18.36 (2014)

There are groups of cardinality at most $2^{\aleph_0}$, even nilpotent of class 2, that cannot be embedded in $S := \text{Sym}(\mathbb{N})$ (V. A. Churkin, Algebra and model theory (Novosibirsk State Tech. Univ.), 5 (2005), 39–43 (Russian)). One condition which might characterize subgroups of $S$ is the following. Consider the metric $d$ on $S$ where for all $x \neq y$ we define $d(x, y) := 2^{-k}$ when $k$ is the least element of the set $\mathbb{N}$ such that $k^x \neq k^y$. Then $(S, d)$ is a separable topological group, and so each subgroup of $S$ (not necessarily a closed subgroup of $(S, d)$) is a separable topological group under the induced metric. Is it true that every group $G$ for which there is a metric $d'$ such that $(G, d')$ is a separable topological group is (abstractly) embeddable in $S$?

Note that any such group $G$ can be embedded as a section in $S$.