# Reduced Residue System under Multiplication forms Abelian Group/Corollary

## Corollary of Reduced Residue System under Multiplication forms Abelian Group

Let $p$ be a prime number.

Let $\Z_p$ be the set of integers modulo $p$.

Let $\struct {\Z'_p, \times}$ denote the multiplicative group of reduced residues modulo $p$.

Then $\struct {\Z'_p, \times}$ is an abelian group.

## Proof

Suppose $p \in \Z$ be a prime number.

From the definition of reduced residue system modulo $p$, as $p$ is prime, $\Z'_p$ becomes:

$\set {\eqclass 1 p, \eqclass 2 p, \ldots, \eqclass {p - 1} p}$

This is precisely $\Z_p \setminus \set {\eqclass 0 p}$ which is what we wanted to show.

The result follows from Reduced Residue System under Multiplication forms Abelian Group.

$\blacksquare$