Cardinal invariants and convergence properties of locally minimal groups

Abstract

If G is a locally essential subgroup of a compact abelian group K, then:Items (i)-(iii) hold when G is a dense locally minimal subgroup of K. We show that locally minimal locally precompact abelian groups of countable tightness are metrizable. In particular, a minimal abelian group of countable tightness is metrizable. This answers a question of O. Okunev posed in 2007.For every uncountable cardinal κ, we construct a Fréchet-Urysohn minimal group G of character κ such that the connected component of G is an open normal ω-bounded subgroup (thus, G is locally precompact). We also build a minimal nilpotent group of nilpotency class 2 without non-trivial convergent sequences having an open normal countably compact subgroup.All topological groups are assumed to be Hausdorff.