Definition:Dihedral Group D3

Example of Dihedral Group

The dihedral group $D_3$ is the symmetry group of the equilateral triangle:

Let $\triangle ABC$ be an equilateral triangle.

We define in cycle notation the following symmetries on $\triangle ABC$:

\(\ds e\)

\(:\)

\(\ds \tuple A \tuple B \tuple C\)

Identity mapping

\(\ds p\)

\(:\)

\(\ds \tuple {ABC}\)

Rotation of $120 \degrees$ anticlockwise about center

\(\ds q\)

\(:\)

\(\ds \tuple {ACB}\)

Rotation of $120 \degrees$ clockwise about center

\(\ds r\)

\(:\)

\(\ds \tuple {BC}\)

Reflection in line $r$

\(\ds s\)

\(:\)

\(\ds \tuple {AC}\)

Reflection in line $s$

\(\ds t\)

\(:\)

\(\ds \tuple {AB}\)

Reflection in line $t$

Note that $r, s, t$ can equally well be considered as a rotation of $180 \degrees$ (in $3$ dimensions) about the axes $r, s, t$.

Then these six operations form a group.

This group is known as the symmetry group of the equilateral triangle.

Group Presentation

$D_3 = \gen {a, b: a^3 = b^2 = e, a b = b a^{-1} }$

Cayley Table

$\begin{array}{c|cccccc} & e & a & a^2 & b & a b & a^2 b \\ \hline e & e & a & a^2 & b & a b & a^2 b \\ a & a & a^2 & e & a b & a^2 b & b \\ a^2 & a^2 & e & a & a^2 b & b & a b \\ b & b & a^2 b & a b & e & a^2 & a \\ a b & a b & b & a^2 b & a & e & a^2 \\ a^2 b & a^2 b & a b & b & a^2 & a & e \\ \end{array}$