Cardinal invariants and convergence properties of locally minimal groups

Abstract

Let G be a locally essential subgroup of a locally compact abelian group K. Then:(i)t(G)=χ(G)=χ(K), where t(G) and χ(G) are the tightness and the character of G, respectively;(ii)if G is radial, then K must be metrizable;(iii)if G is non-discrete, then G contains a super-sequence S converging to 0 such that |S|=χ(G)=χ(K); in particular, G has non-trivial convergent sequences.Items (i)–(iii) hold when G is a dense locally minimal subgroup of K. It follows 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 nilpotent group G of nilpotency class 2 and character κ such that the connected component of G is an open normal ω-bounded subgroup of G (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.