Prove or disprove: There exists an infinite group with only one subgroup of infinite index.
By Lagrange's theorem, a group $G$ for some $H \leqslant G$ is partitioned into $[G:H]$ many subsets, each with cardinality $|H|$. If $|H|$ is finite, then the index of $H$ must be infinite by this theorem. If $|H| = \infty$, then we could have either a finite index or an infinite index.
Since $G$ contains infinitely many subgroups, I can't think of case where this would be true, since the above implies that this would be a group with a single finite subgroup. Is the above proposition false?