# Algorithms for computing with nilpotent matrix groups over infinite domains

## TL;DR

This paper introduces practical algorithms for analyzing nilpotent matrix groups over infinite domains, including testing nilpotency and answering structural questions, with implementations in GAP and MAGMA.

## Contribution

It presents new algorithms for computing properties of nilpotent matrix groups over infinite fields, including a nilpotency test and structural analysis tools.

## Key findings

- Algorithms successfully implemented in GAP and MAGMA.
- Practical methods for nilpotency testing over infinite fields.
- Structural questions about nilpotent matrix groups can be answered efficiently.

## Abstract

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.