Question 106
Write $S_{n}$ for the set $\{1,2, \ldots, n\}$. Determine all positive integers $n$ for which there exist functions $f: S_{n} \rightarrow S_{n}$ and $g: S_{n} \rightarrow S_{n}$ such that for every $x$ exactly one of the equalities $f(g(x))=x$ and $g(f(x))=x$ is true.