Let $H$ and $K$ be subgroups of a group $G$ satisfying the following two properties.

(i) All elements of $G$ can be written in the form $h k$ for some $h \in H$ and some $k \in K$.

(ii) $H \cap K=\{e\}$.

Prove that $H$ and $K$ are normal subgroups of $G$ if and only if all elements of $H$ commute with all elements of $K$.

State and prove Cauchy's Theorem.

Let $p$ and $q$ be distinct primes. Prove that an abelian group of order $p q$ is isomorphic to $C_{p} \times C_{q}$. Is it true that all abelian groups of order $p^{2}$ are isomorphic to $C_{p} \times C_{p}$ ?