#############################################################################
##
#W cong.gi                 The Congruence package                   Ann Dooms
#W                                                               Eric Jespers
#W                                                         Olexandr Konovalov
##
##
#############################################################################


#############################################################################
##
## Constructors of congruence subgroups

java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 0
    [ IsPosInt ],

    , ;
    type :                      and 
                     IsGroup and 
                     IsAttributeStoringRep and 
                     IsFinitelyGeneratedGroup and 
                     IsMatrixGroup and
                     oup);
                         IsCongruenceSubgroup)
    ( G ,
      DimensionOfMatrixGroup ( ,type,
      , 2java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
      ,truejava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
      ,falsejava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
      LevelOfCongruenceSubgroup, n,
   IsPrincipalCongruenceSubgroup, true,
      IsIntersectionOfCongruenceSubgroups, IsCongruenceSubgroupGamma1, false, IsCongrjava.lang.StringIndexOutOfBoundsException: Range [74, 72) out of bounds for length 80
      
      (CongruenceSubgroupGamma0for"
IsCongruenceSubgroupGammaMN)
    (njava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 end);


InstallMethod( CongruenceSubgroupGamma0, "for                      and 
    [                   and
    function(n)
    local type, G;
    type := NewType( FamilyObj([                     );
                     IsGroup and 
                    and 
                      and 
                     IsMatrixGroup and
                     );
G : (;                  
    IsIntegerMatrixGroup, true,
      DimensionOfMatrixGroup, 2,
      IsFinite false,
     ,true,
      IsFinite false,
      LevelOfCongruenceSubgroup, n,
   IsPrincipalCongruenceSubgroupIsIntersectionOfCongruenceSubgroups,false,
   IsIntersectionOfCongruenceSubgroups, false,
      IsCongruenceSubgroupGamma0IsCongruenceSubgroupGamma1 , IsCongruenceSubgroupGammaUpper1, false,
      IsCongruenceSubgroupGamma1, false, IsCongruenceSubgroupGammaUpper1, false,
      java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 13
     G;
 end);
 
    [ ]
    function()
    function()
    local type, G;
    type := NewType( FamilyObj    type :=NewType([[,][,]],
                     IsGroup and 
                      and 
                      andjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
                     IsMatrixGroup and
                     oup)java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
     :=rec)                   
       (G,type,
DimensionOfMatrixGroup,
      OneImmutable, [[1,0],[0,1]],
true,
      IsFinite,       OneImmutable, [1,],01,
            IsIntegerMatrixGroup, tru, true,
ceSubgroup, false,
IsIntersectionOfCongruenceSubgroups,
      IsCongruenceSubgroupGamma0, false, IsCongruenceSubgroupGammaUpper0,  , false,
      IsCongruenceSubgroupGamma1, false, IsCongruenceSubgroupGammaUpper1, false,
      IsCongruenceSubgroupGamma0false,IsCongruenceSubgroupGammaUpper0 truejava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
    return G;
 end); 
 

InstallMethod( CongruenceSubgroupGamma1, "for java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 13
    [ ],
    function(n)
    local type, G;
    type := NewType( java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 17
                     IsGroupjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
                     java.lang.StringIndexOutOfBoundsException: Range [47, 42) out of bounds for length 47
                       
                     IsMatrixGroup and
                     );
     : rec)                   
    ObjectifyWithAttributes( G, type, 
      IsCongruenceSubgroup);
      OneImmutable,      :=) java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
IsIntegerMatrixGroup ,
      IsFinite, false,DimensionOfMatrixGroup2,
,n,
   IsPrincipalCongruenceSubgroupIsIntegerMatrixGroup true,
   , ,
      IsCongruenceSubgroupGamma0, false, IsCongruenceSubgroupGammaUpper0, false
        java.lang.StringIndexOutOfBoundsException: Range [34, 32) out of bounds for length 40
       ,java.lang.StringIndexOutOfBoundsException: Range [73, 71) out of bounds for length 79
turnG;
 java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
 

