Definition:Symmetry Group

Definition

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

• Symmetry Group is Group

• Results about symmetry groups can be found here.

• Do not confuse with the symmetric group on $n$ elements.

Linguistic Note

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

Sources

• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 5$: Groups $\text{I}$
• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 34$. Examples of groups: $(5)$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(iv)}$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.9$
• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Where to begin...