A NOTE ON SERRE’S THEOREM IN GROUP COHOMOLOGY

ERGÜN YALÇIN

Abstract. In [7], Serre proves that if G is a p-group which is not elementary abelian, then a product of Bocksteins of one dimensional classes is zero in the mod p cohomology algebra of G, provided that the product includes at least one nontrivial class from each line in H^{1}(G, 𝔽_{p}). For p = 2, this gives that (σ_{G})^{2} = 0 where σ_{G} is the product of all nontrivial one dimensional classes in H^{1}(G, 𝔽_{2}). In this note, we prove that if G is a nonabelian 2-group, then σ_{G} is also zero.

1. Introduction

Let G be a finite p-group and let H^{∗}(G, 𝔽_{p}) denote the cohomology algebra of G in mod p coefficients. One of the most important theorems about the algebra structure of H^{∗}(G, 𝔽_{p}) is Serre’s theorem [6] which states that if G is not an elementary abelian group, then there exist nontrivial one dimensional classes x_{1}, x_{2}, …, x_{m} ∈ H^{1}(G, 𝔽_{p}) such that β(x_{1})β(x_{2}) ⋯ β(x_{m}) = 0. The original proof by Serre is an existence proof giving no indication how such a vanishing product should be chosen. In his second proof, Serre gives an explicit set of one dimensional classes which has a vanishing product. He proves the following:

Theorem 1.1 (Serre [7]). Let G be a p-group which is not elementary abelian. Then, the product ∏ β(x) is zero if the product includes at least one nontrivial class from each line in H^{1}(G, 𝔽_{p}).

When G is a 2-group, we have β(x) = x^{2} for each x ∈ H^{1}(G, 𝔽_{2}), so Theorem 1.1 gives (σ_{G})^{2} = 0 where

σ_{G} = ∏_{x∈H^{1}(G,𝔽_{2})−{0}}x.

The class σ_{G} itself is zero for many 2-groups, but there are also examples of 2-groups where σ_{G} is not zero. For example, if G ≅ ℤ/4 × ℤ/2, then σ_{G} is not zero. In this note, we show that such examples are limited to abelian 2-groups. Our main result is the following:

Theorem 1.2. If G is a nonabelian 2-group, then σ_{G} = 0.

This theorem has many applications in group cohomology. Our main motivation for studying this version of Serre’s theorem is a calculation related to the Stiefel-Whitney classes of the regular representation of a 2-group. As a consequence of Theorem 1.2, we obtain that the top Stiefel-Whitney class of the augmentation module for the regular representation of a nonabelian 2-group is zero. We also give some other applications related to essential cohomology. All these applications are explained in Section 4.

1991 Mathematics Subject Classification. Primary: 20J06.

Key words and phrases. cohomology of groups, Stiefel-Whitney classes, essential cohomology.

The author is partially supported by TÜBİTAK-BDP and TÜBA-GEBİP/2005-16.

The proof of Theorem 1.2 is very similar to the proof of Theorem 1.1 although there are some subtle differences. In the proof of Theorem 1.1, Serre uses the fact that if G is a p-group which is not elementary abelian, then there exist some two dimensional relations of the form

(1)  ∑_{k} a_{k}β(x_{k}) + ∑_{i<j} c_{ij}x_{i}x_{j} = 0

where {x_{1}, …, x_{n}} is a basis for H^{1}(G, 𝔽_{p}). But the existence of such relations are no longer enough to prove Theorem 1.2. Some complications arise when G is a nonabelian 2-group which has no two dimensional relations with some of the c_{ij}’s nonzero. For example, the modular group

M_{2^{n}} = ⟨a, b | a^{2} = b^{2^{n−1}} = 1, aba = b^{2^{n−2}+1}⟩

of order 2^{n} with n ≥ 4 is such a group (see Remark 3.3). To overcome this problem, we prove the following:

Proposition 1.3. Let G be a 2-group and let {x_{1}, …, x_{n}} be a basis for H^{1}(G, 𝔽_{2}). If G is nonabelian, then there exists a nontrivial relation of the form

(2)  ∑_{i<j} c_{ij}(x_{i}^{2}x_{j} + x_{j}^{2}x_{i}) = 0

in H^{3}(G, 𝔽_{2}).

