To appear in Groups and Computation III Deterministic algorithms for management of matrix groups

We report some results on deterministic algorithms for efficient management of matrix groups over finite fields. Our results generalize Luks’s work on solvable matrix groups to wider classes of matrix groups. In this article, we present algorithms for membership-testing and its closely related problems. The algorithms run deterministically in time polynomial in these three parameters: the input length of given generators for a matrix group G, the largest prime dividing jGj other than the field characteristic, and the maximum order of a nonabelian composition factor of G. In a future paper, these algorithms will be key procedures in an efficient divide-and-conquer algorithm that will solve matrix-group problems resembling graph-isomorphism testing in the class d: finite groups all of whose nonabelian composition factors lie in Sd.