homotopy classes of maps {BG_{+}, BG′_{+}} contains Z = {S^{0}, S^{0}} as a summand. By the Segal conjecture [A,C,LMM], first shown by G. Carlsson, if Q is a finite p–group and G a finite group then the complementary summands are p-complete and Ẑ_{p} ⊗ {BQ_{+}, BG_{+}} is generated as a free Ẑ_{p} (the p–adic integers) module by equivalence classes of maps of the form

(2.1)  BQ_{+} →^{tr} BR_{+} →^{Bρ} BG_{+}

where tr is the transfer, R a subgroup of Q, and ρ : R → G is a group homomorphism. Two such maps are equivalent if there exists x ∈ Q, y ∈ G such that the following diagram commutes:

BQ_{+} ⟶^{tr} BR_{+} ⟶^{Bρ} BG_{+}
c_{x}↓  ^{c_{x}}↓  ^{c_{y}} ↓
BQ_{+} ⟶^{tr} BR_{+}^{x} ⟶^{Bσ} BG_{+}

where c_{x} and c_{y} are conjugation by x and y.

Since BG is connected, we have

{BQ, BG} = {BQ_{+}, BG_{+}}/{BQ_{+}, S^{0}}

which has the effect of identifying all maps of the form BQ_{+} →^{tr} BR_{+} →^{0} BG_{+} with the zero element.

Let J(Q, G) be the submodule of {BQ, BG} generated by maps which factor through BR where R is a proper subgroup of Q. Then it follows that

F_{p} ⊗ {BQ, BG}/J(Q, G) ≈ F_{p}Inj(Q, G)

and in particular if G equals Q, J(Q) = J(Q, Q) is an ideal and

F_{p} ⊗ {BQ, BQ}/J(Q) ≈ F_{p}Out(Q)

where Out(Q) is the outer automorphism group of Q.

If X is an indecomposable stable summand of BQ then X corresponds to a conjugacy class of primitive idempotents e ∈ {BQ, BQ}. If e ∈ J(Q) then X is a summand of BR for some R a proper subgroup of Q. Otherwise e gives rise to a conjugacy class of primitive idempotents ē ∈ F_{p}Out(Q).

Definition 2.2. If X is an indecomposable stable summand of BQ with corresponding idempotent e ∉ J(Q) then X originates in BQ (or X is an original summand of BQ).

Remark 2.3. The idea of an original summand has previously been referred to as a dominant summand [N]. The summands originating in BQ are in one-to-one correspondence with simple modules of F_{p}Out(Q). In [MP1] the authors exploited this idea to determine the complete stable splitting of BG. A different approach to the same problem was taken by D. Benson and M. Feshbach [BF].

Let J_{τ} (Q, G) be the Ẑ_{p} module generated by pointed maps of form 2.1 which begin with the transfer to BR_{+} for R ≨ Q. Then J_{τ} (Q, Q) ⊆ J(Q) and

(2.4)  F_{p} ⊗ {BQ_{+}, BG_{+}}/J_{τ} (Q, G) ≈ F_{p}Rep(Q, G)

Let \overline{J}_{τ} (Q) be the ideal generated by J_{τ} (Q, Q) in \{BQ, BQ\}. If e ∈ \{BQ, BQ\} is an idempotent for a summand X originating in BQ and ē ∈ F_{p}Out(Q) is the corresponding idempotent then viewing ē ∈ {BQ, BQ} we can decompose ē into a sum of primitive orthogonal idempotents

ē = e_{0} + e_{1} + ⋯ + e_{k}

where e_{0} = e and e_{i} ∈ J(Q), i = 1, …, k. We say the summand Y_{i} = e_{i}BQ is associated to X.

Definition 2.5. Using the notation in the above paragraph if e_{i} ∉ \overline{J}_{τ} (Q) then Y_{i} is linked to X in BQ. In particular an original summand X is linked to itself.

Remark 2.6. An alternative definition of linkage is the following. X is linked in BQ if any unit of the form

(2.7)  X → BQ →^{tr} BR → X

must have R equal to Q (in which case tr is the identity). The two definitions are equivalent since if (2.7) can be written with R ≨ Q then an idempotent e ∈ \{BQ, BQ\} representing X will be in \overline{J}_{τ} (Q) and conversely if e ∈ \overline{J}_{τ} (Q) then, since {X, X} is a local ring [Mr], there will be a unit of form (2.7). That is, if X is linked and does not originate in BQ then X must come from a retraction (see Proposition 2.9). Unfortunately not every summand which comes from a retraction is linked (see Remark 2.10). This alternate definition was used in [Ma1,MP1].

We referred to Y_{i} as linked because of:

Proposition 2.8 [Ma1,MP1]. Let Q be a p–subgroup of a finite group G and suppose Q is not a subretract of a Sylow p–subgroup of G. If Y is a summand which originates in BR, R ≨ Q which is linked to a summand X which originates in BQ then if X is a summand of BG so is Y. Moreover, the multiplicity of X in BG is a lower bound for the multiplicity of Y in BG.

Proposition 2.9. For Q a finite p–group and R ≨ Q. If Y is a summand which originates in BR and is linked to a summand in BQ then R is a retract of Q.