 integer",
    [ functionnjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
    function(               and
    local typeIandjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
    type: (FamilyObj[[,][0,1]]java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
                     IsGroup and 
                     IsAttributeStoringRep 
                           DimensionOfMatrixGroupjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
                     IsMatrixGroupand
                     IsFinite,false,
     : )                   
    ObjectifyWithAttributes( G, type, 
   IsPrincipalCongruenceSubgroup,false,
      java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 34
      , ,
      ,falsejava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
      ,njava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
   InstallMethod ," ;
  IsIntersectionOfCongruenceSubgroups, false,
      IsCongruenceSubgroupGamma0, false, IsCongruenceSubgroupGammaUpper0, false,
                     IsGroup and 
              IsAttributeStoringRep and 
java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 13
 end)
 

llMethod(CongruenceSubgroupGammaMN"two positive "
        ObjectifyWithAttributes ,type,
    function(m,n)
    local type, G;
    type :=     OneImmutable,[1,][,]]
                     IsGroup,true,
                     IsAttributeStoringRepLevelOfCongruenceSubgroup *,
                     IsFinitelyGeneratedGroup and 
                     IsMatrixGroupIsIntersectionOfCongruenceSubgroups falsejava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
                     );
    : (;                  
    returnG;
      , 2java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
      ,1]]java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
     , true,
           IsFinite,,
      LevelOfCongruenceSubgroup, m*n,
      LevelOfCongruenceSubgroupGammaMN, [m,n]IsGroup and 
   IsPrincipalCongruenceSubgroup, false,
   IsAttributeStoringRep and 
      IsCongruenceSubgroupGamma0, false, IsCongruenceSubgroupGammaUpper0,  IsFinitelyGeneratedGroup
      IsMatrixGroup
      IsCongruenceSubgroupGammaMNIsCongruenceSubgroup;
    ifnot ForAll( arg,IsCongruenceSubgroup)then
end;
  

InstallGlobalFunction( IntersectionOfCongruenceSubgroups,
        # First create list toeliminateevident  of .
    #java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
     = NewType ([[0,01])
                     IsGroup and 
                     IsAttributeStoringRep and 
                     IsFinitelyGeneratedGroup and 
                     IsMatrixGroup and
                     IsCongruenceSubgroup);
    if notForAll(arg,IsCongruenceSubgroupthen
      Error("# - we do not add
    fi;
#
    # Then we eliminate evident inclusions of one subgroup into another:
    # - since intersection is associative, if we can intersect the group T which
    #   is to be added with another subgroup K already contained in alglist, and
 theresultis  ofthe  congruence replace by
    #   the result of intersection of K and T 
    # - we do not add a subgroup T to the list of defining subgroups, if alglist 
    #   already contains another subgroup K such that K is in T. 
              pos:=PositionProperty( arglist, K -> =(, K- 
    #   subgroups K such that T is in K, we add T and remove all these K.
    arglist: ]java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
    H in  do
      if java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 14
          in ( H  java.lang.StringIndexOutOfBoundsException: Range [52, 53) out of bounds for length 52
         pos=(arglist,K >
                 CanReduceIntersectionOfCongruenceSubgroups in[1 .Lengtharglist  
                 (arglist[i],T)then
            pos]( [os] T )
          else
            if ForAll( arglist, K -> not CanEasilyCompareCongruenceSubgroups( K, T ) ) and
               ForAll( arglist, K -> not IsSubgroup( T, K ) ) then
              for i in [ 1 .. Length(arglist) ] do
                if IsSubgroup( arglist[i], T ) then
                  Unbind( arglist[i] );
                java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
              od;
                      ;
              pos:=PositionProperty( arglist, K -> 
            fi;    
          ;
        od;
      else
        pos( arglist,K 
               java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 17
java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 25
arglist[os]=ntersection [pos], );
        else     
( arglist, K - CanEasilyCompareCongruenceSubgroupsK  ) ) 
             ForAll( arglist, K ->              if (arglist[] )then
           for i              fijava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
              if IsSubgroup( arglist[i], H )             Addarglist, H );
                Unbind( arglist[i] );
              fi;
            od;
            arglist    ;
            Add(arglist, ;
          fi;
        fi;  
      fijava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    od;
    # if the list of defining subgroups was reduced 
    # to a single subgroup, we return this subgroup
    if java.lang.StringIndexOutOfBoundsException: Range [39, 33) out of bounds for length 39
      return arglist[1];
    fi; 
    # otherwise we sort the list of defining subgroups:(arglist,
    # types of subgroups are sorted in the following way:
    # - IsCongruenceSubgroupGamma0
      
