Coprime commutators and the nonsoluble length of a finite group

Let G be a finite group. The nonsoluble length λ(G) of G is the number of nonsoluble factors in a shortest normal series of G, each of whose factors either is soluble or is a direct product of nonabelian simple groups. In the present paper we are concerned with bounding λ(G) in terms of coprime commutators, that is, commutators [a, b] with (|a|, |b|) = 1. Let e be a positive integer and 2^f the maximal 2-power dividing e. We show that if x^e = 1 whenever x is a coprime commutator in G, then λ(G) ≤ f.