A polynomial-time reduction algorithm for groups of semilinear or subfield class

In this paper we present an O(d4 log(log d) log q + d3m) Las Vegas algorithm for find- ing a nontrivial reduction of groups that are irreducible with m generators and either lie in the subfield class of matrix or projective groups or are semilinear or have non-absolutely irreducible derived group. We also characterise the absolutely irreducible groups G over arbitrary fields whose derived group consists only of scalars, and prove probabilistic gener- ation results about matrix groups.