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 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.