Proof. Since Y is linked in BQ there is a unit in {Y, Y } factoring through BQ of the form

Y → BR →^{Bf} BQ →^{Bg} BR → Y

where f and g are homomorphisms. Since {Y, Y } is a local ring [Mr], for a unit of the form ∑ ϕ_{i} at least one ϕ_{i} must be a unit. That unit must be of the displayed form since if it involved the transfer then Y would not be linked in BQ by Remark 2.6. Because Y originates in BR we must have that g ∘ f is an isomorphism. □

Remark 2.10. The converse of Proposition 2.9 is not true. For example, Z/2 is a retract of the dihedral group D_{8} but the indecomposable summand BZ/2 is not linked in BD_{8} since there is a unit of the form

BZ/2 → BD_{8} →^{tr} B(Z/2)^{2} → BZ/2.

This was shown in [MiP]. However BZ/2 is linked in B(Z/2)^{2}, in fact if A is abelian all the indecomposable summands of BA are linked in BA (see Remark 3.7). BZ/2 is linked to a summand called L(2) in B(Z/2)^{2}. Since L(2) is a summand of BD_{8} [MiP], Proposition 2.8 gives another reason why BZ/2 must be a summand of BD_{8}.

Let

R(Q) = F_{p} ⊗ \{BQ, BQ\}/\overline{J}_{τ} (Q).

R(Q) is a quotient ring of F_{p}ÕEnd(Q) where ÕEnd(Q) is the semigroup of outer endomorphisms where the zero homomorphism is identified with the zero element. If X is an indecomposable summand which is linked in BQ then X corresponds to a conjugacy class of primitive idempotents ê ∈ R(Q) and thus to a simple module in R(Q).

We will exploit this correspondence in the next section. To do so we need the following lemma.

Lemma 2.11 (Replacement Lemma). Let X be an indecomposable summand of BG. Suppose there is a unit in {X, X} of the form

(2.12)  X → BG →^{tr} BR → X

where R is a p-subgroup of G and X is not linked in BR. Then there is a unit of the form

(2.13)  X → BG →^{tr} BS → X

where S ≨ R and X is linked in BS.

Proof. Using the unit (2.12), we can write an idempotent e for X in the form

BG →^{tr} BR →^{α} BG

for some map α. Therefore an idempotent e′ for X in BR is given by

e′ : BR →^{α} BG →^{e} BG →^{tr} BR

and e can be rewritten as e = αe′tr. By hypothesis X is not linked in BR, thus e is a linear combination of maps of the form

BG →^{tr} BS → BG

where S ≨ R. We now insert e into the unit (2.12)

X → BG →^{e} BG →^{tr} BS → X.

We then get a unit which is a linear combination of maps of form (2.13) and since {X, X} is a local ring [Mr] one of these maps must be a unit. If X still is not linked in BS repeat the procedure. □

3. The multiplicity of a stable summand in BG.

In this section we determine the multiplicity of a stable summand X in the stable decomposition of the p–localization of BG. To do this we develop a matrix A_{τ} which is a modified version of the matrix we used in our paper [MP1].

Let X be an indecomposable, stable summand of BG. Define the matrix a_{uv} to have entries

X →^{i} BP_{u} →^{Bi} BG →^{tr} BP_{v} →^{π} X

where i is an inclusion map, tr is the reduced transfer, π is a projection onto the summand X. P_{u} and P_{v} are representatives of conjugacy classes of p–subgroups of G in which X is a summand of the classifying space. The maps i and π need only be chosen so as to detect distinct copies of X. Hence the dimension of the matrix a_{kl} is the multiplicity of X in BP_{u} by the multiplicity of X in BP_{v}.

Since {X, X} is a local ring, k = {X, X}/Rad({X, X}) is a finite field [Mr]. For our purposes we consider the possibly larger ideal I_{uv} generated by Rad({X, X}) together with maps of the form

X → BP_{u} →^{δ} BP_{u} → BP_{v} → X

where δ ∈ \overline{J}_{τ} (P_{u}). Then \{X, X\}/I_{uv} is either k or 0 ∈ k ; in either case it is a vector space over k. Define ā_{uv} to be the matrix a_{uv} with entries taken modulo I_{uv}.

Definition 3.1.

A_{τ} (X) = (ā_{uv})

Thus A_{τ} (X) is a square, block matrix over k composed of submatrices ā_{uv}.

Proposition 3.2.

a) ā_{uv} = 0 unless P_{u} ≤_{G} P_{v}.

b) If P_{u} ≤_{G} P_{v} then any row in ā_{uv} is a row in ā_{vv}.
c) ā_{uu} = 0 unless X is linked in BP_{u}.

Proof. (a) follows immediately from the double coset formula and the fact that the entries in ā_{uv} are modulo \overline{J}_{τ} (P_{u}). (b) follows from the fact that up to conjugation the inclusion of P_{u} in G factors through P_{v}. (c) follows since the map π : BP_{u} → X can be written π = π ∘ e where e ∈ {BP_{u}, BP_{u}} is an idempotent for the summand X. If X is not linked in BP_{u} then by definition e ∈ \overline{J}_{τ} (P_{u}). □

Let m(X, BG) be the multiplicity of X in BG then:

Theorem 3.3.

m(X, BG) = rank_{k}A_{τ} (X)

Proof. Let r = rank_{k}A_{τ} (X) and s = m(X, BG). We will first show that r ≤ s. This follows immediately since the identity on X^{∨r} can be written

X^{∨r} → X^{∨n} ⟶^{A_{τ} (X)} X^{∨n} → X^{∨r}

where n is the size of A_{τ} (X). All the entries in A_{τ} (X) factor through BG so we have the identity factoring through BG.