To prove Proposition 1.3, we consider group cohomology in integer coefficients and show that the inflation map

inf : H^{3}(G/Φ(G), ℤ) → H^{3}(G, ℤ)

has a nontrivial kernel when G is a nonabelian p-group (here Φ(G) denotes the Frattini subgroup of G). We can also state this in the following way:

Proposition 1.4. Let G be a p-group. Then, the following are equivalent:

(i) G is abelian. (ii) The map inf : H^{2}(G/Φ(G), ℚ/ℤ) → H^{2}(G, ℚ/ℤ) is injective.

This is similar to a criterion given by Serre in [6] for a p-group to be elementary abelian (see Proposition 3.4).

This note is organized as follows: In Section 2, we go through Serre’s arguments to show how Theorem 1.2 follows from Proposition 1.3. Propositions 1.3 and 1.4 are proved in Section 3. In Section 4, we describe some consequences of Theorem 1.2.

2. Proof of the Main Theorem

Let G be a 2-group and let {x_{1}, …, x_{n}} be a basis for H^{1}(G, 𝔽_{p}). We can express σ_{G} as the product

σ_{G} = ∏_{(a_{1},...,a_{n})∈(𝔽_{2})^{n}−{0}}(a_{1}x_{1} + ⋯ + a_{n}x_{n}).

Note that σ_{G} can also be expressed as the determinant of the matrix A defined by

A_{ij} = x_{j}^{2^{i−1}} for 1 ≤ i, j ≤ n.

This is because both σ_{G} and det(A) have the same degree and the same zero set as polynomials in x_{1}, …, x_{n} over the algebraic closure of 𝔽_{2}. Note that both polynomials are zero at a point α ∈ (\overline{𝔽}_{2})^{n} if and only if the coordinates of α satisfy a linear equation of the form a_{1}x_{1} + ⋯ + a_{n}x_{n} = 0 for some (a_{1}, …, a_{n}) ∈ (𝔽_{2})^{n} − {0}.

Let M denote the n-dimensional free module over the cohomology algebra H^{∗}(G, 𝔽_{2}). For i = 1, …, n, let e_{i} denote the vector

e_{i} = (x_{i}, x_{i}^{2}, …, x_{i}^{2^{n−1}})

in M. Since the column vector given by e_{i} is the i-th column of A, we have

det(A) = e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{n}

in the n-th exterior power ∧^{n}M ≅ H^{∗}(G, 𝔽_{p}). Suppose there is a nontrivial relation

(3)  ∑_{i<j} c_{ij}(e_{i} ∧ e_{j}) = 0

in M ∧ M, say with c_{12} ≠ 0, then by taking exterior product with e_{3} ∧ ⋯ ∧ e_{n}, we get det(A) = 0. So, to complete the proof of Theorem 1.2, we need to show that there is a nontrivial relation in M ∧ M.

Lemma 2.1 (cf. Serre [7]). Let G be a 2-group. If there is a nontrivial relation of the form

∑_{i<j} c_{ij}(x_{i}^{2}x_{j} + x_{i}x_{j}^{2}) = 0

in H^{3}(G, 𝔽_{2}), then there exists a nontrivial relation also in M ∧ M.

Proof. Let c_{ij} be the coefficients of a nontrivial relation in H^{3}(G, 𝔽_{2}). Define

L_{kl} = ∑_{i<j} c_{ij}(x_{i}^{2^{k}}x_{j}^{2^{l}} + x_{i}^{2^{l}}x_{j}^{2^{k}})

for k, l ≥ 0. Note that there is a relation in M ∧ M of the form given in Equation 3 if and only if L_{kl} = 0 for all k, l ≥ 0. The class L_{00} is zero by trivial reasons and L_{10} is zero by assumption. For m ≥ 2, we have

L_{m0} = Sq^{2^{m−1}} ⋯ Sq^{4}Sq^{2}(L_{10}).

So, L_{m0} = 0 for all m ≥ 0. Now, suppose k and l are such that k = l + m for some m ≥ 0. Then, L_{kl} = (L_{m0})^{2^{l}} = 0. Since L_{kl} = L_{lk}, we conclude that L_{kl} = 0 for all k, l ≥ 0. □

This shows that the proof of Theorem 1.2 is complete once we prove Proposition 1.3.

