Definable Sets of Generators in Maximal Cofinitary Groups
by
Yi Zhang
Department of Mathematics, Sun Yat-Sen University, P.R.China; and Department of Mathematics, University of Michigan, Ann Arbor, MI, USA
Coauthors: Su Gao (Deaprtment of Mathematics, University of North Texax, Denton, TX, USA)

g in Sym(N) is cofinitary iff g has finitely many fixed points. A subgroup G in Sym(N) is cofinitary iff for all g in G but identity, g is a cofinitary permutation. It is easily seen that Zorn's Lemma implies the existence of maximal cofinitary permutation groups. However, various questions can be asked about the very existence of maximal cofinitary group. For example, instead of saying such groups existence is the consequence of Zorn's Lemma, does there exist any concrete maximal cofinitary group? Moreover, can we construct a maximal cofinitary group without using Axiom of Choice, e.g. a Borel one? ... etc.

In my talk, I will present the first attempt towards solving such problems. I will out line the proof of the following results:

Theorem 1 (CH) There is a recursive algorithm to construct a maximal cofinitary group. 

Using the idea in the proof of Theorem 1, we can prove the following:

Theorem 2 V=L implies that there is a \Pi₁¹ set X in Sym(N) such that <X >is a maximal cofinitary group. 

We will also ask several open questions arround this area.

Date received: April 22, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cajy-30.