# finite abelian groups tensor product. [closed]

Is the following question obvious ?

Let $$G$$ be an abelian group, such that for any finite abelian group $$A$$, we have $$G\otimes_{\mathbf{Z}}A=0$$, does it mean that $$G$$ is a $$\mathbf{Q}$$-vector space ?

## closed as off-topic by Eevee Trainer, Cesareo, YiFan, Song, AntinousMar 12 at 13:36

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You have $$G\otimes(\Bbb Z/n\Bbb Z)=0$$ for all $$n\in \Bbb N$$. That means $$G/nG=0$$, so $$G=nG$$, that is all elements of $$G$$ are divisible by $$n$$. Then $$G$$ is a divisible Abelian group. Conversely if $$G$$ is a divisible Abelian group, then $$G\otimes(\Bbb Z/n\Bbb Z)=0$$ and so $$G\otimes A=0$$ for all finitely generated Abelian groups.
But not all divisible Abelian groups are $$\Bbb Q$$-modules: they may have torsion. As an example, let $$G=\Bbb Q/\Bbb Z$$.