$κ$ -极大余有限群的结构
The Structure of $κ$-Maximal Cofinitary Groups
摘要：
研究了$κ$正则不可数的极大余有限群$κ= κ^{<κ}$ .回顾早期的工作卡斯特曼斯和建筑上最近获得的更高的模拟贝尔定理，我们表明：1。在$S(κ)$对$κ$的自然群作用下，任何$κ$ -极大余有限群都有${<}κ$许多轨道. 2.如果$\mathfrak{p}(κ) = 2^κ$，则任何将$κ$划分为小于$κ$的集合都可以实现为$κ$ -极大余有限群的轨道. 3.对于任何正则$λ> κ$，一致的是存在一个$κ$ -极大共穷群，它对于大小为${<}2^κ= λ$的群是通用的。如果我们只要求组对于大小为$κ$的组是通用的，那么这可以从$\mathfrak{p}(κ) = 2^κ$得出。
英文摘要：
We study $κ$-maximal cofinitary groups for $κ$ regular uncountable, $κ= κ^{<κ}$. Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell's theorem, we show that: 1. Any $κ$-maximal cofinitary group has ${<}κ$ many orbits under the natural group action of $S(κ)$ on $κ$. 2. If $\mathfrak{p}(κ) = 2^κ$ then any partition of $κ$ into less than $κ$ many sets can be realized as the orbits of a $κ$-maximal cofinitary group. 3. For any regular $λ> κ$ it is consistent that there is a $κ$-maximal cofinitary group which is universal for groups of size ${<}2^κ= λ$. If we only require the group to be universal for groups of size $κ$ then this follows from $\mathfrak{p}(κ) = 2^κ$.
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