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ehrenfest
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[SOLVED] free abelian group
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.
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 ...
?
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 ...
?
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