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

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).
