slender group

《Lecture Notes on Bayesian Estimation and Classification》 · 第 120 页 · Section 4.4「Torsion-Free Groups」，Exercises 4.4，题 8

Let $G$ be a countable reduced torsion-free abelian group and let $C$ be a cartesian sum of countably many infinite cyclic groups. If $\alpha\colon C\to G$ is a homomorphism, prove that $e_i\alpha=0$ for almost all $i$ where $e_i$ is the element of $C$ whose $i$th component is $1$ and other components are $0$. Remark: Groups with the property just established for $G$ are called slender groups.