Finite Groups In Which Subgroups With Given Indices And C-Semipermutable Properties

The theory of groups is the oldest branch of modern algebra. The concept of a group is surely one of the central ideas of mathematics. The theory of finite groups is an important research field in group theory. In the past thirty years we have seen a number of significant advances in the theory of finite groups. In February 1981, the classification of the finite simple groups was completed, representing one of the most remarkable achievements in the history of mathematics. The relationship between the properties of subgroups of a finite group G and the structure of G has been studied extensively.The aim of this Ph.D.dissertation is to study the structure of finite groups in which subgroups have given indices and c-semipermutable properties, and make use of the theory of classes of groups and formation to discuss the property of local formation.First, we intruduce some research background and main development of this paper.Second, we study the influence of t(G) on the structure and property of a finite solvable group G. In particular, the bounds of the nilpotent length(derived length, p-length) of a soluble group G with t(G) ≤ 2 are found.Third, we define and study the concept of the (completely) c-semipermutable subgroups. By using some special subgroups (such as Sylow subgroups, maximal subgroups, minimal subgroups, and so on) (completely) c-semipermutable in a group G, we determine the structure of finite groups. A series of new results have been obtained.Finally, we research into the construction of finite groups with p-nilpotent supplement and give a sufficient and necessary condition for p-nilpotent subgroups of soluble groups by using p-nilpotent supplement subgroups.All groups in this paper are finite. The research results in this paper are used the theory and the method of the class of groups.