Let G be an abelian group, and let H be some subgroup.

Every subgroup of an abelian group is normal, so H is normal. This means you can factor it out.

Look at G/H and let p be the homomorphism that maps G to G/H.

Basically what this means is that p(g1) = p(g2) only if g1 and g2 are in the same coset of H.

Want to show that G/H is abelian.

Take two elements in G/H. Why not p(g1) and p(g2)?

Play with them to see if they commute. Recall with a homomorphism you can manipulate them as follows:

p(g1)*p(g2) = p(g1*g2) = p(g2*g1) = p(g2)*p(g1)

They do commute, so G/H is abelian.