    # - IsCongruenceSubgroupGamma1
    # - IsCongruenceSubgroupGammaUpper1
    # - IsPrincipalCongruenceSubgroup
    # and subgroups of the same type are sorted by ascending level
    Sort( arglist, 
          function(X,Y) CongruenceSubgroup)<LevelOfCongruenceSubgroup)) 
          local f, t;
f=IsCongruenceSubgroupGamma0IsCongruenceSubgroupGammaUpper0,,IsPrincipalCongruenceSubgroup
          return     n := Lcm( List( arglist, H -java.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 88
               (     ObjectifyWithAttributes ,java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
                 LevelOfCongruenceSubgroup)<LevelOfCongruenceSubgroupY) );
          end );
H(H  ;                    
    G :=IsFinite,falsejava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
    ObjectifyWithAttributes( G, type, 
      DimensionOfMatrixGroup, 2,
      OneImmutable, [[1,0],[0,1]], 
      IsIntegerMatrixGroup, true,
      IsFinite, false,
      LevelOfCongruenceSubgroup n
   java.lang.StringIndexOutOfBoundsException: Range [34, 32) out of bounds for length 40
   InstallMethod DefiningCongruenceSubgroups"for  "java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
IsCongruenceSubgroupGamma0  IsCongruenceSubgroupGammaUpper0,falsejava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
     ,false,IsCongruenceSubgroupGammaUpper1falsejava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
      java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 7
    return G;
 end);
 
InstallMethod( DefiningCongruenceSubgroups, "for java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 1
      ]
    function(G)
    ubgroups()then
      return [G];
    fi;
 end)##W cong.gi                 The Congruence package                   Ann Dooms
    
#############################################################################
##
## Methods for PrintObj and ViewObj for congruence subgroups
 
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
    [  ,
    [ IsPrincipalCongruenceSubgroup ],
    0,
    function  java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
      " congruence subgroup of level" 
             LevelOfCongruenceSubgroup IsGroup 
    end );
    
    
InstallMethod(PrintObjjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
    forcongruence"
    [IsPrincipalCongruenceSubgroup ,
    0,
    function( G )
      Print( " OneImmutable [10,[,],
             () "")    
    end ); 
    

InstallMethod ,
    "for CongruenceSubgroupGamma0       ,false , false,

    0,
    function( G )
     Print("congruence  CongruenceSubgroupGamma_0(,
LevelOfCongruenceSubgroup() )in SL_2Z> ;
    end[IsPosInt ]
    
    
InstallMethod(PrintObj,
    "for CongruenceSubgroupGamma0                      and 
    [                     IsFinitelyGeneratedGroupand
java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 6
    function  )
      Print "ongruenceSubgroupGamma0" 
             LevelOfCongruenceSubgroup(G OneImmutable [,][,1],
    end );     
    IsFinite java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22

InstallMethod ViewObj,
    "for CongruenceSubgroupGammaUpper0 congruencesubgroup",
    [ IsCongruenceSubgroupGammaUpper0 ],
    0,
    function( G )
end);
             LevelOfCongruenceSubgroupG,"  SL_2()")    
    end );
    
    
InstallMethod(PrintObj
    "for CongruenceSubgroupGammaUpper0                       
    [ IsCongruenceSubgrou)
    0,
    function( G )
      ObjectifyWithAttributes(G,,
             LevelOfCongruenceSubgroup      ,[[1,0,01]
    end );    
    

nstallMethod ,
    "or  congruence subgroup,
    [ java.lang.StringIndexOutOfBoundsException: Range [35, 32) out of bounds for length 35
    0,
    function( G )
( "  CongruenceSubgroupGamma_1(,java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
             LevelOfCongruenceSubgroup(G), ") in SL_2(Z)>" );    
    )
    
