Reduced Residue System under Multiplication forms Abelian Group/Proof 3

Proof
Taking the finite group axioms in turn:

$\text {FG} 0$: Closure
From Modulo Multiplication on Reduced Residue System is Closed:


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

$\text {FG} 1$: Associativity
We have that Modulo Multiplication is Associative.

$\text {FG} 2$: Finiteness
The order of $\struct {\Z'_m, \times}$ is $\map \phi n$ by definition, where $\map \phi n$ denotes the Euler $\phi$ function.

As $\map \phi n < n$ it follows that $\struct {\Z'_m, \times}$ is of finite order.

$\text {FG} 3$: Cancellability
We have that Modulo Multiplication on Reduced Residue System is Cancellable.

Thus all the finite group axioms are fulfilled, and $\struct {\Z'_m, \times}$ is a group.

It remains to note that Modulo Multiplication is Commutative to confirm that $\struct {\Z'_m, \times}$ is abelian.