The homomorphic image of an abelian group is abelian

Question

We learnt about Group Homomorphisms and Abelian Groups, but never have we been shown how to tackle such question....and I have an exam on this tomorrow.

The question says:

Let $\phi : G \rightarrow H$ be a group epimorphism. Prove that if $G$ is Abelian, then $H$ is Abelian.

You don't necesseraly have to give me the answer. Just how to go about it.

Answer 1

Use the fact that any $h_1,h_2\in H$ can be written $h_1 = \phi(g_1),h_2 = \phi(g_2)$ for some $g_1,g_2\in G$.

Answer 2

Given $f:G\to H$ be an epimorphism, let $a,b$ be two elements of $H$ then there exists $x,y$ in $G$ such that:

$a=f(x) , b=f(y)$. Therefore, $ab = f(x)f(y) =f(xy) =f(yx) =f(y)f(x) =ba$

Therefore $ab = ba$

Hence $H$ is also abelian.