  A Hausdorff topological group $(G,\tau)$ is called locally minimal if there
exists a neighborhood $U$ of 0 in $\tau$ such that $U$ fails to be a
neighborhood of zero in any Hausdorff group topology on $G$ which is strictly
coarser than $\tau.$ Examples of locally minimal groups are all subgroups of
Banach-Lie groups, all locally compact groups and all minimal groups.
  Motivated by the fact that locally compact NSS groups are Lie groups, we
study the connection between local minimality and the NSS property,
establishing that under certain conditions, locally minimal NSS groups are
metrizable.
  A symmetric subset of an abelian group containing zero is said to be a GTG
set if it generates a group topology in an analogous way as convex and
symmetric subsets are unit balls for pseudonorms on a vector space. We consider
topological groups which have a neighborhood basis at zero consisting of GTG
sets. Examples of these locally GTG groups are: locally pseudo--convex spaces,
groups uniformly free from small subgroups (UFSS groups) and locally compact
abelian groups. The precise relation between these classes of groups is
obtained: a topological abelian group is UFSS if and only if it is locally
minimal, locally GTG and NSS. We develop a universal construction of GTG sets
in arbitrary non-discrete metric abelian groups, that generates a strictly
finer non-discrete UFSS topology and we characterize the metrizable abelian
groups admitting a strictly finer non-discrete UFSS group topology. We also
prove that a bounded abelian group $G$ admits a non-discrete locally minimal
and locally GTG group topology iff $|G|\geq \cont$.
