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# Congruence

## Congruence subgroups of \(SL_2(ℤ)\)

Version 1.2.7

28 August 2024

Ann Dooms
Email: andooms@vub.ac.be
Homepage: http://homepages.vub.ac.be/~andooms
Address:
Department of Mathematics, Vrije Universiteit Brussel
Pleinlaan 2, Brussels, B-1050 Belgium

Eric Jespers
Email: efjesper@vub.ac.be
Homepage: http://homepages.vub.ac.be/~efjesper
Address:
Department of Mathematics, Vrije Universiteit Brussel
Pleinlaan 2, Brussels, B-1050 Belgium

Olexandr Konovalov
Email: obk1@st-andrews.ac.uk
Homepage: https://olexandr-konovalov.github.io/
Address:
School of Computer Science
University of St Andrews
Jack Cole Building, North Haugh,
St Andrews, Fife, KY16 9SX, Scotland

Helena Verrill
Email: verrill@math.lsu.edu
Homepage: http://www.math.lsu.edu/~verrill/
Address:
Department of Mathematics
Louisiana State University
Baton Rouge, Louisiana, 70803-4918
USA

### Abstract

The GAP package Congruence provides functionality to work with congruence subgroups of \(SL_2(ℤ)\).

### Copyright

© 2006-2024 by Ann Dooms, Eric Jespers, Olexandr Konovalov and Helena Verrill.

Congruence is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the FSF's own site https://www.gnu.org/licenses/gpl.html.

If you obtained Congruence, we would be grateful for a short notification sent to one of the authors.

If you publish a result which was partially obtained with the usage of Congruence, please cite it in the following form:

A. Dooms, E. Jespers, O. Konovalov and H. Verrill. Congruence --- Congruence subgroups of \(SL_2(ℤ)\), Version 1.2.7; 2024 (https://gap-packages.github.io/congruence/).

### Acknowledgements

We are very grateful to Mong-Lung Lang, Chong-Hai Lim and Ser Peow Tan for their comments provided while implementing algorithms from [LLT95a] and [LLT95b], and to Francqui Stichting (Belgium) for the support of the third author.

### Contents

1 Introduction
1.1 General aims of Congruence package
1.2 Installation and system requirements
2 Construction of congruence subgroups
2.1 Construction of congruence subgroups
2.1-1 PrincipalCongruenceSubgroup
2.1-2 CongruenceSubgroupGamma0
2.1-3 CongruenceSubgroupGammaUpper0
2.1-4 CongruenceSubgroupGamma1
2.1-5 CongruenceSubgroupGammaUpper1
2.1-6 IntersectionOfCongruenceSubgroups
2.2 Properties of congruence subgroups
2.2-1 IsPrincipalCongruenceSubgroup
2.2-2 IsCongruenceSubgroupGamma0
2.2-3 IsCongruenceSubgroupGammaUpper0
2.2-4 IsCongruenceSubgroupGamma1
2.2-5 IsCongruenceSubgroupGammaUpper1
2.2-6 IsIntersectionOfCongruenceSubgroups
2.3 Attributes of congruence subgroups
2.3-1 LevelOfCongruenceSubgroup
2.3-2 IndexInSL2Z
2.3-3 DefiningCongruenceSubgroups
2.4 Operations for congruence subgroups
2.4-1 Random
`2.4-2 \in`
2.4-3 CanEasilyCompareCongruenceSubgroups
2.4-4 IsSubset
2.4-5 Index
3 Farey symbols and their properties
3.1 Construction of Farey symbols
3.1-1 FareySymbolByData
3.1-2 IsValidFareySymbol
3.2 Properties of Farey symbols
3.2-1 GeneralizedFareySequence
3.2-2 NumeratorOfGFSElement
3.2-3 DenominatorOfGFSElement
3.2-4 LabelsOfFareySymbol
4 Farey symbols for congruence subgroups
4.1 Computation of the Farey symbol for a finite index subgroup
4.1-1 FareySymbol
4.2 Computation of generators of a finite index subgroup from its Farey symbol
4.2-1 MatrixByEvenInterval
4.2-2 MatrixByOddInterval
4.2-3 MatrixByFreePairOfIntervals
4.2-4 GeneratorsByFareySymbol
4.2-5 GeneratorsOfGroup
4.3 Other properties derived from Farey symbols
4.3-1 IndexInPSL2ZByFareySymbol
5 Service functions of the Congruence package
5.1 Additional information displayed by Congruence algorithms
5.1-1 InfoCongruence
References
Index
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