Subgroups of Z/One generator/Fact

The subgroups of ${\displaystyle {}\mathbb {Z} }$ are precisely

the subsets of the form

${\displaystyle {}\mathbb {Z} d={\left\{kd\mid k\in \mathbb {Z} \right\}}\,,}$

with a uniquely determined nonnegative number ${\displaystyle {}d}$.