We now want to show that s ≤ r. Since X^{∨s} is a summand of BG localized at p, we can write the s × s identity matrix as a matrix of maps of the form

(1)  X →^{f} BG →^{g} X.

If P is a Sylow p-subgroup then

α : BG →^{tr} BP →^{Bi} BG

is an equivalence. Thus id : BG → BG is homotopic to

BG →^{tr} BP →^{βBi} BG

where β is a homotopy inverse for α. It follows that (1) factors as

(2)  X →^{f} BG →^{tr} BP →^{gβBi} X.

We now use the Replacement Lemma 2.11 to replace a one of form (2) in the identity matrix by a linear combination of entries from A_{τ} (X) of the form

X →^{f} BG →^{tr} BQ → X.

Consequently the space spanned by such maps is a subspace of the row space of A_{τ} (X) and so s ≤ r. □

We have reduced the problem of determining m(X, BG) to determining the rank of A_{τ} (X).

Recall from Section 2 that if X is linked in BP_{i} then X corresponds to a simple R_{i}–module M_{i} where R_{i} = R(Q_{i}) = F_{p} ⊗ \{BP_{i}, BP_{i}\}/\overline{J}_{τ} (P_{i}). Further End_{R_{i}}(M_{i}) = k = {X, X}/Rad. Let {P_{i}} be a set of representatives of G–conjugacy classes of subgroups where X is linked in BP_{i}. Which leads us to the next theorem.

Theorem 3.4.

rank_{k}A_{τ} (X) = ∑ dim_{k} \overline{W}_{i}M_{i}

where the sum ranges over {P_{i}} such that X is linked in BP_{i} and C_{G}(P_{i})/Z(P_{i}) is a p′-group. \overline{W}_{i} = ∑ c_{x} where x ∈ N_{G}(P_{i})/P_{i}. Here C_{G}, N_{G} refer to the centralizer and the normalizer in G respectively.

Proof. By Proposition 3.2, A_{τ} (X) can be reduced to the diagonal matrix diag(ā_{ii}). Thus

(3.5)  rank_{k}A_{τ} (X) = ∑ rank_{k}ā_{ii}

where the sum can be taken over representatives of G–conjugacy classes of P_{i}’s where X is linked in BP_{i}, also by Proposition 3.2.

Let m = m(X, BP_{i}) then by the double coset formula ā_{ii} equals

X^{∨m} → BP_{i} ⟶^{\overline{W}_{i}} BP_{i} → X^{∨m}

modulo \overline{J}_{τ} (P_{i}). Let π : \{X, X\} → End(M_{i}) be reduction by the radical then we get π(ā_{ii}) = \overline{W}_{i} ∈ Mat_{m}(M_{i}). It is now clear from representation theory that

rank_{k}ā_{ii} = dim_{k} \overline{W}_{i}M_{i}.

However if p divides |C_{G}(P_{i})/Z(P_{i})| then \overline{W}_{i} = 0 modulo p. So the sum in (3.5) may be reduced to the sum in the statement of the theorem. □

Remark 3.6. If P_{i} is not a Sylow p–subgroup of G then \overline{W} _{i}^{2} = 0 modulo p thus the Jordan canonical form of the matrix A_{τ} (X) must consist of block matrices of the form

( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )

followed by zeroes down the diagonal. Thus

rank_{k}ā_{ii} ≤ \frac{1}{2} dim_{k} M_{i}.

Remark 3.7. In calculating the rank of A_{τ} (X), for an indecomposable summand X, we need only consider p–subgroups P_{i} which are maximal with respect to X being linked in BP_{i}. For if X is linked in BP_{i} and

X → BG →^{tr} BP_{i} → X

is a unit then by definition X cannot be linked to any supergroup of P_{i}.

This phenomenon is of particular importance if the Sylow p–subgroup P of G is abelian. In that case F_{p} ⊗ \overline{J}_{τ} (Q) is nilpotent for every p–subgroup Q [HK] and hence all summands are linked in BP. So that each indecomposable summand of BP corresponds to a simple module of the endomorphism semigroup algebra of P. The case of P being abelian has been throughly explored by J. Harris and N. Kuhn [HK]. Also the indecomposable summands of BG depend only on the Weyl group W_{G}(P) and we have as a corollary of Theorem 3.4 the well-known theorem of R. Swan [Sw]

H^{∗}(BG; F_{p}) = H^{∗}(BP; F_{p})^{W_{G}(P)}.

4. P Hom and the proof of Theorem 1.1.

We begin by preparing some preliminaires for the implication 3) ⇒ 1), the most difficult of the three. Our strategy is to show that BG and BG′, localized at p, have the same indecomposable stable summands. The result follows by the uniqueness (up to order and equivalence) of splittings into indecomposable summands [H,N]. We will need the following lemma as it applies to permutation modules. This result, which is of interest in its own right, seems not to have been known to the experts.

