SSL cong.gi   Sprache: Shell

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##
#W cong.gi                 The Congruence package                   Ann Dooms
#W                                                               Eric Jespers
#W                                                         Olexandr Konovalov
##
##
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##
## Constructors of congruence subgroups

InstallMethod( PrincipalCongruenceSubgroup, "for positive integer",
    [IsPosInt]java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
    function(    function(G )
    local type, G;
    type := NewType( FamilyObj([[[1,0],[0,1]] Print("principalcongruence    ,
                     and
                     IsAttributeStoringRep java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 10
                     (,
                     IsMatrixGroup and
                     IsCongruenceSubgroup);
    " principal congruence subgroup"
    ObjectifyWithAttributes( G    [IsPrincipalCongruenceSubgroup]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
      DimensionOfMatrixGroup, 2,
     ,[1,][,],
      IsIntegerMatrixGroup, true,
      IsFinite,             LevelOfCongruenceSubgroupG) ) )    
      java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 11
   IsPrincipalCongruenceSubgroup
      IsIntersectionOfCongruenceSubgroups(ViewObjjava.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
      IsCongruenceSubgroupGamma0 ,IsCongruenceSubgroupGammaUpper0false,
      IsCongruenceSubgroupGamma1, false, IsCongruenceSubgroupGammaUpper1
      java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 6
    return G;
  Print < subgroupCongruenceSubgroupGamma_0" 


             (G,"  SL_2()>")    
     IsPosInt ]
    function(n)
    java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 4
    InstallMethod java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
                     IsGroup  
                     IsAttributeStoringRep and 
                     IsFinitelyGeneratedGroup and 
                     IsMatrixGroup and
                     
    G := rec();                   
    ObjectifyWithAttributes( G    (G
            Print( ( C(,java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
     ,[10,0,1]java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
      IsIntegerMatrixGroup, true,
      ,false,
      LevelOfCongruenceSubgroup, n,
   IsPrincipalCongruenceSubgroup, false,
   IsIntersectionOfCongruenceSubgroups, false
      IsCongruenceSubgroupGamma0(,
      IsCongruenceSubgroupGamma1,     "for CongruenceSubgroupGammaUpper0 CongruenceSubgroupGammaUpper0  "java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
      java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 6
    return G;
 end;
 
InstallMethod(             LevelOfCongruenceSubgroup() )inSL_2Z> ;java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
    [ IsPosInt ],
    function(n)
    local 
    type := NewTypeInstallMethod( ,
                     IsGroup and 
IsAttributeStoringRep and 
                     IsFinitelyGeneratedGroup and 
                     IsMatrixGroup and
                     p;
    G :    0java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
     G,type,
      DimensionOfMatrixGroup, 2,
      OneImmutable [[1,0][,],
      IsIntegerMatrixGroup, true,
      IsFinite, false,
      java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 14
   IsPrincipalCongruenceSubgroup,
      IsIntersectionOfCongruenceSubgroups,(ViewObj
      IsCongruenceSubgroupGamma0,     "or CongruenceSubgroupGamma1congruence subgroup"
      IsCongruenceSubgroupGamma1 ],
      IsCongruenceSubgroupGammaMN, false );
    return G;
 end); 
 

InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 6
    [       Print"congruencesubgroup CongruenceSubgroupGamma_1( 
    function(n)
    local type, G;
    type := NewType(              LevelOfCongruencejava.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 65
                     IsGroup andend ;
                     IsAttributeStoringRep and 
                      and
                     IsMatrixGroup"  congruence subgroup,
                      IsCongruenceSubgroupGamma1
G: rec(;java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
    Attributes , type 
      DimensionOfMatrixGroup, 2,
      OneImmutable, [[1,0],[0,1]],
      IsIntegerMatrixGroup, true,
      IsFinite, false,
      LevelOfCongruenceSubgroup, nLevelOfCongruenceSubgroup(G ) )java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
ongruenceSubgroup falsejava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
   java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 0
      IsCongruenceSubgroupGamma0, false"or CongruenceSubgroupGammaUpper1java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 60
      IsCongruenceSubgroupGamma1 ,,false,
      IsCongruenceSubgroupGammaMN, false );
    return G;
 end);
 

