The integral cohomology rings of some p-groups I. J. Leary Trinity College, Cambridge

1. Introduction. We determine the integral cohomology rings of an infinite family of p-groups, for odd primes p, with cyclic derived subgroups. Our method involves embedding the groups in a compact Lie group of dimension one, and was suggested independently by P. H. Kropholler and J. Huebschmann. This construction has also been used by the author to calculate the mod-p cohomology of the same groups and by B. Moselle to obtain partial results concerning the mod-p cohomology of the extra special p-groups [7], [9].

2 The method and the groups. Given a finite group G and a central cyclic subgroup C, we fix an embedding of C into S^{1}, and define a Lie group G̃ as the product of S^{1} and G amalgamating C, that is

G̃ = S^{1} × G/{(c^{−1}, c) : c ∈ C}

Then we have a commutative diagram:

C ⟶ G ⟶ Q
↓  ↓  ↓
S^{1} ⟶ G̃ ⟶ Q.

If M is a G-module on which C acts trivially, we may consider M as a G̃-module by letting S^{1} act trivially, and the Lyndon-Hochschild-Serre spectral sequence for the second extension is often simpler than that for the first. To find H^{∗}(BG; M), given H^{∗}(BG̃; M), we use the Serre spectral sequence of the fibration

S^{1}/C ≅ G̃/G⟶BG⟶BG̃.

This spectral sequence has E_{2}^{i,j} = 0 for j > 1, so the only possible non-zero differential is d_{2}. The above was first suggested to the author by P. Kropholler. A similar idea occurs in J. Huebschmann’s papers [5] and [6]. In the case where M is a commutative ring on which G acts trivially, it appears that we may obtain another filtration of H^{∗}(BG; M) by examining the Eilenberg-Moore spectral sequence for the pullback square:

BG ⟶ BG̃
↓  ↓
{∗} ⟶ BG̃/G,

but it can be shown that the two filtrations are identical. These spectral sequences are just alternative ways to view the Gysin sequence for the S^{1}-bundle BG over BG̃.

It is possible for non-equivalent extensions of C by Q to yield equivalent extensions of S^{1} by Q. In fact this happens if and only if their extension classes in H^{2}(Q; C) map to the same element of H^{3}(Q; ℤ) under the Bockstein associated with the coefficient sequence ℤ ↣ ℤ ↠ C.

The groups we shall consider are central extensions of C_{p^{n−2}} by C_{p} ⊕ C_{p} where p is an odd prime, and n ≥ 3. They may be presented as

P (n) = ⟨A, B, C | A^{p} = B^{p} = C^{p^{n−2}} = [A, C] = [B, C] = 1, [A, B] = C^{p^{n−3}}⟩.

We shall let P̃ be the corresponding central extension of S^{1} by C_{p} ⊕ C_{p}, that is the group obtained from S^{1} ×P (n) by amalgamating the subgroup of P (n) generated by ⟨C⟩ and the C_{p^{n−2}} subgroup of S^{1}. There are four central extensions of C_{p^{n−2}} by C_{p} ⊕ C_{p}; two abelian ones, P (n), and a metacyclic group M(n) containing a cyclic subgroup of index p. This may be checked by verifying that the action of Aut(C_{p} ⊕C_{p}) on H^{2}(B(C_{p} ⊕C_{p}); C_{p^{n−2}}) has only four orbits, and then explicitly constructing four non-isomorphic groups. There are however only two central extensions of S^{1} by C_{p} ⊕C_{p}; the direct product, which is abelian, and P̃ which is not. This follows from the fact that Aut(C_{p} ⊕ C_{p}) acts transitively on the non-zero elements of H^{3}(B(C_{p} ⊕ C_{p}); ℤ), which may be identified with H^{2}(B(C_{p} ⊕ C_{p}); S^{1}) via the Bockstein associated with the coefficient sequence ℤ ↣ ℝ ↠ S^{1}. Hence we see that BM(n) is also an S^{1}-bundle over BP̃, and in fact H^{∗}(BM(n); ℤ) could easily be determined from the results of this paper. This cohomology ring has already been calculated using other methods [11].

