Strong conciseness of coprime commutators in profinite groups

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