# Algorithms for arithmetic groups with the congruence subgroup property

## TL;DR

This paper introduces practical algorithms for computing with arithmetic groups in SL(n,Q) for n>2, including membership testing, subgroup analysis, and orbit-stabilizer problems, implemented in GAP.

## Contribution

It presents new methods for effective computation with arithmetic groups with the congruence subgroup property, focusing on constructing principal congruence subgroups.

## Key findings

- Algorithms successfully tested in GAP
- Effective membership testing developed
- Subnormal structure analysis enabled

## Abstract

We develop practical techniques to compute with arithmetic groups $H\leq \mathrm{SL}(n,\mathbb{Q})$ for $n>2$. Our approach relies on constructing a principal congruence subgroup in $H$. Problems solved include testing membership in $H$, analyzing the subnormal structure of $H$, and the orbit-stabilizer problem for $H$. Effective computation with subgroups of $\mathrm{GL}(n,\mathbb{Z}_m)$ is vital to this work. All algorithms have been implemented in GAP.