Notation for "permutation moves an element"

Question

Let $P$ be a permutation on a set $S$ and let $s \in S$. What is the notation (if any) to indicate that $P$ moves $s$? What about $P$ fixes $s$?

Answer 1

There's not any widely used special notation for this, but you can just write $P(s)\neq s$ and $P(s)=s$, respectively.