Let R be a quasi-frobenius artinian ring, e.g. R = F_{p}Out(Q). In what follows all R-modules are finitely generated. Let N be a left R–module. Since R is artinian, N has a projective cover P_{N} which comes equipped with a quotient homomorphism P_{N} ⟶^{π} N of R-modules. For M, N R–modules we define P Hom_{R}(M, N) to be the set of R-module homomorphisms ϕ : M → N which factor as

P_{N}
ψ ↗ ↘ ^{π}
M ⟶_{ϕ} N

Alternately,

P Hom_{R}(M, N) = Hom_{R}(M, P_{N})/Hom_{R}(M, Ker π)

This follows from the exact sequence

0 ⟶ Hom_{R}(M, Ker π) ⟶ Hom_{R}(M, P_{N}) ⟶ Hom_{R}(M, N)

since P Hom_{R}(M, N) is by definition the image of the rightmost arrow. It is straightforward to verify that

(4.1)  P Hom_{R}(M ⊕ M′, N) ≈ P Hom_{R}(M, N) ⊕ P Hom_{R}(M′, N).

Lemma 4.2. Let M be a left R–module and x ∈ R then

P Hom_{R}(Rx, M) ≈ xM.

Proof. Since R is quasi-frobenius, P_{M} is also an injective R-module. Thus there is a map ψ′ of R-modules completing the diagram

P_{M}
ψ ↗ ↖ ^{ψ′}
Rx ↪  R

Now define a homomorphism

Γ : P Hom_{R}(Rx, M) ⟶ xM

by

Γ(ϕ) = ϕ(x).

Γ is an isomorphism of abelian groups. Since ϕ(x) = πψ′(x1) = xπψ′(1) we have ImΓ ⊂ xM. Also Γ is one to one since ϕ is determined by ϕ(x)

ϕ(rx) = rϕ(x).

To see that Γ is surjective suppose m ∈ M and choose m̃ ∈ P_{M} such that π(m̃) = m. Define an R-module homomorphism ψ′ : Rx ⟶ M by ψ′(rx) = rxm̃. Let ϕ = πψ′ then Γ(ϕ) = ϕ(x) = πψ′(x) = π(xm̃) = xm. □

Corollary 4.3. Suppose x, y ∈ R. Then Rx ≈ Ry as R–modules implies xM ≈ yM.

Proof. : P Hom_{R}(Rx, M) is a functor of Rx as an R–module. □

Proof of Theorem 1.1.

3)⇒ 1). Let S be a finite p-group and fix an original summand X of BS. We will show X has the same multiplicity in BG and BG′, i.e. m(X, BG) = m(X, BG′). Condition 3) immediately implies G and G′ have isomorphic Sylow p-subgroups P, say. We recall from Section 3, that {P_{i}} is a set of representatives for the Gconjugacy classes of subgroups of P_{i} such that S is isomorphic to a retract of P_{i}. Then X corresponds to a simple R_{i}-module M_{i} where R_{i} = \{BP_{i}, BP_{i}\}/\overline{J}_{τ} (P_{i}). By Theorems 3.3, 3.4

m(X, BG) = ∑ dim_{k}\overline{W}_{i}M_{i}

where the sum runs over those conjugacy clases such that C_{G}(P_{i})/Z(P_{i}) is a p′-group and where

(4.4)  \overline{W}_{i} = ∑_{x∈N_{G}(P_{i})/P_{i}} c_{x}

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. Then as Out(Q) modules

F_{p}Cen(Q, G) ≈ ⨁ 1_{W_{G}(Q′)}^{Out(Q)}

where the sum runs over conjugacy classes of Q′ ≤ P such that Q′ ≈ Q and C_{G}(Q′)/Z(Q′) is a p′-group.

Proposition 4.5. If F_{p}Inj(Q, G) ≈ F_{p}Inj(Q, G′) as Out(Q) modules for all pgroups Q then F_{p}Cen(Q, G) ≈ F_{p}Cen(Q, G′) as Out(Q) modules for all Q.

If C_{G}(Q)/Z(Q) is a p′-group then, as an Out(Q) module, 1_{W_{G}(Q)}^{Out(Q)} = F_{p}Out(Q)\overline{W} where \overline{W} = ∑_{x∈N_{G}(Q)/Q} c_{x}. Hence for Q = P_{i}, Prop. 4.5, (4.1) and Cor. 4.3 imply

⨁ \overline{W}_{i}M_{i} = ⨁ \overline{W′}_{i}M_{i}

where W′_{i} = W_{G′}(P_{i}) and \overline{W′}_{i} is defined as in (4.4). Thus

∑ dim_{k}\overline{W}_{i}M_{i} = ∑ dim_{k}\overline{W′}_{i}M_{i}

and so m(X, BG) = m(X, BG′). It remains to prove 4.5.

