  If G is a locally essential subgroup of a compact abelian group K, then: (i)
t(G)=w(G)=w(K), where t(G) is the tightness of G; (ii) if G is radial, then K
must be metrizable; (iii) G contains a super-sequence S converging to 0 such
that |S|=w(G)=w(K).
  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 kappa, we construct a Frechet-Urysohn minimal
group G of character kappa such that the connected component of G is an open
normal omega-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.
