D-independent topological groups

Abstract

A topological group G with |G|>1 is called d-independent if for every subgroup S of G with |S|<2ω, one can find a countable dense subgroup H of G such that S∩H={e}. Therefore, d-independent groups are separable and have cardinality at least 2ω. Our main result is a purely algebraic characterization of d-independence in the class of compact metrizable abelian groups. We prove that a compact metrizable abelian group G with |G|>1 is d-independent if and only if for every integer m≥1, either |mG|=2ω or |mG|=1. This characterization implies that a compact metrizable abelian group is d-independent if and only if it is maximally fragmentable [Comfort and Dikranjan (2014) [4]] iff G an M-group as defined by Dikranjan and Shakhmatov (2016) in [7].Also we present a characterization of separable metrizable d-independent abelian groups and show that products of separable topological groups can often be d-independent, even if the factors fail to be d-independent.