Limited Group Of Several Structural Problems

In this paper, we study the properties of some subgroups affect on the structure of finite p-groups, and characterized finite simple groups with their quantitative properties.the ultimate aim of researching of finite groups is classifying all of finite groups. The complete classification of finite p-groups as a basic group class should be fin-ished primarily, but it is impossible. Under the realistic condition, the classification problem concerns focus on the classification of finite p-groups which have particular subgroups or quotient groups. In fact, it is much easier to finish means in aspects of classifying finite p-groups at present. Z.Janko thought that classifying finite p-groups which have some special subgroups is an important content and direction.Redei gave the definition of inner abelian p-group in [21]. A nature problem is classifying finite p-groups where inner abelian p-group as its maximal subgroup, this is problem 239 in [7]. In this paper, we classify completely finite p-groups which have an abelian maximal subgroup and an inner abelian maximal subgroup.In 1971, Spencer and Armond.E gave firstly the definition of (s-)self-dual group in [32]. In 2008, Y,Berkovich and Z.Janko gave a public problem with classifying finite self-dual group in [6].In 2010, Lijian An gave the specific structure of (s-)self-dual p-group. A na-ture problem is whether classifying finite p-groups that have "maximum" (s-)self-dual subgroup. In this paper, we give the definition of inners-self-dual p-group, and classify such p-groups.After the sensational success in classifying of the finite simple groups, the struc-ture of finite simple group becomes very important. Over the past 30 years, the rise in characterizing finite simple groups with their quantity properties at home and abroad. In this paper, we use the set of the orders of maximal abelian subgroups and the order of finite group to study a series of finite simple groups, and we get their characterizations or classification. The article is divided into five chapters.As an introduction of this paper in the chapter 1, we introduce main results and the research background. We also set out some basic concepts and theorems which are closely related to our results.In chapter 2, we discuss the complete classification of finite p- groups having an abelian maximal subgroup and an inner abelian maximal subgroup.In chapter 3, we define inner (s-)self-dual p- group, minimal non-(s-) self-dual p- group, and classify them completely.In chapter 4, we characterize the simple groups 2F4(22n+1), A1(pn), where pn> 2, n≥1 with the set of the orders of maximal abelian subgroups, and we also characterize the simple groups F4(22n+1), where n≥1 with the set of the orders of maximal abelian subgroups and the order of group.In chapter 5, we give the simple group, whose order is 2n·32·p1·p2···pm.