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

Abstract

A permutation group $G$ on a set $A$ is $κ$-homogeneous iff for all $X,Y\in [A]^κ$ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$. $G$ is $κ$-transitive iff for any injective function $f$ with $dom(f)\cup ran(f)\in [A]^{\le κ}$ 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 $ω$-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 $μ\leλ$ with $ω=cf(μ)<{μ}$, or (iii) our model was obtained by adding $ω_1$ 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 $n\ge 1$ provided $V=L$.