Generically computable Abelian groups and isomorphisms

Approximate computability, in the form of generically computable sets introduced by Jockusch and Schupp, was motivated by asymptotic density problems studied by Gromov in combinatorial group theory. More recently, we have defined notions of generically computable structures, and studied in particular equivalence structures and injection structures. We also introduced a graded family of elementarity conditions for substructures, in which the dense substructures more strongly resemble the original structure by being ₙ elementary substructures for a given n. We now return to group theory, as we explore the generic computability of torsion Abelian groups. We show that any Abelian p-group has a generically computable copy, and that, for an important family of Abelian p-groups, a group G has a ₂-generically c.e. copy if and only if it has a computable copy. We also give a partial characterization of the ₁-generically c.e. Abelian p-groups, and give a non-trivial characterization of the generically computable torsion Abelian groups. Coarsely computable and ₙ-coarsely c.e. groups are also studied. It is well known that there are computable Abelian groups that are countable sums of cyclic groups of order p and p² that are not computably isomorphic. We present a notion of generically computable isomorphism and give conditions under which two such groups will be generically computably isomorphic.