The σ-property Of Subgroups,the Number Of Sylow Subgroups And The Research Of Two Open Problems

This Ph.D dissertation mainly studies the influence of σ-embedded,G-boundary factor,G-trace and n-maximal subgroups on the structure of the finite groups and the problem of the number of Sylow subgroups of finite simple groups.All groups men-tioned in this dissertation are finite.The dissertation is divided into five chapters.In chapter I,we mainly introduce the research background and achievements of the dissertation.In chapter Ⅱ,we list some notation,terminology and some known basic results.In chapter Ⅲ,we study the property of σ-embedded and σ-n-embedded.By us-ing σ-permutable and a-subnormal subgroups given by Skiba and combining the s-embedded and n-embedded subgroups to obtain some new subgroups,which are σ-embedded and σ-n-embedded.Then by studying the property of σ-embedded and σ-n-embedded of the maximal and the minimal subgroups of the Hall subgroup of G,we obtain a new discriminant criterion of supersolvability and a group G belonging to a saturated formation,thus popularizing many previous achievements.In chapter IV,we study the number of Sylow subgroups of finite simple groups through the research of number of Sylow subgroups of finite groups by Professor Guo Wenbin and Evgeny Vdovin.In chapter V,we mainly study two open problem.First,we solve the open prob-lem of G-boundary factor,G-trace and p-solvable,which proposed by Professor Guo Wenbin and Skiba.Second,we use the σ-subnormal of n-maximal subgroups to study the structure of finite groups,and solve the open problem about a sufficient condition under which a group is σ-dispersive,which also generalized the earlier result.