Does φ(N) Form a Normal Subgroup in G' When φ is a Homomorphism?

[ThatPinkSock]

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.

 

[micromass]

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??

 

[Deveno]

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?