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Herstein: Homomorphism Proof

by ThatPinkSock
Tags: herstein, homomorphism, proof
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Oct18-11, 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.

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.
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Oct18-11, 08:01 PM
micromass's Avatar
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??
Oct19-11, 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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