InstallMethod( CongruenceSubgroupGammaUpper1 ,
    [ IsPosInt ],
    function(n
    local type, G;
    type :      Print("  ^1(,java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
                     IsGroup and 
                     IsAttributeStoringRep and 
                     IsFinitelyGeneratedGroup and 
                     IsMatrixGroup and
                     java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 4
        "  congruence subgroup",
    ObjectifyWithAttributes( G, type, 
      DimensionOfMatrixGroup, 2,
      OneImmutable, [[1,0],[0,1]],
      IsIntegerMatrixGroup, true,
      IsFinite, false,
      LevelOfCongruenceSubgroup, n,
   IsPrincipalCongruenceSubgroup, false,
   IsIntersectionOfCongruenceSubgroups0,
      java.lang.StringIndexOutOfBoundsException: Range [0, 32) out of bounds for length 17
      IsCongruenceSubgroupGamma1, false, IsCongruenceSubgroupGammaUpper1 LevelOfCongruenceSubgroup() ) );    
      IsCongruenceSubgroupGammaMNend)  
    return java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 end);
 

InstallMethod( CongruenceSubgroupGammaMN, "for two positive integers",
    [IsPosInt IsPosInt ],
    function(m,n)
    local [IsCongruenceSubgroupGammaMN,
    type := java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 6
                     IsGroup and 
                     and 
                     IsFinitelyGeneratedGroup and 
                     IsMatrixGroup and
                     LevelOfCongruenceSubgroupGammaMNG[] "" 
    G :=              (G)] "in SL_2Z> ;    
    ObjectifyWithAttributes( Gend;
      java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 4
      OneImmutable"java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 56
      IsIntegerMatrixGroup,
      function G java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
      LevelOfCongruenceSubgroup, m*n,
      java.lang.StringIndexOutOfBoundsException: Range [41, 38) out of bounds for length 46
          LevelOfCongruenceSubgroupGammaMNG[],"")    
   IsIntersectionOfCongruenceSubgroups ) 
      java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 4
      lMethod ,
      IsCongruenceSubgroupGammaMN,     "or intersection  congruence subgroups"
    return G    [ IsIntersectionOfCongruenceSubgroups ,
 end;
  

