Dihedral Group D4/Matrix Representation/Formulation 2/Examples of Cosets/Subgroup Generated by D/Right Cosets

From ProofWiki
Jump to navigation Jump to search

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:

$H = \gen {\mathbf D}$

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

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

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


The right cosets of $H$ are:

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


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


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


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


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:

\(\ds H \mathbf I\) \(=\) \(\ds \set {\mathbf I, \mathbf D} \mathbf I\)
\(\ds \) \(=\) \(\ds \set {\mathbf I^2, \mathbf D \mathbf I}\)
\(\ds \) \(=\) \(\ds \set {\mathbf I, \mathbf D}\)
\(\ds \) \(=\) \(\ds H\)


\(\ds H \mathbf D\) \(=\) \(\ds \set {\mathbf I, \mathbf D} \mathbf D\)
\(\ds \) \(=\) \(\ds \set {\mathbf I \mathbf D, \mathbf D^2}\)
\(\ds \) \(=\) \(\ds \set {\mathbf D, \mathbf I}\)
\(\ds \) \(=\) \(\ds H\)


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


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


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


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


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


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

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Dihedral_Group_D4/Matrix_Representation/Formulation_2/Examples_of_Cosets/Subgroup_Generated_by_D/Right_Cosets&oldid=386637"