The Influence Of Subgroups Property On The Structure Of Finite Groups

All groups considered in this dissertation are finite.Suppose that group C is(?)-quasipermutable in G,if it has subgroups T and T≤G such that G = HT,T∩H ＝T∩C,then subgroup H is called P(?)-supplemented in G.Let G be a finite p-group,denote the rank of G with r(G)= max{logp|E|,E≤G,and denote the normal rank of G with rn(G)= max{logp|E|,E≤G,E ⊿G},where E is a elementary abelian subgroups of G.If π is some set of prime numbers,1=A0(?)A1(?)...(?)An=G is the main series of group G.{Ai/Ai-1|i∈I} is the all π’ chief fators in main series of group.Suppose that M,(i∈I)is the all maximal subgroups of G,which is the complement of chief factor Ai/Ai-1,then subgroup∩i∈IMi is called π-prefrattini subgroup of G。(If G has not complement of chief chief π’-factors,thenπ-prefrattini subgroups of G is itself).The article is divided into six chapters.Chapter Ⅰ,introduction exposition the background and the main results of this dissertation.Chapter Ⅱ,introduces some basic definitions,lemmas and symbols required for this article.Chapter Ⅲ,we investigate the influence of P(?)supplemented subgroup on p-nilpotency.Chapter Ⅳ,explore the influence of normal rank on Sylow-subgroup of soluble group,we get some relationships between the normal rank,nilpoten length and p-length.Chapter V,the characteristics of π-prefrattini subgroup are studied by the method of minimal counterexample.Chapter VI,we make a conclusion of this article and puts forward some prospects related to the content of this paper.