|Oct18-11, 07:35 PM||#1|
Herstein: Homomorphism Proof
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.
|Oct18-11, 08:01 PM||#2|
Take a in N, you need to prove that
Replace k with [itex]\varphi(n)[/itex], what do you get??
|Oct19-11, 02:57 AM||#3|
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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