Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets/Subgroup Generated by A/Left Cosets
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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:
\(\ds \mathbf I H\) | \(=\) | \(\ds \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf A H\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf B H\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf C H\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds \mathbf D H\) | \(=\) | \(\ds \set {\mathbf D, \mathbf E, \mathbf F, \mathbf G}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf E H\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf F H\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf G H\) |
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:
\(\ds \mathbf I H\) | \(=\) | \(\ds \mathbf I \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf I^2, \mathbf I \mathbf A, \mathbf I \mathbf B, \mathbf I \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds \mathbf A H\) | \(=\) | \(\ds \mathbf A \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf A \mathbf I, \mathbf A^2, \mathbf A \mathbf B, \mathbf A \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf A, \mathbf B, \mathbf C, \mathbf I}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds \mathbf B H\) | \(=\) | \(\ds \mathbf B \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf B \mathbf I, \mathbf B \mathbf A, \mathbf B^2, \mathbf B \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf B, \mathbf C, \mathbf I, \mathbf A}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds \mathbf C H\) | \(=\) | \(\ds \mathbf C \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf C \mathbf I, \mathbf C \mathbf A, \mathbf C \mathbf B, \mathbf C^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf C, \mathbf I, \mathbf A, \mathbf B}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds \mathbf D H\) | \(=\) | \(\ds \mathbf D \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf D \mathbf I, \mathbf D \mathbf A, \mathbf D \mathbf B, \mathbf D \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf D, \mathbf G, \mathbf F, \mathbf E}\) |
\(\ds \mathbf E H\) | \(=\) | \(\ds \mathbf E \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf E \mathbf I, \mathbf E \mathbf A, \mathbf E \mathbf B, \mathbf E \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf E, \mathbf D, \mathbf G, \mathbf F}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf D H\) |
\(\ds \mathbf F H\) | \(=\) | \(\ds \mathbf F \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf F \mathbf I, \mathbf F \mathbf A, \mathbf F \mathbf B, \mathbf F \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf F, \mathbf E, \mathbf D, \mathbf G}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf D H\) |
\(\ds \mathbf G H\) | \(=\) | \(\ds \mathbf G \set {\mathbf I, \mathbf A, \mathbf B, \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf G \mathbf I, \mathbf G \mathbf A, \mathbf G \mathbf B, \mathbf G \mathbf C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {\mathbf G, \mathbf F, \mathbf E, \mathbf D}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf D H\) |
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Exercise $1 \ \text {(a)}$