Suppose that G is a finite Abelian group. Prove that G has order $p^n$, where p is prime, if and only if the order of every element of G is a power of p.
I tried the following route, but got stuck. Using the fundamental theorem of finite Abelian groups, the problem reduces to proving Cauchy's theorem for a cyclic abelian group. If G is a cyclic group, and p divides G, then G has an element of order p whether p is prime or not. If we regard G as the integers mod p, then we can notice that if $|G| = kp$ then the integer k has order p in G.