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

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}$

Generated Subgroup $\gen {\mathbf A}$

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

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

Generated Subgroup $\gen {\mathbf A, \mathbf D}$

The generated subgroup $\gen {\mathbf A, \mathbf D}$ is the whole of $D_4$.

Generated Subgroup $\gen {\mathbf A, \mathbf C}$

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

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

Generated Subgroup $\gen {\mathbf D}$

The generated subgroup $\gen {\mathbf D}$ is the (cyclic) group of order $2$:

$\set {\mathbf I, \mathbf D}$

Generated Subgroup $\gen {\mathbf B, \mathbf F}$

The generated subgroup $\gen {\mathbf D, \mathbf F}$ is the Klein $4$-group:

$\set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}$