Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets/Subgroup Generated by A/Left Cosets

Examples of Left Cosets of 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}$

Let $H \subseteq D_4$ be defined as:
 * $H = \gen {\mathbf A}$

where $\gen {\mathbf A}$ denotes the subgroup generated by $\mathbf A$.

From Dihedral Group D4: Examples of Generated Subgroups: $\gen {\mathbf A}$ we have:


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

The left cosets of $H$ are:

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

Thus: