SSGP topologies on abelian groups of positive finite divisible rank

Abstract

Let G be an abelian group. For a subset A of G, Cyc(A) denotes the set of all elements x of G such that the cyclic subgroup generated by x is contained in A, and G is said to have the small subgroup generating property (abbreviated to SSGP) if the smallest subgroup of G generated by Cyc(U) is dense in G for every neighbourhood U of zero of G. SSGP groups form a proper subclass of the class of minimally almost periodic groups. Comfort and Gould asked for a characterization of abelian groups G which admit an SSGP group topology, and they solved this problem for bounded torsion groups (which have divisible rank zero). Dikranjan and the first author proved that an abelian group of infinite divisible rank admits an SSGP group topology. In the remaining case of positive finite divisible rank, the same authors found a necessary condition on G in order to admit an SSGP group topology and asked if this condition is also sufficient. We answer this question positively, thereby completing the characterization of abelian groups which admit an SSGP group topology.