20.15 (2022)

Let $\kappa$ be an infinite cardinal. If a residually finite group $G$ is embeddable in the full permutation group of a set of cardinality $\kappa$, must it be embeddable in the direct product of $\kappa$ finite groups? See (G. M. Bergman, Indag. Math., 18 (2007), 349–403).

The converse is true: any such direct product is residually finite and embeddable in the indicated permutation group. Also, the statement asked for becomes true if one replaces “direct product of $\kappa$ finite groups” by “direct product of $2^\kappa$ finite groups”, since $G$ has cardinality at most $2^\kappa$, hence homomorphisms to that many finite groups can be chosen which, together, separate each element of $G$ from $e$.