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

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In summary, the conversation revolves around proving that Rank(G/H) = Rank(G) - Rank(H) for a free abelian group G and its subgroup H, without using complex algebraic proofs. The approach suggested is to use localization functors instead of tensoring with Q, which allows for a more direct proof without worrying about flatness. The person asking for help is looking for a more basic and straightforward proof using group theory, since the assumption is that G is both abelian and finitely generated.
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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.
 
  • #3
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.
 

1. What is the meaning of the equation "Rank(G/H) = Rank(G) - Rank(H)"?

The equation represents the rank of a group G with respect to a subgroup H. It states that the quotient of the rank of G and the rank of H is equal to the difference between the rank of G and the rank of H.

2. Why is it important to understand the concept of "Stuck on Rank(G/H) = Rank(G) - Rank(H) Should be trivial(?)"?

Understanding this concept is crucial for analyzing and solving problems related to group theory and abstract algebra. It helps in understanding the structure and properties of groups and subgroups.

3. Can you provide an example of how this equation is used in real-life scenarios?

One example is in cryptography, where the rank of a group can be used to determine the security of a cryptographic system. The equation can also be applied in physics, specifically in quantum mechanics, to calculate the number of states of a system.

4. Is the equation always true for any group and subgroup?

Yes, the equation holds true for all groups and subgroups. However, it is important to note that the rank of a group or subgroup can only be defined for finite groups.

5. How can one prove the validity of the equation "Stuck on Rank(G/H) = Rank(G) - Rank(H) Should be trivial(?)"?

The validity of the equation can be proved using basic properties of groups and subgroups, such as the Lagrange's theorem and the definition of the quotient group. It can also be proved using specific examples and counterexamples.

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