Functorial topologies and finite-index subgroups of abelian groups

TL;DR

This paper explores the relationships between different functorial topologies on abelian groups, especially focusing on the profinite topology, and characterizes groups based on their subgroup structures and topological properties.

Contribution

It characterizes the profinite topology as the infimum of the Bohr and natural topologies and analyzes the structure of finite-index subgroups in abelian groups.

Findings

Profinite topology is the infimum of Bohr and natural topologies.

Characterization of abelian groups where certain subgroup posets are cofinal.

Introduction of the equalizer E(T,S) to classify groups via functorial topologies.

Abstract

In the general context of functorial topologies, we prove that in the lattice of all group topologies on an abelian group, the infimum between the Bohr topology and the natural topology is the profinite topology. The profinite topology and its connection to other functorial topologies is the main objective of the paper. We are particularly interested in the poset C(G) of all finite-index subgroups of an abelian group G, since it is a local base for the profinite topology of G. We describe various features of the poset C(G) (its cardinality, its cofinality, etc.) and we characterize the abelian groups G for which C(G)\{G} is cofinal in the poset of all subgroups of G ordered by inclusion. Finally, for pairs of functorial topologies T, S we define the equalizer E(T,S), which permits to describe relevant classes of abelian groups in terms of functorial topologies.