Linear Representation Of The Power Unity Of The Free Group Of Dimension 9

In recent years,as the research on semi-simple structures has become more complete,the research on the properties of nilpotency has become extremely active.From Lie algebra operators,nilpotent Lie algebra structures,to solvable groups,nilpotent groups,a large number of studies have focused on nilpotency,especially nilpotent matrices.The unipotent structure is the sum of the identity element and the nilpotent element.Obviously,the transposition of such elements must be nilpotent.Therefore,the study of unipotent properties is also an important direction of current algebra research.Especially when the classification of finite simple groups is completely resolved,and the research of groups is about to develop to infinity,the research on finitely generated groups is of great significance.This article is about to study the unipotency of binary generating groups whose demention of representation from low to high.The equivalent condition of the unipotency of the matrix group when its representation demention is no higher than 9,or a counterexamples under certain conditions will be found.The current relatively hot method such as Lie algebra and other theoretical research methods are avoided in this article,we try to use the most basic and most direct method of using the combination of elements to deal with the related problems of nilpotent matrices,to do methodological explorations for the research,and strive to make the use of our conclusion more general.The specific content of this article is that when the dimension is nine,and each primitive elements of the binary generated free group is unipotent and its Jordan block does not exceed the 4th order,then the free group is a unipotent group.The binary generator matrix group whose Jordan block is not higher than the fourth order can be divided into diag(J4,E5),diag(J4,J4,1),diag(J4,J3,E2),diag(J4,J3,J2),diag(J4,J2,E3),diag(J4,J2,J2,E)according to the Jordan standard form of its generator.We assume that a primitive element is in the above form,and then use the combination properties of primitive elements to obtain a set of generators which must be similar to the upper triangle or the quasi-up triangle at the same time through calculations by computer.The conclusions of this paper enrich the conclusions of power unity determination,improve related theories,and make a more in-depth analysis of the properties of nilpotent matrices.