Unipotency Of Free Group Generated By Two Elements Of Linearly Represent With Any Dimensionality

Group is an important basic structure in algebra,and it is also one of the important research directions in algebra.Representing the group as a concrete transformation is an important method for studying the group.The representation theory used in this has an extremely important position in mathematical research.As the mathematician Izrail Moise-evich Gel-Fand said,"All mathematics is some kind of representation theory." Representation theory can transform abstract algebra problems into linear algebra operations,making representation theory one of the most powerful tools for studying groups.In the twentieth century,the classification of finite simple groups has been completed,and the research of group theory in the new century will inevitably develop towards infinite groups.During this period,the finite generation of infinite groups will be a good transition.The purpose of this paper is to find the unipotent condition of free group generated by two elements in the presence of linear resentation.The specific content of this paper is: use the linear representation of free groups and matrix groups to study the properties of free group generated by two elements And we discussed the unipotency of group when the largest Jordan block of matrix groups generated by two elements does not exceed seven or eight times.In the first chapter,a brief description of group representation theory,group combinatorial theory,power unicity development history,current research status at domestic and foreign are given and the definitions,commonly used lemmas and symbols used in this article are declared.In the third chapter,we prove that free group generated by two elements is unipotent when group satisfy that if the largest Jordan block of any primitive element in the group does not exceed seven,the largest Jordan block of one generator does not exceed two,and the other generator meets one of the following conditions:(1)the largest Jordan block of the other generator does not exceed four,(2)the largest Jordan block of the other generator does not exceed five.In the four chapter,It is discussed the unipotent through the computer in the matrix group generated by two elements with 10 dimension when the largest block of any primitive element does not exceed 8 and the largest element of the generator does not exceed 2 and 4.