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I am asked to provide an example for torsion-free but not a cyclic group. Is $\mathbb{Z}\times\mathbb{Z}$ an example?

darij grinberg
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breezeintopl
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1 Answers1

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All free abelian groups are torsion-free, see here. Of course, free ableian groups of rank $>1$ are not cyclic. This generalizes your example. On the other hand, all finitely generated torsion-free abelian groups are free abelian.

Community
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Dietrich Burde
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  • Thank you for your answer! One more question, the converse is not true(torsion free is not all free abelian group), right? As you mentioned in the last sentence, counterexample can be considered among those not finitely generated torsion-free abelian groups. Is $\mathbb{Z}^{\infty}$ a such example(torsion-free but not free abelian group)? – breezeintopl Apr 07 '15 at 13:03
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    Yes, $\prod_{i\ge 0} \mathbb{Z}$ is not free abelian, but $\mathbb{Q}$ is an easier example of a torsion-free group which is not free abelian - see above. – Dietrich Burde Apr 07 '15 at 13:15
  • OK, thank you so much for your help! – breezeintopl Apr 07 '15 at 13:27