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

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.

    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
    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


    User Avatar
    Science Advisor

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