Let $G$ be a group and $H,K \subset G$ be normal subgroups such that $H \cap K = \{e\}$ and $G=HK$. I need to show that the map, which I have denoted $\phi$, $H \times K \longrightarrow G$ given by $(h,k) \mapsto hk$ is a group isomorphism. I have shown that it is a homomorphism: for some $h_1,h_2 \in H, k_1,k_2 \in K, \phi(h_1k_1,h_2k_2)=(h_1k_1)(h_2k_2)=\phi(h_1,k_1)\phi(h_2,k_2)$.
I have also shown that it is injective: take $(h_1,k_1) \in H \times K$ and $(h_2,k_2) \in H \times K$ such that $(h_1,k_1) \neq (h_2,k_2)$. Then we have $\phi(h_1,k_1)=h_1k_1 \neq h_2k_2 =\phi(h_2,k_2)$.
However, I am not sure how to show that $\phi$ is surjective to complete the proof.