Multiplicative Group of Reduced Residues/Examples

Examples of Multiplicative Groups of Reduced Residues

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.