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Herstein: Homomorphism Proof 
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#1
Oct1811, 07:35 PM

P: 3

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
Oct1811, 08:01 PM

Mentor
P: 18,277

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?? 


#3
Oct1911, 02:57 AM

Sci Advisor
P: 906

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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