Symmetry Group of Equilateral Triangle is Group

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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.

SymmetryGroupEqTriangle.png

We define in cycle notation the following symmetries 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

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