Dihedral Groups

Introduction

The dihedral groups are among the simplest and most important examples of finite, non-abelian groups. They are the symmetry groups of regular polygons and provide excellent examples for understanding group theory concepts.

Definition

Geometric Definition

The dihedral group $D_{n}$ is the group of symmetries of a regular $n$ -gon. For $n \geq 3$ , this group consists of:

$n$ rotational symmetries
$n$ reflectional symmetries

The order of $D_{n}$ is $2 n$ .

Algebraic Presentation

The structure of $D_{n}$ can be captured by two generators: a rotation $r$ and a reflection $s$ . The group is defined by the relations these generators satisfy:

D_{n} = ⟨ r, s ∣ r^{n} = s^{2} = e, and s r s^{- 1} = r^{- 1} ⟩

The relation $s r s^{- 1} = r^{- 1}$ (or equivalently, $s r = r^{- 1} s$ ) algebraically encodes the geometric fact that reflecting a rotation is equivalent to rotating in the opposite direction.

Structure

Elements

The elements of $D_{n}$ can be written as:

Rotations: ${e, r, r^{2}, \dots, r^{n - 1}}$
Reflections: ${s, s r, s r^{2}, \dots, s r^{n - 1}}$

Group Operation

The group operation is function composition, which is not commutative. For example, a rotation by $90 °$ followed by a horizontal flip $(s r)$ is not the same as a horizontal flip followed by a $90 °$ rotation $(r s)$ .

Examples

Example 1: The Group $D_{4}$ (Square)

The group $D_{4}$ represents the symmetries of a square and has order $2 \times 4 = 8$ . Its elements are:

Four rotations:

$e$ : identity (rotation by $0 °$ )
$r$ : rotation by $90 °$ counterclockwise
$r^{2}$ : rotation by $180 °$ counterclockwise
$r^{3}$ : rotation by $270 °$ counterclockwise

Four reflections:

$s$ : reflection across the horizontal axis
$s r$ : reflection across the main diagonal
$s r^{2}$ : reflection across the vertical axis
$s r^{3}$ : reflection across the anti-diagonal

Example 2: The Group $D_{3}$ (Triangle)

The group $D_{3}$ represents the symmetries of an equilateral triangle and has order $2 \times 3 = 6$ . Its elements are:

Three rotations:

$e$ : identity (rotation by $0 °$ )
$r$ : rotation by $120 °$ counterclockwise
$r^{2}$ : rotation by $240 °$ counterclockwise

Three reflections:

$s$ : reflection across the altitude from vertex 1
$s r$ : reflection across the altitude from vertex 2
$s r^{2}$ : reflection across the altitude from vertex 3

Example 3: The Group $D_{6}$ (Hexagon)

The group $D_{6}$ has order 12 and represents the symmetries of a regular hexagon.

Cayley Table

Cayley Table for $D_{4}$

$\cdot$	$e$	$r$	$r^{2}$	$r^{3}$	$s$	$s r$	$s r^{2}$	$s r^{3}$
$e$	$e$	$r$	$r^{2}$	$r^{3}$	$s$	$s r$	$s r^{2}$	$s r^{3}$
$r$	$r$	$r^{2}$	$r^{3}$	$e$	$s r^{3}$	$s$	$s r$	$s r^{2}$
$r^{2}$	$r^{2}$	$r^{3}$	$e$	$r$	$s r^{2}$	$s r^{3}$	$s$	$s r$
$r^{3}$	$r^{3}$	$e$	$r$	$r^{2}$	$s r$	$s r^{2}$	$s r^{3}$	$s$
$s$	$s$	$s r$	$s r^{2}$	$s r^{3}$	$e$	$r$	$r^{2}$	$r^{3}$
$s r$	$s r$	$s r^{2}$	$s r^{3}$	$s$	$r^{3}$	$e$	$r$	$r^{2}$
$s r^{2}$	$s r^{2}$	$s r^{3}$	$s$	$s r$	$r^{2}$	$r^{3}$	$e$	$r$
$s r^{3}$	$s r^{3}$	$s$	$s r$	$s r^{2}$	$r$	$r^{2}$	$r^{3}$	$e$

Properties

Non-abelian

Dihedral groups are non-abelian for $n \geq 3$ . For example, in $D_{4}$ :

$r s = s r^{3} \neq s r$

Subgroups

The subgroups of $D_{n}$ include:

The cyclic subgroup of rotations: ${e, r, r^{2}, \dots, r^{n - 1}}$
Various reflection subgroups
The trivial subgroup ${e}$

Center

The center of $D_{n}$ is:

${e}$ if $n$ is odd
${e, r^{n / 2}}$ if $n$ is even

Applications

Geometric Applications

Study of crystal symmetries
Analysis of molecular symmetries in chemistry
Understanding tiling patterns

Algebraic Applications

Examples of non-abelian groups
Understanding group presentations
Study of group actions