Composition of Symmetries is Associative
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Theorem
Let $P$ be a geometric figure.
Let $S_P$ be the set of all symmetries of $P$.
Let $\circ$ denote composition of mappings.
Let $\phi, \psi, \chi$ be symmetries of $P$.
Then:
- $\paren {\phi \circ \psi} \circ \chi = \phi \circ \paren {\psi \circ \chi}$
That is, composition of symmetries is associative.
Proof
From Composition of Symmetries is Symmetry:
- $\paren {\phi \circ \psi} \circ \chi$ is a symmetry
and:
- $\phi \circ \paren {\psi \circ \chi}$ is a symmetry.
It follows from Composition of Mappings is Associative that:
- $\paren {\phi \circ \psi} \circ \chi = \phi \circ \paren {\psi \circ \chi}$
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.5$