ESSENTIAL mod-ppp COHOHOMOLOGY CLASSES OF ppp-GROUPS:

AN UPPER BOUND FOR NILPOTENCY DEGREES

Pham Anh Minh

10 September, 1998

Abstract. Let p be a prime number and let G be a p-group which is not elementary abelian. For every mod-p cohomology class ξ of G which restricts trivially to all proper subgroups, we show that ξ^{p} = 0. This gives upper bounds for nilpotency degrees of such classes of G and of nilpotent mod-p cohomology classes of finite groups.

1. Introduction

Let p be a prime number. For every finite group G, denote by H^{∗}(G) the mod-p cohomology algebra of G. As usual, let Ess(G) be the ideal of H^{∗}(G) consisting of essential elements, i.e. elemens 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. As showed in [1], [4], [5], [7], when H^{∗}(G) is not detected by proper subgroups (see e.g. [2], [13], for the existence of such groups), this ideal plays an important role in the study of this cohomology. The purpose of this paper is to investigate the nilpotency degrees of non-zero elements of Ess(G) (it is known, by Quillen’s theorem, that any element of Ess(G) is nilpotent). The following was given in [16].

Conjecture (Mui [16]). Ess(G) is of nilpotency degree ≤ 2, i.e. Ess(G)^{2} = {0}.

Let p^{m} be the order of G and let n_{G} be the smallest number such that ξ^{n_{G}} = 0, for every ξ ∈ Ess(G). Upper bounds for n_{G} can be found, using Serre’s theorem [19], as follows. Let β be the Bocktein homomorphism. Serre’s theorem stated that, for p = 2 (resp. p > 2), there exist an integer, says m_{G}, and elements t_{1}, …, t_{m_{G}} of H^{1}(G) satisfying t_{1} · . . . · t_{m_{G}} = 0 (resp. βt_{1} · . . . · βt_{m_{G}} = 0). Furthermore, it follows from [18] that, for ξ ∈ Ess(G), ξ^{2} contains any non-zero element of βH^{1}(G) as a factor (note that, for p = 2, it also follows from the Gysin sequence that ξ contains any non-zero element of H^{1}(G) as a factor). Hence, for p = 2 (resp. p > 2), ξ^{m_{G}} = 0 (resp. ξ^{2m_{G}} = 0). Any upper bound of m_{G} provides then an upper one for n_{G}. So,

by [19], n_{G} ≤ 2\frac{p^{m} − 1}{p − 1}. According to [17], we also have n_{G} ≤ 2(p + 1)p^{m−2}. New

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20J06.

proof of Serre’s theorem in [12] also implied that n_{G} ≤ 2(p + 1)p^{[ \frac{m}{2} ]−1}. In this paper, we prove

Theorem A. If ξ ∈ Ess(G), then ξ^{p} = 0.

This gives an upper bound, which is p, for nilpotency degrees of essential cohomology classes of G; especially, if p = 2, every non-zero essential cohomology class of G is of nilpotency degree 2 (parts of this result were given in [10], [11], in the case where G is generated by less than 3 elements, or |G| ≤ 2^{6}). This result might then hopefully contribute to get an answer to Mui’s conjecture, although, for p > 2, our bound is much bigger than the one Mui expects.

Our result can also be applied for getting an upper bound for nilpotency degrees of nilpotent mod-p cohomology classes of an arbitrary finite group, as follows. Let G be a finite group containing G as a Sylow subgroup (which may be elementary abelian). We have

Theorem B. If ξ ∈ H^{∗}(G) is nilpotent, then ξ^{p} = 0 if G is abelian, and ξ^{p^{m−2}} = 0 if G is not abelian.

In order to prove these theorems, we will need some facts related to modular invariant theory and Evens norm map. They will be given in Sections 2 and 3. Section 4 is devoted to the proofs of Theorems A and B.

