Some new characterizations of PST-groups

Let H and B be subgroups of a finite group G such that G=NG(H)B. Then we say that H is quasipermutable (respectively, S-quasipermutable) in G provided H permutes with B and with every subgroup (respectively, with every Sylow subgroup) A of B such that (|H|,|A|)=1. In this paper we analyze the influence of S-quasipermutable and quasipermutable subgroups on the structure of G. As an application, we give new characterizations of soluble PST-groups.