Yi Zhang

We show that it is consistent with ZFC + ¬CH that there is a maximal cofinitary group (or, maximal almost disjoint group) G ≤ Sym(ω) such that G is a proper subset of an almost disjoint family A $\subseteq$ Sym(ω) and |G| < |A|. We also ask several questions in this area

We construct several forcing models in each of which there exists a maximal cofinitary group, i.e., a maximal almost disjoint group, G≤Sym, such that G is also a maximal almost disjoint family in Sym. We also ask several open questions in this area in the fourth section of this paper.

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

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

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

国家社会科学基金项目获“高等学校优秀青年教师教学科研奖励计划资助”(TRAPOYT)