Commutators and commutator subgroups in profinite groups

TL;DR

This paper characterizes when the commutator subgroup of a profinite group is finite-by-procyclic based on the union of countably many procyclic subgroups containing all commutators.

Contribution

It establishes a precise equivalence between the structure of the commutator subgroup and the covering of all commutators by countably many procyclic subgroups in profinite groups.

Findings

The commutator subgroup is finite-by-procyclic if and only if all commutators are contained in a union of countably many procyclic subgroups.

Provides a structural criterion linking the commutator subgroup's properties to the covering of commutators.

Advances understanding of the relationship between subgroup structure and commutator sets in profinite groups.

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.