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 transformations 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.

Its Cayley table can be written:


 * $\begin{array}{c|cccccc}

& e & p & q & r & s & t \\ \hline e & e & p & q & r & s & t \\ p & p & q & e & s & t & r \\ q & q & e & p & t & r & s \\ r & r & t & s & e & q & p \\ s & s & r & t & p & e & q \\ t & t & s & r & q & p & e \\ \end{array}$

... or directly in cycle notation:


 * $\begin{array}{c|cccccc}

& e & (ABC) & (ACB) & (BC) & (AC) & (AB) \\ \hline e & e & (ABC) & (ACB) & (BC) & (AC) & (AB) \\ (ABC) & (ABC) & (ACB) & e & (AC) & (AB) & (BC) \\ (ACB) & (ACB) & e & (ABC) & (AB) & (BC) & (AC) \\ (BC) & (BC) & (AB) & (AC) & e & (ACB) & (ABC) \\ (AC) & (AC) & (BC) & (AB) & (ABC) & e & (ACB) \\ (AB) & (AB) & (AC) & (BC) & (ACB) & (ABC) & e \\ \end{array}$

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.