Group of Reflection Matrices Order 4/Cayley Table

Cayley Table for Group of Reflection Matrices Order $4$

Consider the group of reflection matrices order $4$

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

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

Let:

\(\ds r_0\)

\(=\)

\(\ds \begin {bmatrix} 1 & 0 \\ 0 & 1 \end {bmatrix}\)

\(\ds r_1\)

\(=\)

\(\ds \begin {bmatrix} 1 & 0 \\ 0 & -1 \end {bmatrix}\)

\(\ds r_2\)

\(=\)

\(\ds \begin {bmatrix} -1 & 0 \\ 0 & 1 \end {bmatrix}\)

\(\ds r_3\)

\(=\)

\(\ds \begin {bmatrix} -1 & 0 \\ 0 & -1 \end {bmatrix}\)

Then we have:

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