On profinite groups with commutators covered by nilpotent subgroups

TL;DR

This paper characterizes profinite groups with certain commutator properties, showing how their structure relates to being covered by countably many nilpotent or abelian subgroups, and establishing equivalences involving finiteness and nilpotency.

Contribution

It provides new characterizations of profinite groups based on coverings by nilpotent subgroups and the structure of their commutator subgroups.

Findings

G' is finite iff G is covered by countably many abelian subgroups

G is finite-by-nilpotent iff G is covered by countably many nilpotent subgroups

G's commutator set is covered by countably many nilpotent subgroups iff G' is finite-by-nilpotent

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.