Conditions For Unipotent Of The Free Group Of Rank Two

Abstract algebra is one of important direction of mathematical research. Object of abstract algebra is special structure such as groups, rings, fields, modules and so on. As an important branch of abstract algebra, theory of groups originated in the research of algebraic equations. It opens a new field of research. At the same time, this theory has a tremendous influence for the development of physics, chemistry, Therefore, It has very important significance to study properties of the group.In recent years, linear representation of groups has become one of the important tools widely used in same research fields. Every group is isomorphic image of some free group. Problems of that we studies belong to linear representation of free groups. In this paper, based on linear representation of group, similar and isomorphism between matrix, as well as a lot of calculation, some characteristic of structure of unipotent group will be gotten.Contents of the paper is consisted of three parts. The main results of this paper are as follows:Firstly, there are background, research subjects of purpose and meaning, topic source, and introduces domestic and international research situation and the main contents.Secondly, some basic definitions and relative lemmas are given. The case will be sdudied when standard form of the primitive element did not contain Jordan blocks of orders more than three.At last, we study the case when standard form of the primitive element did not contain Jordan blocks of orders more than four. Our proof will be given only when the dimentions of representation of group is fixed.