  Let $G$ be a countable group, $\operatorname{Sub}(G)$ the (compact, metric)
space of all subgroups of $G$ with the Chabauty topology and
$\operatorname{Is}(G) \subset \operatorname{Sub}(G)$ the collection of isolated
points. We denote by $X!$ the (Polish) group of all permutations of a countable
set $X$. Then the following properties are equivalent: (i)
$\operatorname{Is}(G)$ is dense in $\operatorname{Sub}(G)$, (ii) $G$ admits a
"generic permutation representation". Namely there exists some $\tau^* \in
\operatorname{Hom}(G,X!)$ such that the collection of permutation
representations $\{\phi \in \operatorname{Hom}(G,X!) \ | \ \phi {\text{is
permutation isomorphic to}} \tau^*\}$ is co-meager in
$\operatorname{Hom}(G,X!)$. We call groups satisfying these properties
solitary. Examples of solitary groups include finitely generated LERF groups
and groups with countably many subgroups.
