可逆上三角矩阵群上保幺幂正规子群的双射(英文)

Bijective maps on invertible upper triangular matrices preserving unipotent normal subgroups

设F是一个特征不为2的域,Tn(F)是域F上所有n×n的可逆上三角矩阵组成的群。首先利用矩阵的运算技巧研究了Tn(F)的所有幺幂正规子群的结构,对Tn(F)的任意一个幺幂正规子群给出了一个完全的刻画,即每一个幺幂正规子群都可以由一个元素来生成;然后借助可逆映射在生成元上的作用方式,给出了可逆上三角矩阵群上保幺幂正规子群的双射的具体表达式。

Let F be a field of characteristic not 2,Tn（F） the group consisting of all n×n invertible upper triangular matrices over F.First,the structure of the unipotent normal subgroups of Tn（F） is studied by techniques of matrix calculations.The unipotent normal subgroups of Tn（F） are determined completely,and each unipotent normal subgroup is shown to be generated by one element;Second,using the method of action on the generators by invertible maps,every invertible map on Tn（F） that preserves unipotent normal subgroup is described in an explicit formula.