Computing in Solvable Matrix Groups

We announce methods for e cient management of solvable matrix groups over nite elds. We show that solvability and nilpotence can be tested in polynomial-time. Such e ciency seems unlikely for membership-testing, which subsumes the discrete-log problem. However, assuming that the primes in jGj (other than the eld characteristic) are polynomiallybounded, membership-testing and many other computational problems are in polynomial time. These problems include nding stabilizers of vectors and of subspaces and nding centralizers and intersections of subgroups. An application to solvable permutation groups puts the problem of nding normalizers of subgroups into polynomial time. Some of the results carry over directly to nite matrix groups over algebraic number elds; thus, testing solvability is in polynomial time, as is testing membership and nding Sylow subgroups.