3. Calculations. We now begin our calculation of H^{∗}(BP̃) by examining the spectral sequence with integer coefficients for P̃ considered as an extension of S^{1} by C_{p} ⊕ C_{p}. The E_{2} page is readily seen to be generated by elements α, β ∈ E_{2}^{2,0}, γ ∈ E_{2}^{3,0} and τ ∈ E_{2}^{0,2} subject only to the relations pα = pβ = 0, pγ = 0 and γ^{2} = 0. Note that τ has infinite order. Since E_{2}^{i,j} is trivial for j odd, we see that all the even differentials must vanish. The behaviour of the differentials is summarised in the following lemma.

Lemma 1. In the above spectral sequence there are exactly two non-zero differentials, d_{3} and d_{2p−1}. d_{3}(τ) is a non-zero multiple of γ, and E_{4} is generated by the classes of the elements α, β, pτ, …, pτ^{p−1}, τ^{p} and τ^{p−1}γ. All of these generators are universal cycles except for τ^{p−1}γ, which is mapped by d_{2p−1} to a non-zero multiple of α^{p}β − β^{p}α. The E_{∞} page is generated by the elements α, β, pτ, …, pτ^{p−1}, τ^{p} subject only to the relations they satisfy as elements of E_{2}, and the relation α^{p}β = β^{p}α.

Proof. The derived subgroup of P̃ consists of the subgroup of its central S^{1} of order p, so there can be no homomorphism from P̃ to S^{1} that restricts to an isomorphism from the centre to S^{1}. It follows by considering the natural isomorphism H^{2}(BG; ℤ) ≅ Hom(BG, S^{1}) that the element τ cannot survive to E_{∞}, so we must have d_{3}(τ) a non-zero multiple of γ. This determines d_{3} completely. It may be checked that E_{4} is isomorphic to the subring of E_{2} generated by α, β, pτ, …, pτ^{p−1}, τ^{p} and τ^{p−1}γ. All these elements must be universal cycles, with the possible exception of τ^{p−1}γ, because the groups in which their images under d_{n} lie are already trivial. The only remaining potentially non-zero differential is d_{2p−1}(τ^{p−1}γ). To complete this proof it suffices to show that in the E_{∞} page the relation α^{p}β = β^{p}α must hold.

Let Q be the quotient of P̃ by its S^{1} subgroup, and take generators α′, β′ for H^{2}(BQ; ℤ) and γ′ for H^{3}(BQ; ℤ). The statement that γ does not survive to E_{∞} in the spectral sequence is equivalent to the statement that γ′ is mapped to zero by the inflation map from Q to P̃. Now we calculate ϕ(γ′), where ϕ is the integral cohomology operation δ_{p}P^{1}π_{∗}, where π_{∗} is the map induced by the change of coefficients from ℤ to 𝔽_{p}, P^{1} is a reduced power, and δ_{p} is the Bockstein for the sequence ℤ ↣ ℤ ↠ 𝔽_{p}. Taking y, y′ ∈ H^{1}(BQ; 𝔽_{p}) such that δ_{p}(y) = α′, and δ_{p}(y′) = β′, we see that

ϕ(γ′) = δ_{p}P^{1}π_{∗}(γ′) = δ_{p}P^{1}(β_{p}(y)y′ − β_{p}(y′)y) = δ_{p}(β_{p}(y)^{p}y′ − β_{p}(y′)^{p}y) = α^{′p}β′ − β^{′p}α′.

It follows that

α^{p}β − β^{p}α = inf(α^{′p}β′ − β^{′p}α′) = inf(ϕ(γ′)) = ϕ inf(γ′) = 0.

We note that the result on d_{2p−1} could also be considered as a case of an integral version of Kudo’s transgression theorem. We are now ready to state our theorem on H^{∗}(P̃).

