Subgroups of an infinite abelian group with a given index

Question

This is followed by the question: Subgroups of an infinite group with a given index (with a counterexample of non-abelian groups). Now, the question is:

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

Answer 1

First, if $G$ is countable, you just take $H$ to be the trivial subgroup of $G$, it will have countably infinite index. Thus, I assume that $G$ is uncountable of cardinality $A$. Then by

W. R. Scott, Proceedings of the American Mathematical Society Vol. 5, No. 1 (1954), pp. 19-22,

The Number of Subgroups of Given Index in Nondenumerable Abelian Groups

the group $G$ contains $2^A$ subgroups of every index $\alpha$, $$ \aleph_0\leq \alpha\leq |G|. $$