PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 6, June 1997, Pages 1577–1578 S 0002-9939(97)04129-4

GROUP HOMOMORPHISMS INDUCING mod -p COHOMOLOGY MONOMORPHISMS

PHAM ANH MINH

(Communicated by Ronald Solomon)

Abstract. Let f: G → K be a homomorphism of p-groups such that f^{(n)} : H^{n}(K, Z/p) → H^{n}(G, Z/p) is injective, for 1 ≤ n ≤ 2. We prove that the non-bijectivity of f implies the existence of a quotient L of G containing K as a proper direct factor. This gives a refined proof of a result of Evens, which asserts that f is bijective if f^{(1)} is.

Let p be a prime number and let Z/p be the prime field of p elements. For every p-group K, let us denote by H^{∗}(K) the mod -p cohomology of K.

Let f: G → K be a homomorphism of p-groups such that f^{(n)} : H^{n}(K) → H^{n}(G) is injective, for 1 ≤ n ≤ 2. We shall give a refined proof of the following result of Evens.

Theorem A (Evens [1, Th. 7.2.4]). If f^{(1)} is bijective, then so is f.

Further, we prove

Theorem B. If f is not bijective, then there exists a quotient L of G containing K as a proper direct factor (i.e., L = J × K, with J ≠ {1}).

Note that the maps H^{1}(K) ⟶^{Inf_{1}(L,K)} H^{1}(L) and H^{1}(L) ⟶^{Inf_{1}(G,L)} H^{1}(G) are injective, and Im Inf_{1}(L, K) is a proper subgroup of H^{1}(L) (since J ≠ {1}), so, by Theorem B, the non-bijectivity of f implies that

dim_{Z/p} H^{1}(K) < dim_{Z/p} H^{1}(L) ≤ dim_{Z/p} H^{1}(G),

hence f^{(1)} is not bijective. We obtain then an alternative proof of Theorem A.

Proof of Theorem B. We first prove that f is surjective. Set K′ = Im f. So f factors through G →^{f} K′ ↪ K. Since f^{(1)} = (H^{1}(K) ⟶^{Res} H^{1}(K′) →^{f^{∗}} H^{1}(G)) is injective, it follows that Res: H^{1}(K) → H^{1}(K′) is injective. This implies K = K′, so f is surjective.

Consider the extension

1 → Ker f → G →^{f} K → 1.

Let p^{m} be the order of Ker f (m ≥ 1, since f is not bijective). The above extension

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors September 7, 1995.

1991 Mathematics Subject Classification. Primary 20J05.

can then be obtained by successive central extensions

0 → Z/p → G_{i} ⟶^{p_{i}} G_{i−1} → 1, (G_{i})

1 ≤ i ≤ m, with G_{0} = K, G_{m} = G, p_{1} ∘ ⋯ ∘ p_{m} = f.

Let z ∈ H^{2}(K) be the factor set of the extension (G_{1}), viewed as a cohomology class of H^{2}(K); then z ∈ Ker Inf_{2}(G_{1}, K). Since f^{(2)} = Inf_{2}(G, K) = Inf_{2}(G, G_{1}) ∘ Inf_{2}(G_{1}, K), it follows from the injectivity of f^{(2)} that Inf_{2}(G_{1}, K) is also injective. Hence z = 0, so G_{1} = Z/p × K. The theorem follows.

References

1. L. Evens, Cohomology of groups, Oxford University Press, 1991. MR 93i:20059

Department of Mathematics, Faculty of Science, University of Hue, Dai hoc Khoa hoc, Hue, Vietnam