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 $V=L$.
Categories: math.LO