ON κ-HOMOGENEOUS, BUT NOT κ-TRANSITIVE PERMUTATION GROUPS

A permutation group G on a set A is κ-homogeneous iff for all X, Y (equation presented) ∈ [A]κ with |A \ X | = |A \ Y | = |A| there is a g ∈ G with g[X ] = Y. G is κ-transitive iff for any injective function f with dom(f) ∪ ran(f) ∈ [A^]≤κ and |A \ dom(f)| = |A \ ran(f)| = |A| there is a g ∈ G with f ⊂ g. Giving a partial answer to a question of P. M. Neumann [6] we show that there is an ω-homogeneous but not ω-transitive permutation group on a cardinal λ provided (equation presented) (i) λ < ω_ω, or (ii) 2^ω < λ, and μ^ω = μ⁺ and 2μ hold for each μ ≤ λ with ω = cf (μ) < μ, or (iii) our model was obtained by adding (2^ω )⁺ many Cohen generic reals to some ground model. For κ > ω we give a method to construct large κ-homogeneous, but not κ-transitive permutation groups. Using this method we show that there exist κ⁺-homogeneous, but not κ⁺ -transitive permutation groups on κ⁺ⁿ for each infinite cardinal κ and natural number n ≥ 1 provided V = L.