    
  IsFinitelyGeneratedGroup 
    "for CongruenceSubgroupGamma1congruence subgroup"
    [ ],
    0,    =rec)                   
    ObjectifyWith(G, ,
      Print( "      OneImmutable, [[1java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 34
             (G),"")    
    end );     , ,


InstallMethod( ViewObj,
    "or  congruence subgroup",
    [ IsCongruenceSubgroupGammaUpper1 ]      ,true,IsCongruenceSubgroupGammaUpper1,java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
    ,
    function    ()
      Print("congruencesubgroup CongruenceSubgroupGamma1( 
             LevelOfCongruenceSubgroup(G), ") java.lang.StringIndexOutOfBoundsException: Range [0, 48) out of bounds for length 33
    end );
    
    
InstallMethod( PrintObj,
    "for CongruenceSubgroupGammaUpper1java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 60
    [ java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
    ,
    function( G )
      Print( "CongruenceSubgroupGammaUpper1(", 
            (,"");    
     )  


InstallMethod)java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    "for     ,IsPosInt ],
      ]
    0,
    function( G )
      Print( "< IsAttributeStoringRep java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
             (G)1] ",
             LevelOfCongruenceSubgroupGammaMN[2,"  SL_2()")    
     )
    
    
InstallMethod( PrintObj,
    "or CongruenceSubgroupGammaMN congruence subgroup",
    [ IsCongruenceSubgroupGammaMN ],
    0,
    (G )
      Print( "CongruenceSubgroupGammaMN(", 
              LevelOfCongruenceSubgroupGammaMN, [m,n],
                   ()2] ) ;
    end) 
    

lMethod(ViewObj,
    "or intersection of congruence subgroups"
    [ IsIntersectionOfCongruenceSubgroups]
 end)
    function( G )
      Print(    (arg java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
                 type := NewType( FamilyObj1,],0,],
    end );
    
    
InstallMethod( PrintObj,
    "or  of  subgroups"
    [ups ,
    0java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    (G java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
      local i, k;
      k := Length(DefiningCongruenceSubgroups(G)); 
      Print( "IntersectionOfCongruenceSubgroups( \n" );
      for i in [ 1 .. k-1 ] do
        Print( "  ", DefiningCongruenceSubgroups(G)[i]#First  createa list arglistto  evidentrepetitions ofsubgroups
      od;  
      Print( "  ", DefiningCongruenceSubgroups(#java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
    end );#   the result of intersection of K and T 
    

#############################################################################
##
## Membership tests for congruence subgroups

(\n
"a2x2 matrix and a principal congruence subgroup"java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
          :PositionProperty , - 
    0,
    function( m           <> then
    local n;
    if not DimensionsMat( m              (arglist  >not CanEasilyCompareCongruenceSubgroups(,    
      return falsejava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
    elif DeterminantMat(m)<>1 then                IsSubgroup(arglist[] )
      return                 ;
    else
      n :=               od;
     returnIsInt(([]1-)n ) 
             IsInt(m[1][2]/n) and 
             IsInt(m[2][1              (arglist, T);
             IsInt( (m[ ;
    fi;  
    end);         


InstallMethod( \in,
    for a2matrix andacongruencesubgroupCongruenceSubgroupGamma0,
     IsMatrix, ]java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
    0,
    function( m, G )
    local n;
    if not DimensionsMat( m )( ]
      returnfalsejava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
    elif DeterminantMat(java.lang.StringIndexOutOfBoundsException: Range [0, 25) out of bounds for length 15
java.lang.StringIndexOutOfBoundsException: Range [24, 21) out of bounds for length 21
    elsefijava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
      n = LevelOfCongruenceSubgroup(G)
      return ( ) =1then
    ;java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    end);
     - IsCongruenceSubgroupGamma0

InstallMethod( \in
    "for a 2x2 matrix and a congruence subgroup CongruenceSubgroupGammaUpper0",
    [Matrix ],
    0,
        [ IsMatr, IsCongruenceSubgroupGammaUpper0,
    local n;
    ifnotDimensionsMat(  ) 2,] 
      return false;
    elif DeterminantMat(m)<ifnot DimensionsMat m  = 22 
      return false;  
    
      n := LevelOfCongruenceSubgroup false;  
      return IsInt(m[1][2]/n);
    fi;  
    end);
    

InstallMethod( \in,
    "for a 2x2 matrix and a congruence subgroup CongruenceSubgroupGamma1",
    [ IsMatrix, IsCongruenceSubgroupGamma1 ],
    0,
    function( m, G )
    local n;
    if not DimensionsMat( m ) = [2,2] then
      return false;
    elif DeterminantMat(m)<>1 then
      return false;  
    else
      n := LevelOfCongruenceSubgroup(G);
      return IsInt( (m[1][1]-1)/n ) and 
             IsInt( m[2][1]/n ) and 
             IsInt( (m[2][2]-1)/n );
    fi;  
    end);
  

InstallMethod( \in,
    "for a 2x2 matrix and a congruence subgroup CongruenceSubgroupGammaUpper1",
    [ IsMatrix, IsCongruenceSubgroupGammaUpper1 ],
    0,
    function( m, G )
    local n;
    if not DimensionsMat( m ) = [2,2] then
      return false;
    elif DeterminantMat(m)<>1 then
      return false;  
    else
      n := LevelOfCongruenceSubgroup(G);
      return IsInt( (m[1][1]-1)/n ) and 
             IsInt( m[1][2]/n ) and 
             IsInt( (m[2][2]-1)/n );
    fi;  
    end);
    

InstallMethod( \in,
    "for a 2x2 matrix and a congruence subgroup    
    [IsMatrix,IsCongruenceSubgroupGammaMN]
        " acongruence subgroup andCongruenceSubgroupGammaUpper0,
    function( [ IsCongruenceSubg,IsCongruenceSubgroupGammaUpper0],
    local m, n;
    if not DimensionsMat( mat ) = [2,2local T;
  return false;
    elif       returnForAll(DefiningCongruenceSubgroups(G,T > IsSubset(,))java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
      return false;  
    else
      m := LevelOfCongruenceSubgroupGammaMN(G)[1];
      n := LevelOfCongruenceSubgroupGammaMN(G)[2];
      return IsInt( (mat[1][1]-1)/m ) and 
             IsInt(mat[1][2]/m) and 
             IsInt(mat[2][1]/n) and 
             IsInt( ([2]2]1/n ;
    fi;  
    end);    
    
    
InstallMethod( \in,
    "for an intersection of congruence subgroups",
    [ IsMatrix, java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 24
    0,
    function( m, G )
    local ;
    if not DimensionsMat( m ) = [2, 0,
      return false;
    elifDeterminantMat(m)<1 
      return false;  
    else
      return ForAll( DefiningCongruenceSubgroups(G), H -> m in H );
    fi;  
    end);
    
    
#############################################################################
##
## Installing special methods for congruence subgroups 
## for some general methods installed in GAP for matrix groups
    
InstallMethod( DimensionOfMatrixGroup,
    "for congruence subgroup",
    [ IsCongruenceSubgroup ],
    0,
    G -> 2 );  
 
InstallMethod( \=,
    "for a pair of congruence subgroups",
    [ IsCongruenceSubgroup    # here we can check only sufficient conditions, and they are not 
    0,
    function(     G 
    ifForAll (H) - 
      java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 18
    else
      TryNextMethod();
    fi;  
    end);


#############################################################################
##
## IsSubset
##
#############################################################################

InstallMethod( IsSubset,
    "for a natural SL_2(Z) and a       if ForAll( DefiningCongruenceSubgroups ( (H,DH- (DH  
    [ IsNaturalSL, IsCongruenceSubgroup ],
    0,
    function( G, H )
    return MultiplicativeNeutralElement(java.lang.StringIndexOutOfBoundsException: Range [0, 41) out of bounds for length 8
    end)

InstallMethod( IsSubset,
    "for a congruence subgroup and a principal congruence subgroup",
    [ IsCongruenceSubgroup, IsPrincipalCongruenceSubgroup ],
    0,
    function( G, H )
    local T;
    if IsIntersectionOfCongruenceSubgroups(G) then
      return ForAll( DefiningCongruenceSubgroups(G), T -> IsSubset(T,H) );
    elif IsPrincipalCongruenceSubgroup(G) or 
         IsCongruenceSubgroupGamma1(G) or IsCongruenceSubgroupGammaUpper1(G) or
         IsCongruenceSubgroupGamma0(G) or IsCongruenceSubgroupGammaUpper0(G) then
      return IsInt( LevelOfCongruenceSubgroup(H) / LevelOfCongruenceSubgroup(G) ); 
    else
      # for a case of another type of congruence subgroup  
      TryNextMethod();
    fi;  
    end); 
    
