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

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

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

  3. 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}
}