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
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 $\triangle ABC$:
\(\ds e\) | \(:\) | \(\ds \tuple A \tuple B \tuple C\) | Identity mapping | |||||||||||
\(\ds p\) | \(:\) | \(\ds \tuple {ABC}\) | Rotation of $120 \degrees$ anticlockwise about center | |||||||||||
\(\ds q\) | \(:\) | \(\ds \tuple {ACB}\) | Rotation of $120 \degrees$ clockwise about center | |||||||||||
\(\ds r\) | \(:\) | \(\ds \tuple {BC}\) | Reflection in line $r$ | |||||||||||
\(\ds s\) | \(:\) | \(\ds \tuple {AC}\) | Reflection in line $s$ | |||||||||||
\(\ds t\) | \(:\) | \(\ds \tuple {AB}\) | 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 $4.10$