Matrix Generators for the Orthogonal Groups

1. Introduction Generators for the groups SL(l; q), Sp(2m; q), U(l; q) and Sz(q) have been available in computer algebra systems for some time 1, 4, 12]. Until recently it has only been practical to work with these groups for small dimensions and small elds. This covered small orthogonal groups (but not in their natural representation) because the orthogonal groups up to dimension 6 are isomorphic to other linear groups. For a complete description of the isomorphisms see 13, Chapters 11 and 12]. However, recent advances in computing speed and memory, as well as better algorithms, make it possible to work with larger groups. Hence there has been an increasing need for matrix generators for the orthogonal groups, particularly in dimensions beyond 6. This demand comes from several sources. For example, those working directly with orthogonal groups as well as those wishing to test new linear group recognition algorithms 10, 3] now need generators for all classical groups. The recent paper of Ishibashi and Earnest 7, 8] provides generators for O(l; q), but not for SO(l; q) nor its derived group (l; q). The matrices of Ishibashi and Earnest have been implemented in GAP by Celler. In 1962 Steinberg gave pairs of generators for all nite simple groups of Lie type. Steinberg's generators are given in terms of root elements and generators for the Weyl group. In this paper we describe the corresponding generators for the nite orthogonal groups (l; q). These generators are presented as matrices and are equal to Steinberg's generators modulo the centre of the group. The purpose is to provide explicit constructions for the orthogonal groups which can be used within computer algebra packages such as Magma or GAP 1, 4]. Our methods are easily adapted to provide generators for SO(l; q) and O(l; q). In the rst part of the paper we outline the (well known) connection between the orthogonal groups and the Chevalley groups of types B m ,