The dihedral group is the group of symmetries of the $n$ -vertex regular_polygon.

Presentation

The dihedral groups have very simple presentations: $D_{2 n} ≅ ⟨ a, b ∣ a^{n}, b^{2}, b a b^{- 1} = a^{- 1} ⟩$ The element $a$ represents a rotation, and the element $b$ represents a reflection in any fixed axis.

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Properties

The dihedral groups $D_{2 n}$ are all non-abelian for $n > 2$ . (Proof.)

The dihedral group $D_{2 n}$ is a subgroup of the symmetric group $S_{n}$ , generated by the elements $a = (123 \dots n)$ and $b = (2, n) (3, n - 1) \dots (\frac{n}{2} + 1, \frac{n}{2} + 3)$ if $n$ is even, $b = (2, n) (3, n - 1) \dots (\frac{n - 1}{2}, \frac{n + 1}{2})$ if $n$ is odd.

Examples

$D_{6}$ , the group of symmetries of the triangle

diagram

list the elements and Cayley table

Infinite dihedral group

The infinite dihedral group has presentation $⟨ a, b ∣ b^{2}, b a b^{- 1} = a^{- 1} ⟩$ . It is the "infinite-sided" version of the finite $D_{2 n}$ .

We may view the infinite dihedral group as being the subgroup of the group of homeomorphisms of $R^{2}$ generated by a reflection in the line $x = 0$ and a translation to the right by one unit. The translation is playing the role of a rotation in the finite $D_{2 n}$ .

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