On Some Unsolved Problems Of Finite P-groups

Finite p-groups play an basic and important role in the theory of finite groups. After the sensational success in classifying of the finite simple groups, the study of finite p-groups becomes more and more active. Many group specialists turn their attention to the study of finite p-groups, for example, G. Glauberman, Y. Berkovich, Z. Janko and so on.The classification up to isomorphism of finite groups of a fixed order is a major theme in Group Theory. However, a complete classification of all non-isomorphic p-groups does not appear feasible at present. A more feasible endeavor is to classify up to isomorphism all p-groups satisfying certain properties ([40]). Many group specialists have done a lot in this arae, for instance, N. Blackburn ([9]) classified up to isomorphism the minimal non-metacyclic p-groups; L. Redei ([44]) classified up to isomorphism the minimal non-Abelian p-groups; Z. Janko ([31]) gave a character of finite 2-groups with exactly three involutions; Y. Berkovich ([4]) classified the finite p-groups with derived subgroup of order p; Xu and etal. ([69]) classified the finite p-groups all of whose non-Abelian subgroup are generated by two elements, Zhang and etal. ([75]) classified the finite p-groups all of whose non-Abelian subgroup are metacyclic; and so on.In this paper, we study finite p-groups by their derived subgroups, normalizers of their non-normal subgroups, centralizers of their non-central elements, and their Frattini subgroups respectively. We clssify finite p-groups satisfying certain properties, and solve four open questions introduced by group theorists Y. Berkovich.This paper is organized as the following five chapters:In Chapter 1, we introduce some notation, basic concepts, and some results that we often use in the paper.In Chapter 2, we study the finite p-groups all of whose proper subgroups have small derived subgroups. The community of a group is reflected by its derived subgroup. For instance, a group G is Abelian if and only if G'=1. In 1947, Redei in [?] gave a classification of minimal non-Abelian p-groups, which implies that a finite p-group G is minimal non-Abelian if and only if d(G)=2 and│G'│= p. In 2000, Y. Berkovich gave the structure of the p-groups in which the order of their derived subgroups is p in [4] and put the clssification of the p-groups all of whose orders of derived subgroups of proper subgroups is p at most as a problem in [3]. In chapter 2, we give the structure of this finite p-groups, and slove this problem introduced by Y. Berkovich.In Chapter 3, we study the finite p-groups all of whose non-normal subgroups have small normalizers. A classific result of finite groups is:a finite group is nilpotent if and only if it has no self-normalized proper subgroup, which means that a proper subgroup of a nilpotent group is a proper subgroup in its normalizer. Basic examples of nilpotent groups are p-groups, thus all non-normal subgroups of a finite p-group have index p in their normalizers at lest. In chapter 3, we classify the finite p-groups all of whose non-normal subgroups have index exactly p in their normalizers, and hence answer the Question in [6].In chapter 4, we study the p-groups all of whose cyclic subgroups generated by non-central elements have small indexs in their centralizers. It is clear that the structure of a group is closely related with the size of centralizers of its non-central elements, many group specialists are intrested in using centralizers of non-central elements to study the structure of a group. We note that the number of elements commuted to a element is closely related to the indexes of the cyclic subgroups generated by the elements in its centralizer, but the relationship betweem the structure of a group and the index of the cyclic subgroup generated by non-central element in its centralizer is studied little. In chapter 4, we give the structure of finite p-groups all of whose cyclic subgroups generated by non-central elements have indexs at most p2 in their centralizers, which answer a question posed in [3].In chapter 5, we study the finite p-groups all of whose proper subgroups have cyclic Frattini subgroups. Frattini subgroup is a important characteristic subgroup of a finite p-group. In 2000, Y.Berkovich classified the p-groups with cyclic Frattini subgroups in [4], and put the clssification of the p-groups all of whose proper subgroups have cyclic Frattini subgroups as Problem 42 in [3]. In chapter 5, we classify this finite p-groups, and slove this problem.