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]