2. Modular invariant theory

We first recall some basic facts, concerning modular invariant theory. Let n be an integer. As usual, denote by P_{n} = P [y_{1}, …, y_{n}] the polynomial algebra over 𝔽_{p} with variables y_{1}, …, y_{n} (of degree 1 for p = 2, 2 for p > 2); furthermore, for p > 2, denote by E_{n} = E[x_{1}, …, x_{n}] the exterior algebra over 𝔽_{p} with variables x_{1}, …, x_{n} (of degree 1). Set

R_{n} = \{ \begin{matrix}P_{n} & , for p = 2 \\ E_{n} ⊗ P_{n} & , for p > 2. \end{matrix}

R_{n} can be viewed as the mod-p cohomology algebra H^{∗}(V) of an elementary abelian p-group V of rank n; if p > 2, y_{i} can be chosen to be the image of x_{i} via the Bockstein homomorphism β : H^{∗}(V) → H^{∗+1}(V). There is then a natural action of GL_{n} = GL(n, 𝔽_{p}) on R_{n}. The subring of invariants of GL_{n} in this algebra, denoted by R_{n}^{GL_{n}}, is determined by Dickson and Mui as follows. Set

\begin{aligned}L_{n} & = L_{n}(y_{1}, …, y_{n}) \\ & = ∏^{n} ; k=1 ∏_{λ_{i}∈𝔽_{p}}(λ_{1}y_{1} + ⋯ + λ_{k−1}y_{k−1} + y_{k}), \end{aligned}

and define the Dickson’s invariants Q_{n,s} = Q_{n,s}(y_{1}, …, y_{n}), 1 ≤ s ≤ n, recursively by:

\begin{aligned}Q_{n,0} & = L_{n}^{p−1}, \\ Q_{n,s} & = Q_{n−1,s}(L_{n}/L_{n−1})^{p−1} + Q_{n−1,s−1}^{p}, \end{aligned}

ESSENTIAL mod-p COHOMOLOGY CLASSES OF p-GROUPS 3

where Q_{s,s} = 1.

For p > 2 and for 0 ≤ s ≤ n − 1, set

M_{n,s} = | \begin{matrix}x_{1} & x_{2} & ⋯ & x_{n} \\ y_{1} & y_{2} & … & y_{n} \\ ⋯ & ⋯ & ⋯ & ⋯ \\ y_{1}^{p^{s−1}} & y_{2}^{p^{s−1}} & ⋯ & y_{n}^{p^{s−1}} \\ y_{1}^{p^{s+1}} & y_{2}^{p^{s+1}} & ⋯ & y_{n}^{p^{s+1}} \\ ⋯ & ⋯ & ⋯ & ⋯ \\ y_{1}^{p^{n−1}} & y_{2}^{p^{n−1}} & ⋯ & y_{n}^{p^{n−1}} \end{matrix} | .

Let (s_{1}, …, s_{k}) be a sequence of integers with 0 ≤ s_{1} < ⋯ < s_{k} < n. As in [15, Prop. I4.5], the product M_{n,s_{1}} · M_{n,s_{2}} ⋯ M_{n,s_{k}} has the factor L_{n}^{k−1}. We have then Mui’s invariants

\begin{aligned}M_{n,s_{1},...,s_{k}} & = M_{n,s_{1},...,s_{k}}(x_{1}, …, x_{n}, y_{1}, …, y_{n}) \\ & = (−1)^{k(k−1)/2}M_{n,s_{1}} · M_{n,s_{2}} ⋯ M_{n,s_{k}}/L_{n}^{k−1}. \end{aligned}

We have

Theorem C (Dickson [6], Mui [15]). (i) For p = 2, R_{n}^{GL_{n}} = P [Q_{n,0}, …, Q_{n,n−1}], the polynomial algebra generated by Q_{n,0}, …, Q_{n,n−1}.

(ii) For p > 2,

R_{n}^{GL_{n}} =P [Q_{n,0}, …, Q_{n,n−1}]
⊕ ⊕_{k=1}^{n} ⊕ ∑_{0≤s_{1}<···s_{k}≤n−1} ⊕M_{n,s_{1},...,s_{k}} · L_{n}^{p−2}P [Q_{n,0}, …, Q_{n,n−1}]  □

We can now deduce a corollary from this theorem. For every positive integer j and for every x ∈ H^{1}(V), set

z_{x}^{(j)} = \{ \begin{matrix}x^{j} & , if p = 2 \\ (βx)^{k} & , if p > 2 and j = 2k \\ x · (βx)^{k} & , if p > 2 and j = 2k + 1 \end{matrix}

and

S_{j} = ∑_{x∈H^{1}(V)} z_{x}^{(j)}.

