Dihedral group


by Patrick Stevens Jun 15 2016 updated Jun 16 2016

The dihedral groups are natural examples of groups, arising from the symmetries of regular polygons.

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


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]



$~$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]