Group of integers in group of rational numbers

Definition
This page is about the situation where the whole group is the group of rational numbers $$(\mathbb{Q},+)$$ (under addition) and the subgroup involved is the group of integers $$(\mathbb{Z},+)$$.

Note that if we only care about the subgroup up to automorphism, we can pick the cyclic subgroup generated by any nonzero element of the whole group.

The quotient group $$\mathbb{Q}/\mathbb{Z}$$, the group of rational numbers modulo integers, is a group in its own right.