Authors:A. Ballester-Bolinches and R. Esteban-Romero
A characterisation of finite soluble groups in which Sylow permutability is a transitive relation by means of subgroup embedding
properties enjoyed by all the subgroups is proved. The key point is an extension of a subnormality criterion due to Wielandt.
Let G be a finite group. A PT-group is a group G whose subnormal subgroups are all permutable in G. A PST-group is a group G whose subnormal subgroups are all S-permutable in G. We say that G is a PTo-group (respectively, a PSTo-group) if its Frattini quotient group G/Φ(G) is a PT-group (respectively, a PST-group). In this paper, we determine the structure of minimal non-PTo-groups and minimal non-PSTo-groups.