Theorem 2. Let p be an odd prime, and let P̃ be the group defined above. Then H^{∗}(P̃; ℤ) is generated by elements α,β,χ_{1}, …, χ_{p−1}, ζ, with

deg(α) = deg(β) = 2 deg(χ_{i}) = 2i deg(ζ) = 2p,

subject to the following relations:

pα = pβ = 0
α^{p}β = β^{p}α

αχ_{i} = \{ \begin{matrix}0 \\ −α^{p} \end{matrix} βχ_{i} = \{ \begin{matrix}0 & for i < p − 1 \\ −β^{p} & for i = p − 1 \end{matrix}

χ_{i}χ_{j} = \{ \begin{matrix}pχ_{i+j} & i + j < p \\ p^{2}ζ & i + j = p \\ pζχ_{i+j−p} & p < i + j < 2p − 2 \\ pζχ_{p−2} + α^{2p−2} + β^{2p−2} − α^{p−1}β^{p−1} & i = j = p − 1 \end{matrix}

Chern classes of representations of P̃ generate the whole ring. An automorphism of P̃ sends χ_{i} to χ_{i} (resp. (−1)^{i}χ_{i}) and ζ to ζ (resp. −ζ) if it fixes (resp. reverses) S^{1}. The effect of an automorphism on α, β may be determined from their definition. Considered as elements of Hom(P̃, S^{1}), α has kernel ⟨S^{1}, B⟩ and sends A to e^{2πi/p}, and β has kernel ⟨S^{1}, A⟩ and sends B to e^{2πi/p}. If we let H be the subgroup generated by B and elements of S^{1} we may define

χ_{i} = \{ \begin{matrix}Cor_{P} ; H(τ^{′i}) & for i < p − 1 \\ Cor_{P} ; H(τ^{′p−1}) − α^{p−1} & for i = p − 1 \end{matrix}

where τ′ is any element of H^{2}(H; ℤ) restricting to S^{1} as the generator τ. Similarly, ζ = c_{p}(ρ), where ρ is an irreducible representation of P̃ restricting to S^{1} as p copies of the representation ξ with c_{1}(ξ) = τ.

Proof. First we note that in the E_{∞} page of the above spectral sequence all the group extensions that we need to examine are extensions of finite groups by the infinite cyclic group, so are split. The elements α and β defined in the statement above clearly yield generators for E_{∞}^{2,0}, and the relations between them are exactly the relations that hold between the corresponding elements in the spectral sequence. Let β′ in H^{2}(BH) be the restriction to H of β, and take any choice of τ′ as in the statement. We may show by considering β′ and τ′ as homomorphisms from H to S^{1} that conjugation by A^{i} induces the map on H^{2}(BH) that fixes β′ and sends τ′ to τ′ − iβ′. Now applying the formula for Res_{K}^{G}Cor_{H}^{G} (see for example [4]) it follows that χ_{i} restricts to S^{1} as pτ^{i}, so yields a generator for E_{∞}^{0,2i}.

Any irreducible representation of P̃ has degree 1 or p, because P̃ has an abelian subgroup of index p. Let ρ be the representation of P̃ induced from a 1-dimensional representation of H with first Chern class τ′. ρ restricts to S^{1} as p copies of the representation with first Chern class τ, so its total Chern class restricts to S^{1} as (1 + τ)^{p}, and so c_{p}(ρ) yields a generator for E_{∞}^{0,2p}, and c_{i}(ρ) = 1/p\binom{p}{i}χ_{i} + P_{i}(α, β) for some polynomial P_{i}. We shall show later that P_{i} = 0.

The restriction to H of α is trivial, so by Fröbenius reciprocity

αCor_{P} ; H(τ^{′i}) = Cor_{P} ; H(Res_{P} ; H(α)τ^{′i}) = 0,

and the expressions given for αχ_{i} follow. By calculating α(βχ_{i}) = β(αχ_{i}), we may deduce that βχ_{i} = 0 for i < p−1 and βχ_{p−1} = λ(α^{p−1}β −β^{p})−α^{p−1}β for some scalar λ. To show that λ = 1 we use the restriction map to H, and the formula for corestriction followed by restriction.

\begin{aligned}Res_{P} ; H(βχ_{p−1}) & = β′ ∑_{i=0}^{p−1}(τ′ + iβ′)^{p−1} \\ & = β′ ∑_{j=0}^{p−1} τ^{′p−1−j}β^{′j} ∑_{i=0}^{p−1} i^{j} \end{aligned}

Newton’s formula tells us that

∑_{i=1}^{p−1} i^{j} ≡ \{ \begin{matrix}0 (p) & for j ≢ 0 (p − 1) \\ 1 (p) & for j ≡ 0 (p − 1) \end{matrix}

so Res_{P} ; H(βχ_{p−1}) = −β^{′p}, and the required relation follows.

