A note on finite products of nilpotent groups. (Q2467882)

Let \(G=AB\) be a finite group factorized as the product of two nilpotent subgroups \(A\) and \(B\). \textit{H. Wielandt} [Ill. J. Math. 2, 611-618 (1958; Zbl 0084.02904)] and \textit{O. H. Kegel} [Arch. Math. 12, 90-93 (1961; Zbl 0099.01401)] proved that \(G\) is a soluble group. Some years before, Itô had proved that if \(A\) and \(B\) are Abelian then \(G\) is metabelian. The result of Itô yields a natural conjecture: If \(G=AB\) is a finite group factorized as product of two nilpotent subgroups \(A\) and \(B\) of classes \(c_A\) and \(c_B\), respectively, then there exists a function \(f\), depending only on \(c_A\) and \(c_B\) such that the derived length \(d_G\) of \(G\) is bounded by \(f(c_A,c_B)\). For a long time the conjecture \(f(c_A,c_B)=c_A+c_B\) was put. This result is true when the orders of \(A\) and \(B\) are coprime, but in general it is false as \textit{J. Cossey} and \textit{S. Stonehewer} [Bull. Lond. Math. Soc. 30, No. 3, 247-250 (1998; Zbl 0939.20017)] proved. It is interesting to begin analizing the case in which \(A\) is Abelian. In this paper the author proves that if \(G=AB\), \(A\) Abelian and \(B\) a nilpotent subgroup of class \(c\), then \(d_G\leq 2c\).