Writing projective representations over subfields

Let G = (X) be an absolutely irreducible subgroup of GL(d, K), and let F be a proper subfield of the finite field K. We present a practical algorithm to decide constructively whether or not G is conjugate to a subgroup of GL(d, F). K (x), where K (x) denotes the centre of GL(d, K). If the derived group of G also acts absolutely irreducibly, then the algorithm is Las Vegas and costs 0 (vertical bar X vertical bar d(3) + d(2) log vertical bar F vertical bar) arithmetic operations in K. This work forms part of a recognition project based on Aschbacher's classification of maximal subgroups of GL(d, K). (c) 2005 Elsevier Inc. All rights reserved.