We now know Res_{P} ; S^{1}(χ_{i}χ_{j}), αχ_{i}χ_{j}, and βχ_{i}χ_{j}, which together imply the relations given for χ_{i}χ_{j}. To complete the proof of the theorem we must determine the effect of automorphisms of P̃ on the χ_{i}. We know that an automorphism sends c_{i}(ρ) to itself or (−1)^{i} times itself depending whether or not it reverses the sense of S^{1}, so it will suffice to show that χ_{i} = 1/p\binom{p}{i}c_{i}(ρ). The character of ρ is zero except on S^{1}, so if θ is a 1dimensional representation of P̃ restricting trivially to S^{1}, then ρ ⊗ θ is isomorphic to ρ. If we apply the formula expressing c.(ρ ⊗ θ) in terms of c.(ρ) and c.(θ) (see [3]) we obtain

c_{i}(ρ) = c_{i}(ρ ⊗ θ) = ∑_{j=0}^{i} \binom{p − i + j}{j}c_{1}(θ)^{j}c_{i−j}(ρ).

and hence inductively

c_{i}(ρ)c_{1}(θ) = \{ \begin{matrix}0 & for i < p − 1 \\ −c_{1}(θ)^{p} & for i = p \end{matrix} .

Since α and β are possible values for c_{1}(θ) the required result follows. We may show inductively that χ_{i} is in the subring generated by Chern classes because χ_{1} is, and χ_{1}χ_{i−1}, 1/p\binom{p}{i}χ_{i} are coprime multiples of χ_{i}.

We are now ready to state our theorem on the integral cohomology of BP (n).

Theorem 3. Let p be an odd prime and let P (n) be as defined above. Then H^{∗}(BP (n); ℤ) is generated by elements α, β, μ, ν, χ_{1}, …, χ_{p−1}, ζ, with

deg(α) = deg(β) = 2 deg(μ) = deg(ν) = 3 deg(χ_{i}) = 2i deg(ζ) = 2p

subject to the following relations:

pα = pβ = 0 pμ = pν = 0 p^{n−3}χ_{1} = 0 p^{n−2}χ_{i} = 0 p^{n−1}ζ = 0
αμ = βν
α^{p}β = β^{p}α α^{p}μ = β^{p}ν

αχ_{i} = \{ \begin{matrix}0 \\ −α^{p} \end{matrix} βχ_{i} = \{ \begin{matrix}0 & for i < p − 1 \\ −β^{p} & for i = p − 1 \end{matrix}

μχ_{i} = \{ \begin{matrix}0 \\ −β^{p−1}μ \end{matrix} νχ_{i} = \{ \begin{matrix}0 & for i < p − 1 \\ −α^{p−1}ν & for i = p − 1 \end{matrix}

χ_{i}χ_{j} = \{ \begin{matrix}pχ_{i+j} & i + j < p \\ p^{2}ζ & i + j = p \\ pζχ_{i+j−p} & p < i + j < 2p − 2 \\ pζχ_{p−2} + α^{2p−2} + β^{2p−2} − α^{p−1}β^{p−1} & i = j = p − 1 \end{matrix}

μν = \{ \begin{matrix}0 & for n > 3 \\ λχ_{3} & for n = 3, p > 3, λ ∈ ℤ_{p}^{×} \\ 3λζ & for n = 3, p = 3, λ = ±1 \end{matrix}

Chern classes of representations of P (n) generate H^{even}(BP (n); ℤ). Under an automorphism of P (n) which restricts to the centre as C ↦ C^{j}, χ_{i} is mapped to j^{i}χ_{i}, and ζ is mapped to j^{p}ζ. The effect of automorphisms on α and β is determined by the natural isomorphism H^{2}(BG; ℤ) ≅ Hom(G, ℝ/ℤ), under which

\begin{matrix}α : A & ↦ 1/p \\ B & ↦ 0 \\ C & ↦ 0 \end{matrix} \begin{matrix}β : A & ↦ 0 \\ B & ↦ 1/p \\ C & ↦ 0 \end{matrix} \begin{matrix}χ_{1} : A & ↦ 0 \\ B & ↦ 0 \\ C & ↦ 1/p^{n−3}. \end{matrix}

