We consider the possible cardinalities of the following three cardinal invariants which are related to the permutation group on the set of natural numbers: the least cardinal number of maximal cofinitary permutation groups; the least cardinal number of maximal almost disjoint permutation families; the cofinality of the permutation group on the set of natural numbers.We show that it is consistent with that ; in fact we show that in the Miller model.

If F ⊆ NN is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable H ⊆ NN. no member of which is covered by finitely many functions from F, there is f ∈ F such that for all h ∈ H there are infinitely many integers k such that f(k) = h(k). However if V = L then there exists a coanalytic family of pairwise eventually different functions satisfying this strong maximality (...) condition. (shrink)

In this paper we show that it is consistent with ZFC that the cardinality of every maximal cofinitary group of Sym(ω) is strictly greater than the cardinal numbers o and a.