Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets/Subgroup Generated by B, F/Right Cosets
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Examples of Right 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:
- $H = \gen {\mathbf B, \mathbf F}$
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:
- $\gen {\mathbf B, \mathbf F} = \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}$
The right cosets of $H$ are:
\(\ds H \mathbf I\) | \(=\) | \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H \mathbf B\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H \mathbf D\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H \mathbf F\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds H \mathbf A\) | \(=\) | \(\ds \set {\mathbf A, \mathbf C, \mathbf E, \mathbf G}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H \mathbf C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H \mathbf E\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H \mathbf G\) |
Proof
The Cayley table of $D_4$ is presented as:
- $\begin{array}{r|rrrrrrrr} & \mathbf I & \mathbf A & \mathbf B & \mathbf C & \mathbf D & \mathbf E & \mathbf F & \mathbf G \\ \hline \mathbf I & \mathbf I & \mathbf A & \mathbf B & \mathbf C & \mathbf D & \mathbf E & \mathbf F & \mathbf G \\ \mathbf A & \mathbf A & \mathbf B & \mathbf C & \mathbf I & \mathbf E & \mathbf F & \mathbf G & \mathbf D \\ \mathbf B & \mathbf B & \mathbf C & \mathbf I & \mathbf A & \mathbf F & \mathbf G & \mathbf D & \mathbf E \\ \mathbf C & \mathbf C & \mathbf I & \mathbf A & \mathbf B & \mathbf G & \mathbf D & \mathbf E & \mathbf F \\ \mathbf D & \mathbf D & \mathbf G & \mathbf F & \mathbf E & \mathbf I & \mathbf C & \mathbf B & \mathbf A \\ \mathbf E & \mathbf E & \mathbf D & \mathbf G & \mathbf F & \mathbf A & \mathbf I & \mathbf C & \mathbf B \\ \mathbf F & \mathbf F & \mathbf E & \mathbf D & \mathbf G & \mathbf B & \mathbf A & \mathbf I & \mathbf C \\ \mathbf G & \mathbf G & \mathbf F & \mathbf E & \mathbf D & \mathbf C & \mathbf B & \mathbf A & \mathbf I \end{array}$
Thus:
\(\ds H \mathbf I\) | \(=\) | \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F} \mathbf I\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf I^2, \mathbf B \mathbf I, \mathbf D \mathbf I, \mathbf F \mathbf I}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds H \mathbf B\) | \(=\) | \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F} \mathbf B\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf I \mathbf B, \mathbf B^2, \mathbf D \mathbf B, \mathbf F \mathbf B}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf B, \mathbf I, \mathbf F, \mathbf D}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds H \mathbf D\) | \(=\) | \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F} \mathbf D\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf I \mathbf D, \mathbf B \mathbf D, \mathbf D^2, \mathbf F \mathbf D}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf D, \mathbf F, \mathbf I, \mathbf B}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds H \mathbf F\) | \(=\) | \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F} \mathbf F\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf I \mathbf F, \mathbf B \mathbf F, \mathbf D \mathbf F, \mathbf F^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf F, \mathbf D, \mathbf B, \mathbf I}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds H \mathbf A\) | \(=\) | \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F} \mathbf A\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf I \mathbf A, \mathbf B \mathbf A, \mathbf D \mathbf A, \mathbf F \mathbf A}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf A, \mathbf C, \mathbf G, \mathbf E}\) |
\(\ds H \mathbf C\) | \(=\) | \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F} \mathbf C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf I \mathbf C, \mathbf B \mathbf C, \mathbf D \mathbf C, \mathbf F \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf C, \mathbf A, \mathbf E, \mathbf G}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H \mathbf A\) |
\(\ds H \mathbf E\) | \(=\) | \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F} \mathbf E\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf I \mathbf E, \mathbf B \mathbf E, \mathbf D \mathbf E, \mathbf F \mathbf E}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf E, \mathbf G, \mathbf C, \mathbf A}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H \mathbf A\) |
\(\ds H \mathbf G\) | \(=\) | \(\ds \set {\mathbf I, \mathbf B, \mathbf D, \mathbf F} \mathbf G\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf I \mathbf G, \mathbf B \mathbf G, \mathbf D \mathbf G, \mathbf F \mathbf G}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf G, \mathbf E, \mathbf C, \mathbf A}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H \mathbf A\) |
$\blacksquare$