若当块不高于五阶矩阵群的幂单性

The Unipotency of Linear Groups Generated by Matrices with No More than Five Jorden Blocks

针对两个若当块均不高于五阶的幂单矩阵生成的矩阵群是否为幂单群这个问题,以幂零矩阵的基本定义为依据,利用矩阵对数工具得到一些关于自由群生成元的组合性质,再利用所得到的组合性质进行理论推导,最终证明了由两个若当块均不高于三阶矩阵生成的群,在满足本原元幂单且不高于五阶的情况下,所生成的群为幂单群。力求为研究幂单的相关性质提供新的思路和方法。

Aiming at the question whether the group generated by two unipotent matrices whose Jordan blocks are of order no more than five is a unipotent group,based on the fundamental definition of nilpotent matrix,we use the matrix logarithm tool to obtain some combinatorial properties of the generators of free groups,and then use the combinatorial properties obtained to carry out theoretical derivation,and finally prove that the group generated by the matrices whose block are of order no more than five is a unipotent group if every primitive is unipotent and their Jordan blocks are of order no more than five.We also strive to provide new ideas and methods for studying the related properties of unipotency.