Q-Vector space is uniquely divisible abelian group
Proposition
Let 
be a uniquely divisible abelian group.
Then admits a unique 
-vectorspace structure.
Proof
Let 
and 
.
Then there exists an 
such that 
.
Then define
One can then show that this then is a welldefined -vector space.
Here one would use the uniqueness to show (pseudo-)associativity, distributivity etc.
Remark
this is a proof using methods from commutative algebra:
Let 
be an abelian group, 
the ring consisting of group-isomorphism (under pointwise addition and composition).
Then a 
vector space structure is precisely a ring homomorphism 
.
If is uniquely divisible, then the map
is an isomorphism, so in particular lives in the unit 
.
So by the universal property of 
there exists an extension
