Composition of Symmetries is Symmetry

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 $2.5$