Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook