  
  
                                   [1X[5XCongruence[105X[101X
  
  
                        [1XCongruence subgroups of [22XSL_2(ℤ)[122X[101X
  
  
                                 Version 1.2.4
  
  
                                 27 April 2022
  
  
                                   Ann Dooms
  
                                  Eric Jespers
  
                               Olexandr Konovalov
  
                                 Helena Verrill
  
  
  
  Ann Dooms
      Email:    [7Xmailto:andooms@vub.ac.be[107X
      Homepage: [7Xhttp://homepages.vub.ac.be/~andooms[107X
      Address:  [33X[0;14YDepartment of Mathematics, Vrije Universiteit Brussel[133X
                [33X[0;14YPleinlaan 2, Brussels, B-1050 Belgium[133X
  
  
  Eric Jespers
      Email:    [7Xmailto:efjesper@vub.ac.be[107X
      Homepage: [7Xhttp://homepages.vub.ac.be/~efjesper[107X
      Address:  [33X[0;14YDepartment of Mathematics, Vrije Universiteit Brussel[133X
                [33X[0;14YPleinlaan 2, Brussels, B-1050 Belgium[133X
  
  
  Olexandr Konovalov
      Email:    [7Xmailto:obk1@st-andrews.ac.uk[107X
      Homepage: [7Xhttps://alex-konovalov.github.io/[107X
      Address:  [33X[0;14YSchool of Computer Science[133X
                [33X[0;14YUniversity of St Andrews[133X
                [33X[0;14YJack Cole Building, North Haugh,[133X
                [33X[0;14YSt Andrews, Fife, KY16 9SX, Scotland[133X
  
  
  Helena Verrill
      Email:    [7Xmailto:verrill@math.lsu.edu[107X
      Homepage: [7Xhttp://www.math.lsu.edu/~verrill/[107X
      Address:  [33X[0;14YDepartment of Mathematics[133X
                [33X[0;14YLouisiana State University[133X
                [33X[0;14YBaton Rouge, Louisiana, 70803-4918[133X
                [33X[0;14YUSA[133X
  
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe  [5XGAP[105X  package  [5XCongruence[105X provides functionality to work with congruence
  subgroups of [22XSL_2(ℤ)[122X.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y©  2006-2022  by  Ann  Dooms,  Eric  Jespers,  Olexandr Konovalov and Helena
  Verrill.[133X
  
  [33X[0;0Y[5XCongruence[105X  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
  [7Xhttp://www.gnu.org/licenses/gpl.html[107X.[133X
  
  [33X[0;0YIf  you  obtained  [5XCongruence[105X, we would be grateful for a short notification
  sent to one of the authors.[133X
  
  [33X[0;0YIf  you  publish  a  result  which  was partially obtained with the usage of
  [5XCongruence[105X, please cite it in the following form:[133X
  
  [33X[0;0YA. Dooms, E. Jespers, O. Konovalov and H. Verrill. [13XCongruence --- Congruence
  subgroups        of        [22XSL_2(ℤ)[122X,        Version        1.2.4;[113X        2022
  ([7Xhttps://gap-packages.github.io/congruence/[107X).[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YWe  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.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (Congruence)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YGeneral aims of [5XCongruence[105X package[133X
    1.2 [33X[0;0YInstallation and system requirements[133X
  2 [33X[0;0YConstruction of congruence subgroups[133X
    2.1 [33X[0;0YConstruction of congruence subgroups[133X
      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 [33X[0;0YProperties of congruence subgroups[133X
      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 [33X[0;0YAttributes of congruence subgroups[133X
      2.3-1 LevelOfCongruenceSubgroup
      2.3-2 IndexInSL2Z
      2.3-3 DefiningCongruenceSubgroups
    2.4 [33X[0;0YOperations for congruence subgroups[133X
      2.4-1 Random
      2.4-2 \in
      2.4-3 CanEasilyCompareCongruenceSubgroups
      2.4-4 IsSubset
      2.4-5 Index
  3 [33X[0;0YFarey symbols and their properties[133X
    3.1 [33X[0;0YConstruction of Farey symbols[133X
      3.1-1 FareySymbolByData
      3.1-2 IsValidFareySymbol
    3.2 [33X[0;0YProperties of Farey symbols[133X
      3.2-1 GeneralizedFareySequence
      3.2-2 NumeratorOfGFSElement
      3.2-3 DenominatorOfGFSElement
      3.2-4 LabelsOfFareySymbol
  4 [33X[0;0YFarey symbols for congruence subgroups[133X
    4.1 [33X[0;0YComputation of the Farey symbol for a finite index subgroup[133X
      4.1-1 FareySymbol
    4.2 [33X[0;0YComputation of generators of a finite index subgroup from its Farey
    symbol[133X
      4.2-1 MatrixByEvenInterval
      4.2-2 MatrixByOddInterval
      4.2-3 MatrixByFreePairOfIntervals
      4.2-4 GeneratorsByFareySymbol
      4.2-5 GeneratorsOfGroup
    4.3 [33X[0;0YOther properties derived from Farey symbols[133X
      4.3-1 IndexInPSL2ZByFareySymbol
  5 [33X[0;0YService functions of the [5XCongruence[105X package[133X
    5.1 [33X[0;0YAdditional information displayed by [5XCongruence[105X algorithms[133X
      5.1-1 InfoCongruence
  
  
  [32X
