Subgroups of an infinite group with a given index

Question

Let $G$ be an infinite group and $\alpha$ a cardinal number with $\aleph_0\leq \alpha\leq |G|$. Is there a subgroup $H$ of $G$ with $|G:H|=\alpha$ (what about $|H|=\alpha$)?

Answer 1

There are groups of cardinality $\aleph_1$ (the particular group is an instance of a Jonsson group) that only have countable proper subgroups, so index of a subgroup group is either $\aleph_1$ or $1$ (no subgroup of index $\aleph_0$).

This is done in a paper of Shelah: On a problem of Kurosh, Jonsson groups, and applications.