3. Proof of Proposition 1.3

As before, we assume that G is a 2-group. Let {x_{1}, …, x_{n}} be a basis for H^{1}(G, 𝔽_{p}). Recall that the inflation map

inf : H^{∗}(G/Φ(G), 𝔽_{2}) → H^{∗}(G, 𝔽_{2})

is bijective in dimension one. For each x_{i} ∈ H^{1}(G, 𝔽_{2}), we denote the associated class in H^{1}(G/Φ(G), 𝔽_{2}) also by x_{i}. Note that to prove Proposition 1.3, it is enough to show that there is an element of the form

∑_{i<j} c_{ij}(x_{i}^{2}x_{j} + x_{i}x_{j}^{2})

in H^{3}(G/Φ(G), 𝔽_{2}) which lies in the kernel of the inflation map. We show this using integral cohomology. The quotient group G/Φ(G) is an elementary abelian 2-group and the integral cohomology rings of elementary abelian 2-groups are well understood. In particular, it is known that H^{3}(G/Φ(G), ℤ) is generated by {u_{ij} | 1 ≤ i < j ≤ n} where the mod 2reduction of u_{ij} is equal to x_{i}^{2}x_{j} + x_{i}x_{j}^{2} in H^{3}(G, 𝔽_{2}) (see, for example, [2]). Consider the following diagram:

H^{3}(G/Φ(G), ℤ) ⟶^{inf} H^{3}(G, ℤ)
↓ _{mod 2}  ↓ _{mod 2}
H^{3}(G/Φ(G), 𝔽_{2}) ⟶^{inf} H^{3}(G, 𝔽_{2}).

If the kernel of the inflation map in integer coefficients is nonzero, then there is a nontrivial class ∑_{i<j} c_{ij}u_{ij} in H^{3}(G/Φ(G), ℤ) whose mod 2 reduction ∑_{i<j} c_{ij}(x_{i}^{2}x_{j} + x_{i}x_{j}^{2}) lies in the kernel of the mod 2 inflation map. Hence the proof of Proposition 1.3 is complete once we prove the following:

Proposition 3.1. If G is a nonabelian p-group, then the kernel of the inflation map

inf : H^{3}(G/Φ(G), ℤ) → H^{3}(G, ℤ)

is nontrivial.

Proof. The statement is equivalent to the statement that the kernel of the inflation map

inf : H^{2}(G/Φ(G), ℚ/ℤ) → H^{2}(G, ℚ/ℤ)

is nontrivial. Consider the 5-term exact sequence

0 → H^{1}(G/Φ(G)) ⟶^{inf} H^{1}(G) ⟶^{res} H^{1}(Φ(G))^{G} ⟶^{d_{2}} H^{2}(G/Φ(G)) ⟶^{inf} H^{2}(G)

where all the cohomology groups are with coefficients in ℚ/ℤ. Note that

H^{1}(G, ℚ/ℤ) = Hom(G, ℚ/ℤ) = Hom(G/[G, G], ℚ/ℤ) = (G/[G, G])^{∗}

where (G/[G, G])^{∗} denotes the dual group of the abelian group G/[G, G]. The above 5-term exact sequence gives an exact sequence of the form

0 → (G/Φ(G))^{∗} ⟶^{inf} (G/[G, G])^{∗} ⟶ ((Φ(G)/[Φ(G), Φ(G)])^{∗})^{G} ⟶ ker(inf) → 0.

From this it is easy to see that the cokernel of the first inflation map is (Φ(G)/[G, G])^{∗}. Also note that

((Φ(G)/[Φ(G), Φ(G)])^{∗})^{G} = (Hom(Φ(G), ℚ/ℤ))^{G} = Hom_{G}(Φ(G), ℚ/ℤ)
= Hom(Φ(G)/[Φ(G), G], ℚ/ℤ) =(Φ(G)/[Φ(G), G])^{∗}.

Thus, the kernel of inflation map in degree 2 is isomorphic to the group

([G, G]/[Φ(G), G])^{∗}.

Now, the result follows from the following group theoretical lemma. □

Lemma 3.2. If G is a nonabelian p-group, then [G, G] ≠ [Φ(G), G].

Proof. Let G be a minimal counterexample to this lemma. Then, G is a nonabelian p-group with [G, G] = [Φ(G), G]. Let H be a normal subgroup of G such that H ≤ [G, G] with index p. Note that G/H is a nonabelian p-group satisfying

[G/H, G/H] = [G, G]/H = [Φ(G), G]/H = [Φ(G)/H, G/H] = [Φ(G/H), G/H].

So, by the minimality of G, we must have H = 1. This gives that [G, G] = [Φ(G), G] is a cyclic group of order p. In a p-group, a normal subgroup of order p is always central, so we have [G, G] ≤ Z(G). Under these conditions, [a^{p}, b] = [a, b]^{p} = 1 for every a, b ∈ G. Thus, the subgroup G^{(p)} generated by p-th powers is also a central subgroup of G. Note that Φ(G) = [G, G]G^{(p)}, so we obtain that Φ(G) ≤ Z(G) which is in contradiction with our assumption since [Φ(G), G] = [G, G] is nontrivial. □

This completes the proof of Proposition 1.3 and hence the proof of Theorem 1.2.

