Let $G$ be an abelian group.

What does it mean that $G$ is a free abelian group? Does this mean that $G$ is a free group or a free-$\mathbb{Z}$-module with the operation $n•a=a+...+a (n-times)$?

Or are they equivalent?


I just realized that no abelian group is free-group since every nonzero element does not have a unique canonical form under basis.

  • $\begingroup$ It means $G$ has no relations, other than the ones that go into the definition of "abelian". $\endgroup$ – Gerry Myerson Mar 8 '15 at 6:44
  • $\begingroup$ Well, the infinite cyclic group $\;\Bbb Z\;$ is the only free group which is abelian, aside from the trivial free group $\;\{0\}\;$ . Any other free group is non-abelian $\endgroup$ – Timbuc Mar 8 '15 at 7:02
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    $\begingroup$ It means $G$ is an abelian group that is free in the category of abelian groups, or $\Bbb Z$-modules. $\endgroup$ – Pedro Tamaroff Mar 8 '15 at 7:27

There are two free groups that are abelian! Namely: $\mathbb{1}$ and $\mathbb{Z}$.

A free group on at least two generators cannot be abelian, because the nonabelian group $S_3$ is its quotient (as it can be generated by two elements).

  • $\begingroup$ @bof That is indeed what I mean. $\endgroup$ – Slade Mar 8 '15 at 7:35

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