Reduced Residue System under Multiplication forms Abelian Group/Proof 2

From ProofWiki
Jump to: navigation, search

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 group axioms in turn:


G0: Closure

From Modulo Multiplication on Reduced Residue System is Closed:

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

$\Box$


G1: Associativity

We have that Modulo Multiplication is Associative.

$\Box$


G2: Identity

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

$\Box$


G3: Inverses

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

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

$\Box$


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

$\blacksquare$


Sources