# Deciding finiteness of matrix groups in positive characteristic

## TL;DR

This paper introduces algorithms to determine whether finitely generated matrix groups over fields of positive characteristic are finite, completing the solution for all fields and providing computational tools with MAGMA implementations.

## Contribution

It presents the first complete algorithmic solution for deciding finiteness of matrix groups over any field, including positive characteristic, and computes their order over function fields.

## Key findings

- Algorithms successfully decide finiteness for positive characteristic fields.
- MAGMA implementations are publicly available.
- Provides a method to compute the order of finite matrix groups.

## Abstract

We present a new algorithm to decide finiteness of matrix groups defined over a field of positive characteristic. Together with previous work for groups in zero characteristic, this provides the first complete solution of the finiteness problem for finitely generated matrix groups over an arbitrary field. We also give an algorithm to compute the order of a finite matrix group over a function field of positive characteristic. Our MAGMA implementations of these algorithms are publicly available.