Dihedral Groups — Mathematics Knowledge System

Overview

The dihedral group Dₙ (order 2n) is the group of symmetries of a regular n-gon, consisting of n rotations and n reflections. It is the smallest interesting family of non-abelian groups (for n ≥ 3) and provides the standard first example encountered when moving beyond cyclic and abelian groups. Dₙ is generated by a rotation r of order n and a reflection s of order 2, subject to the single relation srs = r⁻¹, which encodes that reflecting then rotating is the same as rotating the opposite way then reflecting.

Intuition

Picture a regular n-gon fixed in the plane: you can rotate it by multiples of 360°/n, or you can flip it over one of its axes of symmetry, and those two families of moves — n rotations, n reflections — are all the rigid symmetries it has. The relation srs = r⁻¹ captures a key asymmetry with rotations alone: performing a reflection, then a rotation, then the same reflection again is the same as rotating the opposite direction, which is exactly why Dₙ (for n ≥ 3) is not abelian — reflecting and rotating don't commute.

Formal Definition

Definition

Dₙ has presentation with generators r (rotation by 2π/n) and s (a reflection):

D_n = \langle r, s \mid r^n = e,\ s^2 = e,\ srs = r^{-1} \rangle

Presentation of the dihedral group

|D_n| = 2n

Order

D_n = \{ e, r, r^2, \ldots, r^{n-1}, s, sr, sr^2, \ldots, sr^{n-1} \}

Elements: n rotations, n reflections

sr^k = r^{-k}s \quad (k = 0, \ldots, n-1)

Reflection–rotation swap relation

Worked Examples

D₃ has 3 rotations (by 0°, 120°, 240°) and 3 reflections (through each vertex/midpoint axis).
$D_3 = \{e, r, r^2, s, sr, sr^2\}$
|D₃| = 2×3 = 6, matching the order formula 2n with n=3.
$|D_3| = 6$
Note D₃ ≅ S₃, since both are the unique non-abelian group of order 6.
$D_3 \cong S_3$

Answer: D₃ = {e, r, r², s, sr, sr²} has 6 elements, and is isomorphic to S₃.

Practice Problems

Difficulty 4/10

What is the order of D₆ (symmetries of a regular hexagon), and how many rotations vs. reflections does it contain?

Difficulty 5/10

Show that Dₙ is non-abelian for n ≥ 3 by comparing rs and sr.

Difficulty 5/10

Show that the subgroup of rotations R = {e, r, ..., r^{n-1}} is normal in Dₙ, and identify Dₙ/R.

Quiz

Summary

Dₙ = ⟨r, s | rⁿ=e, s²=e, srs=r⁻¹⟩ is the symmetry group of a regular n-gon, with order 2n (n rotations, n reflections).
Dₙ is non-abelian for n ≥ 3, since reflecting reverses the direction a rotation is 'read' (sr = r⁻¹s).
The rotation subgroup R ≅ ℤₙ has index 2 in Dₙ and is always normal, with Dₙ/R ≅ ℤ₂.

References

WebsiteWikipedia — Dihedral group