Composition of Symmetries is Associative

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 $2.5$