Reduced Residue System under Multiplication forms Abelian Group/Proof 1
Theorem
Let $\Z_m$ be the set of set of residue classes modulo $m$.
Let $\struct {\Z'_m, \times}$ denote the multiplicative group of reduced residues modulo $m$.
Then $\struct {\Z'_m, \times}$ is an abelian group, precisely equal to the group of units of $\Z_m$.
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.
$\blacksquare$
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields: Example $4$