Definition:Symmetry Group

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Let $P$ be a geometric figure.

Let $S_P$ be the set of all symmetries of $P$.

Let $\struct {S_P, \circ}$ be the algebraic structure such that $\circ$ denotes the composition of mappings.

Then $\struct {S_P, \circ}$ is called the symmetry group of $P$.

Also see

Linguistic Note

The word symmetry comes from Greek συμμετρεῖν (symmetría) meaning measure together.