Definition:Multiplicative Group of Reduced Residues
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: $\S 34$