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Math Help - Torsion in the Free Abelian Group

  1. #1
    MHF Contributor Swlabr's Avatar
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    Torsion in the Free Abelian Group

    Let F be a free group (not necessarily finitely generated). I am trying to prove that [F, F] has infinite index, and I have, so far, shown that this is equivalent to proving that F^{\prime} = F/[F, F] = <x_1, \ldots, x_n, \ldots : [x_i, x_j]> contains no torsion. Which is obvious.

    Except...I can't seem to prove it...
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  2. #2
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    Quote Originally Posted by Swlabr View Post
    Let F be a free group (not necessarily finitely generated). I am trying to prove that [F, F] has infinite index, and I have, so far, shown that this is equivalent to proving that F^{\prime} = F/[F, F] = <x_1, \ldots, x_n, \ldots : [x_i, x_j]> contains no torsion. Which is obvious.

    Except...I can't seem to prove it...
    F^{\prime} is simply a free abelian group. A free abelian group is the internal direct sum of a family of infiinite cyclic subgroups, so it is torsion-free.
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by TheArtofSymmetry View Post
    F^{\prime} is simply a free abelian group. A free abelian group is the internal direct sum of a family of infiinite cyclic subgroups, so it is torsion-free.
    Yeah, that makes sense. I knew it was obvious!

    Thanks.
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