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:

Proof
Let us refer to this group as $D_3$.

Taking the group axioms in turn:

$\text G 0$: Closure
From the Cayley table it is seen directly that $D_3$ is closed.

$\text G 1$: Associativity
Composition of Mappings is Associative.

$\text G 2$: Identity
The identity is $e = (A) (B) (C)$.

$\text G 3$: 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.

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