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Matrix generators for the orthogonal groups. (English)
J. Symb. Comput. 25, No.3, 351-360 (1998).
Generators for the groups $\text{SL}(l,q)$, $\text{Sp}(2m,q)$, $U(l,q)$ and $\text{Sz}(q)$ have been available in computer algebra systems for some time. But until recently it has been impractical to work with these groups for large dimensions and fields. Since the orthogonal groups up to dimension 6 are isomorphic to other linear groups, only small orthogonal groups are available in this way. Hence there has been a need for matrix generators for the orthogonal groups of dimensions beyond 6. In 1962, Steinberg gave pairs of generators for all finite simple groups of Lie type. These generators are given in terms of root elements and generators for the Weyl group. In the paper under review, the authors describe the corresponding generators for the finite orthogonal groups $Ω(l,q)$. In order to provide explicit constructions for the orthogonal groups which can be used within computer algebra packages, these generators are presented as matrices. These generators are equal to Steinberg’s generators modulo centre of the group. Their methods are easily adaptable for finding generators for $\text{SO}(l,q)$ and $O(l,q)$.