Reduced Residue System under Multiplication forms Abelian Group

Theorem
Let $$\Z_m$$ be the set of integers modulo $m$.

Let $$\Z'_m$$ be the set of integers coprime to $$m$$ in $$\Z_m$$.

Then the structure $$\left({\Z'_m, \times}\right)$$ is an abelian group.

Corollary
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
We have that the structure $\left({\Z'_m, +, \times}\right)$‎ forms a ring with unity.

Then we have that the units of a ring with unity form a group.

We also have that all the elements of $\left({\Z'_m, \times}\right)$ have inverses, and are therefore units.

The fact that $$\left({\Z'_m, \times}\right)$$ is abelian follows from Restriction of Operation Commutativity.

Proof of Corollary
Suppose $$p \in \Z$$ be a prime number.

From the definition of Set of Coprime Integers, 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.