Solved: Free Abelian Group Rank R Example

[ehrenfest]

[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 ...

?

 

[HallsofIvy]

What do you mean by the "rank" of a group?

 

[ehrenfest]

http://en.wikipedia.org/wiki/Free_abelian_group

The rank of a free abelian group is the cardinality of a basis.

 

[masnevets]

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.