# Definition:Dihedral Group D3

(Redirected from Dihedral Group/Examples/D3)

Jump to navigation
Jump to search
## 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 symmetry mappings on $\triangle ABC$:

\(\displaystyle e\) | \(:\) | \(\displaystyle \tuple A \tuple B \tuple C\) | Identity mapping | ||||||||||

\(\displaystyle p\) | \(:\) | \(\displaystyle \tuple {ABC}\) | Rotation of $120 \degrees$ anticlockwise about center | ||||||||||

\(\displaystyle q\) | \(:\) | \(\displaystyle \tuple {ACB}\) | Rotation of $120 \degrees$ clockwise about center | ||||||||||

\(\displaystyle r\) | \(:\) | \(\displaystyle \tuple {BC}\) | Reflection in line $r$ | ||||||||||

\(\displaystyle s\) | \(:\) | \(\displaystyle \tuple {AC}\) | Reflection in line $s$ | ||||||||||

\(\displaystyle t\) | \(:\) | \(\displaystyle \tuple {AB}\) | 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

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