The Structure Of Finite Groups Having Subgroups And Sets Of Subgroups With Given Properties

All groups considered in this thesis are finite.In the research of group theory,using the properties of subgroups to study the structure of groups is one of the hot topics.In particular,the quasinormality and embeddability of subgroups have always attracted the attention of many scholars,and many interesting results have been obtained.Meanwihle,some new research topics have emerged.In this thesis,we apply the theory ofσ-groups to study the influence of the weaklyt_σ-embedded subgroups and the weakly n-σ-embedded subgroups on the structure of finite groups by using the method of minimal-order counterexample.The main contents are as follows:In chapter Ⅲ,we study the influence of weaklyt_σ-embedded subgroups on the solvability,supersolvability andπ-nilpotency of finite groups.First,by combining the concept of weaklyt-embedded subgroup with thet_σ-quasinormal subgroups given by Beidleman et al.,we introduce the concept of weaklyt_σ-embedded subgroups.Then by studying the weaklyt_σ-embedded property of the maximal subgroups of the Hall subgroup,some new criteria of solvability,supersolvability andπ-nilpotency of G are obtained.These results unify and generalize many previous achievements.In chapter Ⅳ,we study the relationship between weaklyt_σ-embedded subgroups and hypercyclically embedded subgroups.By considering thet_σ-embedded property of the maximal subgroups and minimal subgroups of subgroups in a complete Hallσ-set,we obtain two new sufficient conditions that a normal subgroup is a hypercyclically embedded.As corollaries to the theorems,we give some new criteria for a group belonging to the saturated formations containing all supersolvable groups.At the end of this chapter,we give some applications of the results.In chapter Ⅴ,we study the influence of weakly n-σ-embedded subgroups on the solvability and supersolvability of finite groups.As the continuation and development of previous work,a new concept of weakly n-σ-embedded subgroup is given.Then,the structural properties of finite groups are studied,and new judgment conditions are obtained that the group is a solvable group and a supersolvable group,which generalizes many existing results.In chapterⅥ,we make a summary of this thesis and the next research work and directions are proposed.