Is there a formula or algorithm to find directly the isotropy groups of a permutation group acting on itself by conjugacy ??

Question

I am not sure if I read all of my algebra course, but do we know if we can find all the permutations p that verify (p) ro (p^-1) = ro for a given permtation ro ? It is clear that if ro and p are disjoint then this hold, what about the other p ? Is it something to check one by one by hand ?

Answer 1

If $\tau$ and $\sigma$ are non-trivial permutations, that is elements of $S_n$, then one of the following is true:

1. $\tau\sigma\tau^{-1}=\sigma,$ and $\tau$ and $\sigma$ are disjoint.

2. $\tau\sigma\tau^{-1}=\sigma_1$ where the only thing we can say without knowing $S_n$ specifically is that $\sigma_1$ and $\sigma$ have the same cycle type.

see: http://math.mit.edu/~mckernan/Teaching/12-13/Spring/18.703/l_6.pdf