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!
The hint in my comment seems to be not enough for you here are more:
First some basic facts about permutations.
FACT 1: Any permutation is a product of disjoint cycles which commute with each other.
FACT 2: Order of a permutation (least power making it identity) is the least common multiple of lengths of all the cycles appearing in the above product.
FACT 3: If $\sigma$ is a cycle of length $r$ then $\sigma ^k$ will be a be a permutation of order $r/\gcd(r,k)$. and product of equal cycles of that length.
Now your $\pi$ is a product of a 3-cycle with a 4-cycle. Hence by Fact 2 it has order 12. So you need to consider only 12 powers of $\pi$. That is $\pi^{12+m}=\pi^m$.
You should be able to now work this out now.