Finite dinilpotent groups of small derived length

A finite dinilpotent group G is one that can be written as the product of t wo finite nilpotent groups, A and B say. A finite dinilpotent group is alwa ys soluble. If A is abelian and B is metabelian, with \A\ and \B\ coprime, we show that a bound on the derived length given by Kazarin can be improved . We show that G has derived length at most 3 unless G contains a section w ith a well defined structure; in particular if G is of odd order, G has der ived length at most 3.