An automorphism of P (n) which sends α to n_{1}α+n_{2}β, β to n_{3}α+n_{4}β and restricts to the centre as C ↦ C^{j} sends μ to j(n_{4}μ+n_{3}ν) and ν to j(n_{2}μ+n_{1}ν). If γ′ in H^{2}(B⟨B, C⟩; ℤ) is such that it maps to the following element of Hom(⟨B, C⟩, ℝ/ℤ)

\begin{matrix}γ′ : B & ↦ 0 \\ C & ↦ 1/p^{n−2} \end{matrix} ,

then χ_{i} is defined as follows:

χ_{i} = \{ \begin{matrix}Cor_{⟨B,C⟩}^{P (n)}(γ^{′i}) & for i < p − 1 \\ Cor_{⟨B,C⟩}^{P (n)}(γ^{′p−1}) − α^{p−1} & for i = p − 1. \end{matrix}

These are, up to scalar multiples, equal to c_{i}(ρ), where ρ is a p-dimensional irreducible representation of P (n), whose restriction to ⟨C⟩ is a sum of p copies of the representation θ, with c_{1}(θ) = Res_{⟨C⟩}^{⟨B,C⟩}(γ′). In fact, c_{i}(ρ) = 1/p\binom{p}{i}χ_{i}. Also, we may define ζ = c_{p}(ρ).

Proof. We examine the spectral sequence for BP (n) as an S^{1}-bundle over BP̃. E_{2}^{∗,0} is isomorphic to H^{∗}(BP (n); ℤ) and E_{2}^{∗,∗} is freely generated by E_{2}^{∗,0} and an element ξ of infinite order in E_{2}^{0,1}. We know that H^{2}(BP (n)) ≅ Hom(P (n), S^{1}) ≅ C_{p^{n−3}} ⊕ C_{p} ⊕ C_{p}, so d_{2}(ξ) must be ±p^{n−3}χ_{1}. If we wanted to calculate the cohomology of the metacyclic groups M(n) described above, the differential in this spectral sequence would send ξ to ±p^{n−3}χ_{1} + γ for some non-zero γ in ⟨α, β⟩. It is now easy to see that E_{∞} is generated by the elements α, β, μ = βξ, ν = αξ, χ_{1}, …, χ_{p−1} and ζ subject to the relations they satisfy as elements of E_{2}^{∗,∗} together with p^{n−3}χ_{1} = 0, p^{n−2}χ_{i} = 0, and p^{n−1}ζ = 0. For each m, the filtration of H^{m}(BP (n)) given by the E_{∞} page is trivial, so we may use the same symbols to denote elements of H^{m}(BP (n)), and the relations that hold in E_{∞} determine all the relations that hold in H^{m}(BP (n)) except for the product of the two odd dimensional generators.

We know that pμν = 0, and the relation αμ = βν implies that αμν = βμν = 0, and so μν must be a multiple of p^{n−3}χ_{3} for p ≥ 5 (resp. 3ζ for p = 3). Note that these elements restrict to zero on all proper subgroups of P (n). In the case of P (3), Lewis [8] shows that μν is not zero by considering the spectral sequence for P (n) considered as an extension of a maximal subgroup by C_{p}. A similar method will work in general, but we offer an alternative proof that involves expressing μ and ν as Bocksteins of elements of H^{2}(BP (n); 𝔽_{p}). This proof is contained in lemma 4 and corollary 5.

The effect of automorphisms on χ_{i} and ζ is easily seen to be as claimed from their alternative definitions as Chern classes. To determine the effect of automorphisms on μ and ν, we note that an automorphism of P (n) restricting to the centre as C ↦ C^{j} extends to an endomorphism of P̃ which wraps the central circle j times around itself, so induces a map of the above spectral sequence to itself sending ξ to jξ. This completes the proof of theorem 3 modulo lemma 4 and its corollary.

