Modulo Multiplication/Cayley Table/Modulo 4

Cayley Table for Modulo Multiplication

The multiplicative monoid of integers modulo $m$ can be described by showing its Cayley table.

This one is for modulo $4$:

$\begin{array} {r|rrrrr}

\struct {\Z_4, \times_4} & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 \\ \hline \eqclass 0 4 & \eqclass 0 4 & \eqclass 0 4 & \eqclass 0 4 & \eqclass 0 4 \\ \eqclass 1 4 & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 \\ \eqclass 2 4 & \eqclass 0 4 & \eqclass 2 4 & \eqclass 0 4 & \eqclass 2 4 \\ \eqclass 3 4 & \eqclass 0 4 & \eqclass 3 4 & \eqclass 2 4 & \eqclass 1 4 \\ \end{array}$

which can also be presented:

$\begin{array} {r|rrrrr}

\times_4 & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 \\ 2 & 0 & 2 & 0 & 2 \\ 3 & 0 & 3 & 2 & 1 \\ \end{array}$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 6$. The Residue Classes