A note on subgroups of Z^N

Abstract

In this note, we investigate homomorphisms from subgroups of Z^N to Z^N. Let H(A) be the following assertion for a subgroup A of Z^N: For any linearly independent a_n ∈ A (n ∈ N) there exists a homomorphism h:A → Z such that n: h(a_n) ≠ 0] is infinite. Of course, H(Z^N) does not hold by Specker’s theorem. Let H be the following assertion: H(A) holds for any subgroup A of Z^N of cardinality less than 2^λ. We show that the continuum hypothesis implies H but H is independent of the negation of the continuum hypothesis. Our terminology is the usual one for elementary linear algebra, abelian group theory ([F]) and set theory ([K] and [J]). Z^N is the group consisting of all functions from the set N of natural numbers to the group Z of integers.