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.
KeywordsPermutation Group Finite Subset Separation Theorem Transitive Group Combinatorial Design
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