Harmonic analysis on discrete Abelian groups

Let $G$ be an Abelian group and let $\mathbb C^G$ denote the linear space of all complex-valued functions defined on $G$ equipped with the product topology. We prove that the following are equivalent.

(i) Every nonzero translation invariant closed subspace of $\mathbb C^G$contains an exponential; that is, a nonzero multiplicative function.

(ii) The torsion free rank of $G$ is less than the continuum.