Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Generated Subgroups/B, F

Examples of Generated Subgroups of Dihedral Group $D_4$
Let the dihedral group $D_4$ be represented by the set of square matrices:


 * $D_4 = \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C, \mathbf D, \mathbf E, \mathbf F, \mathbf G}$

under the operation of conventional matrix multiplication, where:


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

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


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

\qquad \mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} \qquad \mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$

The generated subgroup $\gen {\mathbf D, \mathbf F}$ is the cyclic group of order $4$:


 * $\set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}$

Proof
The Cayley table of $D_4$ is as follows:

We have:

Thus:
 * $\gen {\mathbf B, \mathbf D} = \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}$

and its Cayley table is:


 * $\begin{array}{r|rrrrrrrr}

& \mathbf I & \mathbf B & \mathbf D & \mathbf F \\ \hline \mathbf I & \mathbf I & \mathbf B & \mathbf D & \mathbf F \\ \mathbf B & \mathbf B & \mathbf I & \mathbf F & \mathbf D \\ \mathbf D & \mathbf D & \mathbf F & \mathbf I & \mathbf B \\ \mathbf F & \mathbf F & \mathbf D & \mathbf B & \mathbf I \\ \end{array}$

This is seen to be the Klein $4$-group.