Multiplicative Group of Reduced Residues/Examples/Modulo 5

Example of Multiplicative Group of Reduced Residues

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 $5$.

Cayley Table

$\quad \begin {array} {r|rrrr} \times_5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 3 5 & \eqclass 4 5 \\ \hline \eqclass 1 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 3 5 & \eqclass 4 5 \\ \eqclass 2 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 1 5 & \eqclass 3 5 \\ \eqclass 3 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 4 5 & \eqclass 2 5 \\ \eqclass 4 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 2 5 & \eqclass 1 5 \\ \end {array}$

By arranging the rows and columns into a different order, its cyclic nature becomes clear:

$\quad \begin {array} {r|rrrr} \times_5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 3 5 \\ \hline \eqclass 1 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 3 5 \\ \eqclass 2 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 1 5 \\ \eqclass 4 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 2 5 \\ \eqclass 3 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 4 5 \\ \end {array}$

Also see

• Reduced Residues Modulo 5 under Multiplication form Cyclic 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 $1 \ (\gamma)$