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 $f$ a homomorphism of $H$ into $\mathbf{Z}/n\mathbf{Z}$. (a) Let $x \in G$ and let $q$ be the order of the image of $x$ in $G/H$; then $qx \in H$. Show 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) Show that $f$ extends to a homomorphism of $G$ into $\mathbf{Z}/n\mathbf{Z}$ (use Zorn's Lemma).