Let nCen(Q, G) be the complement of Cen(Q, G) in Inj(Q, G). To study this set we shall consider subgroups H ≤ G such that H ≈ Q and C_{G}(H)/Z(H) is not a p′-group. Let H̃ be a Sylow p-subgroup of H ·C_{G}(H). 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 and let R(Q, Q̃_{j}) be the classes of injections α : Q ⟶ Q̃_{j} satisfying α(Q) = Q_{j}. Similarly let Aut(Q̃_{j}|Q_{j}) ≤ Aut(Q̃_{j}) be the subgroup of classes of automorphisms which send Q_{j} to Q_{j}. Then Aut(Q̃_{j}|Q_{j}) acts on R(Q, Q̃_{j}) and Cen(Q̃_{j}, G) by composition.

Lemma 4.6. There is an isomorphism of Out(Q) sets

ψ : ⨆_{j}R(Q, Q̃_{j})×_{Aut(Q̃_{j}|Q_{j})}Cen(Q̃_{j}, G) ⟶ nCen(Q, G),

natural in G, given by composition, i.e. ψ([β] × [α]) = [αβ].

Proof. It is easily checked that ψ is well defined. To see that ψ is injective let γ_{1} = [β_{1}]×[α_{1}], γ_{2} = [β_{2}]×[α_{2}] and suppose ψ(γ_{1}) = ψ(γ_{2}). Then up to conjugation in G, we may assume im(α_{1}β_{1}) = im(α_{2}β_{2}) ≤ P. Hence also up to conjugation, im(α_{1}β_{1}) · C_{G}(im(α_{1}β_{1})) = im(α_{2}β_{2}) · C_{G}(im(α_{2}β_{2})) and similarly for their Sylow p-subgroups. Thus the targets of β_{1} and β_{2} are isomorphic and hence equal. We denote the common target by Q̃_{j} and consider the diagram

Q ⟶^{β_{1}} Q̃_{j} ⟶^{α_{1}} G
id↓  ^{c_{g}} ↓
Q ⟶^{β_{2}} Q̃_{j} ⟶^{α_{2}} G

which commutes for some g ∈ G, i.e., β_{2} = α_{2}^{−1}c_{g}α_{1}β_{1}. Thus

[β_{2}] × [α_{2}] = [(α_{2}^{−1}c_{g}α_{1})β_{1}] × [α_{2}] = [β_{1}] × [α_{2}(α_{2}^{−1}c_{g}α_{1})] = [β_{1}] × [α_{1}].

For surjectivity we note that any injection f : Q ⟶ G of nCen(Q, G) factors uniquely as

f : Q ⟶^{π} { \atop f(Q)} ↪^{i} G

where π is an injection onto f(Q) and i is inclusion. Choose an equivalence u :

{ \atop f(Q)} ⟶^{≈} Q̃_{j} for some j and set β = uπ, α = iu^{−1}. Then β ∈ R(Q, Q̃_{j}), α ∈

Cen(Q̃_{j}, G) and ψ([β] × [α]) = [f]. □

Proof of Prop. 4.5. We proceed by downward induction on the order of Q. If Q = P then Cen(P, G) = Inj(P, G) and the result holds. If R(Q, Q̃_{j}) is non-empty then |Q̃_{j}| > |Q|, hence by induction we have an isomorphism

\begin{matrix}⨁_{j}F_{p}R(Q, Q̃_{j})⊗_{F_{p}Aut(Q̃_{j}|Q_{j})}F_{p}Cen(Q̃_{j}, G) ⟶^{≈} \\ ⨁_{j}F_{p}R(Q, Q̃_{j})⊗_{F_{p}Aut(Q̃_{j}|Q_{j})}F_{p}Cen(Q̃_{j}, G′). \end{matrix}

and the result follows from Lemma 4.6 since

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

and we have cancellation. □

1)⇒ 2). Let f : BG_{p}^{∧} ⟶ BG′_{p}^{∧} be a stable homotopy equivalence and let

f_{∗} : {BQ, BG_{p}^{∧}} → {BQ, BG′_{p}^{∧}}

be the induced map and let J_{τ} ⊆ {BQ, BG} be the Z_{p} module generated by maps of the form

BQ ⟶^{tr} BR ⟶^{Bh} BG

where R ≨ Q and h : R → G is a homomorphism. It follows from 2.4 that

F_{p} ⊗ {BQ_{+}, BG_{p}^{∧}_{+}}/J_{τ} ≅ F_{p}Rep(Q, G)

By the double coset formula, f_{∗}(J_{τ}) = J′_{τ}. Thus f induces an isomorphism

f_{∗} : F_{p}Rep(Q, G) ⟶^{≈} F_{p}Rep(Q, G′).

2)⇒ 3). Let R denote a set of representatives of the isomorphism classes of finite groups; let Surj(Q, R) ⊆ Rep(Q, R) be the set of conjugacy classes of surjective homomorphisms.

Lemma 4.7.

i) There is a natural isomorphism

ϕ : ⨆_{R∈R}Surj(Q, R)×_{Out(R)}Inj(R, G) ⟶ Rep(Q, G)

given by ϕ([β] × [α]) = [αβ]. ii) If Q is a p-group then R can be taken to be a set of representatives for the

isomorphism classes of finite p-groups.

Proof. i) The argument is entirely analogous to that of Lemma 4.6.

Part ii) is clear. □

Applying the functor F_{p} we have Corollary 4.8. There is a natural isomorphism of F_{p} modules

F_{p}ϕ : ⨁_{R∈R}F_{p}Surj(Q, R)⊗_{F_{p}Out(R)}F_{p}Inj(R, G) ⟶ F_{p}Rep(Q, G).

