# Thread: Torsion in the Free Abelian Group

1. ## Torsion in the Free Abelian Group

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

Except...I can't seem to prove it...

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

Except...I can't seem to prove it...
$\displaystyle F^{\prime}$ 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.

3. Originally Posted by TheArtofSymmetry
$\displaystyle F^{\prime}$ 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.