Dihedral Group D4/Matrix Representation/Formulation 2
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
- Dihedral Group D4 Defined by Matrices where it is shown that these have the appropriate properties.