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:


Let $AB$ be a line segment.

The symmetries 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.


Proof

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.

$\Box$


Group Axiom $\text G 1$: Associativity

Composition of Mappings is Associative.

$\Box$


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

The identity is $e$.

$\Box$


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

Each element is seen to be self-inverse:

$r^{-1} = r$

$\Box$


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

$\blacksquare$


Sources

  • 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26 \eta$
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