I can't believe I've been stuck on this the whole day... I'd appreciate any help. Suppose G is a free abelian group, and even feel free to assume it's finitely generated. H is a subgroup. I'm trying to prove that Rank(G/H) = Rank(G) - Rank(H). Also, I'm looking for the most basic proof. (I know there is a proof out there using flatness of the rational numbers as a module etc, but I'm looking for a more direct approach.) I've been messing about with elements, but not getting anywhere. Any tips/insights?