SERRE’S THEOREM

ON THE COHOMOLOGY ALGEBRA OF A p-GROUP

Pham Anh Minh

1. Introduction

The purpose of this note is to give a proof of a theorem of Serre, which states that if G is a p-group which is not elementary abelian, then there exist an integer m and non-zero elements x_{1}, …, x_{m} ∈ H^{1}(G, Z/p) such that

\begin{aligned}x_{1} · x_{2} · . . . · x_{m} & = 0, if p = 2, \\ β(x_{1}) · β(x_{2}) · . . . · β(x_{m}) & = 0, if p > 2, \end{aligned}

with β the Bockstein homomorphism. Denote by m_{G} the smallest integer n satifying the above property. The theorem was originally proved by Serre [5], without any bound on m_{G}. Later, in [2], Kroll showed that m_{G} ≤ p^{k} − 1, with k =dim_{Z/p}H^{1}(G, Z/p). Serre, in [6], also showed that m_{G} ≤ \frac{p^{k}−1}{p−1}. In [3], using the Evens’ norm map, Okuyama and Sasaki gave a proof with a slight improvement on Serre’s bound; it follows from their proof (see e.g. [1, Th. 4.7.3]) that m_{G} ≤ (p + 1)p^{k−2}. However, the m_{G} can be sharpened more, as we can see below.

For convenience, write H^{∗}(G, Z/p) = H^{∗}(G). For every x_{i} ∈ H^{1}(G), set

y_{i} = \{ \begin{matrix}x_{i} & , for p = 2 \\ β(x_{i}) & , for p > 2. \end{matrix}

Our main result is the following

Theorem. m_{G} ≤ (p + 1)p^{[ \frac{k}{2} ]−1}. Further, there exists a basis x_{1}, …, x_{k} of H^{1}(G) such that

X = ∏_{λ_{i}∈Z/p}(y_{1}+λ_{2}y_{2}+λ_{3}y_{3}+⋯+λ_{[ \frac{k}{2} ]+1}y_{[ \frac{k}{2} ]+1})· ∏_{λ_{i}∈Z/p}(y_{2}+λ_{3}y_{3}+⋯+λ_{[ \frac{k}{2} ]+1}y_{[ \frac{k}{2} ]+1}) = 0.

2. Proof of the theorem

As the Bockstein homomorphism commutes with the inflation map, we can suppose that G is given by the central extension

(G)  0 → Z/p → G → V → 0,

1991 Mathematics subject classifications: 20J06

with V a vector space of dimension k over Z/p. Furthermore, if L is a direct factor of G which is not elementary abelian, then m_{G} ≤ m_{L}. So we may suppose that G does not contain any proper direct factor. It follows that there exists a basis {u_{i}} of H^{1}(V) such that the factor set z (viewed as an element of H^{2}(V)) of the central extension (G) is of one of the following forms:

- for p = 2 and k = 2n:  (a) u_{1}u_{2} + u_{3}u_{4} + ⋯ + u_{2n−1}u_{2n}
(b) u_{1}^{2} + u_{2}^{2} + u_{1}u_{2} + u_{3}u_{4} + ⋯ + u_{2n−1}u_{2n}
- for p = 2 and k = 2n + 1:  (c) u_{0}^{2} + u_{1}u_{2} + u_{3}u_{4} + ⋯ + u_{2n−1}u_{2n}
- for p > 2 and k = 2n:  (d) u_{1}u_{2} + u_{3}u_{4} + ⋯ + u_{2n−1}u_{2n}
(e) v_{1} + u_{1}u_{2} + u_{3}u_{4} + ⋯ + u_{2n−1}u_{2n}
- for p > 2 and k = 2n + 1:  (f) v_{0} + u_{1}u_{2} + u_{3}u_{4} + ⋯ + u_{2n−1}u_{2n},

with v_{i} = βu_{i}. With some abuse of notation, we consider the u_{i}’s and v_{i}’s as elements of H^{∗}(G), via the inflation map from H^{∗}(V) to H^{∗}(G). So z = 0 in H^{2}(G). If z is of the form (e) (resp. (f)), then, in H^{2}(G), v_{1} (resp. v_{0}) is expressed as a sum of the u_{i}u_{j}’s, so v_{1}^{p} (resp. v_{0}^{p}) = 0. Hence we need only consider the cases (a)-(d).

Let {a_{k}} be the set of generators of G satisfying u_{i}(a_{k}) = δ_{i,k}, with δ the Kronecker symbol, and let e be a generator of the Frattini subgroup of G. For convenience, write G = G_{n}. So G is just the central product G_{n} = G_{n−1} • ⟨a_{2n−1}, a_{2n}⟩. Denote by A_{G} (resp. EA_{G}) the set of all maximal (resp. maximal elementary) abelian subgroups of G. For every A ∈ EA_{G}, set Ã = C_{G}(A). Let H(G) be the subalgebra of H^{∗}(G) generated by the y_{i}’s. Following [4], [7], we have

Proposition 1 ([4], [7]). (i) Every element of A_{G} or EA_{G} is of rank n + 1 if z is of one of the forms (a), (c), (d), and is of rank n if z is of the form (b);

(ii) Ker(H(G) →^{Res} ∏_{A∈EA_{G}}H^{∗}(Ã)) = {0}.

Our task is now to prove the following

Proposition 2. For every A ∈ A_{G}, there exist μ_{i} ∈ Z/p, with μ_{1}, μ_{2} not all equal to zero, such that A ⊂ Ker(μ_{1}u_{1} + μ_{2}u_{2} + μ_{3}u_{3} ⋯ + μ_{2n−1}u_{2n−1}).

