Quelle  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 ],
    function(n)
    local type, G;
    type := NewType( FamilyObj([[[1,0],[0,1]]]),
                     local type,G
                     IsAttributeStoringRep
                     java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 33
                     IsMatrixGroup IsCongruenceSubgroup);
                     IsCongruenceSubgroup)
    G :    ObjectifyWithAttributes G,type 
    bjectifyWithAttributesG,  
DimensionOfMatrixGroup2,
      OneImmutable, [[1,0],[0,1]],
      IsIntegerMatrixGroup, true,
      IsIntegerMatrixGroup,true,
      LevelOfCongruenceSubgroup, n,
   IsPrincipalCongruenceSubgroup, true,
            IsFinite ,
      IsCongruenceSubgroupGamma0, false, java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 35
      uenceSubgroupGammaUpper1, false,
      IsCongruenceSubgroupGammaMN, false );
    return G;
 end);


InstallMethod , " positive integer",
          , false ;
    function(n)
    local type, G;
    type := NewType( java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 6
                     IsGroup and 
                     IsAttributeStoringRep and 
                     IsFinitelyGeneratedGroupand 
                     java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 15
                     IsCongruenceSubgroupjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
    G := rec( IsAttributeStoringRep  
     IsFinitelyGeneratedGroupjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
      DimensionOfMatrixGroupIsCongruenceSubgroup;
      OneImmutable, [     : rec);                  
      java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 33
      ,false,
       IsIntegerMatrixGroup ,
       ,falsejava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
   IsIntersectionOfCongruenceSubgroups,,
      IsCongruenceSubgroupGamma0   IsIntersectionOfCongruenceSubgroups, ,
      IsCongruenceSubgroupGamma1,false, IsCongruenceSubgroupGammaUpper1, false,
            IsCongruenceSubgroupGamma1java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 80
    return G;
 end);
 
InstallMethod( CongruenceSubgroupGammaUpper0, "for positive integer"    return G;
 IsPosInt ]
function()
        njava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
type:= FamilyObj([10,0,])java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
IsGroup 
                     IsAttributeStoringRep                      IsFinitelyGeneratedGroup 
                     IsFinitelyGeneratedGroup and 
                     IsMatrixGroup                      IsCongruenceSubgr;
                     IsCongruenceSubgroupG:=rec(;
     ObjectifyWithAttributes   
      , 2java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
            IsIntegerMatrixGroup, 
      OneImmutable,[1,][,]]java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
      IsIntegerMatrixGrouptrue,
      IsFinite, false,ceSubgroup, false,
            , falsejava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
 IsPrincipalCongruenceSubgroup,
      IsIntersectionOfCongruenceSubgroupsjava.lang.StringIndexOutOfBoundsException: Range [34, 32) out of bounds for length 80
      ,  , true,
      IsCongruenceSubgroupGamma1, false, IsCongruenceSubgroupGammaUpper1, false,
      java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 13
    return G;
 end); 
 

 IsPosIntjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
    [ IsPosInt ],
    function(n)
    local type, G                      and 
    type := NewType( IsAttributeStoringRep and 
                     IsGroup and 
                     IsAttributeStoringRep                      IsFinitelyGeneratedGroupand 
                     IsCongruenceSubgroup;
                     G : rec(;java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
                     );
    G :=rec(;                  
    ObjectifyWithAttributes( G,     IsIntegerMatrixGroup,true,
     , 2,
      OneImmutable, [[1,0bgroup, ,
      IsIntegerMatrixGroup,true,
       IsIntersectionOfCongruenceSubgroupsfalsejava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 35
   IsPrincipalCongruenceSubgroup, false,
   IsIntersectionOfCongruenceSubgroups, false,
      IsCongruenceSubgroupGamma0, false, IsCongruenceSubgroupGammaUpper0, false,
      IsCongruenceSubgroupGamma1,true IsCongruenceSubgroupGammaUpper1, false,
      IsCongruenceSubgroupGammaMN, false );
    return G;
     re G;
 

InstallMethod( CongruenceSubgroupGammaUpper1, "for positive integer",
    [ IsPosIntInstallMethod( CongruenceSubgroupGammaUpper1,"for positive integer",
    ()
    local type, G;
    type := NewType( FamilyObj([[[1,0],[0,1]]]),
                     IsGroup  
                     IsAttributeStoringRep and 
                     sFinitelyGeneratedGroup and 
                     IsMatrixGroup and
                     IsCongruenceSubgroup);    type =NewType(FamilyObj([10]]]),
    G := rec();                   
    ObjectifyWithAttributes(                      IsAttribu and 
      , 2,
      OneImmutable, [[1,0],[0,1]],
                           IsMat and
      ,false,
      LevelOfCongruenceSubgroupG=rec(;java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
   IsPrincipalCongruenceSubgroup,,
   OneImmutable, [[1,0],[0,1]],
      IsIntegerMatrixGrouptrue,
      IsFinite ,
      IsCongruenceSubgroupGammaMN, false );
    return GLevelOfCongruenceSubgroup ,
 end);
 

