  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.
