The dihedral group $D_{2n}$ is the group of symmetries of the $n$-vertex [-regular_polygon].

Presentation

The dihedral groups have very simple presentations: $$D_{2n} \cong \langle a, b \mid a^n, b^2, b a b^{-1} = a^{-1} \rangle$$ The element $a$ represents a rotation, and the element $b$ represents a reflection in any fixed axis. [todo: picture]

Properties

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

[todo: diagram] [todo: list the elements and Cayley table]

Infinite dihedral group

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

We may view the infinite dihedral group as being the subgroup of the group of [homeomorphism homeomorphisms] of $\mathbb{R}^2$ 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 $D_{2n}$.

[todo: this section]