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Homework Help: Free abelian group

  1. Jan 18, 2008 #1
    [SOLVED] free abelian group

    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations

    3. 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: Jan 18, 2008
  2. jcsd
  3. Jan 18, 2008 #2


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    Science Advisor

    What do you mean by the "rank" of a group?
  4. Jan 18, 2008 #3
  5. Jan 18, 2008 #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.
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