# Algorithms for the Tits alternative and related problems

## TL;DR

This paper introduces algorithms to determine key structural properties of finitely generated linear groups, such as solvability, nilpotency, and abelian characteristics, with implementations available in MAGMA.

## Contribution

It provides the first computationally effective algorithms for the Tits alternative and related properties in finitely generated linear groups.

## Key findings

- Algorithms successfully decide group properties like solvable-by-finite and nilpotent-by-finite.
- Implementation in MAGMA is publicly available for practical use.
- The algorithms advance computational group theory by making theoretical results algorithmically accessible.

## Abstract

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.