The first question is:
If a group has a subgroup of each possible index, is it cyclic?
(1) For a finite group, if, for every divisor of the order of the group, there exists a unique subgroup of that order, then the group is cyclic.
(2) For an infinite cyclic group, there exists a unique subgroup of each finite index.
Is the converse of (2) true? So here is my second question:
If a group has a unique subgroup of each possible index, is it cyclic?
It is not given that the group is countable (although if the conjecture above is true then it has to be).
This is not a homework problem.