Abstract.

The problem of enumerating the types of Boolean functions under the group of variable permutations and complementations was first stated by Jevons in the 1870s, but not solved in a satisfactory way until the work of Pólya in 1940. This paper explains the details of Pólya's solution, and also the history of the problem from the 1870s to the 1970s.

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.