# Dihedral group

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. 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

di­a­gram 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 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 $$D_{2n}$$.

this section

Children:

Parents:

• Group

The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.