# Definition:Dihedral Group

## Contents

## Definition

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

### Even Polygon

Definition:Dihedral Group/Even Polygon

### Odd Polygon

Definition:Dihedral Group/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:

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

- Dihedral Group is Group where it is shown that $D_n$ is a group

- Order of Dihedral Group where it is shown that $D_n$ is of order $2 n$

- Dihedral Group as Semidirect Product where it is shown that $D_n$ can be defined as the semidirect product $\Z_n \rtimes \Z_2$

- Results about
**dihedral groups**can be found here.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 7$: Example $7.3$ - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.8$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26 \theta$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 5$: Groups $\text{I}$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.9$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $4$: Subgroups: Example $4.10$