Herstein: Homomorphism Proof

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

[tex]k\varphi(a)k^{-1}\in \varphi(N)[/tex]

Replace k with [itex]\varphi(n)[/itex], 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?

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ThatPinkSock	Herstein: Homomorphism Proof Tweet 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 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus	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??

Deveno Recognitions: 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?