Abstract

We consider the possible cardinalities of the following three cardinal invariants which are related to the permutation group on the set of natural numbers: a g ≔ the least cardinal number of maximal cofinitary permutation groups; a p ≔ the least cardinal number of maximal almost disjoint permutation families; c ( Sym ( N ) ) ≔ the cofinality of the permutation group on the set of natural numbers. We show 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 ) ) .

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