# 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): $\S 1.1$: Example $4$