Ordered Groups and Infinite Permutation Groups pp 195-219 | Cite as

# The Separation Theorem for Group Actions

Chapter

## Abstract

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.

## Keywords

Permutation Group Finite Subset Separation Theorem Transitive Group Combinatorial Design
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