# Algorithms for linear groups of finite rank

## TL;DR

This paper introduces algorithms for computing key invariants of finitely generated solvable-by-finite linear groups, enabling decisions about subgroup indices, with implementations in MAGMA for algebraic number fields.

## Contribution

It provides the first algorithms to compute torsion-free rank and Pr"{u}fer rank bounds for such groups, and to determine subgroup finite index.

## Key findings

- Algorithms successfully compute ranks and bounds.
- Implementation in MAGMA demonstrates practical applicability.
- Enables decision procedures for subgroup properties.

## Abstract

Let $G$ be a finitely generated solvable-by-finite linear group. We present an algorithm to compute the torsion-free rank of $G$ and a bound on the Pr\"{u}fer rank of $G$. This yields in turn an algorithm to decide whether a finitely generated subgroup of $G$ has finite index. The algorithms are implemented in MAGMA for groups over algebraic number fields.