InstallGlobalFunction( IntersectionOfCongruenceSubgroups,
    function(arg )
    local type, G, H, K, T, arglist, n, i, pos;
    ([[[10],01]])java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
                     IsGroup
                     IsAttributeStoringRep and 
                     IsFinitelyGeneratedGroup andfintersection congruence,
                     [ IsIntersectionOfCongruenceSubgro]
    ,
    if not ForAll( arg, IsCongruenceSubgroup ) then
function G)
    fi;
      we  a list  toeliminate repetitions  .
    # 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
    #   the result is one of the canonical congruence subgroups, we replace K 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. 
    # - if we add to alglist a subgroup T and alglist already contains one or more
## Membership tests for congruence subgroups
    arglist
    for InstallMethod i,
    for  2 matrixaprincipal ,
        for T in DefiningCongruenceSubgroups( H ) do
pos=(arglist,K - 
                 CanReduceIntersectionOfCongruenceSubgroups( K0,
          ifpos<>ail then
            arglist[pos]:=Intersection n;
          else
            if ForAll ,K-  CanEasilyCompareCongruenceSubgroups(K, T))and
               ForAll( arglist, K -> notreturn false;
              for i in [ 1 .. Length(arglist) ] do
                if  arglisti,T  then
                  Unbind( arglist[i] );
                fi;
              od
              arglist := Compacted( arglist );      ((1[]1) )andjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
              Add arglist, T );
            fi;    
          fi;
        odjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
      else
        pos:=PositionProperty( arglist, K -> 
               java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        if pos<"  x2  and  congruence  CongruenceSubgroupGamma0"
          arglist[IsMatrix,IsCongruenceSubgroupGamma0,
        else     
          if ForAll( arglist, K -> not java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 6
             java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 12
ngtharglist)]do
              if IsSubgroup( arglist[i],       ;
                Unbind( arglist[i] );
              fi;
            od;
            arglist := Compacted( arglist );    
            Add( arglist
          fi;
        fi;  
      ;
    od;
    # if the list of defining subgroups was reduced 
    # to a single subgroup, we return this subgroup: LevelOfCongruenceSubgroup)
(arglist ) =1 
      return arglist[1];
    fifi; 
    # otherwise we sort the list of defining subgroups:
    # types of subgroups are sorted in the following way:;
# - IsCongruenceSubgroupGamma0
    # - IsCongruenceSubgroupGammaUpper0
    # - IsCongruenceSubgroupGamma1
ceSubgroupGammaUpper1
    # - IsPrincipalCongruenceSubgroup
    # and subgroups of the same type are sorted by ascending level
    Sort( arglist, 
          function(X,Y) 
          local f, t;
          f:=[IsCongruenceSubgroupGamma0,IsCongruenceSubgroupGammaUpper0,IsCongruenceSubgroupGamma1,IsCongruenceSubgroupGammaUpper1,IsPrincipalCongruenceSubgroup];
          return PositionProperty(f, t -> t(X)) < PositionProperty(f, t -> t(Y)) or
               ( PositionProperty(f, t -> t(X)) = PositionProperty(f, t -> t(Y)) and
                 LevelOfCongruenceSubgroup(X) < LevelOfCongruenceSubgroup(Y) ); 
          end );
    n := Lcm( List( arglist, H -> LevelOfCongruenceSubgroup(H) ) );                     
    G := rec();                   
    ObjectifyWithAttributes( G, type, 
      DimensionOfMatrixGroup, 2,
      OneImmutable, [[1,0],[0,1]], 
      IsIntegerMatrixGroup, true,
      IsFinite, false,
      LevelOfCongruenceSubgroup, n,
   IsPrincipalCongruenceSubgroup, false,
   IsIntersectionOfCongruenceSubgroups, true,
      IsCongruenceSubgroupGamma0, false, IsCongruenceSubgroupGammaUpper0, false,
      IsCongruenceSubgroupGamma1, false, IsCongruenceSubgroupGammaUpper1, false,
      DefiningCongruenceSubgroups, arglist );
    return G;
 end);
 
InstallMethod( DefiningCongruenceSubgroups, "for congruence subgroups",
    [ IsCongruenceSubgroup ],
    function(G)
    if not IsIntersectionOfCongruenceSubgroups(G) then
      return [G];
    fi;
 end);
    
#############################################################################
##
## Methods for PrintObj and ViewObj for congruence subgroups
 
InstallMethod( ViewObj,
    "for principal congruence subgroup",
    [ IsPrincipalCongruenceSubgroup ],
    0,
    function( G )
      Print( "<principal congruence subgroup of level ", 
             LevelOfCongruenceSubgroup(G), " in SL_2(Z)>" );    
    end );
    
    
InstallMethod( PrintObj,
    "for principal congruence subgroup",
    [ IsPrincipalCongruenceSubgroup ],
    0,
    function( G )
      Print( "PrincipalCongruenceSubgroup(", 
             LevelOfCongruenceSubgroup(G), ")" );    
    end ); 
    

InstallMethod( ViewObj,
    "for CongruenceSubgroupGamma0 congruence subgroup",
    [ IsCongruenceSubgroupGamma0 ],
    0,
    function( G )
      Print( "<congruence subgroup CongruenceSubgroupGamma_0(", 
             LevelOfCongruenceSubgroup(G), ") in SL_2(Z)>" );    
    end );
    
