On nilpotency of higher commutator subgroups of a finite soluble group

Abstract

Let G be a finite soluble group and $G^{(k)}$ the kth term of the derived series of G. We prove that $G^{(k)}$ is nilpotent if and only if $|ab|=|a||b|$ for any $\delta _k$-values $a,b\in G$ of coprime orders. In the course of the proof, we establish the following result of independent interest: let P be a Sylow p-subgroup of G. Then $P\cap G^{(k)}$ is generated by $\delta _k$-values contained in P (Lemma 2.5). This is related to the so-called focal subgroup theorem.