Proof. It is easy to show that the proposition holds if k ≤ 3, for A = ⟨e, a_{1}^{μ_{2}}a_{2}^{μ_{1}}⟩ (or ⟨a_{0}, a_{1}^{μ_{2}}a_{2}^{μ_{1}}⟩ if z is of the form (c)) with μ_{i} ∈ Z/p not all equal to zero, hence A ⊂Ker(μ_{1}u_{1} − μ_{2}u_{2}). Suppose that the proposition holds for G_{n−1}. Let A be an element of A with generators w_{1}, …, w_{ℓ} (by Proposition 1 (i), ℓ = n + 1 or n). Write

w_{ℓ} = x · a_{2n−1}^{μ_{2n−1}}a_{2n}^{μ_{2n}},

with x ∈ G_{n−1}, μ_{i} ∈ Z/p. Consider the following cases separately:

(1) If w_{1}, …, w_{ℓ−1} are elements of G_{n−1}, then [w_{i}, x] = 1, for 1 ≤ i ≤ ℓ − 1. Set B = ⟨w_{1}, …, w_{ℓ−1}, x⟩. By the inductive hypothesis, there exists an element α = μ_{1}u_{1} + μ_{2}u_{2} + μ_{3}u_{3} + ⋯ + μ_{2n−3}u_{2n−3}, with μ_{i} ∈ Z/p and μ_{1}, μ_{2} not all equal to zero, such that B ⊂ Kerα. So w_{ℓ} ∈ Kerα. Hence A ⊂ Kerα;

(2) If there exists 1 ≤ i ≤ ℓ − 1 such that w_{i} ∉ G_{n−1}, without loss of generality, we can suppose that w_{1}, …, w_{ℓ−2} are elements of G_{n−1}, μ_{2n−1} = 1, μ_{2n} = 0, and w_{ℓ−1} = y · a_{2n}, with y ∈ G_{n−1}. It follows that every w_{i}, 1 ≤ i ≤ ℓ − 2, commutes with x, y. Set B = ⟨w_{1}, …, w_{ℓ−2}, y⟩. By the inductive hypothesis, there exists an element α = μ_{1}u_{1} + μ_{2}u_{2} + μ_{3}u_{3} + ⋯ + μ_{2n−3}u_{2n−3}, with μ_{i} ∈ Z/p and μ_{1}, μ_{2} not all equal to zero, such that B ⊂ Kerα. Write α(x) = ν ∈ Z/p and set γ = α − νu_{2n−1}. It is clear that w_{ℓ−1} and w_{ℓ} belong to Kerγ. Hence A ⊂ Kerγ. The proposition is proved. □

Proof of the theorem. Set x_{i} = u_{i} for i ≤ 2, and x_{i} = u_{2i−3} for 3 ≤ i ≤ n + 1 (∗). We prove that ResG ; Ã(X) = 0, for every A ∈ EA_{G}, which implies that X = 0 by Proposition 1 (ii). By Proposition 2, this fact obviously holds if z is not of the form (b), as Ã ∈ A_{G} for every A ∈ EA_{G}.

Assume that z is of the form (b). Note that the theorem holds for k = 2 (in that case, G ≅ Q_{8}, the quaternion group of order 8), as u_{1}u_{2}(u_{1} + u_{2}) = βz = 0 in H^{∗}(G). Suppose that k ≥ 4. Let A be an element of EA_{G} and let w_{1}, …, w_{n−1}, e be a basis of A. There exist then x, y ∈ G with ⟨x, y⟩ ≅ Q_{8} and Ã = ⟨w_{1}, …, w_{n−1}⟩ × ⟨x, y⟩. Assume that every factor of degree 1 of X restricts to a non-zero element of H^{1}(Ã) (∗∗). Let u, v be elements of H^{1}(Ã) satisfying u(A) = v(A) = {0}, u(x) = v(y) = 1 and u(y) = v(x) = 0. As showed above, we have uv(u + v) = 0 in H^{∗}(Ã). By Proposition 2 and by (∗∗), there exist factors α, γ, δ of degree 1 of X satisfying α(A) = γ(A) = δ(A) = {0}, α(x) = γ(y) = δ(xy) = 0, but α, γ, δ restrict to non-zero elements of H^{1}(Ã). This means that α · γ · δ restricts to uv(u + v) = 0 in H^{∗}(Ã). Hence ResG ; Ã(X) = 0. The theorem is proved.

Remark. The theorem can be proved by another way, using the Evens’ norm map, as follows. Let K be the subgroup of G defined by K = ⟨a_{i}|i ≤ 2⟩, and set L = K × ⟨a_{4}, …, a_{2n}⟩ ⊂ G. Then, with the notation given in (∗), Y = y_{1}^{p}y_{2} − y_{2}^{p}y_{1} = 0 in H^{∗}(L), which implies that N_{L→G}(Y) = 0 in H^{∗}(G), with N_{L→G} the Evens’ norm map; we can check that N_{L→G}(Y) is nothing but the element X given in the main theorem.

References

1. D. Benson, Representations and cohomology II, Cambridge Studies in Advanced Mathematics 31, Cambridge University Press, Cambridge.

2. O. Kroll, A representation theoretical proof of a theorem of Serre, Århus Preprint, Maj 1986.

3. T. Okuyama and H. Sasaki, Evens’ norm maps and Serre’s theorem on the cohomology algebra of a p-group, Arch. Math 54 (1990), 331-339.

4. D. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 194 (1971), 197-212.

5. J.P. Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413-420.

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

7. M. Tezuka and N. Yagita, The varieties of the mod p cohomology rings of extra-special p-groups for an odd prime p, Math. Proc. Camb. Phil. Soc. 94 (1983), 449-459.

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