Reduced Residue System under Multiplication forms Abelian Group/Proof 3

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Theorem

Let $\Z_m$ be the set of set of residue classes modulo $m$.


Let $\struct {\Z'_m, \times}$ denote the multiplicative group of reduced residues modulo $m$.


Then $\struct {\Z'_m, \times}$ is an abelian group, precisely equal to the group of units of $\Z_m$.


Proof

Taking the finite group axioms in turn:


FG0: Closure

From Modulo Multiplication on Reduced Residue System is Closed:

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

$\Box$


FG1: Associativity

We have that Modulo Multiplication is Associative.

$\Box$


FG2: 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.

$\Box$


FG3: Cancellability

We have that Modulo Multiplication on Reduced Residue System is Cancellable.

$\Box$


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.

$\blacksquare$


Sources