ESSENTIAL INTEGRAL COHOMOLOGY

CLASSES OF ppp-GROUPS

Pham Anh Minh

Abstract. Let p be a prime number and let G be a p-group which is not elementary abelian. For every integral cohomology class ξ of G which restricts trivially to all proper subgroups, we show that ξ^{p} = 0 if p > 2, and ξ^{4} = 0 if p = 2. This result is applied to get an upper bound, which is \frac{|G|}{p}, for the nilpotence degrees of nilpotent integral cohomology classes of G.

1. Statement of main result

Let p be a prime number. For every finite group G, denote by H^{∗}(G) (resp. H^{∗}(G, ℤ/p)) the integral (resp. mod-p) cohomology algebra of G. Let Ess(G) (resp. Ess_{p}(G)) be the ideal of H^{∗}(G) (resp. H^{∗}(G, ℤ/p)) consisting of essential elements, i.e. elements which restrict trivially to all proper subgroups of G. Assume from now on that G is a p-group which is not elementary abelian, unless otherwise specified. The following was proved in [2].

Theorem A.

(i) If ζ ∈ Ess_{p}(G), then ζ^{p} = 0;

(ii) For every nilpotent element ψ of H^{∗}(G, ℤ/p), ψ^{p^{log_{p} |G|−1}} = 0 for p = 2, and ψ^{p^{log_{p} |G|−2}} = 0 for p > 2. □

The purpose of this note is to get a similar result in the case of integral coefficients, as follows.

Theorem B.

(i) If ξ ∈ Ess(G), then ξ^{p} = 0 for p > 2, and ξ^{4} = 0 for p = 2; (ii) For every nilpotent element ϕ of H^{∗}(G), ϕ^{p^{log_{p} |G|−1}} = 0.

2. Preliminaries

Let Φ(G) = G^{p}[G, G] be the Frattini subgroup of G. V = G/Φ(G) is then a vector space over 𝔽_{p}. Set n = dim_{𝔽_{p}} V. There exists then a set of generators

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20J06.

consisting of n elements a_{1}, …, a_{n} of G. For 1 ≤ i ≤ n, define x_{i} ∈ H^{1}(G, ℤ/p) = Hom(G, ℤ/p) by

x_{i}(a_{j}) = δ_{ij}, 1 ≤ j ≤ n,

and set y_{i} = βx_{i} with β the Bockstein homomorphism. With some abuse of notation, we also denote by x_{i} (resp. y_{i}), 1 ≤ i ≤ n, the preimage of x_{i} (resp. y_{i}) under the inflation map Inf : H^{∗}(V, ℤ/p) → H^{∗}(G, ℤ/p). We have then

H^{∗}(V, ℤ/p) = \{ \begin{matrix}𝔽_{2}[x_{1}, …, x_{n}] & for p = 2, \\ Λ[x_{1}, …, x_{n}] ⊗ 𝔽_{p}[y_{1}, …, y_{n}] & for p > 2. \end{matrix}

Let δ^{∗} : H^{∗}(−, ℤ/p) → H^{∗+1}(−) be the connecting homomorphism corresponding to the exact sequence of coefficients

0 → ℤ →^{p} ℤ →^{pr} ℤ/p → 0

and set v_{i} = δ^{∗}(x_{i}), 1 ≤ i ≤ n. It is known (see e.g. [1]) that pr^{∗}(v_{i}) = y_{i}, 1 ≤ i ≤ n, and H^{2}(V) contains R_{n} = 𝔽_{p}[v_{1}, …, v_{n}] as a subalgebra. We also consider v_{i} as an element of H^{2}(G) via the inflation map.

For j ≥ 1, denote by S_{j} (resp. T_{j}) the element of H^{∗}(V, ℤ/p) (resp. H^{∗}(V)) given by

S_{j} = \{ \begin{matrix}∑_{x∈H^{1}(V,ℤ/2)}x^{j} & for p = 2, \\ ∑_{x∈H^{1}(V,ℤ/p)}(βx)^{k} & for p > 2 and j = 2k, \\ ∑_{x∈H^{1}(V,ℤ/p)}x(βx)^{k} & for p > 2 and j = 2k + 1 \end{matrix}

(resp.  T_{j} = ∑_{y∈R_{n}}y^{j}).

