For each n ≥ 3 let P_n be a regular polygon of n sides (for n = 3, P_n is an equilateral triangle; for n = 4, a square). A Symmetry of P_n is a bijection P_n -P_n that preserves distances and maps adjacent vertices onto adjacent vertices.

(a) The set D_n ^* of all symmetries of P_n is a group under the binary operation of composition of functions.

(b) Every ∫ ϵ D_n^* is completely determined by its action on the vertices of P_n. Number the vertices consecutively 1, 2, ..• , n; then each ∫ ϵ D_n* determines a unique permutation σ_∫ of { 1,2, ... , n}. The assignment ∫|→ σ_∫ defines a monomorphism of groups ϕ: D_n* → s_n.
(c) D_n* is generated by ∫ and g, where ∫ is a rotation of 2π/n degrees about the center of P_n and g is a reflection about the "diameter" through the center and vertex 1
(d)

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