Symmetry Group of Line Segment is Group
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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 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.
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: $\S 26 \eta$