Group of Rotation Matrices Order 4/Cayley Table

Cayley Table for Group of Rotation Matrices Order $4$

Consider the group of rotation matrices order $4$

$R_4 = \set {\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}, \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}, \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} }$

$R_4$ can be described completely by showing its Cayley table.

Let:

$r_0 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

$r_1 = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$

$r_2 = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$

$r_3 = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$

Then we have:

$\begin{array}{r|rrrr} \times & r_0 & r_1 & r_2 & r_3 \\ \hline r_0 & r_0 & r_1 & r_2 & r_3 \\ r_1 & r_1 & r_2 & r_3 & r_0 \\ r_2 & r_2 & r_3 & r_0 & r_1 \\ r_3 & r_3 & r_0 & r_1 & r_2 \\ \end{array}$