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

• Group of Reflection Matrices Order 4 is Klein Four-Group

Sources

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 7$: Isomorphic Groups: Example $1 \ \text{(b)}$