Mediterr. J. Math. DOI 10.1007/s00009-017-0893-y c  Springer International Publishing 2017 On S-Semipermutable Subgroups and Soluble PST-Groups R. A. Hijazi, W. M. Fakieh, A. Ballester-Bolinches and J. C. Beidleman Abstract. All groups presented in this article are finite. Using several permutability embedding properties, a number of new characterisations of soluble PST-groups are studied. Mathematics Subject Classification. 20D10, 20D20, 20F16. Keywords. Finite group, Permutability, S-Semipermutability. 1. Introduction and Statements of Results All groups considered in this article will be finite. An S-permutable subgroup A of a group G is one for which the product AP is a subgroup whenever P is a Sylow p-subgroup of G. S-permutable subgroups have presented some intriguing problems since Kegel showed they are subnormal and nilpotent modulo their core ([3, Theorem 1.2.14]). In recent years there has been a widespread interest in the phenomenon of the transitivity of the S-permutability. A lot of interesting structural results have been discovered, as well as many characterisations by means of several embedding properties. Recall that group G is called a PST-group if S-permutability is a tran- sitive relation in G, that is, if H is S-permutable in K and K is S-permutable in G, then H is S-permutable in G. By the above mentioned result of Kegel, a group G is a PST-group if and only if every subnormal subgroup is S- permutable. The structure of soluble PST-groups was determined by Agrawal (see [3, Theorem 2.1.8]). Theorem 1. G is a soluble PST-group if and only if the nilpotent residual L of G is an abelian Hall subgroup of odd order acted on by conjugation by G as a group of power automorphisms. Note that G/L is Dedekind if and only if G is a T-group, that is, every subnormal subgroup of G is normal, and G/L is modular if and only if G is a PT-group, that is, every subnormal subgroup of G is modular.