Reduced Residue System under Multiplication forms Abelian Group

Theorem
Let $\Z_m$ be the set of integers modulo $m$.

Let $\Z'_m$ be the set of integers coprime to $m$ in $\Z_m$.

Then the structure $\left({\Z'_m, \times}\right)$ is an abelian group, precisely equal to the group of units of $\Z_m$.