Does there exist a infinite group besides $\Bbb Z$ that all its nontrivial subgroups have finite index?
I know $\Bbb Z$ works, but is there any other examples? Or a proof that $\Bbb Z$ is the only one with such properties.
On one hand I can't gave a proof: the best I can do is to observe that every element have infinite order with the infinite cyclic group having finite cosets, therefore any other element must have a power going into the cyclic group, in other words every two element must have some kind of "least common multiple" which is similar to the integers. But I can't proof any commutative thing and was stuck.
On the other hand I doubt maybe there exist some weird free group that satisfy the properties, but also I can't gave a construction.
This has been answered in the comments, but I will make it into a proper answer.
Let $G$ be an infinite group in which every nontrivial subgroup has finite index. We claim that $G$ is an infinite cyclic group, and hence is isomorphic to $({\mathbb Z},+)$.
Let $1 \ne x \in G$ and $H = \langle x \rangle$. Since $H$ has finite index in $G$ and $G$ is infinite, $H$ must be infinite. So $G$ is torsion-free. Note also that the core of $H$ in $G$ has finite index, and so we can assume that $H$ is a normal subgroup of $G$. Choose $H$ such that it is normal in $G$ and has minimal index subject to that condition.
Let $C = C_G(H)$. Then $H \le Z(C)$ and so $|C:Z(C)|$ is finite and hence, by a result of Schur, the derived group $[C,C]$ is finite. But then we must have $[C,C]=1$, so $C$ is abelian, and by the fundamental theorem of abelian groups $C$ is infinite cyclic, and then by the maximal choice of $H$ we have $C=H$.
Suppose that $H \ne G$, and let $y \in G \setminus H$. Then, since $y \not\in C_G(H)$, we must have $y^{-1}xy=x^{-1}$. Then $y^{-2}xy^2=x$, so $y^2 \in C_G(H) = H$, and hence $y^2 = x^k$ for some $k \in {\mathbb Z}$. So $y$ centralizes $x^k$, but $y^{-1}x^ky= x^{-k}$,so $x^{2k}=0$ and hence $k=0$ and $y^2=1$, contradiction, since $G$ is torsion-free.
So $G=H$ as claimed.