On 2-generated subgroups and products of groups

For a non-empty class of groups F, two subgroups A and B of a finite group G are said to be F-connected if < a, b > is an element of F for all a is an element of A and b is an element of B. This paper is a study of F-connection for saturated formations F subset of NA (where NA denotes the class of all finite groups with nilpotent commutator subgroup). The class of all finite supersoluble groups constitutes an example of such a saturated formation. It is shown for example that in a finite soluble group G = AB the subgroups A and B are NA-connected if and only if [A, B] <= F(G), where F(G) denotes the Fitting subgroup of G. Also F-connected finite soluble products for any saturated formation F with F subset of NA are characterized.