Computable torsion abelian groups

Abstract

We prove that c.c. torsion abelian groups can be described by a Π04-predicate that describes the failure of a brute-force diagonalisation attempt on such groups. We show that there is no simpler description since their index set is Π04-complete. The results can be viewed as a solution to a 60 year-old problem of Mal'cev in the case of torsion abelian groups. We prove that a computable torsion abelian group has one or infinitely many computable copies, up to computable isomorphism. The result confirms a conjecture of Goncharov from the early 1980s for the case of torsion abelian groups.