Strong conciseness of coprime and anti-coprime commutators (Q2663144)

Let \(G\) be a profinite group. The symbol \(\pi(G)\) denotes the set of those prime numbers which occur among the prime divisors of numbers \(|G:U|\), where \(U\) runs the set of all open normal subgroups in \(G\). If \(x\in G\), then \(\pi(x)=\pi\left(\overline{\langle x\rangle}\right)\). A coprime commutator in \(G\) is an element of the form \([x, y]\), where \(\pi(x)\cap\pi(y)=\emptyset\) and an anti-coprime commutator is a commutator \([x, y]\) such that \(\pi(x)=\pi(y)\). In the paper under review, the authors prove the following two theorems. 1. A profinite group \(G\) is finite-by-pronilpotent if and only if the cardinality of the set of coprime commutators in \(G\) is less than \(2^{\aleph_0}\). 2. A profinite group \(G\) has a finite commutator subgroup \(G'\) if and only if the cardinality of the set of anti-coprime commutators in \(G\) is less than \(2^{\aleph_0}\).