According to [2], we have

Proposition A. For j ≥ 1, S_{j} (resp. T_{j}) contains

∏^{n} ; k=1 ∏_{λ_{i}∈𝔽_{p}}(λ_{1}y_{1} + ⋯ + λ_{k−1}y_{k−1} + y_{k})

(resp.  ∏^{n} ; k=1 ∏_{λ_{i}∈𝔽_{p}}(λ_{1}v_{1} + ⋯ + λ_{k−1}v_{k−1} + v_{k}))

as a factor. □

ESSENTIAL INTEGRAL COHOMOLOGY CLASSES OF p p p-GROUPS 3

Since G is not elementary abelian, by Serre’s theorem (see [1, Th. 6.4.1]), we have Proposition B. For j ≥ 1, Inf_{G}^{V}(S_{j}) = 0 in H^{∗}(G, ℤ/p) and Inf_{G}^{V}(T_{j}) = 0 in H^{∗}(G). □

Denote by C_{p} the cyclic group of order p and set G = G × C_{p}. We have then

Proposition C (Künneth formula).

\begin{aligned}H^{∗}(G, ℤ/p) & = H^{∗}(G, ℤ/p) ⊗ H^{∗}(C_{p}, ℤ/p), \\ H^{∗}(G) & = H^{∗}(G) ⊗ H^{∗}(C_{p}) ⊕ T \end{aligned}

with T = Tor_{1}(H^{∗}(G), H^{∗}(C_{p})). □

Fix a generator a of C_{p} and let x ∈ H^{1}(C_{p}, ℤ/p) = Hom(C_{p}, ℤ/p) be the dual of a. Define y = βx and v = δ^{∗}(x). We have then

H^{∗}(C_{p}, ℤ/p) = \{ \begin{matrix}𝔽_{2}[x] & for p = 2, \\ Λ[x] ⊗ 𝔽_{p}[y] & for p > 2, \end{matrix}

and  H^{∗}(C_{p}) = ℤ[v]/(pv).

For 0 ≠ λ ∈ ℤ/p, let G_{λ} be the maximal subgroup of G generated by a_{1}a^{λ}, a_{2}, …, a_{n}. It is clear that the homomorphism f_{λ} : G → G_{λ} given by

f_{λ}(a_{i}) = \{ \begin{matrix}a_{1}a^{λ} & for i = 1, \\ a_{i} & for i ≥ 2, \end{matrix}

is an isomorphism. So f_{λ} induces isomorphisms f_{λ}^{∗} : H^{∗}(G_{λ}) → H^{∗}(G) and f_{λ,p}^{∗} : H^{∗}(G_{λ}, ℤ/p) → H^{∗}(G, ℤ/p). We have

Lemma A. Given ϕ ∈ H^{∗}(G) and ψ ∈ H^{∗}(G, ℤ/p), then

f_{λ}^{∗} ∘ Res_{G_{λ}}^{G}(ϕ) = \{ \begin{matrix}λv_{1} & if ϕ = 1 ⊗ v, \\ ζ & if ϕ = ζ ⊗ 1 with ζ ∈ H^{∗}(G), \end{matrix}

and  f_{λ,p}^{∗} ∘ Res_{G_{λ}}^{G}(ψ) = \{ \begin{matrix}λx_{1} & if ψ = 1 ⊗ x, \\ λy_{1} & if ψ = 1 ⊗ y, \\ ζ & if ψ = ζ ⊗ 1 with ζ ∈ H^{∗}(G, ℤ/p). \end{matrix}

Proof. The case of mod-p coefficients was treated in [2]. Since

f_{λ,p}^{∗} ∘ Res_{G_{λ}}^{G}(1 ⊗ x) = λx_{1},

we have

f_{λ}^{∗} ∘ Res_{G_{λ}}^{G}(1 ⊗ v) = f_{λ}^{∗} ∘ Res_{G_{λ}}^{G}(δ^{∗}(1 ⊗ x))
= δ^{∗}(f_{λ,p}^{∗} ∘ Res_{G_{λ}}^{G}(1 ⊗ x))
= δ^{∗}(1 ⊗ λx_{1})
= 1 ⊗ λv_{1}.

