The Influence Of Maximal Subgroups And TI-subgroups On The Structure Of Groups

In finite groups,non-nilpotent maximal subgroup is a class of special maximal subgroups,and TI-subgroup is an important generalization of normal subgroup,all of which have a very important influence on the structure of finite groups.In this paper,we mainly investigate nonnilpotent maximal subgroups and TI-subgroups,which are divided into three chapters.The specific contents are as follows.In chapter 1,we introduce definitions?notations and related theorems that are used in this paper,and survey the research progress on non-nilpotent maximal subgroups and TI-subgroups.In chapter 2,we mainly discuss the influence of non-nilpotent maximal subgroups on the structure of finite groups.In section 2.2,we give an elementary proof to show that a finite group with all non-nilpotent maximal subgroups being normal is solvable,and prove that this group must have normal Sylow subgroups.In section 2.3,we characterize a finite group in which all non-nilpotent maximal subgroups of even order are normal,and show that a finite group with all non-nilpotent maximal subgroups of even order being normal is solvable.Moreover,we prove that this group has Sylow tower.In section 2.4,as a generalization of Huppert's theorem,it is shown that is supersolvable,if every maximal subgroup of a finite group which contains the normalizer of some Sylow subgroup has a prime index.Without applying the solvability of the group ,we give a new proof of its supersolvability.We also give a new proof of the solvability of the above group by applying a result of the non-abelian simple group having maximal subgroups of prime index.In chapter 3,combining the TI-property and the subnormality of subgroups,we characterize finite groups in which some special subgroups are TI-subgroups or subnormal subgroups,and generalize and improve several known results.In sections 3.2,for the non-nilpotent subgroups,we prove that every non-nilpotent subgroup of must be subnormal in ,if every non-nilpotent subgroup of a finite group is a TI-subgroup or a subnormal subgroup.In sections 3.3 and 3.4,we characterize a finite group in which all subgroups of non-prime-power order are TI-subgroups or subnormal subgroups and a finite group in which all non-metacyclic subgroups are TI-subgroups or subnormal subgroups respectively.