Problem
ALG-B3-M08-P012 Without Two-Cycles
#12
★★★★☆ Level 4 of 5
A permutation of a set with \(6\) elements satisfies \(f^4(x)=x\). Prove that if it has no fixed points and no cycles of length \(2\), then this is impossible.
Cycle lengths divide \(4\).
Cycle lengths can be \(1,2,4\). If there are no cycles of length \(1\) or \(2\), then all cycles have length \(4\). Hence the number of elements must be divisible by \(4\), but \(6\) is not divisible by \(4\). Contradiction.
Slightly harder than the \(f^3\) cycle task.