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Let $H$ and $K$ be normal subgroups of $G$ such that $G = HK$. Prove that $(G/H \cap K) \cong (G/H) \times (G/K)$.

To be honest, I'm not sure where to approach this. I know that the statement that $H$ and $K$ are normal are to allow the quotient groups to occur. I have studied the three isomorphic theorems and the correspondence theorem, but I do not really have a complete grasp on the concept.

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Hint: Define a homomorphism $G\to (G/H)\times (G/K)$ in an obvious way. What is it's kernel?

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    Is the kernel the identity? Or is it more complex than that? I believe the homomorphism $\phi$ would map $g \rightarrow gH \times gK$ and that if $g \in ker\phi$ then $\phi (g) = e$. – 2017-02-26