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 N-connected if every cyclic subgroup of A is cosubnormal with every cyclic subgroup of B (N denotes the class of nilpotent groups). Though the concepts of strong cosubnormality and N-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 N-connected subgroups A and B, then A and B are strongly cosubnormal.