TITLE: Is there a proper subgroup $H$ of $(\mathbb{R},+)$ such that $[\mathbb{R}:H]$ is finite?
QUESTION [13 upvotes]: I would like to know if there is a proper subgroup $H$  of $(\mathbb{R},+)$ such that $[\mathbb{R}:H]$ is finite.  I know if I pick up $\mathbb{Z}$, then $[\mathbb{R}:\mathbb{Z}]$ is infinite.
Thanks for your help!

REPLY [2 votes]: This observation is implicit in the other answer. 

Let $A$ be a domain which is not a field, and let $M$ be a finitely generated divisible $A$-module. Then $M=0$.

Assume by contradiction $M\neq0$, and let $N$ be a maximal proper submodule of $M$. Such exists because $M$ is nonzero and finitely generated. Then $S:=M/N$ is isomorphic to $A/\mathfrak m$, where $\mathfrak m$ is a maximal ideal of $A$. As $A$ is not a field, $\mathfrak m$ contains a nonzero element $a$, and we have:
$\bullet\ $ $0\neq S=aS$, because $S$ is divisible and $a$ is nonzero, 
$\bullet\ $ $aS=0$, because $a$ is in $\mathfrak m$.