The Power Single Condition Of The Group Of Binary Generator Matrix

In the 20 th century,the classification of finite unipotent group was completed.This classification was a great achievement in algebra.As time goes by,it is time to discuss infinite groups.From finite to infinite,finite generation is an intermediate bridge.This article discusses binary generation groups.According to the existing theorem: the primitive element of the automorphism group on the finitely generated free group is unipotent.Thus there is the presumption that the free group generated by the dual is a unipotent group（solvable group）if the principal element is unipotent.But there is a counter-example that points out that the free group generated by the binary is a complete group,that is,the primitive element is unipotent group,but it is not solvable.Therefore,we need to limit the linear representation of the generators of the binary generative free group like A,B,or limit the dimensionality,study the necessary and sufficient conditions of the unipotent of the group itself of the primitive unipotent group,or find the counterexamples of the primitiveunipotent of other non-unipotent groups.At the same time,this subject also interprets the nature of the nilpotent matrix from another angle.This subject is based on the basic definition of nilpotent matrix and explores in the most primitive way and strives to provide new ideas for studying the related properties of nilpotent.This paper uses the matrix logarithm as a tool to explore the relevant properties of primitive elements and strive to find enough and valuable information and then use the product characteristics of the matrix to construct several independent equations.By comparing and calculating the equations to find the necessity or sufficiency conditions of the matrix nilpotent or unipotent.A large amount of data computation in the text is done with the aid of maple.The details are（1）whether the generated group is unipotent if the group generated when the block is not higher than the second order and not higher than the sixth order is not higher than the sixth order and unipotent in this primitive;（2）whether the generated group is unipotent if the group generated when the block is not higher than the third order is not higher than the fifth order and unipotent in this primitive;（3）some special cases of unipotency of the matrix group when the dimension of linear representation is 10.