We have

Proposition A. S_{j} contains L_{n} as a factor.

Proof. It is clear that S_{j} is invariant under the action of GL_{n}. Consider first the case p = 2, or p > 2 and j = 2k. As setting y_{1} = 0 yields S_{j} = 0, S_{j} contains y_{1}, therefore, every non-zero linear combination of the y_{i}’s, as a factor; so L_{n} is a factor of S_{j}. If p > 2 and j = 2k + 1, it follows from Theorem C that S_{j} contains L_{n}^{p−2}, hence L_{n}, as a factor. The proposition follows. □

3. Evens norm map

Let Γ be a p-group containing G as a subgroup and let N_{G→Γ} be the Evens norm map. Fix an element ξ of H^{r}(G), where r is even for p > 2, so

N_{G→Γ}(ξ) ∈ H^{r(Γ:G)}(G).

As observed in [3], in the case Γ = G×ℤ_{p}, N_{G→Γ} can be expressed via the Steenrod operations, as follows. Fix a generator a of ℤ_{p} and let u ∈ H^{1}(ℤ_{p}) = Hom(ℤ_{p}, 𝔽_{p}) be the dual of a. Define v = βu. We have

H^{∗}(ℤ_{p}) = \{ \begin{matrix}P [u] & , for p = 2 \\ E[u] ⊗ P [v] & , for p > 2 \end{matrix}

and  H^{∗}(Γ) = H^{∗}(G) ⊗ H^{∗}(ℤ_{p}).

There exist then operations D_{j} : H^{r}(G) → H^{pr−j}(G) satisfying

N_{G→Γ}(ξ) = \{ \begin{matrix}∑_{j}D_{j}(ξ) ⊗ u^{j} & , for p = 2 \\ ∑_{j}[D_{2j}(ξ) ⊗ v^{j} + D_{2j+1}(ξ) ⊗ uv^{j}] & , for p > 2. \end{matrix}  (∗)

The D_{i}’s are nothing but, up to constant multiples, the Steenrod operations (composing with the Bockstein homomorphism when p is odd, see e.g. [3] for details).

By setting

w_{i} = \{ \begin{matrix}u & , for i = 1 \\ βu & , for i = 2 \\ w_{2}^{j} & , for i = 2j \\ w_{1}w_{2j} & , for i = 2j + 1. \end{matrix}

we can reformulate (∗) as

N_{G→Γ}(ξ) = ∑_{j} D_{j}(ξ) ⊗ w_{j}.

Set n = dim_{𝔽_{p}} H^{1}(G) and let u_{1}, …, u_{n} be a basis of H^{1}(G). There is then a set of generators of G consisting of n elements, say a_{1}, …, a_{n} satisfying u_{i}(a_{j}) = δ_{ij} (the Kronecker symbol). We also set v_{i} = βu_{i}, 1 ≤ i ≤ n.

For 0 ≠ λ ∈ 𝔽_{p}, let G_{λ} be the maximal subgroup of G × ℤ_{p} 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 an isomorphism f_{λ}^{∗} : H^{∗}(G_{λ}) → H^{∗}(G). We have Proposition B. Given ψ ∈ H^{∗}(G × ℤ_{p}), then

f_{λ}^{∗} ∘ Res_{G_{λ}}^{G×ℤ_{p}}(ψ) = \{ \begin{matrix}λu_{1} & , if ψ = 1 ⊗ u \\ λv_{1} & , if ψ = 1 ⊗ v \\ ζ & , if ψ = ζ ⊗ 1 with ζ ∈ H^{∗}(G). \end{matrix}

Proof. Set j = deg(ψ) and consider ψ as a Bar cocycle of G × ℤ_{p}. Let e_{1}, …, e_{j} be elements of G, we have

f_{λ}^{∗} ∘ Res_{G_{λ}}^{G×ℤ_{p}}(ψ)(e_{1}, …, e_{j}) = ψ(f_{λ}e_{1}, …, f_{λ}e_{j})

