How to find a high power of a given permutation?

Question

Given the permutation $\pi = (4 2 5 3 1)$, I want to find $\pi^{25}$, specifically the value $\pi ^{25}(3)$.

How to approach this problem, without multiplying $\pi$ by itself more than twenty times?

Answer 1

You don't have to: a cycle of length $n$ has order $n$, hence $\pi^5=\operatorname{id}$, and a fortiori $\pi^{25}=\operatorname{id}$, so $\pi^{25}(3)=3$.