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 $\Z'_p = \Z_p \setminus \set {\eqclass 0 p}$ be the set of non-zero residue classes modulo $p$.

Then the structure $\struct {\Z'_p, \times}$ 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.