= \{ \begin{matrix}u(f_{λ}e_{1}) & , if ψ = 1 ⊗ u \\ ζ(e_{1}, …, e_{j}) & , if ψ = ζ ⊗ 1 with ζ ∈ H^{∗}(G). \end{matrix}

So

f_{λ}^{∗} ∘ Res_{G_{λ}}^{G×ℤ_{p}}(ψ) = \{ \begin{matrix}λu_{1} & , if ψ = 1 ⊗ u \\ ζ & , if ψ = ζ ⊗ 1 with ζ ∈ H^{∗}(G). \end{matrix}

The proposition is then proved, by noting that the restriction map and f_{λ}^{∗} commutes with the Bockstein homomorphism. □

As

f_{λ}^{∗}∘Res_{G_{λ}}^{G×ℤ_{p}} ∘ N_{G×ℤ_{p}}(ξ) = f_{λ}^{∗} ∘ Res_{G_{λ}}^{G×ℤ_{p}}(∑_{j} D_{j}(ξ) ⊗ w_{j})

= ∑_{j} f_{λ}^{∗} ∘ Res_{G_{λ}}^{G×ℤ_{p}}(D_{j}(ξ ⊗ 1)) · f_{λ}^{∗} ∘ Res_{G_{λ}}^{G×ℤ_{p}}(1 ⊗ w_{j})

= \{ \begin{matrix}∑_{j}D_{j}(ξ) ⊗ (λu_{1})^{j} & , for p = 2 \\ ∑_{j}[D_{2j}(ξ) ⊗ (λv_{1})^{j} + D_{2j+1}(ξ) ⊗ λu_{1}(λv_{1})^{j}] & , for p > 2 \end{matrix}  (∗∗)

by Proposition B,

we get

Proposition C. If Res_{Keru_{1}}^{G}(ξ) = 0, then

ξ^{p} = \{ \begin{matrix}∑_{j>0}D_{j}(ξ) ⊗ (λu_{1})^{j} & , for p = 2 \\ − ∑_{j>0}[D_{2j}(ξ) ⊗ (λv_{1})^{j}] − ∑_{j≥0}[D_{2j+1}(ξ) ⊗ λu_{1}(λv_{1})^{j}] & , for p > 2 \end{matrix}

for any 0 ≠ λ ∈ 𝔽_{p}.

Proof. Set G′ = ⟨a_{2}, …, a_{n}⟩ = Keru_{1}. As G ∩ G_{λ} = G′ and G × ℤ_{p} = G · G_{λ}, according to [8, Th. 6.1.1], we have

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

Furthermore, as Res_{G′}^{G}(ξ) = 0, we have Res_{G_{λ}}^{G×ℤ_{p}} ∘N_{G→G×ℤ_{p}}(ξ) = 0. The proposition follows from (∗∗) and from the fact that D_{0}(ξ) = ξ^{p}. □

4. Proofs of Theorems A and B

Proof of Theorem A. Let ξ be a non-zero homogeneous element of Ess(G) of degree r. As ξ^{2} = 0 if r is odd and p > 2, we can suppose that r is even when p > 2. Furthermore, if G is abelian, ξ contains any non-zero element of H^{1}(G) as a factor; since there exists such an element of H^{1}(G) which is nilpotent of degree 2 (as G is not elementary abelian), it follows that ξ^{2} = 0. We may then suppose that G is not abelian. As in Section 2, set n = dim_{𝔽_{p}} H^{1}(G) and let u_{1}, …, u_{n} be a basis of H^{1}(G). Since g(u_{1}), …, g(u_{n}) is also a basis of H^{1}(G) for any g ∈ GL_{n}, it follows from Proposition C that, given any non-zero linear combination x of u_{1}, …, u_{n}, we have

ξ^{p} = \{ \begin{matrix}∑_{j>0}D_{j}(ξ) ⊗ x^{j} & , for p = 2 \\ − ∑_{j>0}[D_{2j}(ξ) ⊗ (βx)^{j}] − ∑_{j≥0}[D_{2j+1}(ξ) ⊗ x(βx)^{j}] & , for p > 2. \end{matrix}

