# THE NUMBER AND SIZE OF ORBITS OF SOME PERMUTATION GROUPS

@inproceedings{Bruyns1991THENA, title={THE NUMBER AND SIZE OF ORBITS OF SOME PERMUTATION GROUPS}, author={Peter V. Bruyns}, year={1991} }

Abstract It is shown that Aut ℚ, the group of homeomorphisms of the rational numbers with the usual topology, has 2 No orbits on the power set P(ℚ). We call S ⊆ ℚ a moiety if S and its complement in ℚ are infinite. It is shown that the orbit of any moiety S under Aut ℚ has cardinality 2No while the orbit of S under Aut(ℚ, ≤), the group of order preserving automorphisms of ℚ, has cardinality No if and only if S is a finite union of disjoint rational intervals with rational endpoints.