Solved: Free Abelian Group Rank R Example

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Homework Help Overview

The discussion revolves around the properties of free abelian groups, specifically focusing on the concept of rank and the existence of proper subgroups that share the same rank. The original poster attempts to provide examples to illustrate their understanding of these concepts.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definition of rank in the context of free abelian groups and question whether proper subgroups can have the same rank. The original poster initially doubts the existence of such examples but later suggests potential cases involving infinite generation.

Discussion Status

Some participants have provided clarifications regarding the definition of rank and have confirmed that certain examples, like 2Z and Z, do indeed share the same rank. However, there is a note of caution regarding the validity of the original poster's second example, indicating that further exploration is needed.

Contextual Notes

There is a discussion about the definitions and properties of free abelian groups, particularly concerning finitely-generated versus infinitely-generated groups. The original poster's examples and assumptions are under scrutiny, highlighting the need for precise definitions in this context.

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[SOLVED] free abelian group

Homework Statement


Show by example that is is possible for a proper subgroup of a free abelian group of finite rank r also to have rank r.

Homework Equations


The Attempt at a Solution


I believe that there are no example in the set of finitely-generated free abelian groups. Is that right?

EDIT: I think this is wrong. 2Z is a proper subgroup of Z but they both have the same rank, don't they?

Is this an example in the set of infinitely-generated free abelian groups:

G = Z_1 cross Z_2 cross ...
H = Z_2 cross Z_4 cross ...

?
 
Last edited:
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What do you mean by the "rank" of a group?
 
Yes 2Z has rank 1 and Z has rank 1, so that example works. Your second example does not work because your G is not a free Abelian group.
 

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