Subgroups of Z/One generator/Fact

Theorem about the subgroups of ${}\mathbb {Z}$

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

the subsets of the form

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

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

Proof, Write another proof