# Stuck on Rank(G/H) = Rank(G) - Rank(H) Should be trivial(?)

1. Dec 19, 2013

### nonequilibrium

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?

2. Dec 19, 2013

### jgens

This is essentially the same proof as tensoring with Q, but localization functors are exact, so localizing with respect to the multiplicative set S = Z-0 allows the proof to go through without worrying about flatness.

3. Dec 19, 2013

### nonequilibrium

That's actually the kind of proof I was trying to avoid. I'm wondering whether I can do it without commutative algebra-type stuff. After all, that kind of proof is very general, for any abelian group G, but I'm willing to assume G is (abelian and) (1) free and (2) finitely generated. So there should be a more basic/direct aproach (?)

Basically I'm looking for something that just uses straight-forward group theory.