The Influence Of Conditionally Permutable Subgroups And Supplemented Subgroups On The Structure Of Finite Groups

All groups considered in this dissertation are finite.Let H and Tbe subgroups of G.Then H is called conditionally permutable with T in G if there exists some x?G such that HTx = TxH.A subgroup H of G is called conditionally permutable in G if for every subgroup K of G,there exists some x?G such that HKx = KXH.Suppose H?G and X ?G.Then H is said to be X-quasipermutable(Xs-quasipermutable,respectively)in G provided that G has a subgroup B such that G = NG(H)B and H X-permutes with B and with all subgroups(with all Sylow subgroups,respectively)V of B such that(|H|,|V|)=1.A subgroup A is called PF-supplemented in G if has subgroups T and C?A such that G = AT,T?A=T?C and C is F-quasipermutable in G.The influence of conditionally permutable subgroups,X-quasipermutable(Xs-quasipermutable)subgroups and PF-supplemented subgroups on structure and property of finite groups is investigated in this dissertation.The dissertation consists of six chapters.Chapter ?,introduction introduce the background and the main results of this dissertation.Chapter ?,some commonly used concepts,symbols and some known basic results of this dissertation is introduced in this article.Chapter ?,we use the conditionally permutable subgroups to study the structure and property of finite groups,and obtain one sufficient condition about p-supersolubility of finite groups.Chapter ?,we characterize supersoluble groups and C?-groups by using the property of X-quasipermutable subgroups and Xs-quasipermutable subgroups.Chapter V,the definition of PF-supplemented subgroups is proposed in this section.And we investigate the influence of PF-supplemented subgroups on 2-nilpotency,2'-supersolubility and p-nilpotency.Chapter VI,we make a conclusion of this article and provide some uses of partial conclusions.At the same time,we give some research method and possible innovation on using the property of subgroups to study the structure of the finite group in the future.