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 $(G1/H1)\times(G2/H2)$.

  • 1
    $\begingroup$ Have you tried the obvious map between these two groups? $(a,b)+(H_1\times H_2) \mapsto (a+H_1,b+H_2)$ $\endgroup$ – 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.

  • $\begingroup$ I got it now...thanks. $\endgroup$ – user114873 Mar 7 '14 at 18:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.