Symmetry Group of Line Segment is Group

Theorem
The symmetry group of the line segment is a group.

Definition
Recall the definition of the symmetry group of the line segment:

Proof
Let us refer to this group as $D_1$.

Taking the group axioms in turn:

From the Cayley table it is seen directly that $D_1$ is closed.

Composition of Mappings is Associative.

The identity is $e$.

Each element is seen to be self-inverse:
 * $r^{-1} = r$

No more need be done. $D_1$ is seen to be a group.