On profinite groups with commutators covered by nilpotent subgroups

Abstract

Let G be a profinite group. The following results are proved. 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 commutators in G is covered by countably many nilpotent subgroups.