Symmetry Group of Equilateral Triangle is Group

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.

Group Presentation
Its group presentation is:

Proof
Let us refer to this group as $D_3$.

Taking the group axioms in turn:

G0: Closure
From the Cayley table it is seen directly that $D_3$ is closed.

G1: Associativity
Composition of Mappings is Associative.

G2: Identity
The identity is $e = (A) (B) (C)$.

G3: Inverses
Each element can be seen to have an inverses.

No more need be done. $D_3$ is seen to be a group.