Reduced Residue System under Multiplication forms Abelian Group/Proof 3

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:

$\text {FG} 0$: Closure

From Modulo Multiplication on Reduced Residue System is Closed:

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

$\Box$

$\text {FG} 1$: Associativity

We have that Modulo Multiplication is Associative.

$\Box$

$\text {FG} 2$: 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.

$\Box$

$\text {FG} 3$: 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

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 6$: Examples of Finite Groups: $\text{(iii)}$: Theorem $2$