Symmetry Group of Equilateral Triangle is Group

Theorem
The symmetry group of the equilateral triangle is a group.

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.

G1: Associativity
Composition of Mappings is Associative.

G2: Identity
The identity is $e = (A) (B) (C)$.

G3: Inverses
Each element can be seen to have an inverses.

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