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This is exercise 3.3.7 in Dummit and Foote. I was able to finish this exercise by defining a map $\phi: G \to G/M \times G/N $ by sending $g$ to $(gM, gN)$ and showing the kernel is $M \cap N$ and then conclude the result by the first isomorphism theorem. However, the hint given for the question is "Draw the lattice." so I was wondering if there is another way to do this exercise using the 2nd, 3rd or 4th isomorphism theorems.

Kumar
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abeliangrape
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  • Could you reproduce the text of the exercise? This is certainly not always true, such as when $M=N$. – Matt Samuel Oct 26 '19 at 16:24
  • @Matt Samuel Hi, I edited my question. Previously I forgot to add $G=MN$ and $M, N$ are normal in $G$, – abeliangrape Oct 26 '19 at 16:26
  • I would personally go with your approach. It's almost trivial to show that the kernel is $M\cap N$. It's a little work to show that the homomorphism is surjective, though. But yeah, it's fair to ask here what the hint might be referring to. – Arthur Oct 26 '19 at 16:30
  • Right, I was just curious what the authors had originally in mind though. I wanted to learn about their perspective. Yes, showing surjectivity takes a bit of time, but I was able to do it. – abeliangrape Oct 26 '19 at 16:31
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    Related:https://math.stackexchange.com/questions/254448/showing-that-g-h-cap-k-cong-g-h-times-g-k – Kumar Oct 26 '19 at 16:35
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    This is simply the group theory version of the Chinesre Remainder Theorem, so your approach is quite fine for me. – Bernard Oct 26 '19 at 16:53

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