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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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Concrete examples are usually a good place to start. Your professor gave you a hint, so that's another good place to start. What is the isomorphism theorem? Can you arrange for its hypotheses to be satisfied? Or working backwards, can you arrange for its conclusion to be the thing you're trying to prove? If so, what hypotheses do you need to prove so you can invoke it? – Hurkyl Aug 7 '12 at 7:48
Find a homomorphism $f:G \rightarrow \frac{G}{M} \times \frac{G}{N}$ such that the kernel of $f$ is $M \cap N$. – Mikko Korhonen Aug 7 '12 at 7:59
Given homomorphisms $\alpha:G\to A$ and $\beta:G\to B$ can you think of a map $G\to A\times B\,$? What is the kernel of this map in terms of the kernels of $\alpha$ and $\beta$? Can you think of canonical homomorphisms $G\to G/N$ and $G\to G/M$? – anon Aug 7 '12 at 8:34
The hint is a good one! The only 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 :) – rschwieb Aug 7 '12 at 12:19

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?

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