Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.

The notation for the dihedral group differs in geometry and abstract algebra. In geometry, D_n or Dih_n refers to the symmetries of the n-gon, a group of order 2n. In abstract algebra, D_2n refers to this same dihedral group. This article uses the geometric convention, D_n.

The word "dihedral" comes from "di-" and "-hedron". The latter comes from the Greek word hédra, which means "face of a geometrical solid". Overall, it thus refers to the two faces of a polygon.

A regular polygon with ${\displaystyle n}$ sides has ${\displaystyle 2n}$ different symmetries: ${\displaystyle n}$ rotational symmetries and ${\displaystyle n}$ reflection symmetries; here, ${\displaystyle n\geq 3}$. The associated rotations and reflections make up the dihedral group ${\displaystyle \mathrm {D} _{n}}$. If ${\displaystyle n}$ is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If ${\displaystyle n}$ is even, there are ${\displaystyle n/2}$ axes of symmetry connecting the midpoints of opposite sides and ${\displaystyle n/2}$ axes of symmetry connecting opposite vertices. In either case, there are ${\displaystyle n}$ axes of symmetry and ${\displaystyle 2n}$ elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.

As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group.

The following Cayley table shows the effect of composition in the dihedral group of order 6, ${\displaystyle \mathrm {D} _{3}}$—the symmetries of an equilateral triangle. Here, ${\displaystyle \mathrm {r} _{0}}$ denotes the identity, ${\displaystyle \mathrm {r} _{1}}$ and ${\displaystyle \mathrm {r} _{2}}$ denote counterclockwise rotations by 120° and 240° respectively, as well as ${\displaystyle \mathrm {s} _{0}}$, ${\displaystyle \mathrm {s} _{1}}$, and ${\displaystyle \mathrm {s} _{2}}$ denote reflections across the three lines shown in the adjacent picture.

For example, ${\displaystyle \mathrm {s} _{2}\mathrm {s} _{1}=\mathrm {r} _{1}}$, because the reflection ${\displaystyle \mathrm {s} _{1}}$ followed by the reflection s₂ results in a rotation of 120°. The order of elements denoting the composition is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative.

In general, the group ${\displaystyle \mathrm {D} _{n}}$ has elements ${\displaystyle r_{0},\dots ,r_{n-1}}$ and ${\displaystyle s_{0},\dots ,s_{n-1}}$, with composition given by the following formulae: $${\displaystyle {\begin{aligned}\mathrm {r} _{i}\,\mathrm {r} _{j}=\mathrm {r} _{i+j},&\qquad \mathrm {r} _{i}\,\mathrm {s} _{j}=\mathrm {s} _{i+j},\\\mathrm {s} _{i}\,\mathrm {r} _{j}=\mathrm {s} _{i-j},&\qquad \mathrm {s} _{i}\,\mathrm {s} _{j}=\mathrm {r} _{i-j}.\end{aligned}}}$$