For ϕ = ζ ⊗1, set j = deg(ϕ) and consider ϕ as a Bar cocycle of G. Let e_{1}, …, e_{j} be elements of G, we have

\begin{aligned}f_{λ}^{∗} ∘ Res_{G_{λ}}^{G}(ϕ)(e_{1}, …, e_{j}) & = ϕ(f_{λ}e_{1}, …, f_{λ}e_{j}) \\ & = ζ(e_{1}, …, e_{j}). \end{aligned}

So

f_{λ}^{∗} ∘ Res_{G_{λ}}^{G}(ϕ) = ζ.

The proposition is then proved. □

Set I = Ker(H^{∗}(G, ℤ/p) →^{Res} H^{∗}(G, ℤ/p)). So

I = \{ \begin{matrix}(x) & for p = 2, \\ (x, y) & for p > 2. \end{matrix}

Consider the long exact cohomology sequence

⋯ → H^{∗}(G, ℤ/p) →^{δ^{∗}} H^{∗+1}(G) →^{p} H^{∗+1}(G) →^{pr^{∗}} H^{∗+1}(G, ℤ/p) →^{δ^{∗}} ….

We have

Lemma B. T ⊂ Im(δ^{∗}|_{I} : I → H^{∗}(G)).

Proof. As H^{0}(C_{p}) = ℤ and ∑_{n>0}H^{n}(C_{p}) is a vector space over 𝔽_{p}, pT = {0}. So

T ⊂ Imδ^{∗}. Let u be an element of T. There exists then w ∈ H^{∗}(G, ℤ/p) such that δ^{∗}(w) = u. Write w = w_{1} ⊗ 1 + w_{2} with w_{1} ∈ H^{∗}(G, ℤ/p) and w_{2} ∈ I. So u = δ^{∗}(w) = δ^{∗}(w_{1} ⊗ 1) + δ^{∗}(w_{2}). Since Res_{G}^{G}(u) = 0, it follows that

\begin{aligned}0 & = δ^{∗}(Res_{G}^{G}(w_{1} ⊗ 1)) + δ^{∗}(Res_{G}^{G}(w_{2})) \\ & = δ^{∗}(w_{1}) \end{aligned}

which implies δ^{∗}(w_{1} ⊗ 1) = 0. Hence u = δ^{∗}(w_{2}). The lemma follows. □

Let ξ ∈ H^{∗}(G) be a homogeneous class of even degree and let N : H^{∗}(G) → H^{∗}(G) be the Evens norm map. By Proposition C, N (ξ) can be decomposed as

N (ξ) = η + ∑_{j}ξ_{j} ⊗ v^{j}

with η ∈ T, ξ_{j} ∈ H^{∗}(G). Furthermore, as Res_{G}^{G}N (ξ) = ξ^{p}, we have ξ_{0} = ξ^{p}.

By Lemma B, there exist x_{ξ,i} ∈ H^{∗}(G, ℤ/p), i > 0, such that

η = \{ \begin{matrix}δ^{∗}(∑_{i>0}x_{ξ,i} ⊗ x^{i}) & for p = 2, \\ δ^{∗}(∑_{i>0}[x_{ξ,2i} ⊗ y^{i} + x_{ξ,2i−1} ⊗ y^{i−1}x]) & for p > 2. \end{matrix}

