Exercise 4, § 4

Let G be a finite abelian group. a) For every prime number p, show that the number of elements of order p in G is $p^N - 1$, where $N = \sum_{n=1}^\infty m(p^n)$ (in the notation of Prop. 9 of VII, p. 24). b) Deduce that if the equation $x^p = 3$ has at most p solutions in G, for each prime number p, then G is cyclic.