TITLE: What are all the characteristic subgroups of $(\mathbb Q,+)$?
QUESTION [7 upvotes]: What are all the characteristic subgroups of $(\mathbb Q,+)$? As in, the subgroups $H$ such that $f(H)=H, \forall f \in \operatorname{Aut}(\mathbb Q)$? I have reduced the condition to $qH=H$, $\forall q \in \mathbb Q\setminus \{0\}$.

REPLY [6 votes]: Suppose $h \in H$ is non-zero. Let $g \in \mathbb{Q} \setminus \{0\}$. Then with $q=g/h$ and your reduction you get $g \in H$. Thus, if you have a non-zero element in $H$, then $H$ must be the full group. 
This leaves only two possible subgroups.