Dihedral Group — Definition, Formula & Examples

A dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. It is denoted

D_n

and has

2n

elements, where

n

is the number of sides of the polygon.

The dihedral group $D_n$ (for $n \geq 3$ ) is the group of isometries of the Euclidean plane that map a regular $n$ -gon to itself. It is generated by a rotation $r$ of order $n$ and a reflection $s$ of order $2$ , subject to the relation $srs = r^{-1}$ , giving the presentation $D_n = \langle r, s \mid r^n = s^2 = e,\; srs = r^{-1} \rangle$ .

Key Formula

|D_n| = 2n

Where:

$D_n$ = The dihedral group of order 2n (symmetries of a regular n-gon)
$n$ = Number of sides of the regular polygon (n ≥ 3)
$|D_n|$ = The number of elements (order) of the group

How It Works

Every element of

D_n

can be written uniquely as

r^k

sr^k

for

0 \leq k \leq n-1

. The

n

elements of the form

r^k

are rotations (including the identity

r^0 = e

), and the

n

elements of the form

sr^k

are reflections. To compose two symmetries, you use the key relation

rs = sr^{-1}

(equivalently

srs = r^{-1}

) to rewrite any product into one of these standard forms. For example, in

D_4

the product

sr^2 \cdot sr^3

can be simplified by moving

s

past powers of

r

using the relation repeatedly.

Worked Example

Problem: List all elements of the dihedral group

D_3

(symmetries of an equilateral triangle) and determine whether

D_3

is abelian.

Step 1: Count the elements. Since

n = 3

, the group has

2n = 6

elements.

|D_3| = 2(3) = 6

Step 2: Write out all elements using the generators

r

(rotation by

120°

) and

s

(a reflection).

D_3 = \{e,\; r,\; r^2,\; s,\; sr,\; sr^2\}

Step 3: Check commutativity by computing

rs

and

sr

. The relation

srs = r^{-1} = r^2

gives

sr = r^2 s

, so

rs \neq sr

rs = sr^{-1} = sr^2 \neq sr

Answer:

D_3

has 6 elements:

\{e, r, r^2, s, sr, sr^2\}

. Since

rs \neq sr

, the group is non-abelian. (In fact,

D_3

is isomorphic to the symmetric group

S_3

Why It Matters

Dihedral groups are among the first non-abelian groups students encounter and serve as a testing ground for concepts like normal subgroups, conjugacy classes, and group actions. They appear naturally in chemistry (molecular symmetry), crystallography, and computer graphics whenever rotational and reflective symmetry must be classified.

Common Mistakes

Mistake: Confusing the notation: some textbooks use

D_n

for

2n

elements while others use

D_{2n}

for the same group.

Correction: Always check which convention your textbook uses. If the symmetry group of a square is called

D_4

, it has 8 elements; if it is called

D_8

, it still has 8 elements — same group, different labeling.