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Let $M$ and $N$ be normal subgroups of $G$. Find a homomorphism $f:G \rightarrow \frac{G}{M} \times \frac{G}{N}$ and use this to prove that $\frac{G}{M \cap N}$ is isomorphic to a subgroup of $\frac{G}{M} \times \frac{G}{N}$.

This was an exercise given in class which our professor handwaved the next meeting by saying ``just apply the isomorphism theorem.'' I have no idea where to start with this question as we just started our discussion on this topic.

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    The hint is a good one! The onl$y$ stronger hint is to give you the solution, at which point you are not learning anything. Let's assume good faith from your prof until he *really* gives you handwaving :)2012-08-07

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Some facts.

  • The image of any homomorphism is a subgroup of the target. So $f:U\to V\implies f(U)\le V$.
  • Given maps $\alpha:G\to A$ and $\beta:G\to B$, there is a canonical map $\alpha\times\beta:G\to A\times B$.
  • The kernel of the quotient map $G\to G/N$ is precisely $N$. [What is $\ker(\alpha\times\beta)$ via $\ker\alpha,\ker\beta$?]
  • One of the isomorphism theorems states that if $f:G\to H$ then $G/\ker f\cong \mathrm{im}\,f$.

Can you put these together?