# Integrality and arithmeticity of solvable linear groups

## TL;DR

This paper presents practical algorithms for deciding arithmeticity and integrality of finitely generated solvable linear groups, with implementations in Magma, advancing computational group theory.

## Contribution

It introduces new algorithms for arithmeticity and integrality testing of solvable linear groups, including implementation details.

## Key findings

- Algorithms successfully implemented in Magma
- Efficient procedures for arithmetic subgroup generation
- Effective integrality testing for solvable-by-finite groups

## Abstract

We develop a practical algorithm to decide whether a finitely generated subgroup of a solvable algebraic group $G$ is arithmetic. This incorporates a procedure to compute a generating set of an arithmetic subgroup of $G$. We also provide a simple new algorithm for integrality testing of finitely generated solvable-by-finite linear groups over the rational field. The algorithms have been implemented in {\sc Magma}.