I came across this question in a practice exercise and can't quite understand it.
If f is a homomorphism from a commutative group $(S,*)$ to another group $(T,*')$, then prove that $(T,*') is also commutative.
As far as I understand homomorphic functions don't have to be from a commmutative group to another commutative one. Am I not understanding something about the question?
If the homomorphism is surjective, you can show this directly. Let $f: S \to T$ be a surjective homomorphism and let $a, b$ be elements of $T$. Then there are $a', b'$ in $S$ such that $f(a')=a$ and $f(b')=b$ (this is where you need surjectivity). Thus $f(a'\ast b')=a\ast'b$. But since $S$ is commutative, $a'\ast b'=b' \ast a'$. Thus $f(a'\ast b')=b\ast'a$, which proves that $a\ast' b=b\ast' a$. Hence $T$ is commutative.