Category:Definitions/Disjoint Permutations

This category contains definitions related to Disjoint Permutations.
Related results can be found in Category: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 if and only if:

$(1): \quad i \notin \Fix \pi \implies i \in \Fix \rho$

$(2): \quad i \notin \Fix \rho \implies i \in \Fix \pi$

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

That is, if and only if their supports are disjoint sets.