    
InstallMethod( PrintObj,
    "for CongruenceSubgroupGamma0 congruence subgroup",
    [ IsCongruenceSubgroupGamma0 ],
    0,
    function( G )
      Print( "CongruenceSubgroupGamma0(", 
             LevelOfCongruenceSubgroup(G), ")" );    
    end );     
    

InstallMethod( ViewObj,
    "for CongruenceSubgroupGammaUpper0 congruence subgroup",
    [ IsCongruenceSubgroupGammaUpper0 ],
    0,
    function( G )
      Print( "<congruence subgroup CongruenceSubgroupGamma^0(", 
             LevelOfCongruenceSubgroup(G), ") in SL_2(Z)>" );    
    end );
    
    
InstallMethod( PrintObj,
    "for CongruenceSubgroupGammaUpper0 congruence subgroup",
    [ IsCongruenceSubgroupGammaUpper0 ],
    0,
    function( G )
      Print( "CongruenceSubgroupGammaUpper0(", 
             LevelOfCongruenceSubgroup(G), ")" );    
    end );    
    

InstallMethod( ViewObj,
    "for CongruenceSubgroupGamma1 congruence subgroup",
    [ IsCongruenceSubgroupGamma1 ],
    0,
    function( G )
      Print( "<congruence subgroup CongruenceSubgroupGamma_1(", 
             LevelOfCongruenceSubgroup(G), ") in SL_2(Z)>" );    
    end );
    
    
InstallMethod( PrintObj,
    "for CongruenceSubgroupGamma1 congruence subgroup",
    [ IsCongruenceSubgroupGamma1 ],
    0,
    function( G )
      Print( "CongruenceSubgroupGamma1(", 
             LevelOfCongruenceSubgroup(G), ")" );    
    end );     


InstallMethod( ViewObj,
    "for CongruenceSubgroupGammaUpper1 congruence subgroup",
    [ IsCongruenceSubgroupGammaUpper1 ],
    0,
    function( G )
      Print( "<congruence subgroup CongruenceSubgroupGamma^1(", 
             LevelOfCongruenceSubgroup(G), ") in SL_2(Z)>" );    
    end );
    
    
InstallMethod( PrintObj,
    "for CongruenceSubgroupGammaUpper1 congruence subgroup",
    [ IsCongruenceSubgroupGammaUpper1 ],
    0,
    function( G )
      Print( "CongruenceSubgroupGammaUpper1(", 
             LevelOfCongruenceSubgroup(G), ")" );    
    end );  


InstallMethod( ViewObj,
    "for CongruenceSubgroupGammaMN congruence subgroup",
    [ IsCongruenceSubgroupGammaMN ],
    0,
    function( G )
      Print( "<congruence subgroup CongruenceSubgroupGammaMN(", 
             LevelOfCongruenceSubgroupGammaMN(G)[1], ",", 
             LevelOfCongruenceSubgroupGammaMN(G)[2], ") in SL_2(Z)>" );    
    end );
    
    
InstallMethod( PrintObj,
    "for CongruenceSubgroupGammaMN congruence subgroup",
    [ IsCongruenceSubgroupGammaMN ],
    0,
    function( G )
      Print( "CongruenceSubgroupGammaMN(", 
              LevelOfCongruenceSubgroupGammaMN(G)[1], ",", 
             LevelOfCongruenceSubgroupGammaMN(G)[2], ")" );    
    end ); 
    

InstallMethod( ViewObj,
    "for intersection of congruence subgroups",
    [ IsIntersectionOfCongruenceSubgroups ],
    0,
    function( G )
      Print( "<intersection of congruence subgroups of resulting level ", 
             LevelOfCongruenceSubgroup(G), " in SL_2(Z)>" );    
    end );
    
