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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.

In this paper the structure of finite groups which are the product of two totally permutable subgroups is studied. In fact we can obtain the ^-residual, where J-is a formation, Ti-projectors and %-normalisers, where 7i is a saturated formation, of the group from the corresponding subgroups of the factor subgroups.

Let p be a prime. The class of all p-soluble groups G such that every p-chief factor of G is cyclic and all p-chief factors of G are Gisomorphic is studied in this paper. Some results on T -, PT -, and PST -groups are also obtained.