Procyclic coverings of commutators in profinite groups

TL;DR

This paper investigates the structure of profinite groups where all commutators are contained within finitely many procyclic subgroups, revealing bounds on their derived subgroups and conditions for procyclicity.

Contribution

It establishes bounds and structural properties of the derived subgroup in profinite groups with commutators covered by finitely many procyclic subgroups, including pro-p groups.

Findings

Existence of a finite characteristic subgroup M with bounded order such that G'/M is procyclic.

In pro-p groups, G' is either finite with bounded order or procyclic.

Structural constraints on commutator coverage in profinite groups.

Abstract

We consider profinite groups in which all commutators are contained in a union of finitely many procyclic subgroups. It is shown that if G is a profinite group in which all commutators are covered by m procyclic subgroups, then G possesses a finite characteristic subgroup M contained in G' such that the order of M is m-bounded and G'/M is procyclic. If G is a pro-p group such that all commutators in G are covered by m procyclic subgroups, then G' is either finite of m-bounded order or procyclic.