Strong conciseness of coprime commutators in profinite groups

Abstract

Let G be a profinite group. The coprime commutators γj⁎ and δj⁎ are defined as follows. Every element of G is both a γ1⁎-value and a δ0⁎-value. For j≥2, let X be the set of all elements of G that are powers of γj−1⁎-values. An element a is a γj⁎-value if there exist x∈X and g∈G such that a=[x,g] and (|x|,|g|)=1. For j≥1, let Y be the set of all elements of G that are powers of δj−1⁎-values. The element a is a δj⁎-value if there exist x,y∈Y such that a=[x,y] and (|x|,|y|)=1.In this paper we establish the following results.A profinite group G is finite-by-pronilpotent if and only if there is k such that the set of γk⁎-values in G has cardinality less than 2ℵ0 (Theorem 1.1).A profinite group G is finite-by-(prosoluble of Fitting height at most k) if and only if there is k such that the set of δk⁎-values in G has cardinality less than 2ℵ0 (Theorem 1.2).