Remark 3.3. In the proof of Theorem 1.1 given in [7], Steenrod operations are applied to a relation of the form

∑_{k} a_{k}β(x_{k}) + ∑_{i<j} c_{ij}x_{i}x_{j} = 0

to produce higher dimensional relations which eventually forces ∏ β(x) to be zero. But, the argument with Steenrod operations works only when some of the c_{ij}’s are nonzero. Because of this, the case where all c_{ij}’s are zero is considered separately. In that case, one has ∑_{k} a_{k}β(x_{k}) = 0 which gives β(∑_{k} a_{k}x_{k}) = 0. So, Theorem 1.1 still holds if G has no relations with some c_{ij}’s nonzero. However, this causes a problem with the proof of Theorem 1.2. One may expect that if G is nonabelian, then there should exist at least one relation of the above form with some c_{ij}’s nonzero, but unfortunately this is not true. The modular group

M_{2^{n}} = ⟨a, b | a^{2} = b^{2^{n−1}} = 1, aba = b^{2^{n−2}+1}⟩

of order 2^{n} with n ≥ 4 has no relation of the above form with some c_{ij}’s nonzero. The only two dimensional relation that M_{2^{n}} has in its mod 2 cohomology is β(u) = 0 where u ∈ H^{1}(M_{2^{n}}, 𝔽_{2}) is the class defined by the homomorphism u(a^{i}b^{j}) = j mod 2.

Note that if G is abelian, then [G, G] = [Φ(G), G] = 1, and hence the kernel of the inflation map is trivial. So, the proof of Proposition 1.4 is also complete. Proposition 1.4 has formal similarities to the following result by Serre.

Proposition 3.4 (Proposition 3 in [6]). Let G be a p-group. Then, the following are equivalent:

(i) G is elementary abelian. (ii) The inflation map inf : H^{2}(G/Φ(G), 𝔽_{p}) → H^{2}(G, 𝔽_{p}) is injective.

We can interpret Proposition 1.4 as a similar criterion for being abelian. Another way to interpret Proposition 3.1 is to say that the image of the differential d_{2} is nontrivial in the Hochschield-Serre spectral sequence for the extension

0 → Φ(G) → G → G/Φ(G) → 0

of a nonabelian 2-group G. In this sense, Proposition 3.1 is a generalization of Proposition 7.2 in [9].

Remark 3.5. There is an alternative approach to the proof Proposition 1.3 which was suggested to us by the anonymous referee. Since it gives more insight to this proposition, we briefly explain it here. Consider the quotient group F = G/[Φ(G), G] which fits into a central extension of the form 1 → Φ(F) → F → E → 1 where E = G/Φ(G). By Lemma 3.2, F is nonabelian. Since Φ(F) is central, F has a nonabelian quotient Q which fits into a central extension of the form

E : 1 → C → Q → E → 1

where C is cyclic. By taking further quotients if necessary, we can assume that Q/C_{2} is abelian where C_{2} is the unique subgroup of C of order 2. Let α ∈ H^{2}(E, C) be the extension class for E. Consider the long exact sequence

⋯ → H^{2}(G, ℤ/2) → H^{2}(E, \overline{C}) → H^{2}(E, C) ⟶^{d} H^{3}(E, ℤ/2) → ⋯

associated to the short exact sequence 0 → ℤ/2 → \overline{C} → C → 0 where \overline{C} is a cyclic group of order 2|C|. We claim that d(α) is nonzero. Assume the contrary that d(α) = 0. Then, the extension E will lift to a central extension of the form 1 → \overline{C} → \overline{Q} → E → 1. Note that since the commutator subgroup of \overline{Q} is central, we have [a, b]^{2} = [a^{2}, b] = 1 for every a, b ∈ \overline{Q}. This gives that Q = \overline{Q}/(ℤ/2) is abelian which contradicts our starting assumption. This proves d(α) is nonzero. It is clear that inflation of d(α) to G is zero, because inf_{E}^{G}(α) = 0. So, it remains to show that d(α) is in the desired form. For this it is enough to show that d(α) is in the image of the Bockstein operator. Since Q/C_{2} is abelian, there is a class γ ∈ H^{2}(E, C) such that d(γ) = 0 and that π_{∗}(γ) = π_{∗}(α) under the homomorphism π_{∗} : H^{2}(E, C) → H^{2}(E, C/C_{2}). So, α′ = α − γ is in the image of j_{∗} : H^{2}(E, C_{2}) → H^{2}(E, C) and d(α) = d(α′). If α′ = j_{∗}(u), then an easy diagram chase shows that d(α′) = β(u). This completes the proof that d(α) is in the desired form, and hence completes the argument.

