关于$κ$ -齐性但非$κ$ -传递置换群
On $κ$-homogeneous, but not $κ$-transitive permutation groups
摘要：
集合$A$上的置换群$G$是$κ$ -齐性的当且仅当对所有具有$|A\setminus X|=|A\setminus Y|=|A|$的$X,Y\in [A]^κ$，存在具有$g[X]=Y$的$g\in G$. $G$是$κ$ -传递的当且仅当对任意具有$dom(f)\cup ran(f)\in [A]^{\le κ}$和$|A\setminus dom(f)|=|A\setminus ran(f)|=|A|$的内射函数$f$，存在具有$f\subset g$的$g\in G$.对P.M.的一个问题作了部分回答。Neumann证明了在基数$λ$上存在一个$ω$ -齐次但不是$ω$ -传递的置换群，条件是（i）$λ<ω_ω$，或（ii）$2^ω<λ$，且$μ^ω=μ^+$和$\Box_μ$对每个具有$ω=cf(μ)<{μ}$的$μ\leλ$成立，或（iii）我们的模型是通过将$ω_1$许多Cohen类属实加到某个基模型上而得到的.对于$κ>ω$给出了构造大的$κ$ -齐性但非$κ$ -传递置换群的方法.利用这一方法，我们证明了对于每个无穷基数$κ$和自然数$n\ge 1$，在$κ^{+n}$上存在$κ^+$ -齐次但不存在$κ^+$ -传递的置换群.
英文摘要：
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 $V=L$.
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