# Symmetry Group of Equilateral Triangle is Group

## Contents

## Theorem

The symmetry group of the equilateral triangle is a group.

### 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 $\triangle ABC$:

\(\displaystyle e\) | \(:\) | \(\displaystyle \tuple A \tuple B \tuple C\) | 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**.

## 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.

$\Box$

### G1: Associativity

Composition of Mappings is Associative.

$\Box$

### G2: Identity

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

$\Box$

### G3: Inverses

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.

$\Box$

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

$\blacksquare$

## Sources

- 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.3$: Example $9$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26 \eta$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.9$