Unipotency Of Free Group Generated By Two Elements Of 9 Order With Linear Representation

Since the 21 th century, the research of group theory focuses on to the infinite group naturally, with the solved completely structure problem of limited simple group. During this period, developing rapidly group representation theory gradually become a relatively active branch in the algebra.Among them,it is worth mentioned that Lie group research has had a complete set of professional terms and relatively complete theoretical system. The paper will answer the problem when two unipotent matrix generate nilpotent Lie algebra. In terms of the group theory, this article main research relationship between the unipotency of primitive element and the unipotency of free group generated by two elements in which the standard form of a certain generator don't consist Jordan normal blocks of size at five or higher size.In this thesis, following situations will be studied: the unipoteny of matrix goup generated by two elements under the condition that every generator in this group has Jordan normal blocks of size at most five and there exists at lest a certain generator with Jordan normal blocks of size at most two; the unipoteny of matrix goup generated by two elements under the condition that every generator in this group has Jordan normal blocks of size at most four and there exists at lest a certain generator with Jordan normal blocks of size at most three; the unipoteny of matrix goup generated by two element of 9 order under the condition that every generator in this group has Jordan normal blocks of size at most four. Under these conditions, the sufficient and necessary conditions that matrix group is unipotent is that all primitive element are unipotent. Thus the same conclusion is got when a free group generated by two elements has linear representation. The conclusion is also a generalization of determination conditions that the free group generated by two elements is unipotent.The prove method of thesis contain both purely theoretical proof and derivation based on computer programming. This method provides a new way for the study.Especially, equation for programming not only carry on all equations used earlier, but contain unified description of the structure. Based on these new equations, in thisarticle sevral new conclusions can be got.