The implication 2)⇒ 3) now follows from Cor. 4.8 by upward induction on the order of Q, the case Q = e being trivial. □

A natural question arises if one has a stable map inducing one of the isomorphisms in Theorem 1.1. Is that map a stable homotopy equivalence? The answer is yes.

Theorem 4.9. Let f : BG_{p}^{∧} → BG′_{p}^{∧} be a stable map. Then the following are equivalent:

i) f_{∗} : F_{p}Rep(Q, G) → F_{p}Rep(Q, G′) is an Out(Q) isomorphism for all p– groups Q.

ii) f_{∗} : F_{p}Inj(Q, G) → F_{p}Inj(Q, G′) is an Out(Q) isomorphism for all p–

groups Q. iii) f^{∗} : H^{∗}(BG′; F_{p}) → H^{∗}(BG; F_{p}) is an isomorphism. iv) f is a stable homotopy equivalence.

Proof. (iii) and (iv) are equivalent by the Whitehead Theorem. (i) and (ii) are equivalent by an analogous argument to Corollary 4.8. (iv) implies (i) as in the proof of Theorem 1.1. Thus we need only show that (i) implies (iii).

Assume (i), Theorem 1.1 implies that BG ≃ BG′ at p. Therefore G and G′ have isomorphic Sylow p–subgroups. Let P be this common Sylow p–subgroup. By the Segal conjecture the elements of {BP_{+}, BG_{+}} are linear combinations of maps of the form

BP_{+} →^{tr} BQ_{+} →^{Bρ} BG_{+}

where Q is a p–subgroup of G and ρ a group homomorphism both determined up to conjugacy class. If we filter {BP_{+}, BG_{+}} by Q then the associated graded group satisfies

gr(F_{p} ⊗ {BP_{+}, BG_{+}}) ≈ ⨁_{(Q)} F_{p}(W_{P} (Q)\Rep(Q, G)).

where (Q) is a set of G–conjugacy-class representatives of p–subgroups. Therefore

f_{∗} : {BP_{+}, BG_{+}} → {BP_{+}, BG′_{+}}

is a bijection since by (i) gr f_{∗} is a bijection.

There exists an α ∈ {BP_{+}, BG_{+}} such that f_{∗}(α) = f ∘α = incl : BP_{+} → BG′_{+}. Since incl^{∗} is a monomorphism in cohomology so is f^{∗}. But BG ≃ BG′ at p from which it follows that the Poincaré series for H^{∗}(BG; F_{p}) and H^{∗}(BG′; F_{p}) are equal. Hence f^{∗} is an isomorphism. □

5. Groups with normal Sylow p-subgroups.

In this section we prove Theorem 1.5 and give an example illustrating the theorem. We also discuss the relationship between Theorems 1.1 and 1.5.

Suppose G has a normal Sylow p-subgroup P. By a theorem of Zassenhaus we can write G = P ⋊ H for a p′-group H. Thus we have

W_{G}(P) = P · H/P · C_{G}(P) ≈ H/H ∩ C_{G}(P)

and a p-local equivalence

(5.1)  BG ≃ B(G/H ∩ C_{G}(P)) = B(P ⋊ W_{G}(P)).

Proof of Theorem 1.5. For convenience set G_{1} = G, G_{2} = G′, P_{1} = P, P_{2} = P′. If BG_{1} ≃ BG_{2} then by Theorem 1.1 (3) we can assume P_{1} ≈ P_{2} (≈ P, say). Then for i = 1, 2, G_{i} = P ⋊ H_{i}, where H_{i} is a p′ group. Thus setting W_{i} = W_{G_{i}}(P) we have p-local equivalences BG_{i} ≃ B(P ⋊ W_{i}) by (5.1).

Let

e_{i} = ( ∑_{w∈W_{i}} w)/|W_{i}|

then e_{i} is an idempotent and B(P ⋊ W_{i}) ≃ e_{i}BP. Since BG_{1} ≃ BG_{2}, e_{1} ∼ e_{2} (e_{1} is conjugate to e_{2}) in {BP, BP }, the stable homotopy classes of self maps. Let R = Ẑ_{p} Out(P), by [MP1, Prop. 1.2] the inclusion

R ↪^{i} {BP, BP }

is split by a ring homomorphism retraction. Thus e_{1} ∼ e_{2} in R.

Now let S = ⟨ab − ba | a, b ∈ R⟩. Then R. Brauer has shown that

S = ⟨ ∑_{g∈Out(P)} a_{g}g | ∑_{g∈K_{j}} a_{g} = 0, for every j⟩

where K_{1}, ..., K_{m} are the conjugacy classes of elements in Out(P). It is immediate that x ∼ y in R implies x ≡ y mod S. Thus e_{1} − e_{2} ∈ S. Hence w_{1} ∼ w_{2}, for some w_{i} ∈ W_{i} for otherwise ∑_{g∈K_{j}} a_{g} ≠ 0 for all j. Further, since 1 ∈ Out(P) is in a singleton conjugacy class, |W_{1}| = |W_{2}|. Thus (e_{1} − w_{1}/|W_{1}|) − (e_{2} − w_{2}/|W_{2}|) ∈ S and by downward induction W_{1} is pointwise conjugate to W_{2} in Out(P).