4. Consequences of the main theorem

The converse of Theorem 1.2 does not hold since there are some abelian groups with σ_{G} = 0. For example, when G = ℤ/4 × ℤ/4, we have

H^{∗}(G, 𝔽_{2}) ≅ ∧(x_{1}, x_{2}) ⊗ 𝔽_{2}[y_{1}, y_{2}],

so σ_{G} = x_{1}x_{2}(x_{1} + x_{2}) = 0. In general, if G is an abelian group with more than two factors with exponent greater or equal to 4, then σ_{G} = 0. On the other hand, it is easy to see that σ_{G} ≠ 0 when G ≅ ℤ/2^{k} × (ℤ/2)^{n} for some n ≥ 0 and k ≥ 1. Thus, we conclude:

Corollary 4.1. Let G be a 2-group. Then, σ_{G} ≠ 0 if and only if G ≅ ℤ/2^{k} × (ℤ/2)^{n} for some n ≥ 0 and k ≥ 1.

This gives a complete description of 2-groups with σ_{G} ≠ 0. The element σ_{G} is also defined for p-groups with p > 2, but it is easy to see that σ_{G} = 0 for all p-groups in this case.

Corollary 4.1 suggests another way to prove Theorem 1.2. Since abelian 2-groups with σ_{G} ≠ 0 are in a special form, a minimal counterexample to Theorem 1.2 would be a nonabelian 2-group G which fits into an extension of the form

1 → ℤ/2 → G → ℤ/2^{k} × (ℤ/2)^{n} → 1

for some n ≥ 0 and k ≥ 1. One can argue directly that such groups have some nontrivial three dimensional relations of the form given in Proposition 1.3. The proof we give in Section 3 is more conceptual and explains why such three dimensional relations exist.

4.1. The top Stiefel-Whitney class of an augmented regular representation. Our main motivation for obtaining the vanishing product in Theorem 1.2 is a calculation related to the Stiefel-Whitney classes of regular representations. Let

I_{G} = ker{ε : ℝG ⟶ ℝ}

denote the augmentation ideal of the regular representation of G over real numbers. We are interested in calculating the top Stiefel-Whitney class of I_{G}.

This calculation is important for two reasons. First, this calculation is related to a question asked by Reiner and Webb. In [5], Reiner and Webb define a cohomology class ζ_{G} as the extension class of the subset complex of G and ask for which groups ζ_{G} is nonzero. We showed in [3] that the mod 2-reduction of ζ_{G} is the top-Stiefel Whitney class of I_{G}. So, calculation of the top Stiefel-Whitney class of I_{G} has direct consequences for the class ζ_{G}. Another reason for calculating the top Stiefel-Whitney class of I_{G} is a theorem of Turygin which says that there are Borsuk-Ulam type theorems for group actions on products of spheres if the top Stiefel-Whitney class of I_{G} pulls back to a nonzero class in the associated equivariant cohomology (see [8] for details).

Now, we explain how Theorem 1.2 applies to the calculation of Stiefel-Whitney classes. Let w_{top}(I_{G}) denote the top Stiefel-Whitney class of I_{G}. Note that

I_{G} = I_{G/Φ(G)} ⊕ ind_{Φ(G)}^{G}(I_{Φ(G)}).

So, by the multiplicative property of the top Stiefel-Whitney class, we have

w_{top}(I_{G}) = w_{top}(I_{G/Φ(G)}) · w_{top}(ind_{Φ(G)}^{G}(I_{Φ(G)})).

Recall that

I_{G/Φ(G)} = ⨁_{ξ∈Hom(G,ℝ^{×})−{0}}ξ

and the first Stiefel-Whitney class w_{1}(ξ) ∈ H^{1}(G, 𝔽_{2}) is the class associated to ξ under the following isomorphisms

H^{1}(G, 𝔽_{2}) ≅ Hom(G, ℤ/2) ≅ Hom(G, ℝ^{×}).

So, we have

w_{top}(I_{G/Φ(G)}) = ∏_{ξ∈Hom(G,ℝ^{×})−{0}}w_{1}(ξ) = σ_{G}.

