Commutativity properties in groups of permutations

In the group Sym(S) of permutations on a nonempty set S, fixed points and transient points are defined. Preliminary results on fixed and transient points are developed. Disjoint pairs of permutations and disjoint collections of permutations are then defined in terms of transient points. Several commutativity results for disjoint permutations are established. Semidisjoint pairs of permutations and semidisjoint collections of permutations are also defined in terms of transient points. It is shown that the collection of disjoint pairs of permutations in Sym(S) is (properly) contained in the collection of semidisjoint pairs of permutations in Sym(S). Two main commutativity results for semidisjoint permutations are established. A counterexample is provided to verify that these two results cannot be combined to produce a more general commutativity result for semidisjoint permutations which is established for disjoint permutations.