Inspired by the fact that a compact topological group is
hereditarily normal if and only if it is metrizable, we prove that various
levels of compactness-like properties imposed on a topological group G
allow one to establish that G is hereditarily normal if and only if G is
metrizable (among these properties are locally compactness, local minimality
and \omega-boundedness). This extends recent results from [4] in the
case of countable compactness