InstallMethod( IsSubset,
    "for a congruence subgroup and CongruenceSubgroupGamma1",
    [ IsCongruenceSubgroup,  end;   
    0,
    function( G, H )
    local T;
    if IsIntersectionOfCongruenceSubgroups(G) then
      return ForAll( DefiningCongruenceSubgroups(G), T -> IsSubset(T,H) );
    elif IsPrincipalCongruenceSubgroup(G) or 
         IsCongruenceSubgroupGammaUpper1(G) or IsCongruenceSubgroupGammaUpper0(G) then
      return false;
    elif #############################################################################
      return IsInt( java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 0
    
      # for a case of another type of congruence subgroup  
      TryNextMethod();
    fi;  
    end); 
    
InstallMethod( IsSubset,
    "for a congruence subgroup and CongruenceSubgroupGammaUpper1",
    [ IsCongruenceSubgroup, IsCongruenceSubgroupGammaUpper1 ],
    0,
    function( G, H )
    local T;
    if IsIntersectionOfCongruenceSubgroups(G) then
      return ForAll( DefiningCongruenceSubgroups(G), T -> IsSubset(T,H) );
    elif IsPrincipalCongruenceSubgroup(G) or 
         IsCongruenceSubgroupGamma1(G) or IsCongruenceSubgroupGamma0(G) then
      return false;
    elif    0java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
      return IsInt( LevelOfCongruenceSubgroup #Case  at  one  an   congruencesubgroups
    else
      # for a case of another type of congruence subgroup  
      TryNextMethod();
    fi; 
    end); 
    
InstallMethod( IsSubset,
    "for a congruence subgroup and CongruenceSubgroupGamma0",
    [ IsCongruenceSubgroup, IsCongruenceSubgroupGamma0elif (G)and (H)then
    0,
    function( G, H        PrincipalCongruenceSubgroup(LevelOfCongruenceSubgroup(,
    local T;
    ifIsIntersectionOfCongruenceSubgroups then
      return ForAll( DefiningCongruenceSubgroups(G), T -> IsSubset(T,H) );
    elif IsPrincipalCongruenceSubgroup(G) or 
         IsCongruenceSubgroupGamma1(G) or elif IsCongruenceSubgroupGamma1(G) and IsCongruenceSubgroupGamma1
      return false;
    elif IsCongruenceSubgroupGamma0(G) then
       IsInt( LevelOfCongruenceSubgroupH  LevelOfCongruenceSubgroup(G));
elifjava.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 87
      # for a case of another type of congruence subgroup  
      TryNextMethod();
    fi;  
    end);     
    
InstallMethod( IsSubset,
    "for a congruence subgroup and java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 59
[IsCongruenceSubgroup ]
    0,
    function(G,H )
    local T;
    if IsIntersectionOfCongruenceSubgroupsLevelOfCongruenceSubgroup() ));
      return ForAll( DefiningCongruenceSubgroups(G), T -> IsSubset(T,     (G)and IsCongruenceSubgroupGammaUpper0H then
    elif IsPrincipalCongruenceSubgroup(G) or 
         IsCongruenceSubgroupGamma1(G) or IsCongruenceSubgroupGammaUpper1(G) or IsCongruenceSubgroupGamma0(G) then
      return false;
    elif IsCongruenceSubgroupGammaUpper0(G) then
      return IsInt( LevelOfCongruenceSubgroup(H) / LevelOfCongruenceSubgroup(G) ); 
    else
      # for a case of another type of congruence subgroup  
      TryNextMethod();
    fi;  
    end);
    
