Dihedral Group D4/Matrix Representation/Formulation 2

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Matrix Representation of Dihedral Group $D_4$

Let $\mathbf I, \mathbf A, \mathbf B, \mathbf C, \mathbf D, \mathbf E, \mathbf F, \mathbf G$ denote the following $8$ elements of the matrix space $\map {\MM_\Z} 2$:

$\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 set:

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

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


Cayley Table

Its Cayley table is given by:

$\begin{array}{r|rrrrrrrr}
         & \mathbf I & \mathbf A & \mathbf B & \mathbf C & \mathbf D & \mathbf E & \mathbf F & \mathbf G \\

\hline \mathbf I & \mathbf I & \mathbf A & \mathbf B & \mathbf C & \mathbf D & \mathbf E & \mathbf F & \mathbf G \\ \mathbf A & \mathbf A & \mathbf B & \mathbf C & \mathbf I & \mathbf E & \mathbf F & \mathbf G & \mathbf D \\ \mathbf B & \mathbf B & \mathbf C & \mathbf I & \mathbf A & \mathbf F & \mathbf G & \mathbf D & \mathbf E \\ \mathbf C & \mathbf C & \mathbf I & \mathbf A & \mathbf B & \mathbf G & \mathbf D & \mathbf E & \mathbf F \\ \mathbf D & \mathbf D & \mathbf G & \mathbf F & \mathbf E & \mathbf I & \mathbf C & \mathbf B & \mathbf A \\ \mathbf E & \mathbf E & \mathbf D & \mathbf G & \mathbf F & \mathbf A & \mathbf I & \mathbf C & \mathbf B \\ \mathbf F & \mathbf F & \mathbf E & \mathbf D & \mathbf G & \mathbf B & \mathbf A & \mathbf I & \mathbf C \\ \mathbf G & \mathbf G & \mathbf F & \mathbf E & \mathbf D & \mathbf C & \mathbf B & \mathbf A & \mathbf I \end{array}$


Generated Subgroups

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:

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


Cosets

Generated Subgroup $\gen {\mathbf A}$

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:

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


Left Cosets

The left cosets of $H$ are:

\(\ds \mathbf I H\) \(=\) \(\ds \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}\)
\(\ds \) \(=\) \(\ds \mathbf A H\)
\(\ds \) \(=\) \(\ds \mathbf B H\)
\(\ds \) \(=\) \(\ds \mathbf C H\)
\(\ds \) \(=\) \(\ds H\)


\(\ds \mathbf D H\) \(=\) \(\ds \set {\mathbf D, \mathbf E, \mathbf F, \mathbf G}\)
\(\ds \) \(=\) \(\ds \mathbf E H\)
\(\ds \) \(=\) \(\ds \mathbf F H\)
\(\ds \) \(=\) \(\ds \mathbf G H\)


Right Cosets

The right cosets of $H$ are:

\(\ds H \mathbf I\) \(=\) \(\ds \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}\)
\(\ds \) \(=\) \(\ds H \mathbf A\)
\(\ds \) \(=\) \(\ds H \mathbf B\)
\(\ds \) \(=\) \(\ds H \mathbf C\)
\(\ds \) \(=\) \(\ds H\)


\(\ds H \mathbf D\) \(=\) \(\ds \set {\mathbf D, \mathbf E, \mathbf F, \mathbf G}\)
\(\ds \) \(=\) \(\ds H \mathbf E\)
\(\ds \) \(=\) \(\ds H \mathbf F\)
\(\ds \) \(=\) \(\ds H \mathbf G\)


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

Let $H \subseteq D_4$ be defined as:

$H = \gen {\mathbf B, \mathbf F}$

where $\gen {\mathbf B, \mathbf F}$ denotes the subgroup generated by $\set {\mathbf B, \mathbf F}$.

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

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


Left Cosets

The left cosets of $H$ are:

\(\ds \mathbf I H\) \(=\) \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\)
\(\ds \) \(=\) \(\ds \mathbf B H\)
\(\ds \) \(=\) \(\ds \mathbf D H\)
\(\ds \) \(=\) \(\ds \mathbf F H\)
\(\ds \) \(=\) \(\ds H\)


\(\ds \mathbf A H\) \(=\) \(\ds \set {\mathbf A, \mathbf C, \mathbf E, \mathbf G}\)
\(\ds \) \(=\) \(\ds \mathbf C H\)
\(\ds \) \(=\) \(\ds \mathbf E H\)
\(\ds \) \(=\) \(\ds \mathbf G H\)


Right Cosets

The right cosets of $H$ are:

\(\ds H \mathbf I\) \(=\) \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\)
\(\ds \) \(=\) \(\ds H \mathbf B\)
\(\ds \) \(=\) \(\ds H \mathbf D\)
\(\ds \) \(=\) \(\ds H \mathbf F\)
\(\ds \) \(=\) \(\ds H\)


\(\ds H \mathbf A\) \(=\) \(\ds \set {\mathbf A, \mathbf C, \mathbf E, \mathbf G}\)
\(\ds \) \(=\) \(\ds H \mathbf C\)
\(\ds \) \(=\) \(\ds H \mathbf E\)
\(\ds \) \(=\) \(\ds H \mathbf G\)


Generated Subgroup $\gen {\mathbf D}$

Let $H \subseteq D_4$ be defined as:

$H = \gen {\mathbf D}$

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

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

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


Left Cosets

The left cosets of $H$ are:

\(\ds \mathbf I H\) \(=\) \(\ds \set {\mathbf I, \mathbf D}\)
\(\ds \) \(=\) \(\ds \mathbf D H\)
\(\ds \) \(=\) \(\ds H\)


\(\ds \mathbf A H\) \(=\) \(\ds \set {\mathbf A, \mathbf E}\)
\(\ds \) \(=\) \(\ds \mathbf E H\)


\(\ds \mathbf B H\) \(=\) \(\ds \set {\mathbf B, \mathbf F}\)
\(\ds \) \(=\) \(\ds \mathbf F H\)


\(\ds \mathbf C H\) \(=\) \(\ds \set {\mathbf C, \mathbf G}\)
\(\ds \) \(=\) \(\ds \mathbf G H\)


Right Cosets

The right cosets of $H$ are:

\(\ds H \mathbf I\) \(=\) \(\ds \set {\mathbf I, \mathbf D}\)
\(\ds \) \(=\) \(\ds H \mathbf D\)
\(\ds \) \(=\) \(\ds H\)


\(\ds H \mathbf A\) \(=\) \(\ds \set {\mathbf A, \mathbf G}\)
\(\ds \) \(=\) \(\ds H \mathbf G\)


\(\ds H \mathbf B\) \(=\) \(\ds \set {\mathbf B, \mathbf F}\)
\(\ds \) \(=\) \(\ds H \mathbf F\)


\(\ds H \mathbf C\) \(=\) \(\ds \set {\mathbf C, \mathbf E}\)
\(\ds \) \(=\) \(\ds H \mathbf E\)


Also see

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