(i) Prove the first Sylow theorem, that a finite group of order $p^{n} r$ with $p$ prime and $p$ not dividing the integer $r$ has a subgroup of order $p^{n}$.

(ii) State the remaining Sylow theorems.

(iii) Show that if $p$ and $q$ are distinct primes then no group of order $p q$ is simple.