Exercise 19, § 6

Let G be a finite commutative group. (a) Now prove 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 into a product of $p$-groups.) Now prove that the subgroup H generated by such an element is a direct factor in G (apply the preceding exercise to construct a homomorphism $G \to H$ whose restriction to H is the identity). (b) Now prove that G is a direct product of cyclic groups. (Reason by induction on Card(G) using a).) (Cf. VII, §4, No. 6.)