If is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable , no member of which is covered by finitely many functions from , there is such that for all 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.
A cofinitary group is a subgroup of Sym(ℕ) where all nonidentity elements have finitely many fixed points. A maximal cofinitary group is a cofinitary group, maximal with respect to inclusion. We show that a maximal cofinitary group cannot have infinitely many orbits. We also show, using Martin's Axiom, that no further restrictions on the number of orbits can be obtained. We show that Martin's Axiom implies there exist locally finite maximal cofinitary groups. Finally we show that there exists a uniformly computable sequence of permutations generating a cofinitary group whose isomorphism type is not computable.
Abstract. A cofinitary group is a subgroup of the infinite symmetric group in which each element of the subgroup has at most finitely many fixed points. A maximal cofinitary group is a cofinitary group that is maximal with respect to inclusion. We investigate the possible complexities of maximal cofinitary groups. In particular we show that (1) under the axiom of constructibility there exists a coanalytic maximal cofinitary group and (2) there does not exist an eventually bounded maximal cofinitary group. We also suggest some further directions for investigation.
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