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.
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