# SSGP topologies on abelian groups of positive finite divisible rank

## TL;DR

This paper characterizes abelian groups with positive finite divisible rank that admit an SSGP topology, completing the classification for all abelian groups based on their divisible rank.

## Contribution

It provides a positive answer to whether the necessary condition for positive finite divisible rank groups is also sufficient for admitting an SSGP topology.

## Key findings

- Complete characterization of abelian groups with positive finite divisible rank admitting SSGP topology.
- Confirmed the necessary condition is also sufficient for these groups.
- Extended the classification to all abelian groups based on 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.