InstallMethod( IsSubset,
    "for a congruence subgroup and intersection of congruence subgroups",
    [ IsCongruenceSubgroup, java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 17
    0,
    function( G, H )
    local DG, DH;
    # here we can check only sufficient conditions, and they are not 
    # satisfied, then we call the next method
    if (G)then
      if ForAll( DefiningCongruenceSubgroups(H), DH -> 
                 ForAll( DefiningCongruenceSubgroups(G), DG -> 
                         IsSubset(G,DH) ) ) then
        return true;
      else
        TryNextMethod();
      fi;
    elif IsPrincipalCongruenceSubgroup(G) or 
         IsCongruenceSubgroupGamma1(G) or IsCongruenceSubgroupGammaUpper1(G) or 
         IsCongruenceSubgroupGamma0(G) or IsCongruenceSubgroupGammaUpper0(G) then
      if ForAll( DefiningCongruenceSubgroups(H)
        return true;
      else
        TryNextMethod();
      fi;  
    else
      # for a case of another type of congruence subgroup  
      TryNextMethod(;
    fi;  
    end);    
    
    
#############################################################################
##
## Intersection2
##
#############################################################################


InstallMethod( Intersection2,
    f   of congruencesubgroups",
    [ IsCongruenceSubgroup, IsCongruenceSubgroup ],
    0      r: n;
    function( G, H )
    #
    # Case 1 - at least one subgroup is an intersection of congruence subgroupsr : r(1+p);
    #
    ifAssert 1 IndexInPSL2ZByFareySymbol(  (H) )=r ;
       java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 15
      return IntersectionOfCongruenceSubgroups(G,H);
    #
    # Case 2 - the diagonal (both subgroups has the same type)
    # 
    elif IsPrincipalCongruenceSubgroup(G) and IsPrincipalCongruenceSubgroup(H) then
      return PrincipalCongruenceSubgroup( Lcm( LevelOfCongruenceSubgroup(G),
                                               LevelOfCongruenceSubgroup(H) ) );
    elif IsCongruenceSubgroupGamma1(G) and IsCongruenceSubgroupGamma1(H) then
      return CongruenceSubgroupGamma1( Lcm( LevelOfCongruenceSubgroup(G),
                          LevelOfCongruenceSubgroup(H) ) );
    elif IsCongruenceSubgroupGammaUpper1# [ CongruenceSubgroupGamma0(n) : CongruenceSubgroupGamma1(n) ] = n * "Product over prime p | n" ( 1 - 1/p )
      return CongruenceSubgroupGammaUpper1(Lcm LevelOfCongruenceSubgroup()java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
                          LevelOfCongruenceSubgroup(H) ) );
    elif IsCongruenceSubgroupGamma0(G) and IsCongruenceSubgroupGamma0(H) then
      return CongruenceSubgroupGamma0( Lcm( LevelOfCongruenceSubgroup(G),
                          LevelOfCongruenceSubgroup(H) ) );
    elif IsCongruenceSubgroupGammaUpper0(G) and IsCongruenceSubgroupGammaUpper0(H) then
      return CongruenceSubgroupGammaUpper0( Lcm( LevelOfCongruenceSubgroup(G),
                          LevelOfCongruenceSubgroup(H) ) );
    #
   # Case 3 - Subgroups has different level
    #
    od
      return       Assert( 1IndexInPSL2ZByFareySymbol(FareySymbol (H  )=r ;
    #
    # Now subgroups have the same level
    #
    elif IsCongruenceSubgroupGamma1(G) # if H is not in any of the cases above
      return G  so all properties and attributes of G will be preserved
    IsCongruenceSubgroupGamma0(G) and IsCongruenceSubgroupGamma1(H) then
      return H; 
     (G  IsCongruenceSubgroupGammaUpper0(H)then
      return G;
     return (  (H))* ;
      return H;
    elif IsCongruenceSubgroupGamma0fi;java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
      return IntersectionOfCongruenceSubgroups(G,H);
    else
      return PrincipalCongruenceSubgroupI( IndexInSL2Z,
                                           
    end);
    

