The Structure Of Finite Groups With Given Subgroup Properties

In the research of group theory,it is one of the hot topics to study the structure of groups by using the properties of subgroups.Among them,the permutabilit and supplementary properties of subgroups have always been concerned by group theorists.Using the permutability and supplementary properties of maximal subgroups and minimal subgroups of Sylow subgroups to study the structure of finite groups,we have obtained a series of important research results.At the same time,new research methods and new topics have been produced.In this dissertation,based on the analysis and summary of some existing structures,we use the new concept of m-S-supplemented to study the structure of finite groups:On the one hand,we study the influence of m-S-supplemented subgroups on p-nilpotent group.By studying the m-S-supplementary properties of the primary subgroups of a given order,a new sufficient condition for the group to be a p-nilpotent group is given(Theorem 3.2.1),and the application of the conclusions is also given(Corollary3.3.1 and 3.3.2).On the other hand,we study the influence of m-S-supplemented subgroups on supersoluble group.By studying the m-S-supplementary properties of the primary subgroups of a given order,a conclusion about supersoluble hypercenter is given(Theorem 4.2.1),and a new sufficient condition that the group is a supersoluble group is also given(Theorem 4.2.2).Also,we give the application of the conclusions(Corollary4.3.1,4.3.2 and 4.3.3).At last,the main work of this paper is summarized.Some new characterizations of p-nilpotency and supersolubility of finite groups are obtained.Then the σ-group theory is introduced,we will explore combining the related properties of subgroups with σ-group theory to study the structure and properties of groups.In particular,some recent developments of our research group are introduced.