Given a divisible group $G$, I wonder why $G$ has no nontrivial subgroup of finite index.
Suppose $H$ is a subgroup (of $G$) of finite index. Then there exists a normal subgroup $K$ of $G$ which is contained in $H$ and also has finite index. Given an element $g$ of $G$, I need to show that $g$ is in $K$. But I don't know how to continue... Could you explain it for me?