Finite Prime Element P-Groups

The order of element is one of the basic arithmetical invariants in group theory,it has an important impact on the structure of groups.A finite p-group of exponent p is said to be a finite prime element p-group.In this thesis,we study finite prime element p-groups with nilpoteney class 2.Specifically,a criterion for two finite prime element p-groups with nilpotency class 2 to be isomorphism is given by using matrix.As by-products,some finite prime element p-groups are classified,and it is also given that a bound of the number of non-isomorphic type of finite prime element p-groups.This thesis consists of four chapters.The first chapter is the introduction,mainly introduces the research background,methods and main results.The second chapter is the preliminaries,we introduce a new concept about matrix multiplication,list some lemmas to be used in this thesis,and prove some basic properties of the finite prime element pgroups.The third chapter studies finite prime element p-groups with nilpotency class 2.It is divided into three sections.The first section gives a sufficient and necessary condition for two finite prime element p-groups with nilpotency class 2 to be isomorphism.The second section gives a classification of finite prime element p-groups under certain constraints by using this sufficient and necessary condition.In the third section,a bound of the number of non-isomorphic type of finite prime element p-groups is given.The forth chapter is a summary and prospect.