Symmetry Group of Line Segment is Group

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The symmetry group of the line segment is a group.


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

Let $AB$ be a line segment.


The symmetry mappings of $AB$ are:

The identity mapping $e$
The rotation $r$ of $180 \degrees$ about the midpoint of $AB$.

This group is known as the symmetry group of the line segment.


Let us refer to this group as $D_1$.

Taking the group axioms in turn:

Group Axiom $\text G 0$: Closure

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


Group Axiom $\text G 1$: Associativity

Composition of Mappings is Associative.


Group Axiom $\text G 2$: Existence of Identity Element

The identity is $e$.


Group Axiom $\text G 3$: Existence of Inverse Element

Each element is seen to be self-inverse:

$r^{-1} = r$


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