Reduced Residue System under Multiplication forms Abelian Group/Proof 2

Proof
Taking the group axioms in turn:

G0: Closure
From Modulo Multiplication on Reduced Residue System is Closed:


 * $\left({\Z'_m, \times}\right)$ is closed.

G1: Associativity
We have that Modulo Multiplication is Associative.

G2: Identity
From Modulo Multiplication has Identity, $\left[\!\left[{1}\right]\!\right]_m$ is the identity element of $\left({\Z'_m, \times}\right)$.

G3: Inverses
From Multiplicative Inverse in Monoid of Integers Modulo m, $\left[\!\left[{k}\right]\!\right]_m \in \Z_m$ has an inverse in $\left({\Z_m, \times_m}\right)$ $k$ is coprime to $m$.

Thus every element of $\left({\Z'_m, \times}\right)$ has an inverse.

All the group axioms are thus seen to be fulfilled, and so $\left({\Z'_m, \times}\right)$ is a group.