The Influence Of The Quasi-prime Subgroups Of A Finite Group On The Structure Of The Group

The main object in this paper is to study the influence of the primary subgroups on the structure of finite groups.Two subjects are researched:the first,structure of the finite group is researched by using the X-s-semi-permutability of some primary subgroups;the second,?-normality is studied to describe the structure of finite groups.The paper is divided into four chapters.In Chapter ?,we introduce the background and research status of the paper and some main research methods.In Chapter ?,we introducesome definitions,symbols and properties.In Chapter ?,we obtain some new criteria of supersolvability of groups by X-s-semi-permutability of some primary subgroups of Op(G).In chapter IV,we give some new judgment of groups to be quasi-(?)*-group.The main results in this paper is the following:Theorem 3.3.1 Let G be a finite group and X a soluble normal subgroup of G.Letp be a prime.Let P ? Sylp(G)withp be a prime and let d be a power of p such that 1?d<|P|.Write U = Op(G),and assume that H ? Op(G)is X-s-semipermutable in G for all subgroups H(?)P with |H|=d.Then either G is p-supersoluble,or else | P ? U>d.Theorem 4.2.1 Let P be a prime divisor of the order of a group G and(|G|,P-1)=1.Let(?)be a saturated formation contains all nilpotent groups.Assume that N is a normal subgroup of G with G/N ?(?)*.If every cyclic subgroup of order P or 4(if p = 2 and Sylow P-subgroup of Fp*(N)is nonabelian)of Fp*(N)not having a quasi-(?)-complement in G is ?-normal in G,then G is a quasi-(?)-group.Theorem 4.2.2 Let P be a prime divisor of the order of a group G and(|G|,p-1)= 1,Let(?)be a saturated formation contains all nilpotent groups.Assume that N is a normal subgroup of G with G/N ?(?)*.If there exists a normal subgroup X such that Fp*(N)?X?N and all the maximal subgroups of the Sylow P-subgroup of X are ?-normal in G,then G is a quasi-(?)-group.