Exercise 18, § 6

Let $n$ be an integer $\geq 1$ and let G be a commutative group, written additively, such that $nx = 0$ for all $x \in G$. Let H be a subgroup of G and let $f$ be a homomorphism from H into $\mathbf{Z}/n\mathbf{Z}$. (a) Let $x \in G$ and let q be the order of the image of $x$ in G/H; one has $qx \in H$. Prove that there exists $\alpha \in \mathbf{Z}/n\mathbf{Z}$ such that $q\alpha = f(x)$. Deduce the existence of an extension of $f$ to the subgroup of G generated by H and $x$. (b) Prove that $f$ extends to a homomorphism from G into $\mathbf{Z}/n\mathbf{Z}$ (use Zorn's theorem).