Problem
ALG-B3-M08-P005 Cycles of Length Three
#5
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A permutation \(f\) of a set with \(10\) elements satisfies \(f^3(x)=x\) for all \(x\). Prove that it has a fixed point.
Possible cycle lengths divide \(3\).
The cycles of the permutation have lengths \(1\) or \(3\). If there are no fixed points, all cycles have length \(3\), so the number of elements is divisible by \(3\). But \(10\) is not divisible by \(3\). Hence there is a cycle of length \(1\), i.e. a fixed point.
Basic finite-cycle task.