Multiplicative Group of Reduced Residues/Examples/Modulo 7
Example of Multiplicative Group of Reduced Residues
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.
Cayley Table
- $\begin{array}{r|rrrr}
\times_5 & \eqclass 1 7 & \eqclass 2 7 & \eqclass 3 7 & \eqclass 4 7 & \eqclass 5 7 & \eqclass 6 7 \\ \hline \eqclass 1 7 & \eqclass 1 7 & \eqclass 2 7 & \eqclass 3 7 & \eqclass 4 7 & \eqclass 5 7 & \eqclass 6 7 \\ \eqclass 2 7 & \eqclass 2 7 & \eqclass 4 7 & \eqclass 6 7 & \eqclass 1 7 & \eqclass 3 7 & \eqclass 5 7 \\ \eqclass 3 7 & \eqclass 3 7 & \eqclass 6 7 & \eqclass 2 7 & \eqclass 5 7 & \eqclass 1 7 & \eqclass 4 7 \\ \eqclass 4 7 & \eqclass 4 7 & \eqclass 1 7 & \eqclass 5 7 & \eqclass 2 7 & \eqclass 6 7 & \eqclass 3 7 \\ \eqclass 5 7 & \eqclass 5 7 & \eqclass 3 7 & \eqclass 1 7 & \eqclass 6 7 & \eqclass 4 7 & \eqclass 2 7 \\ \eqclass 6 7 & \eqclass 6 7 & \eqclass 5 7 & \eqclass 4 7 & \eqclass 3 7 & \eqclass 2 7 & \eqclass 1 7 \\ \end{array}$