A finite group G is said to be a PST -group if every subnormal subgroup of G permutes with every Sylow subgroup of G. We shall discuss the normal structure of soluble PST -groups, mainly defining a local version of this concept. A deep study of the local structure turns out to be crucial for obtaining information about the global property. Moreover, a new approach to soluble PT -groups, i.e., soluble groups in which permutability is a transitive relation, follows naturally from our vision of PSTgroups. Our techniques and results provide a unified point of view for T -groups, PT -groups, and PST -groups in the soluble universe, showing that the difference between these classes is quite simply their Sylow structure.

We investigate the structure of finite groups that are the mutually permutable product of two supersoluble groups. We show that the supersoluble residual is nilpotent and the Fitting quotient group is metabelian. These results are consequences of our main theorem, which states that such a product is supersoluble when the intersection of the two factors is core-free in the group.

It is well known that a group G = AB which is the product of two supersoluble subgroups A and B is not supersoluble in general. Under suitable permutability conditions on A and B, we show that for any minimal normal subgroup N both AN and BN are supersoluble. We then exploit this to establish some sufficient conditions for G to be supersoluble.

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