
Two subgroups A and B of a group G are cosubnormal if A and B are subnormal in their join $\langle A, B\rangle$ 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 hA, Bi and, if Z is the hypercentre of $G = \langle A, B\rangle$, 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 Nconnected if every cyclic subgroup of A is cosubnormal with every cyclic subgroup of B. Though the concepts of strong cosubnormality and Nconnectedness 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 cosubnor mal.
