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 $m$.

Let $\Z'_p = \Z_p \setminus \left\{{\left[\!\left[{0}\right]\!\right]}\right\}$ be the set of non-zero residue classes modulo $p$.

Then the structure $\left({\Z'_p, \times}\right)$ 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:
 * $\left\{{\left[\!\left[{1}\right]\!\right]_p, \left[\!\left[{2}\right]\!\right]_p, \ldots, \left[\!\left[{p - 1}\right]\!\right]_p}\right\}$

This is precisely $\Z_p \setminus \left\{{\left[\!\left[{0}\right]\!\right]_p}\right\}$ which is what we wanted to show.

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