( CongruenceSubgroupGammaMN for two positive integers",
    [ IsPosInt, IsPosInt ],
    function(m,n)
    local type,G
    type java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 46
                     and 
             IsAttributeStoringRep and
                     
                     IsMatrixGroup and
                     IsCongruenceSubgroup;
    Insta(, for  positive integers,
    ObjectifyWithAttributes(G , 
      java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 17
     OneImmutable,[10]01]]
      IsIntegerMatrixGroup ,
      IsFinite, false,
      , m*,
      LevelOfCongruenceSubgroupGammaMN, [m,n],
   IsPrincipalCongruenceSubgroup, false,
   ,false,
      IsCongruenceSubgroupGamma0, false, IsCongruenceSubgroupGammaUpper0, false,
      IsCongruenceSubgroupGamma1,  IsCongruenceSubgroup);
       G =rec(;                  
    return ;
 end DimensionOfMatrixGroup2,
  

InstallGlobalFunction(       OneImmutable, [[1,0],[0,
    function( arg )
    local type,IsIntegerMatrixGroup true,
          IsFinite,falsejava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
                     java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 33
                     java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 47
                      and 
                      and
                     )
      ForAll( arg,   
      Error("Usage :);
    fijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 we  a arglist  evident repetitions  subgroups
     Then we eliminate evident inclusions of one subgroup into another:
    # - since intersection is associative, if we can intersect the group T whichtype: (FamilyObj[[[1,][,]],
    #   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 byjava.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 50
    #   the result of intersection of K and T      ) then
     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
    #   subgroups K such that T is in K, we add T and remove all these K.
    arglist := [];
        #    isone canonical subgroups,we  K
      if IsIntersectionOfCongruenceSubgroups# - 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 -java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
                 CanReduceIntersectionOfCongruenceSubgroupsarglist =[;
          if pos<>fail then
            arglist for H in arg 
          else
            if ForAll( arglist, K -        forTDefiningCongruenceSubgroups )do
               ForAll( arglist,         pos:PositionProperty ,K - 
              for i  1. ()]do
                ifIsSubgroup [i] )
                  Unbind(arglist[]=Intersection( arglistpos]  )
                fi;
              od;fi;
              arglist := Compacted( arglist );    
              Add( arglist, T );
            fi;    
          fi;
        od
      else
        java.lang.StringIndexOutOfBoundsException: Range [29, 11) out of bounds for length 45
fi
        if        od;
          :=PositionProperty,K ->
        else     
          if ForAll( arglist, K -> not CanEasilyCompareCongruenceSubgroups( 
                       [os:I(arglist[pos],H );
           for i           if ForAllarglist,  -not ( ,H ) )) and
              ifIsSubgroup []H  
                Unbind( arglist[i] );
              ;
            od;
            arglist := Compacted( arglist );    
           ( java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 30
          fi;
        fi;  
      fi;
    od
    # if the list of defining subgroups was reduced (,H)
    # to a single subgroup, we return this subgroup
    if       ;
      return arglist[1];
    fi; 
    # otherwise we sort the list of defining subgroups:
    # types of subgroups are sorted in the following way:
    # - IsCongruenceSubgroupGamma0
    # - IsCongruenceSubgroupGammaUpper0
    # - IsCongruenceSubgroupGamma1
    # - IsCongruenceSubgroupGammaUpper1
    # - IsPrincipalCongruenceSubgroup
    # and subgroups of the same type are sorted by ascending level
    Sort ,
          function(X,Y) 
          local f, t;
          f:#-IsCongruenceSubgroupGammaUpper0
          return java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
               ( java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 19
CongruenceSubgroup(X  (Y) )java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
          end )          :[IsCongruenceSubgroupGamma0,,IsCongruenceSubgroupGamma1IsCongruenceSubgroupGammaUpper1];
> LevelOfCongruenceSubgroup(H) ) );                     
    G := rec();                   
    ( G,type, 
      DimensionOfMatrixGroup, 2,
      OneImmutable, [[1,0],                 (X  (Y) );
      IsIntegerMatrixGroup,   -> LevelOfCongruenceSubgroupH)));                    
     ,,
      LevelOfCongruenceSubgroup, n,
   IsPrincipalCongruenceSubgroup, java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
   java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 35
      IsCongruenceSubgroupGamma0, false, java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 33
      IsCongruenceSubgroupGamma1, n,
      IsPrincipalCongruenceSubgroup, false,
    return G;
 end);
 
