So we have two permutations, composed of $k$ and $s$ disjoint cycles with increasing lengths (number of elements in each cycle). We want to prove:
They have the SAME amount of cycles. The distance of each cycle is the same.
As long as $\alpha$ and $\beta$ are conjugate.
This is my interpretation of the problem. However to get started, I need to understand it better, especially the conjugate part and how it relates to my problem.
Any help appreciated, thanks!