Dihedral groups

The dihedral group Dₙ is the symmetry group of a regular n-gon: the n rotations and n reflections, 2n elements in all. It is the split extension Cₙ ⋊ C₂ in which the order-2 generator inverts every rotation (s r s⁻¹ = r⁻¹).

Definition

For n ≥ 1,

Dₙ = ⟨ r, s |  rⁿ = s² = 1,  s r s⁻¹ = r⁻¹ ⟩,    |Dₙ| = 2n.

r is rotation by 2π/n, generating a cyclic subgroup ⟨r⟩ ≅ Cₙ of index 2; s is any reflection. The catalog names D₃, D₄, …; the degenerate D₁ ≅ C₂ and D₂ ≅ C₂ × C₂ are abelian and usually excluded.

The Cₙ ⋊ C₂ family

Dₙ is the split cyclic-by-C₂ extension with s acting by inversion. Keeping the inversion action but changing how s closes up, or moving to a 2-power cyclic part, gives its close relatives:

• Dicyclic / generalized quaternion — the non-split extension with the same inversion action: s² equals the unique central involution instead of 1. Dicₘ has order 4m; its 2-power members are the generalized quaternion groups Q₂ⁿ.

• At 2-power order 2ᵏ, Dₙ is one of five non-cyclic groups with a cyclic maximal subgroup ⟨r⟩ of order 2ᵏ⁻¹, separated by how s conjugates r:

  Group	s r s⁻¹	split?
  Dihedral Dₙ	r⁻¹	yes
  Semi-dihedral SD	r2k−2 − 1	yes
  Modular maximal-cyclic M	r2k−2 + 1	yes
  Generalized quaternion Q	r⁻¹	no (s² central)
  C2k−1 × C₂	r (trivial)	yes

  Dihedral, semi-dihedral and generalized quaternion have maximal class k − 1; the modular group has class 2; the last is abelian.

Every group above is metacyclic — a cyclic normal subgroup with cyclic quotient — hence supersolvable.

Structure

• Order 2n, with a normal cyclic subgroup ⟨r⟩ ≅ Cₙ of index 2, so Dₙ is metacyclic, supersolvable, and solvable.
• Center Z(Dₙ): trivial when n is odd; ⟨rn/2⟩ ≅ C₂ when n is even.
• Commutator subgroup [Dₙ, Dₙ] = ⟨r²⟩: this is all of ⟨r⟩ ≅ Cₙ for n odd, and Cn/2 for n even — so the abelianization is C₂ (n odd) or C₂ × C₂ (n even).
• Conjugacy classes: (n + 3)/2 for n odd, (n + 6)/2 for n even.
• Nilpotent exactly when n is a power of 2 — the dihedral 2-groups, of maximal class.
• Dₙ ≅ Dn/2 × C₂ when n ≡ 2 (mod 4), e.g. D₆ ≅ D₃ × C₂.

Examples

• D₃ — order 6, the smallest non-abelian group; it is the symmetric group S₃ on 3 points.
• D₄ — order 8, the symmetries of a square; one of the two non-abelian groups of that order, alongside Q₈.
• D₆ — order 12, the smallest decomposable dihedral group: D₆ ≅ D₃ × C₂ (as 6 ≡ 2 mod 4).

See also

• Cyclic group — the rotation subgroup ⟨r⟩.
• Dicyclic / generalized quaternion group — the non-split inversion cousin.
• Semi-dihedral, modular maximal-cyclic — the other 2-power groups with a cyclic index-2 subgroup.
• Metacyclic group, symmetric group.
• Wikipedia: Dihedral group