As a consequence of Theorem 1.2, we obtain the following

Corollary 4.2. If G is a nonabelian 2-group, then w_{top}(I_{G}) = 0.

This was also proved in [3] using different methods.

4.2. Inflated essential classes. The essential cohomology ideal of a p-group G is defined as the ideal

Ess^{∗}(G) = ker{ ∏_{H<G} res_{H}^{G} : H^{∗}(G, 𝔽_{p}) → ∏_{H<G} H^{∗}(H, 𝔽_{p})}.

Note that if N is a normal subgroup of G such that N ≤ Φ(G), then

inf_{G/N}^{G}(Ess^{∗}(G/N)) ⊆ Ess^{∗}(G).

We would like to know if these inflated essential classes give some interesting essential classes in Ess^{∗}(G). We consider the case where G is 2-group and N = Φ(G). Note that

H^{∗}(G/Φ(G), 𝔽_{2}) ≅ 𝔽_{2}[x_{1}, …, x_{k}]

where k = dim(H^{1}(G, 𝔽_{2})). It is easy to see that the essential ideal of an elementary abelian 2-group is the principle ideal generated by the product of all nontrivial one dimensional classes. So, inf_{G/Φ(G)}^{G}(u) is divisible by σ_{G}. Another corollary of Theorem 1.2 is the following:

Corollary 4.3. If G is a nonabelian 2-group, then

inf_{G/Φ(G)}^{G}(Ess^{∗}(G/Φ(G))) = 0.

This corollary has some further applications in terms of essential classes coming from relative cohomology classes. For a G-set X, there is a X-relative version of the usual group cohomology. The usual cohomology algebra of a p-group G with coefficients in a field k = 𝔽_{p} is isomorphic to the algebra Ext_{kG}^{∗}(k, k) of the equivalence classes of k by k extensions of kG-modules. If we require the extensions to be X-split (i.e., split after tensoring with permutation module kX), then we get a X-relative version of the usual Ext_{kG}^{∗}(k, k) which is usually denoted by XExt_{kG}^{∗}(k, k). There is an obvious map

φ : XExt_{kG}^{∗}(k, k) → Ext_{kG}^{∗}(k, k)

which takes an X-relative extension to a usual kG-module extension. It is shown in [1] that if G is p-group and X is the disjoint union of all G-sets of the form G/H with H a maximal subgroup, then the image of φ lies in inf_{G/Φ(G)}^{G}(Ess^{∗}(G/Φ(G))). As a consequence of Corollary 4.3, it is obtained in [1] that φ is a zero map when G is a nonabelian 2-group.

4.3. Minh’s theorem on essential cohomology. After proving Theorem 1.2, we noticed that in [4] Minh uses the statement of Theorem 1.2 to prove the p = 2 case of the following theorem:

Theorem 4.4 (Minh [4]). Let G be a p-group. If u ∈ Ess^{∗}(G), then u^{p} = 0.

Minh refers to Serre’s Theorem for the proof of Theorem 1.2 but, as we explained in Remark 3.3, the proof of Theorem 1.2 does not directly follow from Serre’s arguments in [7]. So, the p = 2 case of the above theorem can be considered as another application of Theorem 1.2.

Acknowledgements: I would like to thank David Green and Peter Symonds for reading and commenting on earlier versions of this note. I also thank the referee for careful reading of the manuscript and for suggesting the alternative proof for Proposition 1.3 described in Remark 3.5.

References

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[3] A. Güçlükan and E. Yalçın, The Euler class of a subset complex, preprint 2007.

[4] P. A. Minh, Essential mod-p cohomology classes of p-groups: an upper bound for nilpotency degrees, Bull. London Math. Soc. 32 (2000), 285–291.

[5] V. Reiner and P. Webb, The combinatorics of the bar resolution in group cohomology, J. Pure Appl. Algebra 190 (2004), 291–327.

[6] J. P. Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413–420.

[7] J. P. Serre, Une relation dans la cohomologie des p-groupes, C.R. Acad. Sci. Paris 304 (1987), 587–590.

[8] Y. A. Turygin, A Borsuk-Ulam theorem for (ℤ_{p})^{k}-actions on products of (mod p) homology spheres, Topology Appl. 154 (2007), 455–461.

[9] E. Yalçın, Group actions and group extensions, Trans. Amer. Math. Soc. 352 (2000), 2689–2700.

Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey

E-mail address: yalcine@fen.bilkent.edu.tr