ESSENTIAL INTEGRAL COHOMOLOGY CLASSES OF p p p-GROUPS 5

Hence, by setting

w_{i} = \{ \begin{matrix}x & for i = 1, \\ y & for i = 2, \\ w_{2}^{j} & for i = 2j, \\ w_{1}w_{2j} & for i = 2j + 1, \end{matrix}

we have proved Proposition D. There exist x_{ξ,i} ∈ H^{∗}(G, ℤ/p) (i > 0) and ξ_{j} ∈ H^{∗}(G) (j ≥ 0 with ξ_{0} = ξ^{p}) such that

N (ξ) = δ^{∗}(∑_{i>0}x_{ξ,i} ⊗ w_{i}) + ∑_{j}ξ_{j} ⊗ v^{j}.  □

As, by Lemma A,

(∗)

f_{λ}^{∗} ∘ Res_{G_{λ}}^{G}(∑_{j}ξ_{j} ⊗ v^{j}) = ∑_{j}[f_{λ}^{∗} ∘ Res_{G_{λ}}^{G}(ξ_{j} ⊗ 1) · f_{λ}^{∗} ∘ Res_{G_{λ}}^{G}(1 ⊗ v^{j})]

= ∑_{j}ξ_{j} · (λv_{1})^{j}

and

(∗∗)

f_{λ}^{∗} ∘ Res_{G_{λ}}^{G}δ^{∗}(∑_{i>0}x_{ξ,i} ⊗ w_{i}) = δ^{∗}(f_{λ,p}^{∗} ∘ Res_{G_{λ}}^{G}(∑_{i>0}x_{ξ,i} ⊗ w_{i}))

= ∑_{i>0}δ^{∗}(f_{λ,p}^{∗} ∘ Res_{G_{λ}}^{G}(x_{ξ,i} ⊗ 1) · f_{λ,p}^{∗} ∘ Res_{G_{λ}}^{G}(1 ⊗ w_{i}))

= \{ \begin{matrix}∑_{i>0}δ^{∗}(x_{ξ,i} · (λx_{1})^{i}) & for p = 2, \\ ∑_{i>0}δ^{∗}(x_{ξ,2i} · (λy_{1})^{i} + x_{ξ,2i−1} · λx_{1}(λy_{1})^{i−1}) & for p > 2, \end{matrix}

we get Proposition E. If Res_{Kerx_{1}}^{G}(ξ) = 0, then

−ξ^{p} = \{ \begin{matrix}∑_{i>0}ξ_{i} · (λv_{1})^{j} + δ^{∗}(x_{ξ,i} · (λx_{1})^{i}) & for p = 2, \\ ∑_{i>0}ξ_{i} · (λv_{1})^{i} + δ^{∗}(x_{ξ,2i} · (λy_{1})^{i} + x_{ξ,2i−1} · λx_{1}(λy_{1})^{i−1}) & for p > 2, \end{matrix}

for any 0 ≠ λ ∈ 𝔽_{p}.

Proof. Set G′ = ⟨a_{2}, …, a_{n}⟩ = Kerx_{1}. As G ∩ G_{λ} = G′ and G = G · G_{λ}, according to [1, Th. 6.1.1], we have

Res_{G_{λ}}^{G} ∘ N_{G→G} = N_{G′→G_{λ}} ∘ Res_{G′}^{G}.

Furthermore, as Res_{G′}^{G}(ξ) = 0, we have Res_{G_{λ}}^{G} ∘ N (ξ) = 0. The proposition follows from (∗), (∗∗), Proposition D and from the fact that ξ_{0} = ξ^{p}. □

3. Proof of Theorem B

Let ξ be a non-zero homogeneous element of Ess(G). Note that ξ is of order a power of p (as G is a p-group). If p > 2 and ξ is of odd degree, as ξ^{2} = −ξ^{2}, we have 2ξ^{2} = 0, so ξ^{2} = 0. We may then suppose that ξ is of even degree if p > 2.

