On κ-homogeneous, but not κ-transitive permutation groups

2020/03/04 by Saharon Shelah, Shelah, Saharon, Lajos Soukup +1
Mathematics · #03E35 #20B22 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Finite Group Theory Research #Logic (math.LO) #Mathematical and Theoretical Analysis #math.LO #msc:03E35 #msc:20B22

paper · pdf · doi:10.48550/arxiv.2003.02023

published as J. Symb. Log. 88 No. 1 (2023) 363--380 · 16 pages

arxiv created 2020/03/04 · openalex publication_date 2020/03/04 · arxiv updated 2020/03/05 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28

Abstract

A permutation group G on a set A is κ-homogeneous iff for all X,Y∈ [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 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 \Box_μ hold for each μ≤λ with ω=cf(μ)<μ, or (iii) our model was obtained by adding ω₁ 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 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.

Related

• A new notion of transitivity for groups and sets of permutations
• A criteria for a finite permutation group to be transitive
• The classification of (3/2)-transitive permutation groups and (1/2)-transitive linear groups
• A permutation group acting transitively on certain collections of models
• Separating subsets from their images