This is a problem I'm trying to solve. Given a permutation $ \sigma $ on a finite set $ \mathcal{A} $ of order $ n $, show that there exists a positive integer $ 0< k \leq n $ such that
$$\bigl( \begin{smallmatrix} a_1&a_2&...&a_n \\ b_1&b_2&...&b_n \end{smallmatrix} \bigr)^k = \bigl( \begin{smallmatrix} a_1&a_2&...&a_n \\ a_1&a_2&...&a_n \end{smallmatrix} \bigr) $$