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.

Proof
We have that the structure $\left({\Z'_m, +, \times}\right)$‎ forms a (commutative) ring with unity.

Then we have that the units of a ring with unity form a group.

We also have that all the elements of $\left({\Z'_m, \times}\right)$ have inverses, and are therefore units.

The fact that $\left({\Z'_m, \times}\right)$ is abelian follows from Restriction of Operation Commutativity.