Let $H$ and $K$ be normal subgroups of $G$ such that $H \cap K = \langle e \rangle$ and $HK = G$. Prove that $G$ is isomorphic to $H \times K$.
$G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$