P-Supersoluble Hypercenter And S-Semipermutability Of Subgroups Of A Finite Group

Let G be a finite group and H a subgroup of G.We say H is s-semipermutable in G ifHG_p?28?G_p H for any Sylow p-subgroupG_p of G with?p,|H|??28?1.In the investigation of the structure of finite groups,it is an effective method to use some embedding properties of subgroups to investigate the structure of a finite group.In this thesis,we consider some properties of s-semipermutable subgroups and investigate the influence of s-semipermutability of prime-power order subgroups on the structure of finite groups and prove the following results which extends some recent results in the literatures.The thesis contains three chapters.In the first chapter,we mainly introduce the relevant investigative background,notions and known results.In the second chapter,we mainly introduce the basic notions and lemmas.In the third chapter,we use s-semipermutable subgroups to investigate the structure of finite groups and get some main results as follows:Theorem 1 Let p???p?G?,P???Syl_p?G?.Then G is p-supersoluble if P satisfies:1.H?40?O^p(G^?9?)is s-semipermutable in G for all subgroups H?P with|H|?28?d,where d is a power of p with 1?27?d?27?|P|.2.2.If p?28?d?28?2 and P is non-abelian,we further supposeH?40?O^p(G^?9?)is s-semipermutable in G for H?P all cyclic subgroups of order 4.Theorem 2 Suppose that p is a prime dividing the order of a finite group G and E is a normal subgroup of G.ThenE?Z_Up?G?if there exist a normal subgroup X of G such that_pF^?9??E??X?E,and a Sylow p-subgroup P of G satisfies:1.H?40?O^p(G^?9?)is s-semipermutable in G for all subgroups H?P with|H|?28?d,where d is a power of p with 1?27?d?27?|P|.2.If p?28?d?28?2 and P is non-abelian,we further supposeH?40?O^p(G^?9?)is s-semipermutable in G for H?P all cyclic subgroups of order 4.Theorem 3 Suppose that p???p?G?and E?27?G.ThenE?Z_Up?G?if there exists a normal subgroupX of G such that_pF^?9??E??X?E,and a non-cyclic Sylow p-subgroup P of X satisfies:1.|P|?29?p^e3p³2.H?40?O^p(G^?9?)is s-semipermutable in G for all non-cyclic subgroups H?P with|H|?28?p^e.