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Show they are isomorphic

  1. Sep 20, 2011 #1
    Let M and N be normal subgroups of G such that G=MN.
    Prove that G/(M[itex]\bigcap[/itex]N)[itex]\cong[/itex](G/M)x(G/N).

    I tried coming up with an isomorphism from G to (G/M)x(G/N) such that the kernel is M[itex]\bigcap[/itex]N, so that I can use the fundamental homomorphism theorem.
    I tried f(a) = (aM, aN). It is an homomorphism and M[itex]\bigcap[/itex]N is the kernel but I'm having a hard time showing it is onto.

    I would appreciate any help.
    Thank you
     
  2. jcsd
  3. Sep 20, 2011 #2

    CompuChip

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    So you need to show that for all g, g' in G, there is an a in G such that: (g M, g' N) = (a M, a N).

    I haven't worked this out in detail, but: since G = MN, you can write g = m n, g' = m' n'. I suspect that a = m' n might do the trick.
    You will need that M and N are normal, so in particular h M = M h, h N = N h for all h in G.
     
  4. Sep 20, 2011 #3
    Thank you Compuchip
     
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