The Unipotency Of Linear Groups Generated By Matrices With Jordan Normal Blocks Of Size Not Contain More Than Six

Representation theory of groups has become one of the most rapidly developed and active branches in modern algebra,and has become the mainstream trend of the present mathematics domestic and abroad.The representation theory of finite groups has already been used not only to solve some common mathematical problems,but also to slove problems from physics and chemical systems with certain symmetry.The representation theory of groups has developed well in the present,especially there have been a lot ofterminologies and theoretical system in the theory of character and the structure of Lie group.In the case of that the linear representation of free groups exists,the study of free group is manifested as the study of matrix groups.In this paper,we consider the unipotency of matrix groups which primitive element has whose order is not greater than six in its Jordan standard norm when the dimension of representation is fixed unknown or is fixed integer.The main contents include: the matrix group must be unipotent(solvable groups)if it’s generated by two elements which have Jordan normal blocks of size two or five and every primitive element of this group has Jordan normal blocks of size at most six.The matrix group must be unipotent if every primitive element in this group which is unipotent and has Jordan normal blocks of size six in 9 order with linear representation.In this paper matrix logarithmic has been used to analyse the combinatorial properties of primitive element deeply.And the product characteristics of matrices is also used to construct more independent equations.We deduce the necessity and sufficiency of the nilpotency or unipotency of matrices through these equations,then a computer program has been used to solve these equations.