Problem 26 of L. Fuchs

Abstract:

This solves the following problem: Which Abelian groups are the inverse limits of Abelian groups, each of which is a finite direct sum of quasi-cyclic and bounded Abelian groups? (Here quasi-cyclic means isomorphic to some $Z({p^\infty })$.) A necessary and sufficient condition for an Abelian group to be such is that it takes the form ${A_r} \oplus \Pi _p\operatorname {Hom}_z({A_p},Z({p^\infty }))$ where ${A_r}$ is complete and reduced, the ${A_p}$ are torsion-free and the direct product is taken over the set of prime numbers.