Now assuming W_{1} and W_{2} are pointwise conjugate we claim M_{1} = 1_{W_{1}}^{Out(P)} and M_{2} = 1_{W_{2}}^{Out(P)} have the same Brauer character. This follows from the Mackey formula:

χ_{M_{i}} (s) = \frac{1}{|W_{i}|} ∑_{\begin{matrix}g^{−1}sg∈W_{i} \\ g∈Out(P) \end{matrix}} χ_{1_{W_{i}}} (g^{−1}sg) = \frac{|(s) ∩ W_{i}|}{|W_{i}|} s ∈ (Out(P))_{reg}

Since W_{i} is a p′-group, M_{i} = e_{i}R and is therefore projective. Hence e_{1}R is isomorphic to e_{2}R and thus e_{1} ∼ e_{2} in R and in {BP, BP }. Thus e_{1}BP ≃ e_{2}BP and so BG_{1} ≃ BG_{2}. □

Example 5.2. Let p, l be different primes and let V be an elementary abelian pgroup of rank l^{n}. For i = 1, 2, let H_{i} be non-isomorphic l-groups of exponent l and order l^{n} (e.g. for n = 3, l > 2, let H_{1} = (Z/l)^{n}, H_{2} = U_{3}(F_{l}), the 3 × 3 upper triangular matrices with 1’s on the diagonal). These groups act on V by means of the regular representation

H_{i} ↪^{reg} Σ_{l^{n}} ≤ GL_{l^{n}}(F_{p})

It follows that

G_{i} = V ⋊ H_{i}

are not isomorphic and satisfy O_{p′}(G_{i}) = 1. Furthermore H_{1} is pointwise conjugate to H_{2} in Σ_{l^{n}} and hence in GL(V) = Out(V). In fact the elements of H_{1} and H_{2} are all conjugate in Σ_{l^{n}} since each is an l^{n−1} fold product of disjoint l cycles and the conjugacy of elements is determined by their cycle structure. By Theorem 1.5, BG_{1} is stably equivalent to BG_{2} at p. It also follows that B(G_{1} × H_{2}) and B(G_{2} × H_{1}) are stably homotopy equivalent over the integers.

However, the completions BG_{1}_{p}^{∧}, BG_{2}_{p}^{∧} have different unstable homotopy types. We are indebted to Hans-Werner Henn and Nick Kuhn for pointing out this fact. To prove it we recall the following result of Adams, Miller-Wilkerson, and Lannes [L,HLS]: For G a compact Lie group and V an elementary abelian p-group, the correspondence f ↦ (Bf)^{∗} induces a bijection of End(V) sets

Rep(V, G) ≈ Hom_{K}(H^{∗}BG, H^{∗}BV)

where Rep(V, G) = Hom(V, G)/G with G acting by conjugation. Thus an isomorphism of unstable algebras H^{∗}BG_{i} would imply an isomorphism of End(V) sets Rep(V, G_{i}) = End(V)/H_{i} and thus an isomorphism of GL(V) sets GL(V)/H_{i}. However, this is false since H_{1} ≉ H_{2}.

Remark 5.3. We now examine how Theorem 1.5 follows from Theorem 1.1. In the former only a condition involving the Sylow p-subgroup is given whereas in the latter all Q ≤ P must be considered. This can be explained as follows:

Suppose G has a normal Sylow p-subgroup P then by (5.1) we can assume G = P ⋊ W for some W ≤ Out(P). Thus Rep(Q, G) = Rep(Q, P)/W, and we have

(5.4.) F_{p}(Inj(Q, G)) = F_{p}(Inj(Q, P)/W) = F_{p}(Inj(Q, P)) ⊗_{Out(P)} 1_{W}^{Out(P)}.

If both G and G′ have normal Sylow p-subgroups then Theorem 1.1(3) implies G ≈ P ⋊ W_{G}(P) and G′ ≈ P ⋊ W_{G′}(P). Setting Q = P in (5.4) we have the further implication

1_{W_{G}(P)}^{Out(P)} ≈ 1_{W_{G′} (P)}^{Out(P)}

which is equivalent to the pointwise conjugacy of W_{G}(P) and W_{G′}(P) by the proof of Theorem 1.5. Thus in this case, the hypothesis of Theorem 1.5 is equivalent to statement (3) of Theorem 1.1.

Proposition 5.7. For G, G′ arbitrary finite groups, if BG and BG′ are stably homotopy equivalent at p then the Sylow p–subgroups are isomorphic (say ≈ P) and the W_{G}(P) and W_{G′}(P) are pointwise conjugate in Out(P).

Proof. The Sylow p–subgroups are isomorphic by Nishida’s result (Theorem 1.2) and the Weyl groups are pointwise conjugate by the same argument as in the proof of Theorem 1.5. □

6. Cyclic Mod p Groups.

In this section we show that cyclic mod p groups are determined by their classifying spaces, and that in a sense these classifying spaces determine all other classifying spaces.

