# Symmetry Group of Equilateral Triangle is Group

## Theorem

### Definition

Recall the definition of the symmetry group of the equilateral triangle:

Let $\triangle ABC$ be an equilateral triangle.

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

 $\ds e$ $:$ $\ds \tuple A \tuple B \tuple C$ 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.

## Proof

Let us refer to this group as $D_3$.

Taking the group axioms in turn:

### Group Axiom $\text G 0$: Closure

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

$\Box$

### Group Axiom $\text G 1$: Associativity

$\Box$

### Group Axiom $\text G 2$: Existence of Identity Element

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

$\Box$

### Group Axiom $\text G 3$: Existence of Inverse Element

Each element can be seen to have an inverse:

$p^{-1} = q$ and so $q^{-1} = p$
$r$, $s$ and $t$ are all self-inverse.

$\Box$

Thus $D_3$ is seen to be a group.

$\blacksquare$