We study \(\kappa \)-maximal cofinitary groups for \(\kappa \) regular uncountable, \(\kappa = \kappa ^{<\kappa }\). Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell’s theorem, we show that:

1. (1)

   Any \(\kappa \)-maximal cofinitary group has \({<}\kappa \) many orbits under the natural group action of \(S(\kappa )\) on \(\kappa \).

2. (2)

   If \(\mathfrak {p}(\kappa ) = 2^\kappa \) then any partition of \(\kappa \) into less than \(\kappa \) many sets can be realized as the orbits of a \(\kappa \)-maximal cofinitary group.

3. (3)

   For any regular \(\lambda > \kappa \) it is consistent that there is a \(\kappa \)-maximal cofinitary group which is universal for groups of size \({<}2^\kappa = \lambda \). If we only require the group to be universal for groups of size \(\kappa \) then this follows from \(\mathfrak {p}(\kappa ) = 2^\kappa \).

我们研究\(\kappa \) -正则不可数的\(\kappa \)最大共有限群，\(\kappa = \kappa ^{<\kappa }\)。回顾卡斯特曼斯的早期工作并基于最近获得的贝尔定理的更高模拟，我们证明：

1. (1)

   任何\(\kappa \)最大共陪群在 \(\kappa \)上\(S(\kappa )\)的自然群作用下都有\({<} \kappa \ ) 个轨道。

2. (2)

   如果\(\mathfrak {p}(\kappa ) = 2^\kappa \)则将\(\kappa \)划分为少于\(\kappa \)个集合可以实现为\( \kappa \) -最大共尾群。

3. (3)

   对于任何正则\(\lambda > \kappa \) ，一致的是存在一个\(\kappa \)最大共穷群，该群对于大小为\({<}2^\kappa = \lambda \) 的群是通用的。如果我们只要求该群对于大小为\(\kappa \)的群是通用的，那么这可以从\(\mathfrak {p}(\kappa ) = 2^\kappa \)得出。