The Influence Of ?-Permutable Subgroups On The Structure Of Finite P?T-Groups

subgroups play an important role in the research of group theory.In 2015,Skiba gave a new definition of ?-permutable subgroups:A subgroup A of G is said to be ?-permutable or ?-quasinormal in G if G has a complete Hall ?-set H such that AHx = Hx A for all x ? G and all H?H.In this paper,we give some new structures of P?T-groups by using the a-permutability of ?-primary subgroups.In particular,we obtain the following result:Theorem 3.1.Suppose that G has a generalized Wielandt ?-set.Let D = Gn?.If every ?i-subgroup of G is a-permutable in G for all ?i ??(D),then G is a soluble P?T-group,where(a)D is an abelian Hall subgroup of odd order of G;(b)G = D×M,every element of M induces a power automorphism in D;(c)G/O?i(D)is a special PaT-group for all ?i??(D).Conversely,if G is a soluble P?T-group,then every ?i-subgroup of G permutes with every Hall ?j-subgroup of G for all ?i??(D)and ?j ??(G)with ?i??j =(?).