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 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

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 $1.9$