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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