Symmetry Group of Square is Group

Theorem
The symmetry group of the square is a non-abelian group.

Definition
Recall the definition of the symmetry group of the square:

Proof
Let us refer to this group as $D_4$.

Taking the group axioms in turn:

G0: Closure
From the Cayley table it is seen directly that $D_4$ is closed.

G1: Associativity
Composition of Mappings is Associative.

G2: Identity
The identity is $e$ as defined.

G3: Inverses
Each element can be seen to have an inverse:
 * $r^{-1} = r^3$ and so $\left({r^3}\right)^{-1} = r$
 * $r^2$, $t_{AC}$, $t_{BD}$, $t_x$ and $t_y$ are all self-inverse.

Thus $D_4$ is seen to be a group.

Note that from the Cayley table it can be observed directly that:
 * $r \circ t_x = t_{BD}$
 * $t_x \circ r = t_{AC}$

thus illustrating by counterexample that $D_4$ is not abelian.