(i) What is a Sylow subgroup? State Sylow's Theorems.

Show that any group of order 33 is cyclic.

(ii) Prove the existence part of Sylow's Theorems.

[You may use without proof any arithmetic results about binomial coefficients which you need.]

Show that a group of order $p^{2} q$, where $p$ and $q$ are distinct primes, is not simple. Is it always abelian? Give a proof or a counterexample.