Coprime commutators in profinite groups

TL;DR

This paper analyzes coprime commutators in profinite groups and proves that if their set is covered by countably many procyclic subgroups, the pronilpotent residual $\gamma_\infty(G)$ is finite-by-procyclic, which in turn implies $G$ is finite-by-pronilpotent-by-abelian. It also establishes the converse: a finite-by-procyclic $\gamma_\infty(G)$ forces the coprime commutators into countably many procyclic subgroups. The authors develop a framework combining coprime action theory, Baire category arguments, and Engel-sink analysis, and handle reductions to prosoluble and metapronilpotent cases, culminating in a sharp structural description of $G$. These results extend conciseness phenomena for verbal subgroups to the setting of coprime commutators in profinite groups and provide strong structural constraints with potential applications in understanding pronilpotent and solvable factors.

Abstract

By a coprime commutator in a profinite group $G$ we mean any element of the form $[x, y]$, where $x,y\in G$ and $(|x|,|y|)=1$. It is well-known that the subgroup generated by the coprime commutators of $G$ is precisely the pronilpotent residual $γ_\infty(G)$. There are several recent works showing that finiteness conditions on the set of coprime commutators have strong impact on the properties of $γ_\infty(G)$ 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 $γ_\infty(G)$ is finite-by-procyclic. In particular, it follows that $G$ is finite-by-pronilpotent-by-abelian.

Theorems & Definitions (23)

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