# Definition:Dihedral Group

## Definition

The dihedral group $D_n$ of order $2 n$ is the group of symmetries of the regular $n$-gon.

### Even Polygon ### Odd Polygon ## Group Presentation

The dihedral group $D_n$ has the group presentation:

$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$

That is, the dihedral group $D_n$ is generated by two elements $\alpha$ and $\beta$ such that:

$(1): \quad \alpha^n = e$
$(2): \quad \beta^2 = e$
$(3): \quad \beta \alpha = \alpha^{n - 1} \beta$

## Examples

### Dihedral Group $D_1$

The dihedral group $D_1$ is the symmetry group of the line segment:

Let $\triangle AB$ be a line segment. The various symmetry mappings of $AB$ are:

The identity mapping $e$
The rotation $r$ of $180^\circ$ about the center of $AB$.

This group is known as the symmetry group of the line segment.

### Dihedral Group $D_2$

The dihedral group $D_2$ is the symmetry group of the rectangle:

Let $\mathcal R = ABCD$ be a (non-square) rectangle. The various symmetry mappings of $\mathcal R$ are:

The identity mapping $e$
The rotation $r$ (in either direction) of $180^\circ$
The reflections $h$ and $v$ in the indicated axes.

### Dihedral Group $D_3$

The dihedral group $D_3$ is the symmetry group of the equilateral triangle:

Let $\triangle ABC$ be an equilateral triangle. We define in cycle notation the following symmetry mappings on $\triangle ABC$:

 $\displaystyle e$ $:$ $\displaystyle \tuple A \tuple B \tuple C$ Identity mapping $\displaystyle p$ $:$ $\displaystyle \tuple {ABC}$ Rotation of $120 \degrees$ anticlockwise about center $\displaystyle q$ $:$ $\displaystyle \tuple {ACB}$ Rotation of $120 \degrees$ clockwise about center $\displaystyle r$ $:$ $\displaystyle \tuple {BC}$ Reflection in line $r$ $\displaystyle s$ $:$ $\displaystyle \tuple {AC}$ Reflection in line $s$ $\displaystyle t$ $:$ $\displaystyle \tuple {AB}$ Reflection in line $t$

Note that $r, s, t$ can equally well be considered as a rotation of $180 \degrees$ (in $3$ dimensions) about the axes $r, s, t$.

Then these six operations form a group.

This group is known as the symmetry group of the equilateral triangle.

### Dihedral Group $D_4$

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

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

### Dihedral Group $D_6$

The dihedral group $D_6$ is the symmetry group of the regular hexagon:

Let $\mathcal H = ABCDEF$ be a regular hexagon. The various symmetry mappings of $\mathcal H$ are:

The identity mapping $e$
The rotations through multiples of $60 \degrees$
The reflections in the indicated axes.

Let $\alpha$ denote rotation of $\mathcal H$ anticlockwise through $\dfrac \pi 3$ radians ($60 \degrees$).

Let $\beta$ denote reflection of $\mathcal H$ in the $AD$ axis.

The symmetries of $\mathcal H$ form the dihedral group $D_6$.

## Also see

• Results about dihedral groups can be found here.