| In recent years,using some special properties of subgroups to study the structure and properties of finite groups is an important direction.In finite groups,self-centralizing subgroups are a class of special subgroups,and TI-subgroups are a class of generalized normal subgroups,all of which have a very controlling influence on the structure of finite groups.In this paper,we continue to investigate the structure of finite groups by using the TI-subgroups and self-centralizing subgroups.Many new results have been obtained,and they generalize some significative conclusions.This paper is divided into five chapters.In the first chapter,we introduce the research background of this paper and the previous research results.In the second chapter,we introduce some basic concepts and basic lemmas.The third chapter is divided into two parts,we portray the finite groups whose some special subgroups are subnormal subgroups or TI-subgroups.In section 3.1,we discuss EST-groups whose subgroups of even order are subnormal subgroups or TI-subgroups.And the results are as follows:Theorem 3.1.1 If G is an EST-group,then G is solvable.Theorem 3.1.2 If G is an EST-group,then one of following statements holds:（1）Every subgroup of G of even order is a subnormal subgroup of G;（2）G=Zp×H is a Frobenius group with complement H,where p is an odd prime and H is a cyclic subgroup of G of even order.Theorem 3.1.3 Let G be a group of even order.If every non-nilpotent maximal subgroup of G of even order is a TI-subgroup,then G is solvable and every non-nilpotent maximal subgroup of G of even order is a normal subgroup of G.In section 3.2,we mainly study the NST-groups whose non-nilpotent self-centralizing subgroups are subnormal subgroups or TI-subgroups.If G is an NST-group,then every subgroup of G is an NST-group.And some conclusions have been obtained:Theorem 3.2.1 Let G be an NST-group.Then every non-nilpotent subgroup of G is a subnormal subgroup of G and G is solvable.Theorem 3.2.4 Let G be a non-nilpotent group.If non-nilpotent self-centralizing subgroups of G are TI-subgroups of G,then non-nilpotent self-centralizing subgroups of G are normal subgroups of G.The fourth chapter is divided into two parts,we mainly discuss the influence of selfcentralizing subgroups on the structure of finite groups.In section 4.1,we investigate the SCSN-groups whose self-centralizing subgroups are all s-normal in G.And some sufficient conditions of solvable groups and p-nilpotent groups are obtained:Theorem 4.1.1 Let G be an SCSN-group.Then following statements hold:（1）If H is a normal subgroup of G,then G/H is an SCSN-group.（2）If H is a normal self-centralizing subgroup of G,then every subgroup of G/H is a s-normal subgroup of G/H;（3）G is a solvable group.Theorem 4.1.4 Let G be a group,p be a prime factor of |G| and P be a Sylow p-subgroup of G.If every minimal subgroup of P is a s-normal self-centralizing subgroup of G and Op（G）=1,then G is a p-nilpotent group.In section 4.2,we study the influence of the number of the same order class of the self-centralizing subgroups on the structure of finite groups.And some conclusions have been obtained:Theorem 4.2.1 Let G be a finite group.Then G is an abelian group if and only ifΩ（G）=1.Theorem 4.2.2 There are no a group with two classes of self-centralizing subgroups of the same order.Theorem 4.2.3 Let G be a non-abelian solvable group.If G has a Sylow p-subgroup which is a self-centralizing subgroup of G,then Ω（G）≥2|π（G）|-1.The fifth chapter is summary and prospect. |
