Solved: Free Abelian Group Rank R Example

In summary, it is possible for a proper subgroup of a free abelian group of finite rank r to also have rank r. This can be seen in the example of 2Z being a proper subgroup of Z, both with rank 1. However, the proposed example of G = Z_1 cross Z_2 cross ... and H = Z_2 cross Z_4 cross ... does not work as G is not a free abelian group. The rank of a free abelian group is defined as the cardinality of a basis.
  • #1
ehrenfest
2,020
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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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  • #2
What do you mean by the "rank" of a group?
 
  • #4
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.
 

1. What is a free abelian group?

A free abelian group is a mathematical structure that consists of a set of elements and operations of addition and subtraction, satisfying the axioms of an abelian group. It is considered "free" because the elements can be freely combined without any restrictions.

2. What does the rank R refer to in a free abelian group?

The rank R refers to the number of elements in a basis for the free abelian group. This means that any element in the group can be uniquely expressed as a linear combination of the elements in the basis.

3. Can you provide an example of a free abelian group with rank R?

Yes, an example of a free abelian group with rank R would be the group of integers (Z) with the standard operations of addition and subtraction. The rank in this case would be the number 1, as any integer can be expressed as a linear combination of 1.

4. How is a free abelian group with rank R different from other types of groups?

A free abelian group with rank R differs from other types of groups in that it has a basis of R elements, which allows for unique representations of elements as linear combinations. Other types of groups may have different structures and restrictions on their elements and operations.

5. What are some real-world applications of free abelian groups?

Free abelian groups have various applications in mathematics, physics, and computer science. In mathematics, they are used to study algebraic structures and topology. In physics, they are used to describe symmetries in quantum mechanics. In computer science, they are used in error-correcting codes and in the construction of cryptographic systems.

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