Homomorphisms Group Theory

Question

Does anybody know how I would go about proving this question ?

Let G and H be groups and let φ : G −→ H be a group homomorphism. Suppose that G is abelian and φ is a surjection. Prove that H is abelian.

Answer 1

$$\phi(a)\phi(b)=\phi(ab)=\phi(ba)=\phi(b)\phi(a)$$

Question 1: Where did I use the fact that $\phi$ is a homomorphism?

Question 2: Where did I use the fact that $G$ is abelian?

Question 3: How can you use surjectivity of $\phi$ to show that the above is enough to prove that $H$ is abelian?

Answer 2

Another approach: As $G$ is abelian, $G/\ker(\varphi)$ is also abelian. But $H \cong G/\ker(\varphi)$ by the isomorphism theorem.

This also conveys why we are interested in maps - they preserve properties.