# Symmetry Group of Equilateral Triangle is Group

## 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 $\triangle ABC$:

\(\ds e\) | \(:\) | \(\ds \tuple A \tuple B \tuple C\) | 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

Composition of Mappings is Associative.

$\Box$

### 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.

$\blacksquare$

## 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 $1.9$