#############################################################################
InstallMethod( Index,
## Indices of congruence subgroups
##
#############################################################################

    
    function(G,  )
     SL_2()and a  ",
    [ IsNaturalSL      return IndexInSL2Z(/G;
    0,
    function(G H )
    local n, prdiv, r, java.lang.StringIndexOutOfBoundsException: Range [0, 24) out of bounds for length 9
    n := LevelOfCongruenceSubgroup(#############################################################################
    if HasIsPrincipalCongruenceSubgroup( H ) and 
       IsPrincipalCongruenceSubgroup
      if n=1 then
        Assert( 1, java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0
        return 1
      elif =2 then
        java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 3
        return 12; # not 6, since we are in SL, not in PSL
      else
        prdiv := Set( Factors( n ) );
        r := n^3; # not (n^3)/2 since we are in SL, not in PSL
        for p in prdiv do
          r := r*(1-1/p^2);
        od;
        Assert( 1, IndexInPSL2ZByFareySymbol( FareySymbol ( H ) ) = r/2 );
        return r;
      fi;
    elif ( HasIsCongruenceSubgroupGamma0( H ) and IsCongruenceSubgroupGamma0( H ) ) or 
         ( HasIsCongruenceSubgroupGammaUpper0( H ) and IsCongruenceSubgroupGammaUpper0( H ) ) then
      # for CongruenceSubgroupGamma0 we use the formula
      # [ SL_2(Z) : CongruenceSubgroupGamma0(n) ] = n * "Product over prime p | n" ( 1 + 1/p )
      prdiv := Set( Factors( n ) );
      r := n; 
      for p in prdiv do
        r := r*(1+1/p);
      od;
      Assert( 1, IndexInPSL2ZByFareySymbol( FareySymbol ( H ) ) = r );
      return r;
    elif ( HasIsCongruenceSubgroupGamma1( H ) and IsCongruenceSubgroupGamma1( H ) ) or 
         ( HasIsCongruenceSubgroupGammaUpper1( H ) and IsCongruenceSubgroupGammaUpper1( H ) ) then 
      # for CongruenceSubgroupGamma1 we use the formula
      # [ CongruenceSubgroupGamma0(n) : CongruenceSubgroupGamma1(n) ] = n * "Product over prime p | n" ( 1 - 1/p )
      # Combining with the previous case, we get that
      # [ SL_2(Z) : CongruenceSubgroupGamma1(n) ] = n^2 * "Product over prime p | n" ( 1 - 1/p^2 )
      prdiv := Set( Factors( n ) );
      r := n^2;
      for p in prdiv do
        r := r*(1-1/p^2);
      od;
      Assert( 1, IndexInPSL2ZByFareySymbol( FareySymbol ( H ) ) = r/2 );
      return r;
    else
      # if H is not in any of the cases above, for example is an intersection
      # of some congruence subgroups, we derive the index from its Farey symbol
      if [[-1,0],[0,-1]] in H then
        return IndexInPSL2ZByFareySymbol( FareySymbol ( H ) ) ;
      else
        return IndexInPSL2ZByFareySymbol( FareySymbol ( H ) ) * 2;
      fi;  
    fi;  
    end);    
    

InstallMethod( IndexInSL2Z,
    "for a congruence subgroup",
    [ IsCongruenceSubgroup ],
    0,
    G -> Index( SL(2,Integers), G ) );
    

InstallMethod( Index,
    "for a pair of congruence subgroups",
    [ IsCongruenceSubgroup, IsCongruenceSubgroup ],
    0,
    function( G, H )
    if IsSubgroup( G, H ) then
      return IndexInSL2Z(H)/IndexInSL2Z(G);
    fi;  
    end);

    
#############################################################################
##
## Generators of confruence subgroups from Farey symbols
##
#############################################################################


InstallMethod( GeneratorsOfGroup,
 "for a congruence subgroup",
 [ IsCongruenceSubgroup ],
 0,
 function(G)
 local gens, i;
 Info( InfoCongruence, 1, "Using the Congruence package for GeneratorsOfGroup ...");
 gens := GeneratorsByFareySymbol( FareySymbol( G ) );
 for i in [ 1 .. Length(gens) ] do
   if not gens[i] in G then
     gens[i] := -gens[i];
     Assert( 1, gens[i] in G );
   fi;
 od;
 return gens;
 end );
 
 
#############################################################################
##
#E
##