Cardinal invariants related to permutation groups

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; ym ( N ) ) ≔ the cofinality of the permutation group on the set of natural numbers. w that it is consistent with ZFC that a p = a g < c ( Sym ( N ) ) = ℵ 2 ; in fact we show that in the Miller model a p = a g = ℵ 1 < ℵ 2 = c ( Sym ( N ) ) .