Definition:Dihedral Group

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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 $AB$ be a line segment.

The symmetries of $AB$ are:

The identity mapping $e$

The rotation $r$ of $180 \degrees$ about the midpoint 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 $\RR = ABCD$ be a (non-square) rectangle.

The various symmetries of $\RR$ are:

The identity mapping $e$

The rotation $r$ (in either direction) of $180^\circ$

The reflections $h$ and $v$ in the indicated axes.

The symmetries of $\RR$ form the dihedral group $D_2$.

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 symmetries 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 $\SS = ABCD$ 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 $D_4$.

Dihedral Group $D_5$

The dihedral group $D_5$ is the symmetry group of the regular pentagon:

Let $\PP = ABCDE$ be a regular pentagon.

The various symmetries of $\PP$ are:

the identity mapping $e$

the rotations $r, r^2, r^3, r^4$ of $72^\circ, 144^\circ, 216^\circ, 288^\circ$ around the center of $\PP$ anticlockwise respectively

the reflections $t_A, t_B, t_C, t_D, t_E$ in the lines through the center of $\PP$ and the vertices $A$ to $E$ respectively.

This group is known as the symmetry group of the regular pentagon.

Dihedral Group $D_6$

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

Let $\HH = ABCDEF$ be a regular hexagon.

The various symmetries of $\HH$ are:

The identity mapping $e$

The rotations through multiples of $60 \degrees$ anticlockwise about the center of $\HH$

The reflections in the indicated axes.

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

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

The symmetries of $\HH$ 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): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: 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 $4.10$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): dihedral group
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): dihedral group