I'm very new to working with permutations and I have a question that I cannot really understand how to solve.
Suppose that you have the two permutations $\sigma=(17)(264)(35)$ and $\pi=(162)(3574)$ and both belong to the set $\mathrm S_7$. Then the task is to find all the positive integers $k$ such that $\pi^k$ and $\sigma$ are conjugated.
According to the definition of conjugacy between two permutations, both of the permutations must have the same cycle structure. But then I'm trapped. One of the matters I have with this question is that I really do not understand how the permutation $\pi^k$ changes its cycle cycle structure when $k$ goes to infinity. I believe that each possible cycle structure somehow gets repeated depending on the value of the positive integer $k$. But I don't know, I'm simply clueless.
Thank you very much in advance with helping me with this!