Countable recognizability and residual properties of groups

Abstract

A class of groups $\mathfrak{X}$ is said to be countably recognizable if a group belongs to $\mathfrak{X}$ whenever all its countable subgroups lie in $\mathfrak{X}$. It is proved here that the class of groups whose subgroups are closed in the profinite topology is countably recognizable. Moreover, countably detectable properties of the finite residual of a group are studied.