Adjoining Cofinitary Permutations

Yi Zhang

We consider two kinds of closely related mathematical structures, cofinitary permutation groups and almost disjoint permutation families in Sym(N).

We say that two permutations f, g ∈ Sym(N) are almost disjoint (a.d.) iff |f ∩ g| < ω. And an a.d. permutation family A ⊆ Sym(N) is a subset of Sym(N) such that for any f, g ∈ A, f and g are a.d.

A permutation g is cofinitary iff g has only finitely many fixed points. A group G ≤ Sym(N) is cofinitary iff every non-identity elements is cofinitary. It is easily seen that G ≤ Sym(N) is cofinitary iff G is both an almost disjoint set of permutations and a group. For a discussion of different aspects of cofinitary groups, the reader can consult the well-written survey paper [C].

In several papers, some common features had been proved about these two structures, eg, in [BSZ], we proved that any maximal cofinitary group or any maximal a.d. family in Sym(N) cannot be meager in N^{N}.

However, it is also interesting to know the difference between the cardinalities of these two structures. In my talk, I will explain a forcing result in [Z] which is my first attempt to work in this direction. I will also ask several open questions concerning the subject.

[BSZ] J.Brendle, O.Spinas and Y.Zhang: Uniformity of meager ideals and maximal cofinitary groups, Journal of Algebra, to appear

[C] P. Cameron: Cofinitary permutation groups Bulletin of the London Mathematical Society, 28, March, 1996, pp. 113–140

[Z] Y. Zhang: Adjoining Cofinitary Permutations Journal of Symbolic Logic, to appear

Address: Department of Mathematics and Computer Science, Istanbul Bilgi University, Istanbul, 80310, Turkey

E-mail: zhangbilgi.edu.tr