The Separation Theorem for Group Actions
Let G be a group of permutations of a set Ω, and let Γ and Δ be (not necessarily distinct) subsets of Ω. Under what conditions on G, or on Γ and Δ is it possible to separate Γ from Δ by an element of G, that is to have Γ g ⋂Δ = \(\not 0\) for some g ∈ G? In 1976 Peter M. Neumann [25, Lemma 2.3] proved that every finite subset can be separated from itself in this way provided that all the G-orbits in Ω are infinite.
Unable to display preview. Download preview PDF.
- L. Brailovsky, D. V. Pasechnik and C. E. Praeger, Classification of 2-quasi- mvariant sets, Ars Combin., to appear.Google Scholar
- L. Brailovsky, D. V. Pasechnik and C. E. Praeger, Subsets close to invariant subsets under group actions, Proc. Amer. Math. Soc., to appear.Google Scholar
- J. R. Cho, P. S. Kim and C. E. Praeger, The maximum number of orbits of a permutation group with bounded movement, (in preparation).Google Scholar
- P. Dembowski, Finite Geometries, Springer-Verlag, New York, (1967).Google Scholar
- A. Gardiner and C. E. Praeger, Bounds on orbit sizes for permutation groups admitting subsets with restricted movement, (preprint, 1995 ).Google Scholar
- G. B. Khosrovshahi and C. E. Praeger, On the intersection problem for combinatorial designs, (in preparation).Google Scholar
- A. Mann and C. E. Praeger, Transitive permutation groups of minimal movement, J. Algebra, to appear.Google Scholar
- P. M. Neumann, A combinatorial problem in group theory, (unpublished manuscript, 1989 ).Google Scholar
- P. M. Neumann and C. E. Praeger, An inequality for tactical configurations, Bull. London Math. Soc. (to appear).Google Scholar
- C. E. Praeger, Restricted movement for intransitive group actions, Proceedings of Groups Korea 1994, (to appear).Google Scholar
- C. E. Praeger, Movement and separation of subsets of points under group actions, (preprint, 1995 ).Google Scholar