  A compact group with {\bf all dense subspaces} separable is metrizable.
Inspired by this, we conjecture that a compact group with {\bf all dense
subgroups} separable is metrizable. Positive answers are given here for two
elementary cases, say, when the compact group is additionally assumed to be
abelian or connected. However, a locally compact abelian group, even when it
has an open compact subgroup, with all dense subgroups separable, may not be
metrizable. At the end of this note, it is shown that a locally compact group
with {\bf all subgroups} separable is metrizable. Our arguments are formalized
in a more general form, namely, not restricted to the countable case.
