Definition:Dihedral Group D4

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Example of Dihedral Group

The dihedral group $D_4$ is the symmetry group of the square:

Let $\mathcal S = ABCD$ be a square.

SymmetryGroupSquare.png

The various symmetry mappings of $\mathcal S$ are:

The identity mapping $e$
The rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ counterclockwise respectively about the center of $\mathcal S$.
The reflections $t_x$ and $t_y$ are reflections about the $x$ and $y$ axis respectively.
The reflection $t_{AC}$ is a reflection about the diagonal through vertices $A$ and $C$.
The reflection $t_{BD}$ is a reflection about the diagonal through vertices $B$ and $D$.

This group is known as the symmetry group of the square.


Group Presentation

$D_4 = \gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$


Cayley Table

$\begin{array}{l|cccccccc} & e & a & a^2 & a^3 & b & b a & b a^2 & b a^3 \\ \hline e & e & a & a^2 & a^3 & b & b a & b a^2 & b a^3 \\ a & a & a^2 & a^3 & e & b a^3 & b & b a & b a^2 \\ a^2 & a^2 & a^3 & e & a & b a^2 & b a^3 & b & b a \\ a^3 & a^3 & e & a & a^2 & b a & b a^2 & b a^3 & b \\ b & b & b a & b a^2 & b a^3 & e & a & a^2 & a^3 \\ b a & b a & b a^2 & b a^3 & b & a^3 & e & a & a^2 \\ b a^2 & b a^2 & b a^3 & b & b a & a^2 & a^3 & e & a \\ b a^3 & b a^3 & b & b a & b a^2 & a & a^2 & a^3 & e \end{array}$


Matrix Representations

Formulation 1

Let $\mathbf I, \mathbf A, \mathbf B, \mathbf C$ denote the following four elements of the matrix space $\map {\mathcal M_\Z} 2$:

$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \qquad \mathbf B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \qquad \mathbf C = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$


The set:

$D_4 = \set {\mathbf I, -\mathbf I, \mathbf A, -\mathbf A, \mathbf B, -\mathbf B, \mathbf C, -\mathbf C}$

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


Formulation 2

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

$\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf B = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \qquad \mathbf C = \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix}$
$\mathbf D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \qquad \mathbf E = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} \qquad \mathbf F = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf G = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$

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 $D_4$.


Subgroups

The subsets of $D_4$ which form subgroups of $D_4$ are:

\(\displaystyle \) \(\) \(\displaystyle D_4\)
\(\displaystyle \) \(\) \(\displaystyle \set e\)
\(\displaystyle \) \(\) \(\displaystyle \set {e, a, a^2, a^3}\)
\(\displaystyle \) \(\) \(\displaystyle \set {e, a^2}\)
\(\displaystyle \) \(\) \(\displaystyle \set {e, b}\)
\(\displaystyle \) \(\) \(\displaystyle \set {e, b a}\)
\(\displaystyle \) \(\) \(\displaystyle \set {e, b a^2}\)
\(\displaystyle \) \(\) \(\displaystyle \set {e, b a^3}\)
\(\displaystyle \) \(\) \(\displaystyle \set {e, a^2, b, b a^2}\)


Cosets of Subgroups

Generated Subgroup $\gen b$

Let $H \subseteq D_4$ be defined as:

$H = \gen b$

where $\gen b$ denotes the subgroup generated by $b$.

From Subgroups of Dihedral Group D4 we have:

$\gen b = \set {e, b}$


Left Cosets

The left cosets of $H$ are:

\(\displaystyle e H\) \(=\) \(\displaystyle \set {e, b}\)
\(\displaystyle \) \(=\) \(\displaystyle b H\)
\(\displaystyle \) \(=\) \(\displaystyle H\)


\(\displaystyle a H\) \(=\) \(\displaystyle \set {a, b a^3}\)
\(\displaystyle \) \(=\) \(\displaystyle b a^3 H\)


\(\displaystyle a^2 H\) \(=\) \(\displaystyle \set {a^2, b a^2}\)
\(\displaystyle \) \(=\) \(\displaystyle b a^2 H\)


\(\displaystyle a^3 H\) \(=\) \(\displaystyle \set {a^3, b a}\)
\(\displaystyle \) \(=\) \(\displaystyle b a H\)


Right Cosets

The right cosets of $H$ are:

\(\displaystyle H e\) \(=\) \(\displaystyle \set {e, b}\)
\(\displaystyle \) \(=\) \(\displaystyle H b\)
\(\displaystyle \) \(=\) \(\displaystyle H\)


\(\displaystyle H a\) \(=\) \(\displaystyle \set {a, b a}\)
\(\displaystyle \) \(=\) \(\displaystyle H b a\)


\(\displaystyle H a^2\) \(=\) \(\displaystyle \set {a^2, b a^2}\)
\(\displaystyle \) \(=\) \(\displaystyle H b a^2\)


\(\displaystyle H a^3\) \(=\) \(\displaystyle \set {a^3, b a^3}\)
\(\displaystyle \) \(=\) \(\displaystyle H b a^3\)


It is seen that the left cosets do not equal the corresponding right cosets.

It follows by definition that $\gen b$ is not a normal subgroup of $D_4$.


Normal Subgroups

Generated Subgroup $\gen {a^2}$

The subgroup of $D_4$ generated by $\gen {a^2}$ is normal.


Generated Subgroup $\gen a$

The subgroup of $D_4$ generated by $\gen a$ is normal.


Generated Subgroup $\gen {a^2, b}$

The subgroup of $D_4$ generated by $\gen {a^2, b}$ is normal.


Center

The center of $D_4$ is given by:

$\map Z {D_4} = \set {e, a^2}$


Also see

  • Results about the dihedral group $D_4$ can be found here.