So

ξ^{p} = − ∑_{0≠x∈H^{1}(G)} ξ^{p} = \{ \begin{matrix}∑_{j>0}[D_{j}(ξ) ⊗ ∑_{0≠x∈H^{1}(G)}x^{j}] & , for p = 2 \\ ∑_{j>0}[D_{2j}(ξ) ⊗ ∑_{0≠x∈H^{1}(G)}(βx)^{j}] \\ + ∑_{j≥0}[D_{2j+1}(ξ) ⊗ ∑_{0≠x∈H^{1}(G)}x(βx)^{j}] & , for p > 2. \end{matrix}

For i > 0, set

f_{i} = \{ \begin{matrix}∑_{0≠x∈H^{1}(G)}x^{i} & , for p = 2 \\ ∑_{0≠x∈H^{1}(G)}(βx)^{j} & , for p > 2 and i = 2j \\ ∑_{0≠x∈H^{1}(G)}x(βx)^{j} & , for p > 2 and i = 2j + 1. \end{matrix}

Let Φ(G) be the Frattini subgroup of G. V = G/Φ(G) is then a vector space of dimension n over 𝔽_{p}. We now use the notation given in Section 1. For p = 2 (resp. p > 2), the basis y_{1}, …, y_{n} (resp. x_{1}, …, x_{n}) of H^{1}(V) can be chosen such that Inf_{G}^{V}(y_{k}) = u_{k} (resp. Inf_{G}^{V}(x_{k}) = u_{k}). We have then

f_{i} = Inf_{G}^{V}(S_{i}).

By Proposition A, S_{i} contains L_{n} = L_{n}(y_{1}, …, y_{n}) as a factor. Therefore f_{i} contains

Inf_{G}^{V}(L_{n}) = \{ \begin{matrix}L_{n}(u_{1}, …, u_{n}) & , for p = 2 \\ L_{n}(v_{1}, …, v_{n}) & , for p > 2, \end{matrix}

as a factor. By Serre’s theorem, Inf_{G}^{V}(L_{n}) = 0 in H^{∗}(G), which implies that f_{i} = 0. So ξ^{p} = 0. The theorem is proved.

Proof of Theorem B. Since Res_{G}^{G} is injective, we can reduce to the case G = G. If G is abelian, it is clear that ξ^{p} = 0. Suppose that G is not abelian (so p^{m} ≥ p^{3}).

We argue by induction on m. If |G| = p^{3}, G is extraspecial; it follows from [9] and [14] that ξ^{p} = 0. Assume that m > 3 and that the theorem holds for every proper subgroup of G. As every maximal subgroup K of G is order p^{m−1}, it follows from the inductive hypothesis that Res_{K}^{G}(ξ^{p^{m−3}}) = 0 , hence ξ^{p^{m−3}} ∈ Ess(G). Therefore ξ^{p^{m−2}} = 0, by Theorem A. The theorem is proved.

We end the paper by the following

Remark. (1) According to [18], for ξ ∈ Ess(G) and for a given non-zero linear combination ν of the v_{i}’s, ξ^{2} contains ν as a factor. So there exists ξ′ ∈ H^{∗}(G) such that ξ^{2} = ξ′ · ν. However, until now, there is no way to determine ξ′. Besides, it follows from Proposition C that ξ′ depends on both ξ and ν. Hence it seems rather difficult to deduce from this expression of ξ^{2} a proof of Theorem A.

(2) Amost all known essential cohomology classes of G were obtained by means of the transfers tr_{G}^{K}, with K maximal in G (see [2], [9], [13], [14]). It turns out that these elements are of nilpotency degree 2. Furthermore, if we denote by E(G) the intersection of Ess(G) and the ideal generated by images of transfers from maximal subgroups, then E(G)^{2} = 0. In fact, if ξ ∈ E(G) is of form ξ = tr_{G}^{K}(η), with K maximal in G, then for every proper subgroup L of G and for every ζ ∈ Ess(G), we have

ξ · ζ = tr_{G}^{K}(η) · ζ
= tr_{G}^{K}(η · Res_{K}^{G}(ζ))  by Frobenius formula
= 0  since Res_{K}^{G}(ζ) = 0.

So a counterexample, if any, to Mui’s conjecture should consist of essential classes not belonging to the ideal generated by transfers.

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

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