On $\kappa$-homogeneous, but not $\kappa$-transitive permutation groups

Abstract

A permutation group $G$ on a set $A$ is ${\kappa}$-homogeneous iff for all $X,Y\in [A]^{\kappa}$ 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 $dom(f)\cup ran(f)\in [A]^{\le {\kappa}}$ and $|A\setminus dom(f)|=|A\setminus ran(f)|=|A|$ there is a $g\in G$ with $f\subset g$. 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 (i) ${\lambda}<{\omega}_{\omega}$, or (ii) $2^{\omega}<{\lambda}$, and ${\mu}^{\omega}={\mu}^+$ and $\Box_{\mu}$ hold for each ${\mu}\le{\lambda}$ with ${\omega}=cf({\mu})<{{\mu}}$, or (iii) 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 $V=L$.