We know that any $\mathbb{Z}$ module structure on an abelian group is unique, and furthermore the same is true for $\mathbb{Q}$.
Any complex vector space structure is not unique, we can just compose with an automorphism of $\mathbb{C}$ (for example the conjugation map).
However, $\mathbb{R}$ has trivial automorphism group, see:
Is an automorphism of the field of real numbers the identity map?
So my question is, given an abelian group made into an $\mathbb{R}$ vector space, is this the only way we can do this?
We would get another structure if we can embed $\mathbb{R}$ into itself, but I'm not sure if this is possible.
If not, then of all the different $\mathbb{R}$ structures, must they all have the same dimension?
I know that both of these are not true for general fields.
Thanks in advance.
(Related: number of differents vector space structures over the same field $\mathbb{F}$ on an abelian group)
Best Answer
$\Bbb{R}$ and $\Bbb{R}^2$ are isomorphic as vector spaces over $\Bbb{Q}$ (because they have dimensions $c$ and $c^2$ over $\Bbb{Q}$ respectively, where $c$ is the cardinality of the continuum. and $c^2 = c$). Hence $\Bbb{R}$ and $\Bbb{R}^2$ are certainly isomorphic as abelian groups but they not isomorphic as real vector spaces because they have different dimensions over $\Bbb{R}$.