Practical Algorithms for Polycyclic Matrix Groups

In this paper we describe a suite of new algorithms for studying polycyclic matrix groups—algorithms for testing membership and for uncovering the polycyclic structure of the group. We also describe an algorithm for deciding whether or not a group is solvable, which, in the important context of subgroups ofGL (n,Z), is equivalent to deciding whether or not a group is polycyclic. Algorithms were developed in Baumslag et al. (1991. The algorithmic theory of polycyclic-by-finite groups. J. Algebra, 142, 118–149) for all of these problems, but the algorithms in this paper represent a first step toward finding practical algorithms: experiments show that they are fast enough to be useful in studying some reasonably complex examples using current technology.