    
InstallMethod( PrintObj,
    "for intersection of congruence subgroups",
    [ IsIntersectionOfCongruenceSubgroups ],
    0,
    function( G )
      local i, k;
      k := Length(DefiningCongruenceSubgroups(G)); 
      Print( "IntersectionOfCongruenceSubgroups( \n" );
      for i in [ 1 .. k-1 ] do
        Print( "  ", DefiningCongruenceSubgroups(G)[i], ", \n" );
      od;  
      Print( "  ", DefiningCongruenceSubgroups(G)[k], " )" );    
    end );
    

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

InstallMethod( \in,
    "for a 2x2 matrix and a principal congruence subgroup",
    [ IsMatrix, IsPrincipalCongruenceSubgroup],
    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][1]/n) and 
             IsInt( (m[2][2]-1)/n );
    fi;  
    end);         


InstallMethod( \in,
    "for a 2x2 matrix and a congruence subgroup CongruenceSubgroupGamma0",
    [ IsMatrix, IsCongruenceSubgroupGamma0 ],
    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[2][1]/n);
    fi;  
    end); 
    

InstallMethod( \in,
    "for a 2x2 matrix and a java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 8
    fi;
    ,
    function( m, G )
    local n;
    if not DimensionsMat( m ) = [2,2] then
      return false;
    elif DeterminantMat(mInstallMethod(\in,
      return false;
    else
      n =LevelOfCongruenceSubgroup()
      return IsInt(m[0,
    fi;java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    endjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9


InstallMethod( \in,
    "for a    else
[IsMatrix, IsCongruenceSubgroupGamma1,
    0,
   function , java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
    local n;
    if              (2[]1/n )
      return falsejava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
    elif InstallMethod(in,
urnfalse; 
    else
      n: LevelOfCongruenceSubgroupG;
      return IsInt( (m[1][1]-1)/n ) and 
Int( m[]1]n)java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
             IsInt( (m[    ifnotDimensionsMat(m) =[22] then
    fi;  
    )java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
  

InstallMethod( \in,
    "for a 2x2 matrix and a congruence subgroup java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 40
    [IsMatrix,IsCongruenceSubgroupGammaUpper1 ],
    0,
    function( m, G )
    local ;
    java.lang.StringIndexOutOfBoundsException: Range [24, 6) out of bounds for length 42
      return false;
    elif DeterminantMat(m)<>1 
      return false;  
    else
      n := LevelOfCongruenceSubgroup(G);
      return (([]1]1)n)and 
                  ,  ]java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
             IsInt( (m[2][2]- ocal m,njava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
    fi;  
    end);
    

InstallMethod( \in,
    "for a 2x2java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
    [ IsMatrix, IsCongruenceSubgroupGammaMN ],
    0,
    function( mat,      returnIsInt((at[1-)m )andjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    local m, n;
    if not DimensionsMat( mat ) = [2,2] then
      java.lang.StringIndexOutOfBoundsException: Range [18, 12) out of bounds for length 19
t)>1 then
      return false;  
    else
    end);   
      n := java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 4
      for an intersectioncongruencesubgroups"
             mat[]2]) and 
             IsInt    0java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
             IsInt( (mat[2]    local ;
    fi; 
    end);    
    
   
InstallMethod( \in,
    "for an intersection of congruence subgroups",
    [ IsMatrixjava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8

    function( mfi;  
    ocal H
    
      return false;
    elif DeterminantMat(m)<>1 then
      return false;  
    else
      return ForAll( DefiningCongruenceSubgroups(G), H -> 
    fi;  
    end);
    
    
#############################################################################
##
## Installing special methods for congruence subgroups 
## for some general methods installed in GAP for matrix groups
    
    "forfora of  "
    "for congruence subgroup",
    [ ],
    0,
    G -> 2 );  
 
 \=java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
    "for a pairelse
    [TryNextMethod)java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
    0,
    function( G, H )
    if java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
      return true;
    else
      TryNextMethod();
    fi;  
    end);


