If is an isomorphism, prove that is abelian if and only if is abelian. If is only a homomorphism, what additional conditions on (if any) are sufficient to ensure that if is abelian, then so is ?
Let be a group isomorphism.
Suppose is abelian, and let . Since is surjective, there exist such that and . Now we have . Thus and commute; since were arbitrary, is abelian.
Suppose is abelian, and let . Then we have . Since is injective, we have . Since were arbitrary, is abelian.
Note that in the proof of above, we only needed to use the surjectivity of . So in fact the image of any surjective group homomorphism from an abelian group is abelian. Similarly, in the proof of above we only used injectivity; thus the source of any injective group homomorphism to an abelian group is abelian.