# Definition:Multiplicative Group of Reduced Residues

## Contents

## Definition

Let $m \in \Z_{> 0}$ be a (strictly) positive integer.

Let $\Z'_m$ denote the reduced residue system modulo $m$.

Consider the algebraic structure:

- $\struct {\Z'_m, \times_m}$

where $\times_m$ denotes multiplication modulo $m$.

Then $\struct {\Z'_m, \times_m}$ is referred to as **the multiplicative group of reduced residues modulo $m$**.

## Also known as

Some sources refer to this group merely as **the multiplicative group modulo $m$**, glossing over the fact that the underlying set is actually a reduced residue system.

## Examples

### Modulo 5

Consider the reduced residue system $\Z'_5$ modulo $5$ under modulo multiplication:

- $\Z'_5 = \set {\eqclass 1 5, \eqclass 2 5, \eqclass 3 5, \eqclass 4 5}$

$\struct {\Z'_5, \times_5}$ is the **multiplicative group of reduced residues modulo $5$**.

### Modulo 7

Consider the reduced residue system $\Z'_7$ modulo $7$ under modulo multiplication:

- $\Z'_7 = \set {\eqclass 1 7, \eqclass 2 7, \eqclass 3 7, \eqclass 4 7, \eqclass 5 7, \eqclass 6 7}$

$\struct {\Z'_7, \times_7}$ is the **multiplicative group of reduced residues modulo 7**.

### Modulo 8

Consider the reduced residue system $\Z'_8$ modulo $8$ under modulo multiplication:

- $\Z'_8 = \set {\eqclass 1 8, \eqclass 3 8, \eqclass 5 8, \eqclass 7 8}$

$\struct {\Z'_8, \times_8}$ is the **multiplicative group of reduced residues modulo 8**.

## Also see

- Results about
**multiplicative groups of reduced residues**can be found here.

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Examples of Group Structure: $\S 34$