关键词: Group automorphism, field, characteristic

Abstract:

Suppose F is a field of characteristic not 2 and F its multiplicative group.
Let T
n(F) be the multiplicative group of invertible upper triangular n × n matrices over
F and ST±
n (F) its subgroup {(aij ) 2 T
n(F)|aii = ±1, 8i}. This paper proves that f :
T
n(F) ! T
n(F) is a group automorphism if and only if there exist a matrix Q in T
n(F)
and a field automorphism  of F such that either
f(A) =  (A)QAQ−1, 8A = (aij ) 2 T
n(F)
or
f(A) =  (A−1)Q[J(A)−T J]Q−1, 8A = (aij) 2 T
n(F),
where A = ((aij)), A−T is the transpose inverse of A, J =
n
Pi=1
Ei n+1−i, and   :
T
n(F) ! F is a homomorphism which satisfies { (xIn)(x)|x 2 F} = F and {x 2
F| (xIn)(x) = 1} = {1}. Simultaneously, they also determine the automorphisms of
ST±
n (F).

Key words: Group automorphism, field, characteristic