Proof of Theorem 1.7. Theorem 1.7 is a direct corollary of Theorem 1.5. If G and G′ are reduced cyclic mod p groups then G ≈ P ⋊C and G′ ≈ P ⋊C′, where P is the common (up to isomorphism) Sylow p-subgroup and C, C′ are cyclic p′-subgroups of Out(P). Hence C and C′ are pointwise conjugate in Out(P). So their generators are conjugate, and thus C and C′ are conjugate which completes the proof. □

Note that if G and G′ are not reduced then the localization of BG and BG′ at other primes than p will yield the order of the p′-cyclic quotients and the following corollary:

Corollary 6.1. Suppose G, G′ are unreduced cyclic mod p groups. Then BG ≃ BG′ stably (over the integers) if and only if G ≈ G′.

To show the uniqueness of the expression in the Minami-Webb formula we need to quote a result from Harris and Kuhn [HK].

Proposition 6.2 [HK]. Let P be a nontrivial finite p-group, and let {W_{i}}_{i∈I} and {W′_{j}}_{j∈J} be two finite families of p′-subgroups of Out(P). Then ⋁_{i∈I} B(P ⋊ W_{i}) ≃ ⋁_{j∈J} B(P ⋊ W′_{j}) if and only if ⨁_{i∈I} 1_{W_{i}}^{Out(P)} ≈ ⨁_{j∈J} 1_{W′_{j}}^{Out(P)}.

Proposition 6.2 follows from the fact that both equivalences are induced by the same idempotents. In [HK] the proposition is stated only for abelian p-groups, but it is true for all p-groups.

Definition 6.3. Let G be a finite group and p a prime.

(1) Let A_{p}(G) be the Burnside ring generated by isomorphism classes of finite G-sets X such that X^{H} = ∅ if H is not a p′-subgroup of G, with addition given by disjoint union.

(2) Let R_{p}(G) be the modular representation ring with generators [M], where M is a finitely generated F_{p}[G]-module, and relations [M] = [M_{1}] + [M_{2}] if there is a short exact sequence 0 → M_{1} → M → M_{2} → 0.

(3) Let ψ_{G} : A_{p}(G) ⟶ R_{p}(G) be the map defined by ψ_{G}(X) = F_{p}[X].

N.B. A_{p}(G) is generated by G/H where H is a p′-subgroup and ψ_{G}(G/H) = 1_{H}^{G}.

Proposition 6.4. For any finite group G, ψ_{G} is injective on the linear span of ⟨G/H | H is a cyclic p′-subgroup of G⟩

Proof. There is a commutative diagram

A_{p}(G) ⟶^{ψ_{G}} R_{p}(G)
↓^{χ_{1}}  ↓^{χ_{2}}
∏_{(H)≤G} Z ⟶^{Δ} ∏_{(g)∈G} C

where the left product runs over the conjugacy classes of p′-subgroups of G, the right product runs over the conjugacy classes of p-regular elements of G, χ_{2} is the usual Brauer character assignment, χ_{1}(H) = |X^{H}|, and Δ is the trivial map on non-cyclic H, and the diagonal map Z → ∏_{⟨g⟩=H} C if H is cyclic.

Let A(G) be the Burnside ring generated by isomorphisms classes of finite G-sets X and extend χ_{1} by χ_{1}(K) = |X^{K}| where K ≤ G.

χ_{1} ⊗ C : A(G) ⊗ C ⟶^{≈} ∏_{(K)≤G} C,

where the product runs over all conjugacy classes of subgroups of G, is an isomorphism [B; Thm. 5.4.2]. A_{p}(G) is a subring of A(G) so the restriction χ_{1} : A_{p}(G) ⟶ ∏_{(H)≤G} Z is a monomorphism. By definition Δ is injective on the span of conjugacy classes of cyclic p′-subgroups of G. Thus the composite is injective and hence χ_{G} is injective on the span of conjugacy classes of cyclic p′-subgroups of G. □

Proof of Theorem 1.8. If

⋁_{i} α_{i}BH_{i}_{p}^{∧} ≃ ⋁_{j} α′_{j}BH′_{j}_{p}^{∧},

where H_{i}, H′_{j} are reduced cyclic mod p groups. Then by grading the Grothendieck group of classifying spaces by Sylow p-subgroups we may assume that the H_{i}’s and the H′_{j}’s have isomorphic Sylow p-subgroups. By Proposition 6.4 the p′-cyclic permutation modules are linearly independent, and by Proposition 6.2 so are the classifying spaces of reduced cyclic mod p groups G with isomorphic p–subgroups. □

We have now established that BG can be expressed as a unique linear combination of classifying spaces of cyclic mod p groups. So in a very real sense one can say that BG is determined by G’s cyclic mod p subgroups and their inclusion lattice as expressed by the Möbius function.

Example 6.5. The Minami–Webb formula at the prime 2 for the classifying space of the symmetric group on 4 letters is

BΣ_{4} = BD_{8} − \frac{1}{2}B(Z/2)^{2} + \frac{1}{2}BA_{4},

where Σ_{4} is the symmetric group, D_{8} is the dihedral group of order 8, and A_{4} is the alternating group on 4 letters.

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John Martino  Stewart Priddy
Department of Mathematics  Department of Mathematics
University of Virginia  Northwestern University
Charlottesville, VA 22903  Evanston, IL 60208

current address Department of Mathematics and Statistics Western Michigan University Kalamazoo, MI 49008