On the Fitting series of the product of two finite nilpotent groups (Q1332510)

Let the finite group \(G = AB\) be the product of two nilpotent subgroups \(A\) and \(B\). Then \(G\) is soluble by a well-known theorem of Kegel and Wielandt, and the structure of such a group has been the object of many investigations. In particular, it has recently been proved by \textit{S. Franciosi, H. Heineken, M. L. Newell} and the reviewer [Arch. Math. 57, 313-318 (1991; Zbl 0774.20021)] that, if the subgroup \(A\) is abelian and \(F\) denotes the Fitting subgroup of \(G\), then the product \(AF\) is a normal subgroup of \(G\). The theorem proved in this article extends the above result to an arbitrary finite product \(G = AB\) of two nilpotent subgroups. In fact, if \(F_ i(G)\) denotes the \(i\)-th term of the Fitting series of \(G\) (\(i \geq 1\)) and \(d\) is the derived length of \(A\), it is shown that \(AF_{2n - 1} (G)\) is a normal subgroup of \(G\) for every \(n \geq d\). In particular, \(G\) has Fitting length at most \(2d + 1\), and an example proves that this bound is the best possible.