LAUSR

Let G be a finite soluble group and $$G^{(k)}$$ G ( k ) the k th term of the derived series of G . We prove that $$G^{(k)}$$ G ( k ) is nilpotent if and only if $$|ab|=|a||b|$$ | a b | = | a | | b | for any $$\delta _k$$ δ k -values $$a,b\in G$$ a , b ∈ 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 $$P\cap G^{(k)}$$ P ∩ G ( k ) is generated by $$\delta _k$$ δ k -values contained in P (Lemma 2.5 ). This is related to the so-called focal subgroup theorem.