We study\documentclass[12pt]{minimal}
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				\begin{document}$$\kappa $$\end{document}κ-maximal cofinitary groups for\documentclass[12pt]{minimal}
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				\begin{document}$$\kappa $$\end{document}κregular uncountable,\documentclass[12pt]{minimal}
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				\begin{document}$$\kappa = \kappa ^{<\kappa }$$\end{document}κ=κ<κ. Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell’s theorem, we show that:Any\documentclass[12pt]{minimal}
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				\begin{document}$$\kappa $$\end{document}κ-maximal cofinitary group has\documentclass[12pt]{minimal}
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				\begin{document}$${<}\kappa $$\end{document}<κmany orbits under the natural group action of\documentclass[12pt]{minimal}
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				\begin{document}$$S(\kappa )$$\end{document}S(κ)on\documentclass[12pt]{minimal}
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				\begin{document}$$\kappa $$\end{document}κ.If\documentclass[12pt]{minimal}
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				\begin{document}$$\mathfrak {p}(\kappa ) = 2^\kappa $$\end{document}p(κ)=2κthen any partition of\documentclass[12pt]{minimal}
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				\begin{document}$$\kappa $$\end{document}κinto less than\documentclass[12pt]{minimal}
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				\begin{document}$$\kappa $$\end{document}κmany sets can be realized as the orbits of a\documentclass[12pt]{minimal}
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				\begin{document}$$\kappa $$\end{document}κ-maximal cofinitary group.For any regular\documentclass[12pt]{minimal}
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				\begin{document}$$\lambda > \kappa $$\end{document}λ>κit is consistent that there is a\documentclass[12pt]{minimal}
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				\begin{document}$$\kappa $$\end{document}κ-maximal cofinitary group which is universal for groups of size\documentclass[12pt]{minimal}
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				\begin{document}$${<}2^\kappa = \lambda $$\end{document}<2κ=λ. If we only require the group to be universal for groups of size\documentclass[12pt]{minimal}
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				\begin{document}$$\kappa $$\end{document}κthen this follows from\documentclass[12pt]{minimal}
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				\begin{document}$$\mathfrak {p}(\kappa ) = 2^\kappa $$\end{document}p(κ)=2κ.