Exercise 14, § 6

Let G be a finite group. Suppose that, whatever $x, y$ in G may be, the subgroup of G generated by $\{x, y\}$ is nilpotent. Now prove that G is nilpotent (apply the assumption 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).