Definition:Dihedral Group D4
Example of Dihedral Group
The dihedral group $D_4$ is the symmetry group of the square:
Let $\SS = ABCD$ be a square.
The various symmetries of $\SS$ are:
- the identity mapping $e$
- the rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ around the center of $\SS$ anticlockwise respectively
- the reflections $t_x$ and $t_y$ are reflections in the $x$ and $y$ axis respectively
- the reflection $t_{AC}$ in the diagonal through vertices $A$ and $C$
- the reflection $t_{BD}$ in the diagonal through vertices $B$ and $D$.
This group is known as the symmetry group of the square, and can be denoted $D_4$.
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 $\map {\MM_\Z} 2$:
- $\mathbf I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\qquad \mathbf A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \qquad \mathbf B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \qquad \mathbf C = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
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 $8$ 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 $D_4$.
Subgroups
The subsets of $D_4$ which form subgroups of $D_4$ are:
| \(\ds \) | \(\) | \(\ds D_4\) | ||||||||||||
| \(\ds \) | \(\) | \(\ds \set e\) | ||||||||||||
| \(\ds \) | \(\) | \(\ds \set {e, a, a^2, a^3}\) | ||||||||||||
| \(\ds \) | \(\) | \(\ds \set {e, a^2}\) | ||||||||||||
| \(\ds \) | \(\) | \(\ds \set {e, b}\) | ||||||||||||
| \(\ds \) | \(\) | \(\ds \set {e, b a}\) | ||||||||||||
| \(\ds \) | \(\) | \(\ds \set {e, b a^2}\) | ||||||||||||
| \(\ds \) | \(\) | \(\ds \set {e, b a^3}\) | ||||||||||||
| \(\ds \) | \(\) | \(\ds \set {e, a^2, b, b a^2}\) | ||||||||||||
| \(\ds \) | \(\) | \(\ds \set {e, a^2, b a, b a^3}\) |
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:
| \(\ds e H\) | \(=\) | \(\ds \set {e, b}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds b H\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds H\) |
| \(\ds a H\) | \(=\) | \(\ds \set {a, b a^3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds b a^3 H\) |
| \(\ds a^2 H\) | \(=\) | \(\ds \set {a^2, b a^2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds b a^2 H\) |
| \(\ds a^3 H\) | \(=\) | \(\ds \set {a^3, b a}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds b a H\) |
Right Cosets
The right cosets of $H$ are:
| \(\ds H e\) | \(=\) | \(\ds \set {e, b}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds H b\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds H\) |
| \(\ds H a\) | \(=\) | \(\ds \set {a, b a}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds H b a\) |
| \(\ds H a^2\) | \(=\) | \(\ds \set {a^2, b a^2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds H b a^2\) |
| \(\ds H a^3\) | \(=\) | \(\ds \set {a^3, b a^3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds H b a^3\) |
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.