Group Membership for Groups with Primitive Orbits Namita Sarawagi, Gene Cooperman, and 253

This paper considers a permutation group G = 〈S〉 of degree n with t orbits such that the action on each orbit is primitive. It presents a O(tn2 logc(n)) time Monte Carlo group membership algorithm for some constant c. The algorithm is notable for its use of a new theorem showing how to find O(t log n) generators in O (̃|S|n) time under a more general form of the above hypotheses. The algorithm relies on new combinatorial methods for computing with groups [CF92] and previous work of Babai, Luks and Seress [BLS88]. In addition, it makes extensive use of a structure theorem for primitive groups by Cameron [Cam81], which can be derived from results of Kantor [Kan79] and the classification of finite simple groups.