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

Presentation

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

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

Infinite dihedral group

The infinite dihedral group has presentation $⟨a,b\mid b^2,ba{b}^{-1}=a^{-1}⟩$. 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 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 $D_{2n}$.