subgroups of the additive group of Q containing the subgroup Z Find all those subgroups of the additive group Q which contain the subgroup Z.
Here Q means the set of rational numbers and Z refers to the set of integers. This problem comes from algebra written by Saunders Mac Lane and Garrett Birkhoff, the last exercise of chapter III on rings. How to solve it? Thanks.
 A: For any function $f$ from the set of primes to the set $(\mathbb N-\{0\})\cup\{\infty\}$, there is a subgroup $S_f$ of the desired sort consisting of those rational numbers expressible as reduced fractions $\frac mn$ such that, in the prime factorization of the denominator $n$, each prime $p$ occurs fewer than $f(p)$ times.  If $f$ is the constant function $1$, then $S_f=\mathbb Z$. If $f$ is the constant function $\infty$, then $S_f=\mathbb Q$.  In-between choices of $f$ give you in-between groups.  (For eample, if $f$ is the constant function $2$, then $S_f$ is the group of rational numbers with square-free denominators.)  I believe that these subgroups $S_f$ are all of the subgroups between $\mathbb Z$ and $\mathbb Q$, but I don't have a proof at the moment.
A: Andreas Blass is correct. These are all the subgroups in question. To see this first observe that these subgroups are in bijection with the
subgroups of $\newcommand{\Q}{\Bbb Q}\newcommand{\Z}{\Bbb Z}\Q/\Z$.
This is a torsion group, and so is the direct sum of its $p$-power torsion
subgroups:
$$\Q/\Z=\bigoplus_p(\Q/\Z)_{p^\infty}$$
where
$$(\Q/\Z)_{p^\infty}=\bigcup_{n=1}^\infty(\Q/\Z)_{p^n}$$
and $G_{p^n}$ denotes the $p^n$-torsion subgroup of the torsion Abelian
group $G$. If $H$ is a subgroup of $G$ then $H=\bigoplus_p (H\cap G_{p^\infty})$. To see this note that for $h\in H$. Then $nh=0$ for some
integer $n$. Write $n=p^kn'$ for $p\nmid n'$ then $mh$ is the component
of $h$ in $G_{p^\infty}$ where $n'\mid n$ and $m\equiv1\pmod{p^k}$.
To choose a subgroup of $\Q/\Z$ we just need to pick one of
$(\Q/\Z)_{p^\infty}$ for all $p$. But the proper subgroups of
$(\Q/\Z)_{p^\infty}$ consist of the $p^k$-torsion elements for
$k\in\{0,1,2,\ldots\}$. This gives Andreas's classification.
