Algorithms for the Tits alternative and related problems

We present an algorithm that decides whether a finitely generated linear group over an infinite field is solvable-by-finite: a computationally effective version of the Tits alternative. We also give algorithms to decide whether the group is nilpotent-by-finite, abelian-by-finite, or central-by-finite. Our algorithms have been implemented in MAGMA and are publicly available

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 algorithm to test nilpotency of matrix groups over an infinite field. We also provide algorithms that answer a number of structural questions for a given nilpotent matrix group. The algorithms have been implemented in GAP and MAGMA

Linear groups and computation

We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in this class of groups are surveyed. We illustrate the solution of hard mathematical problems by computer experimentation. Possible avenues for further progress are discussed. This article is aimed at a broad mathematical audience, and more particularly at users of group-theoretical methods and computer algebra systems.Comment: 31p