We now examine the spectral sequence with 𝔽_{p} coefficients for the central extension C_{p^{n−2}} ↣ P (n) ↠ C_{p}⊕C_{p}. Take generators so that H^{∗}(BC_{p} ⊕ C_{p}; 𝔽_{p}) ≅ 𝔽_{p}[x, x′]⊗Λ[y, y′], where β_{p}(y) = x, β_{p}(y′) = x′, and H^{∗}(BC_{p^{n−2}}; 𝔽_{p}) ≅ 𝔽_{p}[t]⊗Λ[u], where β_{p}(u) = t for n = 3 (resp. β_{p}(u) = 0 for n ≥ 4). Then the E_{2} page is isomorphic to 𝔽_{p}[x, x′, t] ⊗ Λ[y, y′, u], and the first two differentials are as described in the following lemma.

Lemma 4. With notation as above, identify the elements x, x′, y, y′ in the spectral sequence with their images in H^{∗}(BP (n); 𝔽_{p}) under the inflation map.

1) Let n ≥ 4. Then d_{2} is trivial, and d_{3}(t) is a non-zero multiple of xy′ − x′y. The set {x, x′, yy′, u′y, u′y′} is a basis for H^{2}(BP (n)), where u′ is any element of H^{1}(BP (n)) restricting to C_{p^{n−2}} as u.

2) Let n = 3. Then d_{2}(u) is a non-zero multiple of yy′, d_{2}(t) = 0, and E_{3} is generated by y, y′, x, x′, [uy], [uy′] and t subject to the relation yy′ = 0 and those implied by the relations in E_{2}. In particular [uy]y′ = −[uy′]y but this element is nonzero. As in the case n ≥ 4, d_{3}(t) is a non-zero multiple of xy′ − x′y. Let Y, Y′ be elements of H^{2}(BP (n)) such that {x, x′, Y, Y′} is a basis for H^{2}(BP (3)), and let X = β_{p}(Y), X′ = β_{p}(Y′). Then {yY′, xy, xy′, x′y′, X, X′} is a basis for H^{3}(BP (3)) and {xX, xX′, x′X′, x^{2}y, x^{2}y′, xx′y′, x^{′2}y′, Y X′} is a basis for H^{5}(BP (3)).

Proof. 1). In this case H^{1} has order p^{3}, so u must survive. The element xy′ − x′y is the image under π_{∗} of a generator for H^{3}(B(C_{p} ⊕ C_{p}); ℤ), so must be killed by some differential. We have already shown that it cannot be killed by d_{2}, so the only possibility is that t survives until E_{3} and kills it. The rest of the statement follows easily.

2). In this case H^{1} has order p^{2}, so d_{2}(u) must be non-zero. It is true in general that if G is a central extension of C_{p} by Q, then in the corresponding spectral sequence with 𝔽_{p} coefficients d_{2} : E_{2}^{0,1} → E_{2}^{2,0} must kill the extension class. This follows by naturality, since one may regard the extension class as defining a homotopy class of maps from BQ to K(C_{p}, 2) such that BG is the BC_{p}-bundle induced by the path-loop fibration over K(C_{p}, 2). Since all subgroups of P (3) of order p^{2} are copies of C_{p} ⊕ C_{p}, the extension class of P (3) must restrict to zero on all cyclic subgroups, so must be a multiple of yy′. The transgression commutes with the Bockstein so d_{2}(t) = 0 and d_{3}(t) = β_{p}d_{2}(u).

Given the values of these differentials it is routine to compute the E_{4} page of the spectral sequence. If we write E_{r}^{n} = ⨁_{i+j=n} E_{r}^{i,j}, then {[uy], [uy′], x, x′} forms a basis for E_{4}^{2} = E_{∞}^{2}, and {[ty], [ty′], [uy]y′, xy, xy′, x′y′} forms a basis for E_{4}^{3} = E_{∞}^{3}. The spectral sequence operation _{F} β introduced by Araki [2] and Vasquez [12] maps [uy] to [ty] and [uy′] to [ty′], so if Y and Y′ are chosen to yield the generators for E_{4}^{1,1} their Bocksteins yield generators for E_{4}^{1,2}. A basis for E_{4}^{5} is given by the eight elements of the statement, which we know to be universal cycles, and the elements [t^{2}y], [t^{2}y′]. E_{4}^{4} consists of universal cycles, and the universal coefficient theorem tells us that H^{5} has order p^{8}, so [t^{2}y] and [t^{2}y′] cannot be universal cycles.

