# Prove Abelian group

• Apr 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.
• Apr 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 $ab=\theta(g)\theta(g')=\theta(gg')=\theta(g'g)=\th eta(g')\theta(g)=ba$
• Apr 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 $ab=\theta(g)\theta(g')=\theta(gg')=\theta(g'g)=\th eta(g')\theta(g)=ba$