$(G\times G')/(H\times H')\simeq (G/H)\times(G'/H')$.

Let $G_1$ and $G_2$ be two groups.
Let $H_1$ and $H_2$ be normal subgroups of $G_1$ and $G_2$ respectively.
Then prove that $(G1\times G2)/(H1\times H2)$ is isomorphic to $(G1/H1)\times(G2/H2)$.

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Have you tried the obvious map between these two groups? $(a,b)+(H_1\times H_2) \mapsto (a+H_1,b+H_2)$ – fkraiem Mar 7 '14 at 17:41

You can consider the obvious surjective homomorphism $G_1\times G_2\to (G_1/H_1)\times (G_2/H_2)$. What is its kernel? Then you can use the first isomorphism theorem. This is essentially the same solution as if you just defined a map and checked that it is an bijective homomorphism. But using the 1st isom. theorem you don't need to do this many verifications, they are already encoded.