Corollary 5. In H^{∗}(BP (n); ℤ) the product μν is non-zero if and only if n = 3.

Proof. In the notation of lemma 4 it suffices to determine δ_{p}(u′y)δ_{p}(u′y′) in the case n ≥ 4, and δ_{p}(Y)δ_{p}(Y′) in the case n = 3. In the case when n = 3,

δ_{p}(Y)δ_{p}(Y′) = δ_{p}(Y β_{p}(Y′)) = δ_{p}(Y X′).

The kernel of δ_{p} : H^{5}(BP (3); 𝔽_{p}) → H^{6}(BP (3); ℤ) is equal to π_{∗}(H^{5}(BP (3); ℤ)), which is generated by xX, xX′ and x′X′, so by lemma 4 δ_{p}(Y X′) is non-zero.

In the case when n = 4, H^{i}(BP (4); ℤ) has exponent p for i = 2, 3, so π_{∗} is injective from these groups, and ker β_{p} : H^{2}(BP (4)) → H^{3}(BP (4)) is equal to β_{p}(H^{1}(BP (4))). β_{p}(yy′) = xy′ −x′y = 0, so we may choose the element u′ in lemma 4 so that β_{p}(u′) = λyy′ for some non-zero λ. Then we have

δ_{p}(u′y)δ_{p}(u′y′) = δ_{p}(u′yβ_{p}(u′y′)) = δ_{p}(u′y(λyy′y′ − u′x′)) = 0.

The case when n ≥ 5 is similar but simpler, since u′ may be chosen so that δ_{p}(u′) = p^{n−4}χ_{1}, which implies that β_{p}(u′) = 0.

Remarks. Theorem 3 contains independent proofs of Thomas’ result that the even degree subring of H^{∗}(BP (n); ℤ) is generated by Chern classes [10], and Lewis’ calculation of H^{∗}(BP (3); ℤ). Our notation differs slightly from that of Lewis. We have renumbered the generators χ_{i} (note that χ_{1} vanishes for n = 3). Also our χ_{p−1} and Lewis’ χ_{p−2} are related by the formula

χ_{p−2}^{Lewis} = χ_{p−1} + α^{p−1} + β^{p−1}.

Our result disagrees with that of AlZubaidy [1].

The author wishes to thank his supervisor C. B. Thomas for encouragement with this work, P. H. Kropholler for suggesting the Lie group construction, and J. Huebschmann for helpful comments concerning an earlier version of this work.

REFERENCES

[1] K. ALZUBAIDY. Rank 2 p-groups, p > 3, and Chern classes. Pacific J. Math. 103 (1982), 259–267.

[2] S. ARAKI. Steenrod reduced powers in the spectral sequences associated with a fibering I, II. Mem. Fac. Sci. Kyusyu Univ. Series (A) Math. 11 (1957), 15–64 81–97.

[3] M. F. ATIYAH. Characters and the cohomology of finite groups. Publ. Math. IHES 9 (1961), 23–64.

[4] K. S. BROWN. Cohomology of groups. Springer Verlag, (1982).

[5] J. HUEBSCHMANN. Perturbation theory and free resolutions for nilpotent groups of class 2. J. of Algebra (to appear).

[6] J. HUEBSCHMANN. Cohomology of nilpotent groups of class 2. J. of Algebra (to appear).

[7] I. J. LEARY. The mod-p cohomology rings of some p-groups. In preparation.

[8] G. LEWIS. The integral cohomology rings of groups of order p^{3}. Trans. Amer. Math. Soc. 132 (1968), 501–529.

[9] B. MOSELLE. Calculations in the cohomology of finite groups. Unpublished essay, (1988).

[10] C. B. THOMAS. Riemann-Roch formulae for group representations. Mathematika 20 (1973), 253–262.

[11] C. B. THOMAS. Characteristic classes and the cohomology of finite groups. Cambridge University Press, (1986).

[12] R. VASQUEZ. Nota sobre los cuadrados de Steenrod en la sucesion espectral de un espacio fibrado. Bol. Soc. Mat. Mexicana 2 (1957), 1–8.