\begin{definition}[Definition:Disjoint Permutations]
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\pi, \rho \in S_n$ both be permutations on $S_n$.
Then $\pi$ and $\rho$ are '''disjoint''' {{iff}}:
:$(1): \quad i \notin \Fix \pi \implies i \in \Fix \rho$
:$(2): \quad i \notin \Fix \rho \implies i \in \Fix \pi$
That is, each element moved by $\pi$ is fixed by $\rho$ and (equivalently) each element moved by $\rho$ is fixed by $\pi$.
That is, {{iff}} their supports are disjoint sets.
We may say that:
: $\pi$ is '''disjoint from $\rho$'''
: $\rho$ is '''disjoint from $\pi$'''
: $\pi$ and $\rho$ are '''(mutually) disjoint'''.
Note of course that it is perfectly possible for $i \in \Fix \pi$ and also $i \in \Fix \rho$, that is, there may well be elements fixed by more than one of a pair of '''disjoint permutations'''.
\end{definition}