Algorithms for polycyclic-by-finite matrix groups

Abstract

Let K be a number eld. We present several algorithms for working with polycyclic-by-nite subgroups of GL(n; K). Let G be a subgroup of GL(n; K) given by a nite generatingset of matrices. We describe an algorithm for deciding whether or not G is polycyclic-by-nite. For polycyclic-by-nite G, we describe an algorithm for deciding whether or not a given matrix is an element of G. We also describe an algorithm for deciding whether or not G is solvable-by-nite, providing an alternative to the algorithm proposed by Beals. Preliminary experiments indicate that the algorithms described in this paper are suitable for computer implementation. Further experimentation is needed to determine the range of input for which they are practical. 1. Introduction 1.1. Notation and deenitions. Throughout this article, let Z denote the ring of integers, Q the eld of rationals, and C the eld of complex numbers. Let R denote either Z or a number eld. If p is a prime, then the eld of p-adic numbers is denoted by Q p , its algebraic closure by Q p , and the ring of p-adic integers by Z p. The eld with p elements is denoted by F p .