Some Studies On Finite Groups In Which Special Subgroups Have Given Properties

It is a very effective method to investigate finite groups by considering special subgroups having given properties in finite group theory.The special subgroups that are mainly considered in this thesis include non-nilpotent subgroups,non-abelian subgroups,non-metacyclic subgroups,non-nilpotent maximal subgroups and minimal subgroups.Our studies mainly focus on the following three topics: finite groups in which special subgroups of order divisible by p are TI-subgroups,finite groups in which all non-nilpotent maximal subgroups are normal and finite groups in which special subgroups are cyclic.All content is divided into the following four chapters.In Chapter 1,we mainly introduce some basic definitions,the background and current research and main results that are obtained in this thesis.In Chapter 2,combining TI-subgroups and prime divisor p of|G|(the order of group G)together,we mainly investigate the structure of finite group G in which some special subgroups of order divisible by p are TI-subgroups and our studies generalize some known results in related literature.The main content is as follows: In Section 2.1,we introduce some lemmas that are used in this chapter.In Section 2.2,we obtain a complete characterization of finite group G in which every non-nilpotent subgroup of order divisible by p is a TI-subgroup for any fixed prime divisor p of|G|.In Section 2.3,we characterize the structure of finite group G in which every non-abelian subgroup of order divisible by p is a TI-subgroup,where p is a fixed prime divisor of|G|.In Section 2.4,let p be the smallest prime divisor of|G|,we give a complete description of finite group G in which every non-metacyclic subgroup of order divisible by p is a TI-subgroup.In Chapter 3,suppose that G is a finite group in which every non-nilpotent maximal subgroup is normal.In Section 3.1,we introduce some lemmas that are used in this chapter.In Section 3.2,without using the solvability of G,we give an elementary proof to show that G possesses a Sylow tower,which improves the proof provided in related literature.In Chapter 4,we first introduce some lemmas that are used in this chapter in Section4.1.In Section 4.2,we mainly investigate the structural properties of finite group G in which every minimal subgroup of the derived subgroup is either normal or has cyclic normalizer.Moreover,we characterize the structure of finite groups in which every minimal subgroup either is normal or has cyclic centralizer in Section 4.3.In Section 4.4,we provide two new characterizations on cyclic groups.Our main results generalize some known results in related literature.