2025/10/04 by Acciarri, Cristina, Shumyatsky, Pavel
#20E18 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2510.03754
By a coprime commutator in a profinite group G we mean any element of the form [x, y], where x,y∈ G and (|x|,|y|)=1. It is well-known that the subgroup generated by the coprime commutators of G is precisely the pronilpotent residual γ_∞(G). There are several recent works showing that finiteness conditions on the set of coprime commutators have strong impact on the properties of γ_∞(G) and, more generally, on the structure of G. In this paper we show that if the set of coprime commutators of a profinite group G is covered by countably many procyclic subgroups, then γ_∞(G) is finite-by-procyclic. In particular, it follows that G is finite-by-pronilpotent-by-abelian.