Group classes and mutually permutable products

Let G=AB be the mutually permutable product of the nontrivial subgroups A and B of the group G. Then A or B contains a nontrivial normal subgroup of G. It is also established that S(G)∩A=S(A), where S(U) is the solvable radical of U. These facts lead to some generalized results about mutually permutable products of SC-groups. It is also shown that if G is a PTS-group, then A and B are PST-groups.