On profinite groups with commutators covered by nilpotent subgroups

Abstract

The following results about a profinite group $G$ are obtained. The commutator subgroup $G’$ is finite if and only if $G$ is covered by countably many abelian subgroups. The group $G$ is finite-by-nilpotent if and only if $G$ is covered by countably many nilpotent subgroups. The main result is that the commutator subgroup $G’$ is finite-by-nilpotent if and only if the set of all commutators in $G$ is covered by countably many nilpotent subgroups.