Definition:Disjoint Permutations

Definition

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:

$(1): \quad i \notin \operatorname{Fix} \left({\pi}\right) \implies i \in \operatorname{Fix} \left({\rho}\right)$

$(2): \quad i \notin \operatorname{Fix} \left({\rho}\right) \implies i \in \operatorname{Fix} \left({\pi}\right)$

That is, each number moved by $\pi$ is fixed by $\rho$ and (equivalently) each number moved by $\rho$ is fixed by $\pi$.

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 $i \in \operatorname{Fix} \left({\pi}\right)$ and also $i \in \operatorname{Fix} \left({\rho}\right)$, that is, there may well be elements fixed by more than one disjoint permutation.

Sources

• John F. Humphreys: A Course in Group Theory (1996): $\S 9$: Definition $9.4$