Countable recognizability and residual properties of groups (Q2414657)

Summary: 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.