Definition:Symmetry Group of Equilateral Triangle

Group Example
Let $\triangle ABC$ be an equilateral triangle.


 * SymmetryGroupEqTriangle.png

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

Note that $r, s, t$ can equally well be considered as a rotation of $180^\circ$ (in three 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.

Cycle Notation
It can also be presented in cycle notation as:

Group Presentation
Its group presentation is:

Also see

 * Symmetry Group of Equilateral Triangle is Group
 * Symmetry Group of Equilateral Triangle is Isomorphic to Symmetric Group on 3 Letters