Computing generators of groups preserving a bilinear form over residue class rings.
Let \(R=\mathbb Z_q(=\mathbb Z/q\mathbb Z)\) be a residue class ring, and \(J\in M_n(\mathbb Z_q)\) a matrix for which \(J^T=\pm J\). Let \(F_n(R)\) be the group of matrices preserving the form given by \(J\), that is \[ F_n(R)=\{A\in M_n(R)\mid A^TJA=J\}. \] This group \(F_n(R)\) is called an orthogonal group if \(J\) is symmetric (\(J=J^T\)); it is called a symplectic group if \(J\) is alternating (\(J=-J^T\)). If \(q\) is prime, generators for symplectic groups have been given by Taylor (1987). Similarly generators for orthogonal groups are given by Rylands and Taylor (1998). This paper is to show how to extend this result to obtain generators for these groups over the ring \(R=\mathbb Z_q\) if \(q\) is a proper odd prime power. These generators have been implemented and are available in the computer algebra system GAP.
scientific article; zbMATH DE number 6124232