# 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 $$D_{2n}$$.

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• Group

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