摘要
设G是群,φ:G→G为自同构.若对任意的x∈G,有φ(x)x=xφ(x),则称φ为G上的交换自同构.设Tn是域F上所有n×n阶可逆上三角矩阵全体按矩阵乘法构成的群,n≥3,F*为F中非零元全体组成的乘法群.证明了映射φ:Tn→Tn为Tn的交换自同构当且仅当存在群同态σi:F*→F*,1≤i≤n,使得φ(A)=(∏ni=1σi(aii))A,对A=(aij)n×n∈Tn,并且对任意的k=1,2,…,n,以及任意的a∈Imσk,方程xσ1(x)σ2(x)…σn(x)=a在F*中存在唯一解.
Let G be a group,an automorphism φ: G → G is called a commuting automorphism,if for all x ∈ G,φ(x) x = xφ(x). Let n be a positive integer larger than 3 and F a field. Denote by Tnthe multiplicative group consisting of all n × n invertible upper triangular matrices over F.Prove that a map φ: Tn→ Tnis a commuting automorphism on Tnif and only if there exist group honmomorphisms σi: F^*→ F^*,1 ≤ i ≤ n,such that φ(A) =(∏ni=1σi(aii)) A,for A =(aij)n × n∈Tn,and for any k = 1,2,…,n and any a ∈ Imσk,the equation xσ1(x) σ2(x) …σn(x) = a has a unique solution in F*.
出处
《福建师范大学学报(自然科学版)》
CAS
CSCD
北大核心
2015年第3期1-6,共6页
Journal of Fujian Normal University:Natural Science Edition
基金
国家自然科学基金资助项目(11101084)
福建省自然科学基金资助项目(2013J01005)
关键词
交换自同构
可逆上三角矩阵
群同态
commuting automorphism
invertible upper triangular matrix
group homomorphism