Exercise 14, § 6

Let G be a finite group. Suppose that, for all $x, y \in G$, the subgroup of G generated by $\{x, y\}$ is nilpotent. Show that G is nilpotent (apply the hypothesis to x and y of orders $p_1^{n_1}$ and $p_2^{n_2}$, where $p_1$ and $p_2$ are distinct prime numbers).