Abstract

We study \(\kappa \) -maximal cofinitary groups for \(\kappa \) regular uncountable, \(\kappa = \kappa ^{. Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell’s theorem, we show that: Any \(\kappa \) -maximal cofinitary group has \({ many orbits under the natural group action of \(S(\kappa )\) on \(\kappa \). 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. For any regular \(\lambda > \kappa \) it is consistent that there is a \(\kappa \) -maximal cofinitary group which is universal for groups of size \({. If we only require the group to be universal for groups of size \(\kappa \) then this follows from \(\mathfrak {p}(\kappa ) = 2^\kappa \).