Group of Reflection Matrices Order 4

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Definition

Consider the algebraic structure $S$ of reflection matrices:

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

under the operation of (conventional) matrix multiplication.


$R_4$ is the group of reflection matrices of order $4$.


Cayley Table

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


Also see


Sources