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 symmetry mappings 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.