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Homework Help: Subgroups and normal subgroups

  1. Jan 19, 2009 #1
    Let G be a group, let N be normal in G and let H be a subgroup of G. Assume that G/N and H are finite and that gcd(|G/N|,|H|)=1. Prove that H is a subgroup of N.

    I was thinking about using Lagrange Theorem. and maybe using the fact that G may act on the set of left cosets (G/N) by conjugation.
    and the find the kernel of that action and then maybe use the first isomorphism theorem.

    But I don't get very far with that.
     
  2. jcsd
  3. Jan 19, 2009 #2

    Dick

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    H also acts on G/N by left multiplication. Furthermore, every element of H is in one of the the left cosets of N.
     
  4. Jan 19, 2009 #3

    Hurkyl

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    Maybe looking at G isn't the right way to go. What can you learn about H by studying the group G/N? What about N?
     
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