Anzahl Theorems in Symmetric Matrices over Finite Local Rings (I)

Abstract:

      Let $A_{n}(R)$ be the set of symmetric matrices over $Z/p^{k}Z$ with order $n$, where $n\geq 2$, $p$ is a prime, $p>2$ and $p\equiv1({\rm mod}4)$, $k>1$. By determining the normal form of $n$ by $n$ symmetric matrices over $Z/p^{k}Z$, we compute the number of the orbits of $ A_{n}(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.