TITLE: Terminology for infinite groups, all of whose subgroup have finite index.
QUESTION [9 upvotes]: Is there a name for (infinite) groups such that every non-trivial, proper subgroup has finite index (e.g. $\mathbb{Z}$)?

REPLY [9 votes]: The only group with this property is the infinite cyclic group $\mathbb{Z}$.
Let $G$ be an infinite group where every nontrivial subgroup has finite index. Then $G$ must be torsionfree, so $G$ contains a subgroup $H \cong \mathbb{Z}$ of finite index. Then $H$ will contain a normal subgroup $N$ of $G$ such that $N \cong \mathbb{Z}$ and $G/N$ is finite, so $G$ is finitely generated. 
Let $g_1, \ldots, g_n$ be a set of generators for $G$, where each $g_i \neq 1$. Now each $C_G(g_i)$ has finite index, so $\cap C_G(g_i) = Z(G)$ has finite index. By Schur's theorem, the commutator subgroup of $G$ is finite. Thus it must be trivial and $G$ is abelian. Classification of finitely generated abelian groups shows that $G \cong \mathbb{Z}$.