Problem
ALG-B3-M11-P006 Fourteen Elements
#6
★★★★☆ Level 4 of 5
A permutation of a set with \(14\) elements satisfies \(f^3(x)=x\). Prove that it has a fixed point.
Cycle lengths divide \(3\).
Cycles have length \(1\) or \(3\). If there are no fixed points, all cycles have length \(3\), so \(14\) is divisible by \(3\), false. Therefore a fixed point exists.
Finite cycles.