A.S. Detinko

Algorithms for computing with nilpotent matrix groups over infinite domains

We develop methods for computing with matrix groups defined over a range of infinite domains, and apply those methods to the design of algorithms for nilpotent groups. In particular, we provide a practical nilpotency testing algorithm for matrix groups over an infinite field. We also provide algorithms to answer a number of structural questions for a nilpotent matrix group.The main algorithms have been implemented in GAP, for groups over the rational number field.

Computing in Nilpotent Matrix Groups

We present algorithms for testing nilpotency of matrix groups over finite fields, and for deciding irreducibility and primitivity of nilpotent matrix groups. The algorithms also construct modules and imprimitivity systems for nilpotent groups. In order to justify our algorithms, we prove several structural results for nilpotent linear groups, and computational and theoretical results for abstract nil-potent groups, which are of independent interest.

On deciding finiteness of matrix groups

We provide a new, practical algorithm for deciding finiteness of matrix groups over function fields of zero characteristic. The algorithm has been implemented in GAP. Experimental results and extensions of the algorithm to any field of zero characteristic are discussed.

ON DECIDING FINITENESS FOR MATRIX GROUPS OVER FIELDS OF POSITIVE CHARACTERISTIC

The author considers the development of algorithms for deciding whether a finitely generated matrix group over a field of positive characteristic is finite. A deterministic algorithm for deciding the finiteness is presented for the case of a field of transcendence degree one over a finite field.

Zariski density and computing in arithmetic groups

For

NILPOTENT PRIMITIVE LINEAR GROUPS OVER FINITE FIELDS

n > 2

CLASSIFICATION OF NILPOTENT PRIMITIVE LINEAR GROUPS OVER FINITE FIELDS

, let

Algorithms for Experimenting with Zariski Dense Subgroups

\Gamma

Integrality and arithmeticity of solvable linear groups

denote either

Linear groups and computation

SL(n, Z)