(i) Let $p$ be a prime number. Show that a group $G$ of order $p^{n}(n \geqslant 2)$ has a nontrivial normal subgroup, that is, $G$ is not a simple group.

(ii) Let $p$ and $q$ be primes, $p>q$. Show that a group $G$ of order $p q$ has a normal Sylow $p$-subgroup. If $G$ has also a normal Sylow $q$-subgroup, show that $G$ is cyclic. Give a necessary and sufficient condition on $p$ and $q$ for the existence of a non-abelian group of order $p q$. Justify your answer.