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

InstallMethod( IsSubset,
    "for a     0,
    [,IsCongruenceSubgroup ,
    0,
    function( G, H )
MultiplicativeNeutralElement()[[1,0 , [0   ;
)

InstallMethod( IsSubset,
    "for a subgroup and aprincipal  "java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
    [ IsCongruenceSubgroup,    ,
    0,
    function( G, H )

    if     local ;
      (G),  ->T,H))
    elif IsPrincipalCongruenceSubgroup(      returnForAll( (G) T >IsSubset(,)java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
         IsCongruenceSubgroupGamma1()or (G)
         IsCongruenceSubgroupGamma0(G) or IsCongruenceSubgroupGammaUpper0(G) then
      return       return (LevelOfCongruenceSubgroup() / LevelOfCongruenceSubgroupG ;java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
    else
      # for a case of another type of congruence subgroup  
      TryNextMethodfi  
        end); 
    end); 
    
InstallMethod( IsSubset,
    "for a congruence subgroup and CongruenceSubgroupGamma1",
     "for a congruence subgroup  CongruenceSubgroupGamma1"java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
    0,
    (G,H)
    local T;
    ifIsIntersectionOfCongruenceSubgroupsG 
      (DefiningCongruenceSubgroupsG)  >IsSubset(,))
    elif    IsPrincipalCongruenceSubgroupG)or 
         IsCongruenceSubgroupGammaUpper1(G) or IsCongruenceSubgroupGammaUpper0(G) then
      false
     (G)or (G)then
      return IsInt( LevelOfCongruenceSubgroup(H) /      return  (H)/LevelOfCongruenceSubgroup()) 
     else
      # for a case of another type of congruence subgroup  
()
    fi;  
    end); 
    
InstallMethod IsSubset,
    "for a congruence subgroup and CongruenceSubgroupGammaUpper1",
[,IsCongruenceSubgroupGammaUpper1 ]java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
    0,
    function( G,  function( G, H)
    local T;
    if     local ;
oups(G,T- (T,))java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
    elifIsPrincipalCongruenceSubgroupG) or 
         IsCongruenceSubgroupGamma1elifIsPrincipalCongruenceSubgroup()or 
      return false;
    elif IsCongruenceSubgroupGammaUpper1(G) or IsCongruenceSubgroupGammaUpper0(G)       return false;
      (H) (G) ;java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
    else
      # for a case of another type of congruence subgroup  
      TryNextMethod();
    fi;  
    end); 
    
InstallMethod( IsSubset,
     nd) 
    [ IsCongruenceSubgroup
    0,
    function( G, H )
    ;
     IsIntersectionOfCongruenceSubgroupsG)java.lang.StringIndexOutOfBoundsException: Range [50, 51) out of bounds for length 50
      return ForAll( DefiningCongruenceSubgroups(G    local ;
    elifIsPrincipalCongruenceSubgroup) java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
IsCongruenceSubgroupGamma1() (G)or G then
      return falseelif java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 45
    elif IsCongruenceSubgroupGamma0(G) then
      return IsInt      return false;
    else
      # for a case of another type of congruence subgroup  
      TryNextMethod();
    fi;  
    end);     
    
InstallMethod( IsSubset,
    "fi;  
enceSubgroup,  ],
    0,
     )
    local T;
    if IsIntersectionOfCongruenceSubgroups(G) then
       ForAll(DefiningCongruenceSubgroupsG) T - (,));
    elif IsPrincipalCongruenceSubgroup(G) or 
             #
      false;
    elif IsCongruenceSubgroupGammaUpper0(G) then
      return (LevelOfCongruenceSubgroup()/LevelOfCongruenceSubgroup()); 
    else
      # for a case of another type of congruence subgroup  
      java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 5
    fi;  
    end)java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    
InstallMethod( IsSubset,
    "for a congruence subgroup and intersection of congruence    elif ()and ()then
    [, IsIntersectionOfCongruenceSubgroups,
    0 IsCongruenceSubgroupGammaUpper0()and (H 
    function( G, H       H;
     DG ;
    # here we can check only sufficient conditions, and they are not 
    # satisfied, then we call the next method
    if java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 8
      if ForAll( DefiningCongruenceSubgroups(H), DH -fi                                             
                 ForAll( java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 4
                         IsSubset(G,DH) ) ) then
        return true;
      else
        TryNextMethod();
      
