Abstract
Let G be a profinite group. We prove that the commutator subgroup G′ is finite-by-procyclic if and only if the set of all commutators of G is contained in a union of countably many procyclic subgroups.
Highlights
Several recent results indicate that if the set of all commutators is covered by finitely, or countably, many subgroups with certain specific properties, the structure of G′ is somehow similar to that of the covering subgroups. It was shown in [2] that if G is a profinite group that has finitely many periodic subgroups whose union contains all commutators, G′ is locally finite
In [1] similar results were obtained for the case where commutators are covered by countably many subgroups: if G is a profinite group that has countably many periodic subgroups whose union contains all commutators, G′ is locally finite
The commutator subgroup G′ is finite-by-procyclic if and only if the set of all commutators of G is contained in a union of countably many procyclic subgroups
Summary
Several recent results indicate that if the set of all commutators is covered by finitely, or countably, many subgroups with certain specific properties, the structure of G′ is somehow similar to that of the covering subgroups It was shown in [2] that if G is a profinite group that has finitely many periodic subgroups (respectively, subgroups of finite rank) whose union contains all commutators, G′ is locally finite (respectively, G′ is of finite rank). In [1] similar results were obtained for the case where commutators are covered by countably many subgroups: if G is a profinite group that has countably many periodic subgroups (respectively, subgroups of finite rank) whose union contains all commutators, G′ is locally finite (respectively, G′ is of finite rank).
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