Reduced Residue System under Multiplication forms Abelian Group/Proof 1

Proof
From Ring of Integers Modulo m is Ring, $\struct {\Z_m, +, \times}$‎ 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 $\struct {\Z'_m, \times}$ are precisely those that have inverses, and are therefore the units of $\struct {\Z_m, +, \times}$‎.

The fact that $\struct {\Z'_m, \times}$ is abelian follows from Restriction of Commutative Operation is Commutative.