摘要
Let An（R） be the set of symmetric matrices over Z/p^kZ with order n, where n 〉 2, p is a prime, p 〉 2 and p≡1（mod4）, k 〉 1. By determining the normal form of n by n symmetric matrices over Z/p^kZ, we compute the number of the orbits of An（R） and then compute the order of the orthogonal group determined by the special symmetric matrix. Finally we get the number of the symmetric matrices which are in the same orbit with the special symmetric matrix.
设A_n(R)是有限局部环Z/p^k Z上n阶对称矩阵的集合,这里n≥2．p是大于2素数,p≡1(mod4)且k＞1．通过确定有限局部环Z/p^k Z上对称矩阵的标准型,计算出A_n(R)在线性群GL_n(R)作用下的轨道数,从而计算出由特定对称矩阵确定的正交群的阶以及与特定对称矩阵在同一轨道的对称矩阵的阶．
基金
the Key Project of Chinese Ministry of Education (03060)
