# Recognizing finite matrix groups over infinite fields

## TL;DR

This paper introduces a comprehensive method for computing with finitely generated matrix groups over infinite fields, solving the finiteness decision problem and enabling structural analysis through finite field isomorphisms.

## Contribution

It provides a novel uniform approach for matrix group computations over infinite fields, including finiteness decision and finite field isomorphism construction algorithms.

## Key findings

- Decided finiteness for finitely generated matrix groups over infinite fields
- Developed algorithms to construct isomorphic copies over finite fields
- Implemented algorithms successfully in MAGMA

## Abstract

We present a uniform methodology for computing with finitely generated matrix groups over any infinite field. As one application, we completely solve the problem of deciding finiteness in this class of groups. We also present an algorithm that, given such a finite group as input, in practice successfully constructs an isomorphic copy over a finite field, and uses this copy to investigate the group's structure. Implementations of our algorithms are available in MAGMA.