Composition of Symmetries is Symmetry
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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 $\phi$ and $\psi$ be symmetries of $P$.
Then $\phi \circ \psi$ is also a symmetry of $P$.
Proof
By definition of composition of mappings:
- $\phi \circ \psi$ is a mapping.
We have by definition of symmetry that:
- $\map \phi P$ is congruent to $P$
and:
- $\map \psi {\map \phi P}$ is congruent to $\map \phi P$
Therefore:
- $\phi \circ \psi$ is congruent to $P$
Thus $\phi \circ \psi$ is a symmetry of $P$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.5$