Dihedral Group D4/Matrix Representation/Formulation 1

Let $\mathbf I, \mathbf A, \mathbf B, \mathbf C$ denote the following four elements of the matrix space $\map {\mathcal M_\Z} 2$:


 * $\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

\qquad \mathbf A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \qquad \mathbf B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \qquad \mathbf C = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

The set:
 * $D_4 = \set {\mathbf I, -\mathbf I, \mathbf A, -\mathbf A, \mathbf B, -\mathbf B, \mathbf C, -\mathbf C}$

under the operation of conventional matrix multiplication, forms the dihedral group $D_4$:

Cayley Table
Its Cayley table is given by:

Also see

 * Dihedral Group D4 Defined by Matrices where it is shown that these have the appropriate properties.