Soluble products of connected subgroups

The main result in the paper states the following: For a finite group G = AB, which is the product of the soluble subgroups A and B, if 〈a, b〉 is a metanilpotent group for all a ∈ A and b ∈ B, then the factor groups 〈a, b〉F (G)/F (G) are nilpotent, F (G) denoting the Fitting subgroup of G. A particular generalization of this result and some consequences are also obtained. For instance, such a group G is proved to be soluble of nilpotent length at most l + 1, assuming that the factors A and B have nilpotent length at most l. Also for any finite soluble group G and k ≥ 1, an element g ∈ G is contained in the preimage of the hypercenter of G/Fk−1(G), where Fk−1(G) denotes the (k − 1)th term of the Fitting series of G, if and only if the subgroups 〈g, h〉 have nilpotent length at most k for all h ∈ G.