# Herstein: Homomorphism Proof

by ThatPinkSock
Tags: herstein, homomorphism, proof
 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.
 Emeritus Sci Advisor Thanks PF Gold P: 15,876 Take a in N, you need to prove that $$k\varphi(a)k^{-1}\in \varphi(N)$$ Replace k with $\varphi(n)$, what do you get??
 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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