Reduced Residue System under Multiplication forms Abelian Group/Proof 2

Proof
Taking the group axioms in turn:

From Modulo Multiplication on Reduced Residue System is Closed:


 * $\struct {\Z'_m, \times}$ is closed.

We have that Modulo Multiplication is Associative.

From Modulo Multiplication has Identity, $\eqclass 1 m$ is the identity element of $\struct {\Z'_m, \times_m})$.

From Multiplicative Inverse in Monoid of Integers Modulo m, $\eqclass k m \in \Z_m$ has an inverse in $\struct {\Z_m, \times_m}$ $k$ is coprime to $m$.

Thus every element of $\struct {\Z'_m, \times_m}$ has an inverse.

All the group axioms are thus seen to be fulfilled, and so $\struct {\Z'_m, \times_m}$ is a group.