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Preprint Number 1740

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1740. Saharon Shelah and Lajos Soukup
On κ-homogeneous, but not κ-transitive permutation groups

Submission date: 4 March 2020


A permutation group G on a set A is κ-homogeneous iff for all X,Y in [A]^κ with |A ∖ X|=|A ∖ Y|=|A| there is a g in G with g[X]=Y. G is κ-transitive iff for any injective function f with dom(f) ∪ ran(f) in [A]^{≤ κ} and |A ∖ dom(f)|=|A ∖ ran(f)|=|A| there is a g in G with f ⊂ g.
Giving a partial answer to a question of P. M. Neumann we show that there is an ω-homogeneous but not ω-transitive permutation group on a cardinal λ provided
(i) λ < ω_ω, or
(ii) 2^ω < λ, and μ^ω = μ^+ and □_μ hold for each μ ≤ λ with ω = cf(μ) < μ, or
(iii) our model was obtained by adding ω_1 many Cohen generic reals to some ground model.
For κ > ω we give a method to construct large {\kappa}-homogeneous, but not κ-transitive permutation groups. Using this method we show that there exists κ^+-homogeneous, but not κ^+-transitive permutation groups on κ^{+n} for each infinite cardinal κ and natural number n ≥ 1 provided V=L.

Mathematics Subject Classification: 03E35, 20B22

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Full text arXiv 2003.02023: pdf, ps.

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