Exercise 19, § 6

Let $G$ be a finite commutative group. (a) Show that there exists an element $x \in G$ whose order $n$ is a multiple of the orders of the elements of $G$. (Use the decomposition of $G$ as a product of $p$-groups.) Show that the subgroup $H$ generated by such an element is a direct factor of $G$ (apply the above exercise to construct a homomorphism $G \to H$ whose restriction to $H$ is the identity). (b) Show that $G$ is a direct product of cyclic groups. (Argue by induction on $\mathrm{Card}(G)$ using (a).) (Cf. VII, § 4, no. 6.)