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f(f(k)) = g(g(k)) = k , g(f(k)) = k +1, prove or disprove

Source: Polish second round 1999 p1

January 19, 2020
functioncomposite functionFunctional Equationsfunctional equationalgebra

Problem Statement

Let An={1,2,...,n}A_n = \{1,2,...,n\}. Prove or disprove: For all integers n≥2n \ge 2 there exist functions f,g:An→Anf,g : A_n \to A_n which satisfy f(f(k))=g(g(k))=kf(f(k)) = g(g(k)) = k for 1≤k≤n1 \le k \le n, and g(f(k))=k+1g(f(k)) = k +1 for 1≤k≤n−11 \le k \le n -1.
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