Counting Theorem (i), Finite Local Ring Of Symmetric Matrices

Let An(R) be the set of symmetric matrices over Z/pkZ with order n,where n > 2, p is a prime ,p > 2and p = I(mod4),k > 1. By determining the normal form of symmetric matrices over Z/pkZ,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 .At last we obtain the number of the symmetric matrices which are in the same orbit with the special symmetric matrix.