This paper is a compilation of results originating in the author's master thesis. We give a useful characterization of the generalized bounding and dominating numbers, and. We show that when. And we prove a higher analogue of Bell's theorem stating that is equivalent to ‐centered).

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 \(\mathfrak {p}(\kappa ) = 2^\kappa \). (shrink)

Assuming that every set is constructible, we find a${\text{\Pi }}_1^1 $maximal cofinitary group of permutations of$\mathbb{N}$which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans’ result that there exists a${\text{\Pi }}_1^1 $maximal cofinitary group inL.