Suppose $G$ is an abelian group closed under scalar multiplication with elements in the field $F$. Is $G$ always a vector space over $F$?
I have been trying to find a counter-example, but failing. In such counter-example, at least one of the following needs not to hold for some $u,v \in G$, $a,b \in F$:
- $a(u+v)=au+av$
- $(a+b)u=au+bu$
- $a(bu)=(ab)u$
Thoughts/pointers appreciated.