State and prove the first isomorphism theorem. [You may assume that kernels of homomorphisms are normal subgroups and images are subgroups.]

Let $G$ be a group with subgroup $H$ and normal subgroup $N$. Prove that $N H=\{n h: n \in N, h \in H\}$ is a subgroup of $G$ and $N \cap H$ is a normal subgroup of $H$. Further, show that $N$ is a normal subgroup of $N H$.

Prove that $\frac{H}{N \cap H}$ is isomorphic to $\frac{N H}{N}$.

If $K$ and $H$ are both normal subgroups of $G$ must $K H$ be a normal subgroup of $G$ ?

If $K$ and $H$ are subgroups of $G$, but not normal subgroups, must $K H$ be a subgroup of $G$ ?

Justify your answers.