     (G)or
         IsCongruenceSubgroupGamma1(InstallMethod(Index,
         IsCongruenceSubgroupGamma0(G) or IsCongruenceSubgroupGammaUpper0(G) then
      if ForAll( DefiningCongruenceSubgroups(H), DH -> IsSubset(G,DH) ) then
        return true;
      else
        TryNextMethod();
      fi;  
    else
      # for a case of another type of congruence subgroup  
      TryNextMethod();
    fi;  
    end);    
    
    
#############################################################################
##
## Intersection2
##
#############################################################################


InstallMethod(local  prdiv ,p;
    "n :=LevelOfCongruenceSubgroupH;
    [],
    0,
    function( G, H )
    #
    # Case 1 - at least one subgroup is an intersection of congruence subgroups
    #
    if IsIntersectionOfCongruenceSubgroups(G) or
       IsIntersectionOfCongruenceSubgroups(H) then
      return IntersectionOfCongruenceSubgroups(G,H);
    #
    # Case 2 - the diagonal (both subgroups has the same type)
    # 
    elif IsPrincipalCongruenceSubgroup(G) and IsPrincipalCongruenceSubgroup(H) then
      return PrincipalCongruenceSubgroup( Lcm  Congruencepackage GeneratorsOfGroup ..);
                                             java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 80
    elif IsCongruenceSubgroupGamma1(G) and if i]in G 
 gens[]: -ensi]java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
                          LevelOfCongruenceSubgroup(H) ) );
    elif IsCongruenceSubgroupGammaUpper1(G) and IsCongruenceSubgroupGammaUpper1(H) then
      return CongruenceSubgroupGammaUpper1( Lcm( LevelOfCongruenceSubgroup(G),
                          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
    #
    elif LevelOfCongruenceSubgroup(G) <> LevelOfCongruenceSubgroup(H) then
      return IntersectionOfCongruenceSubgroups(G,H);
    #
    # Now subgroups have the same level
    #
    elif IsCongruenceSubgroupGamma1(G) and IsCongruenceSubgroupGamma0(H) then
      return G; # so all properties and attributes of G will be preserved
    elif IsCongruenceSubgroupGamma0(G) and IsCongruenceSubgroupGamma1(H) then
      return H; 
    elif IsCongruenceSubgroupGammaUpper1(G) and IsCongruenceSubgroupGammaUpper0(H) then
      return G;
    elif IsCongruenceSubgroupGammaUpper0(G) and IsCongruenceSubgroupGammaUpper1(H) then
      return H;
    elif IsCongruenceSubgroupGamma0(G) and  end ;
      return IntersectionOfCongruenceSubgroups(G,H);
    else
      return PrincipalCongruenceSubgroup(LevelOfCongruenceSubgroup(G)##
    ##
    end);
    

#############################################################################
##
## Indices of congruence subgroups
##
#############################################################################

    
InstallMethod( Index,
    "for a natural SL_2(Z) and a congruence subgroup",
    [ IsNaturalSL, IsCongruenceSubgroup ],
    0,
    function( G, H )
    local n, prdiv, r, p;
    n := LevelOfCongruenceSubgroup(H);     
    if HasIsPrincipalCongruenceSubgroup( H ) and 
       IsPrincipalCongruenceSubgroup( H ) then
      if n=1 then
        Assert( 1, IndexInPSL2ZByFareySymbol( FareySymbol ( H ) ) = 1 );
        return 1;
      elif n=2 then
        Assert( 1, IndexInPSL2ZByFareySymbol( FareySymbol ( H ) ) = 6 );
        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
##