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.