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 $\Z_m$.

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.

By Multiplicative Inverse in Ring of Integers Modulo m we have that the elements of $\left({\Z'_m, \times}\right)$ are precisely those that have inverses, and are therefore the units of $\left({\Z_m, +, \times}\right)$‎.

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