Multiplicative Group of Reduced Residues/Examples/Modulo 8

Example of Multiplicative Group of Reduced Residues

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.

Cayley Table

$\begin{array}{r|rrrr} \times_8 & \eqclass 1 8 & \eqclass 3 8 & \eqclass 5 8 & \eqclass 7 8 \\ \hline \eqclass 1 8 & \eqclass 1 8 & \eqclass 3 8 & \eqclass 5 8 & \eqclass 7 8 \\ \eqclass 3 8 & \eqclass 3 8 & \eqclass 1 8 & \eqclass 7 8 & \eqclass 5 8 \\ \eqclass 5 8 & \eqclass 5 8 & \eqclass 7 8 & \eqclass 1 8 & \eqclass 3 8 \\ \eqclass 7 8 & \eqclass 7 8 & \eqclass 5 8 & \eqclass 3 8 & \eqclass 1 8 \\ \end{array}$

Also see

• Multiplicative Group of Reduced Residues Modulo 8 is Klein Four-Group

Sources

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 7$: Isomorphic Groups: Example $2 \ \text{(c)}$