Multiplicative Group of Reduced Residues Modulo 5/Cayley Table

Cayley Table for Multiplicative Group of Reduced Residues Modulo 5
The multiplicative group of reduced residues modulo $5$:
 * $\Z'_5 = \left\{ {\left[\!\left[{1}\right]\!\right]_5, \left[\!\left[{2}\right]\!\right]_5, \left[\!\left[{3}\right]\!\right]_5, \left[\!\left[{4}\right]\!\right]_5}\right\}$

can be described completely by showing its Cayley table:
 * $\begin{array}{r|rrrr}

\times_5 & \left[\!\left[{1}\right]\!\right]_5 & \left[\!\left[{2}\right]\!\right]_5 & \left[\!\left[{3}\right]\!\right]_5 & \left[\!\left[{4}\right]\!\right]_5 \\ \hline \left[\!\left[{1}\right]\!\right]_5 & \left[\!\left[{1}\right]\!\right]_5 & \left[\!\left[{2}\right]\!\right]_5 & \left[\!\left[{3}\right]\!\right]_5 & \left[\!\left[{4}\right]\!\right]_5 \\ \left[\!\left[{2}\right]\!\right]_5 & \left[\!\left[{2}\right]\!\right]_5 & \left[\!\left[{4}\right]\!\right]_5 & \left[\!\left[{1}\right]\!\right]_5 & \left[\!\left[{3}\right]\!\right]_5 \\ \left[\!\left[{3}\right]\!\right]_5 & \left[\!\left[{3}\right]\!\right]_5 & \left[\!\left[{1}\right]\!\right]_5 & \left[\!\left[{4}\right]\!\right]_5 & \left[\!\left[{2}\right]\!\right]_5 \\ \left[\!\left[{4}\right]\!\right]_5 & \left[\!\left[{4}\right]\!\right]_5 & \left[\!\left[{3}\right]\!\right]_5 & \left[\!\left[{2}\right]\!\right]_5 & \left[\!\left[{1}\right]\!\right]_5 \\ \end{array}$

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


 * $\begin{array}{r|rrrr}

\times_5 & \left[\!\left[{1}\right]\!\right]_5 & \left[\!\left[{2}\right]\!\right]_5 & \left[\!\left[{4}\right]\!\right]_5 & \left[\!\left[{3}\right]\!\right]_5 \\ \hline \left[\!\left[{1}\right]\!\right]_5 & \left[\!\left[{1}\right]\!\right]_5 & \left[\!\left[{2}\right]\!\right]_5 & \left[\!\left[{4}\right]\!\right]_5 & \left[\!\left[{3}\right]\!\right]_5 \\ \left[\!\left[{2}\right]\!\right]_5 & \left[\!\left[{2}\right]\!\right]_5 & \left[\!\left[{4}\right]\!\right]_5 & \left[\!\left[{3}\right]\!\right]_5 & \left[\!\left[{1}\right]\!\right]_5 \\ \left[\!\left[{4}\right]\!\right]_5 & \left[\!\left[{4}\right]\!\right]_5 & \left[\!\left[{3}\right]\!\right]_5 & \left[\!\left[{1}\right]\!\right]_5 & \left[\!\left[{2}\right]\!\right]_5 \\ \left[\!\left[{3}\right]\!\right]_5 & \left[\!\left[{3}\right]\!\right]_5 & \left[\!\left[{1}\right]\!\right]_5 & \left[\!\left[{2}\right]\!\right]_5 & \left[\!\left[{4}\right]\!\right]_5 \\ \end{array}$