Number of times we have to compose a permutation in order to have exactly k fixed points

Question

Let $f = (1 4 6)(2 7 5 8 10)(3 9)$ in $S_{10}$. Find an integer $n$ such that $f^n$ has exactly $7$ fixed points.

I provided the exact numbers, but would welcome a more general solution.

Answer 1

You need to look at the cycle lengths and find a set of them that adds up to the number of fixed points required. Here you have cycles of $2,3,5$, so you need the $2$ and $5$ to be back at start and the $3$ not to be. The first time the $2$ and $5$ will be back at start together is at $\operatorname{lcm} (2,5)=10$. As $3$ does not divide into $10$, it will not be back at start, so $10$ works. Any multiple of $10$ that is not a multiple of $3$ will also work.