TITLE: Groups such that every finitely generated subgroup is  isomorphic to the integers
QUESTION [6 upvotes]: What are examples of groups such that every finitely generated subgroup is isomorphic to $\mathbb{Z}$?

REPLY [12 votes]: Suppose any nontrivial finitely generated subgroup of $G$ is isomorphic to $\Bbb{Z}$, and that $G$ itself is nontrivial. Choose some nonzero $e \in G$. For any $g \in G$, the subgroup $\langle e, g\rangle$ is isomorphic to $\Bbb{Z}$, so in particular $ne=mg$ for some integers $m,n$. Define a map $f:G \to \Bbb{Q}$ by $g \mapsto n/m$. Then $f$ is a well-defined group homomorphism; since $G$ is torsion-free and $e$ nonzero, it is straightforward to check that $f$ is injective. So $G$ is isomorphic to a nontrivial subgroup of $\Bbb{Q}$, which are classified, e.g., in this paper.