$G/H$ is a finite group so $G\cong\mathbb Z$ 
Let $G$ is an abelian infinte group such that for all nontrivial subgroups $H$ $$\forall H\leq G, \left|\frac{G}{H}\right|<\infty$$ Prove that $G\cong\mathbb Z$.

What I have done:
Clearly, it is enough to show that $G$ is a cyclic group. Moreover, I see as $G/H$ is a finite group for all subgroups $H$, then $G$ cannot have any elements with the finite order. This means to me that if $H \leq G$ and $H$ is cyclic then $H$ cannot be finite. Help your friend for the rest. Thanks. :)
 A: Show that such a group must be finitely generated. Using the fundamental theorem of finitely generated abelian groups, you have that 
$G \simeq \mathbb{Z}^r \times \mathbb{Z} / n_1 \mathbb{Z} \times ... \times \mathbb{Z} / n_k \mathbb{Z}$
where $n_i \in \mathbb{Z}$. There is an isomorphic copy of each factor of the direct product in $G$. Quotienting $G$ by $\mathbb{Z}^s$ for various $1 \leq s \leq r$ and the $\mathbb{Z} / n_i \mathbb{Z}$, you see that $r = 1$ ($r \geq 1$ for the group to be infinite and $r \leq 1$ for the finite quotient property) and $k = 0$ (for the finite quotient property).
A: Your group is torsion-free (otherwise it has a finite subgroup of finite index, contradicting $G$ being infinite).  Now let $g\in G$, and consider $H=\langle g\rangle$, which is of some finite index $n=[G:H]$.  The map $G\rightarrow H$, $g\mapsto g^n$, then maps $G$ into an infinite cyclic group.  The kernel, if non-trivial, would have finite index, impossible because $H$ is torsion-free.  Thus the kernel is trivial and $G$ embeds in a cyclic group, so is itself cyclic.
