# Prove Abelian group

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• April 12th 2010, 08:44 AM
rainyice
Prove Abelian group
If G is an abelian group and the group G' is a homomorphic image of G, prove that G' is abelian.
• April 12th 2010, 04:22 PM
Drexel28
Quote:

Originally Posted by rainyice
If G is an abelian group and the group G' is a homomorphic image of G, prove that G' is abelian.

I just helped someone with this in another thread.

Just note that if $a,b\in G'$ then $a=\theta(g),b=\theta(g')$ and so $ab=\theta(g)\theta(g')=\theta(gg')=\theta(g'g)=\th eta(g')\theta(g)=ba$
• April 12th 2010, 05:52 PM
rainyice
Quote:

Originally Posted by Drexel28
I just helped someone with this in another thread.

Just note that if $a,b\in G'$ then $a=\theta(g),b=\theta(g')$ and so $ab=\theta(g)\theta(g')=\theta(gg')=\theta(g'g)=\th eta(g')\theta(g)=ba$

do you mind linking me to that thread? thank you ~!
• April 12th 2010, 09:52 PM
Drexel28
Quote:

Originally Posted by rainyice
do you mind linking me to that thread? thank you ~!

I said less there than I did here, so I don't think it would help.