Set

φ = \{ \begin{matrix}ξ & for p > 2, \\ ξ^{2} & for p = 2. \end{matrix}

φ is then of even degree and φ ∈ Ess(G). For 0 ≠ u ∈ H^{1}(G, ℤ/p), set ũ = δ^{∗}(u) ∈ H^{2}(G). As Res_{Keru}^{G}(φ) = 0, it follows from Proposition E that

−φ^{p} = \{ \begin{matrix}∑_{i>0}φ_{i} · ũ^{i} + δ^{∗}(x_{φ,i} · u^{i}) & for p = 2, \\ ∑_{i>0}φ_{i} · ũ^{i} + δ^{∗}(x_{φ,2i} · (βu)^{i} + x_{φ,2i−1} · u(βu)^{i−1}) & for p > 2. \end{matrix}

Hence, by taking the sum over all non-zero elements u of H^{1}(G, ℤ/p), we have

(1 − p^{n})φ^{p} = −∑_{0≠u}φ^{p}

= \{ \begin{matrix}∑_{0≠u} ∑_{i>0}φ_{i} · ũ^{i} + δ^{∗}(x_{φ,i} · u^{i}) & for p = 2, \\ ∑_{0≠u} ∑_{i>0}φ_{i} · ũ^{i} + δ^{∗}(x_{φ,2i} · (βu)^{i} + x_{φ,2i−1} · u(βu)^{i−1}) & for p > 2, \end{matrix}

= \{ \begin{matrix}∑_{i>0}φ_{i} · T_{i} + δ^{∗}(x_{φ,i} · S_{i}) & for p = 2, \\ ∑_{i>0}φ_{i} · T_{i} + δ^{∗}(x_{φ,2i} · S_{2i} + x_{φ,2i−1} · S_{2i−1}) & for p > 2, \end{matrix}

= 0

by Proposition B. Since φ^{p} is of order a power of p, (1 − p^{n})φ^{p} = 0 implies φ^{p} = 0.

(i) is proved.

(ii) Let ϕ be a non-zero nilpotent element of H^{∗}(G). We may suppose that ϕ is of even degree if p is odd.

Consider the following cases:

- G is elementary abelian. If G is of order p, G has no non-zero nilpotent cohomology class. If G is of rank > 1, as pr^{∗}(ϕ) is nilpotent in H^{∗}(G, ℤ/p), pr^{∗}(ϕ^{p}) = pr^{∗}(ϕ)^{p} = 0. So ϕ^{p} belongs to the image of the map p which vanishes on positive degrees, hence ϕ^{p} = 0;

- |G| = p^{2}: G is either isomorphic to C_{p^{2}} or C_{p} × C_{p}. As each of C_{p^{2}}, C_{2} × C_{2} has no non-zero nilpotent cohomology class, we may suppose that G = C_{p} × C_{p} with p > 2. So ϕ^{p} = 0;

- |G| ≥ p^{3}: we may then assume that, for every proper subgroup K of G and for τ ∈ Ess(K), τ^{p^{log_{p} |K|−1}} = 0. So ϕ^{p^{log_{p} |G|−2}} ∈ Ess(G). As ϕ^{p^{log_{p} |G|−2}} is of even degree, by (i), ϕ^{p^{log_{p} |G|−1}} = (ϕ^{p^{log_{p} |G|−2}})^{p} = 0. (ii) is then proved.

Acknowledgements

This result was obtained during my visit to the University of Southampton in Autumn 1999. I would like to thank the London Mathematical Society for supporting the visit under a Scheme 5 grant. Many thanks to Ian Leary for helpful comments, and without whom the paper would not have been done, and to AnhMaoChiThu & NaVi for the generous help and for the encouragement during my stay in the UK.

References

1. L. Evens, The cohomology of groups, Oxford Mathematical Monographs, Clarendon Press, 1991.

2. P.A. Minh, Essential mod-p cohomology classes of p-groups: an upper bound for the nilpotency degree, to appear in Bull. London Math. Soc.

Department of Mathematics, College of Sciences, University of Hue, Dai hoc Khoa hoc, Hue, Vietnam

E-mail address: paminh@bdvn.vnmail.vnd.net