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

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The discussion centers on proving the equation Rank(G/H) = Rank(G) - Rank(H) for a free abelian group G and its subgroup H. The user seeks a basic proof that avoids advanced concepts like flatness of rational numbers as a module. They express a preference for a straightforward group theory approach, emphasizing the need for a proof that is accessible without delving into commutative algebra. The user is open to assuming G is finitely generated and free to facilitate the proof.

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  • Understanding of free abelian groups
  • Knowledge of group rank and its properties
  • Familiarity with quotient groups
  • Basic concepts of module theory (optional for advanced understanding)
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  • Research the properties of free abelian groups and their ranks
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This discussion is beneficial for mathematicians, particularly those focused on group theory, algebraists, and students seeking to understand the rank of abelian groups and their subgroups.

nonequilibrium
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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?
 
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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.
 
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 approach (?)

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

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