Semipermutability and Sylow permutability of subnormal subgroups in finite groups

Abstract

Let G be a finite group. A subgroup H of a group G is called (S-permutable) in G if H permutes with all Sylow p-subgroups of G. a group G is called PST-groups if every subnormal subgroups are S-permutable. A subgroup H of G is called semipermutable (S-semipermutable) in G if H permutes with every subgroup (Sylow p-subgroup) K of G with (│H│,│K│) ꞊1. A group G is called SP-group (SPS-group) if every subnormal subgroup is semipermutable (S-semipermutable) in G. In this paper we study semipermutable subgroups and we present some elementary results about this kind of subgroups. Also, we gave a direct proof for the equivalence between the classes PST-groups and SP-groups using subnormal subgroups properties of these groups. In particular, we proved that Sylow permutability and semipermutability of the subnormal subgroups are coincide in the solvable universe.