Take $G$ a group such that ${\rm Hom}(G,\mathbb{R})=0$. Now I have to prove that $G$ is a finite group.
But I don't know if it's true.
In fact I think that the "only" thing that I can say about $G$ is that every element of $G$ is of finite order but this doesn't imply that $G$ is finite. Any suggest or countrexample?
I was thinking about this and I think that maybe the right way is to prove that if $G$ is infinite so ${\rm Hom}(G, \mathbb{R})$.
Thanks in advance.