TITLE: number of differents vector space structures over the same field $\mathbb{F}$ on an abelian group
QUESTION [5 upvotes]: My question here raised another one. How many differents vector space structures over a field $\mathbb{F}$ we may have on an abelian group?  I know that there are abelian groups that we can not endow it with a structure of vector space over any field, for example $\mathbb{Z}_{6}$. But if an abelian group has a structure of a vector space over a field $\mathbb{F}$, is there an upper bound for ways we can define a different structure? Such as the number of automorphisms of the field $\mathbb{F}$. I am conjecturing according to the answer given before.

REPLY [5 votes]: A $k$-vector space structure on an abelian group $A$ is nothing more than a ring homomorphism $k \to \text{End}(A)$. From this it follows that if $k'$ is another field and $k' \to k$ a field homomorphism, then the composition $k' \to k \to \text{End}(A)$ gives $A$ the structure of a $k'$-vector space. In particular, $k'$ may be any subfield of $k$, and we can also compose with any automorphism of $k'$. 
More importantly, since a ring homomorphism $k \to \text{End}(A)$ is necessarily injective, the set of vector space structures on $A$ can be identified (non-canonically) with the set of pairs 
$$(\text{subring of } \text{End}(A) \text{ that is a field } k, \text{automorphism of } k).$$