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Isomorphism theorems

  1. Dec 13, 2007 #1
    Does anybody know of a nice, intuitive way to remember the second and third isomorphism theorems?
     
  2. jcsd
  3. Dec 14, 2007 #2

    quasar987

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    For the second, with hypothesis "If H,K<G, K<N_G(H), then blahblah", draw a picture. Then notice how it "makes sense" that K/(HnK) ~ HK/K, in the sense that HnK is to K what H is to HK.

    The third is easy. It says that if N,M and normal in G and N is in M, then the "fraction" (G/N)/(M/N) can be "simplified": (G/N)/(M/N) ~ G/M.

    Actally you just have to remember the formula (G/N)/(M/N) ~ G/M because for it to make any sense, we must have that N and M are normal in G and that N in in M, otherwise G/M, G/N and M/n are not defined.
     
  4. Dec 14, 2007 #3

    morphism

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    For 2, draw the lattice - for groups it looks something like this:
    Code (Text):

         G
         |
        HK
       /  \\
      H    K
      \\  /
       H[itex]\cap[/itex]K
        |
       {1}
    where I'm using \\ to indicate the isomorphism you get when you collapse the line \, namely HK/K =~ H/H[itex]\cap[/itex]K. (Note that when you collapse the other two lines, you get a corresponding statement about indices; what is it?)

    For 3, remember how fractions work: (a/b)/(c/b) = a/c.
     
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