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Herstein: Homomorphism Proof

  1. Oct 18, 2011 #1
    From Herstein's Abstract Algebra. Section 2.7 #7



    If φ is a homomorphism of G onto G' and N ◅ G, show that φ(N) ◅ G.



    Attempt:
    I want to prove that if k ∈ G' then kφ(N)k-1 = φ(N), but k = φ(n) for some n... then idk what.
     
  2. jcsd
  3. Oct 18, 2011 #2

    micromass

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    Take a in N, you need to prove that

    [tex]k\varphi(a)k^{-1}\in \varphi(N)[/tex]

    Replace k with [itex]\varphi(n)[/itex], what do you get??
     
  4. Oct 19, 2011 #3

    Deveno

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    the fact that φ is onto is important.

    this means that EVERY k in G' is the image of some g in G:

    k = φ(g). now use the fact that φ is a homomorphism.

    what can we say about kφ(n)k-1?
     
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