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**Markeur** If $\displaystyle H$ and K are normal subgroups of a group $\displaystyle G$ with $\displaystyle HK=G$, prove that $\displaystyle G/(H \cap K) \simeq (G/H) \times (G/K)$.

Hint: If $\displaystyle \phi:G \rightarrow (G/H) \times (G/K)$ is defined by $\displaystyle x \mapsto (xH,xK)$, then $\displaystyle ker \: \phi=H \cap K$; moreover, we have $\displaystyle G=HK$, so that $\displaystyle \displaystyle\cap_{a}aH=HK=\displaystyle\cap_{b}bK$.

I have managed to prove that $\displaystyle ker \: \phi=H \cap K$. I'm having difficulty in proving that $\displaystyle \phi$ is surjective. This is crucial in proving the above isomorphism using 1st Isomorphism Theorem.