# Sh:1193

- Shelah, S., & Soukup, L. (2023).
*On \kappa-homogeneous, but not \kappa-transitive permutation groups*. J. Symb. Log.,**88**(1), 363–380. arXiv: 2003.02023 DOI: 10.1017/jsl.2021.63 MR: 4550395 -
Abstract:

A permutation group G on a set A is*{\kappa}-homogeneous*iff for all X,Y\in\bigl[ {A} \bigr]^ {\kappa} with |A\setminus X|=|A\setminus Y|=|A| there is a g\in G with g[X]=Y. G is*{\kappa}-transitive*iff for any injective function f with dom(f) \cup ran(f)\in \bigl[ {A} \bigr]^ {\le \kappa} and |A\setminus dom(f)|=|A\setminus ran(f)|=|A| there is a g\in G with f\subseteq g.Giving a partial answer to a question of P. M. Neumann we show that there is an {\omega}-homogeneous but not {\omega}-transitive permutation group on a cardinal {\lambda} provided

{\lambda}<{\omega}_{\omega}, or

2^{\omega}<{\lambda}, and {\mu}^{\omega}={\mu}^+ and \Box_{\mu} hold for each {\mu}\le{\lambda} with {\omega}=cf ({\mu})<{{\mu}}, or

our model was obtained by adding {\omega}_1 many Cohen generic reals to some ground model.

For {\kappa}>{\omega} we give a method to construct large {\kappa}-homogeneous, but not {\kappa}-transitive permutation groups. Using this method we show that there exists {\kappa}^+-homogeneous, but not {\kappa}^+-transitive permutation groups on {\kappa}^{+n} for each infinite cardinal {\kappa} and natural number n\ge 1 provided V=L.

- Version 2021-07-07 (19p)

@article{Sh:1193, author = {Shelah, Saharon and Soukup, Lajos}, title = {{On {$\kappa $}-homogeneous, but not {$\kappa $}-transitive permutation groups}}, journal = {J. Symb. Log.}, fjournal = {The Journal of Symbolic Logic}, volume = {88}, number = {1}, year = {2023}, pages = {363--380}, issn = {0022-4812}, mrnumber = {4550395}, mrclass = {03E35 (20B22)}, doi = {10.1017/jsl.2021.63}, note = {\href{https://arxiv.org/abs/2003.02023}{arXiv: 2003.02023}}, arxiv_number = {2003.02023} }