On finite groups generated by strongly cosubnormal subgroups

Two subgroups A and B of a group G are cosubnormal if A and B are subnormal in their join A,B and are strongly cosubnormal if every subgroup of A is cosubnormal with every subgroup of B. We find necessary and sufficient conditions for A and B to be strongly cosubnormal in A,B and, if Z is the hypercentre of G= A,B , we show that A and B are strongly cosubnormal if and only if G/Z is the direct product of AZ/Z and BZ/Z. We also show that projectors and residuals for certain formations can easily be constructed in such a group. Two subgroups A and B of a group G are -connected if every cyclic subgroup of A is cosubnormal with every cyclic subgroup of B ( denotes the class of nilpotent groups). Though the concepts of strong cosubnormality and -connectedness are clearly closely related, we give an example to show that they are not equivalent. We note, however, that if G is the product of the -connected subgroups A and B, then A and B are strongly cosubnormal.