一类Euclidean环上二阶线性群的自同构

AUTOMORPHISMS OF THE TWO-DIMENSIONAL LINEAR GROUPS OVER EUCLIDEAN RINGS R_d

域上线性群的自同构问题已经解决,在[4]中对Euclidean环R5={m+n(51/2)/2|m,n是奇、偶相同的整数}上二阶线性群的自同构给出了具体结果。习知,d>0时,

Let R_d={m+d^(1/n)|=5,13, 21, 29; m, n have the same parity}. these Euclidean rings R_d, it is proved that the commutator subgroups of the two-dimensional general linear groups over R_d are just the two-dimensionai special linear groups over R_d. Using the concept “-automorphism of R_d”, it is also proved that the only automorphisms of these linear groups and their proiective groups are identity and automorphism induced by automor- phism of ring R_d, except the general linear group.