( DefiningCongruenceSubgroups, "for congruencesubgroups,
    [ IsCongruenceSubgroup ],
    function      ,false, ,false,
    if not IsIntersectionOfCongruenceSubgroups(G IsCongruenceSubgroupGamma1 ,IsCongruenceSubgroupGammaUpper1, false,
      return [G];
    fi;
 end);
    
#############################################################################
##
## Methods for PrintObj and ViewObj for congruence subgroups
 
InstallMethod( ViewObj,
    "for principal congruence subgroup[IsCongruenceSubgroup,
    [ if not IsIntersectionOfCongruenceS()
    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,
    " Is, IsCongruenceSubgroupGammaUpper0],
ix ]
    0,
    functionif not  m)=[2,]then
    local n;
      ( m)= [,]then
      return false;
    elif DeterminantMat(m)<>else
      return ;  
    else
      n := LevelOfCongruenceSubgroup(G);
      return IsInt(m[1][2]/n);
    ; 
    0java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    

InstallMethod \,
    "for a        false; 
    [ IsMatrix      n: LevelOfCongruenceSubgroupG;
    0,
    function( m, fi  
    local n;
    if not DimensionsMat( m ) =);
      return false;
    elif
      return java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
    
      n :=     ,  ],
      return IsInt( (m[1][function( m,G )
             IsInt( m[2][1]/n ) and 
             IsInt((m[]2-);
    fi;  
    endreturnfalse;
  

 \,
    "for  false; 
    [ IsMatrixn : ()java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
    0,
    function(              Is( m[2[]/  and 
    local n;
if  DimensionsMat  )=[2,2]then
      return false;
    elif DeterminantMat(m)<>1end);
      return false;  
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      n := LevelOfCongruenceSubgroup(G);
      return IsInt([ java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 50
             n
             if not DimensionsMat( m ) = [2,2] then
    fi;  

    

InstallMethod( \in,
    "for a 2x2 matrix and areturnIsInt( m1[1]1/n  and 
    [IsMatrix, IsCongruenceSubgroupGammaMN],
    0,
    function( mat, G )
    ocal m ;
    if not DimensionsMat( mat ) = [2,2] then
      return false;
    elif java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 4
      return false;  
    else
      m := LevelOfCongruenceSubgroupGammaMN(G)[1];
      n := LevelOfCongruenceSubgroupGammaMN(G)[2];
      return  (at1][]1/m )and 
             IsInt(mat[1][2]/m) and 
             return false;
             IsInt( (mat[2)1 
    fielse
    );   
    
    
InstallMethod( \in,
    " of congruence ,
             IsInt(1[2/m) and 
    0,
    function( m, G )
    local H
    if not DimensionsMat( m   
      return false;
    elif DeterminantMat(m)<> java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
      return false;  
    else
      return ForAll( DefiningCongruenceSubgroups(G), H -> m in 
    fi;
    end) ocalH;
    
    
#############################################################################
##
      return false;  
## for some general methods installed in GAP for matrix groups

InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 9
    "for 
    [ IsCongruenceSubgroup ],
##
    G -> 2 );  
 
InstallMethod( \=,
    " a pair of congruence subgroups"java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
    [  IsCongruenceSubgroup
    0,
    function( G, H )
    if CanEasilyCompareCongruenceSubgroups( G, HInstallMethod(\,
      return true;
    
      (;
    fi;  
    end    java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6


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

InstallMethod( IsSubset,
    "for a natural SL_2(Z) and a java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 2
    [ java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0
    0,
    function( G, H    [IsNaturalSL,IsCongruenceSubgroup ]java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    return MultiplicativeNeutralElement(G)    return MultiplicativeNeutralElement(G= [1 0],  ,1]]
    end)

 IsSubset,
    "    "for"for congruence subgroupand a principalcongruencesubgroup",
    [ IsCongruenceSubgroup, IsPrincipalCongruenceSubgroup ],
    0java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    function( 
    local T;
    if return ForAll( DefiningCongruenceSubgroups,T->IsSubset(,) ;
      return (DefiningCongruenceSubgroups,T- IsSubsetTH );
    elif IsPrincipalCongruenceSubgroup(G) or 
                  IsCongruenceSubgroupGamma1G IsCongruenceSubgroupGammaUpper1G)or
         java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 81
       IsInt LevelOfCongruenceSubgroup(());
    else
      # for a case of another type of congruence subgroup  
      TryNextMethod();
    ;
end 
    
InstallMethod( IsSubsetjava.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 24
   foracongruence and ,
    [ IsCongruenceSubgroup, IsCongruenceSubgroupGamma1 ],
    0,
    functionfunction(  )
    local T;
 IsIntersectionOfCongruenceSubgroups()then
      return ForAll(       return ForAll(DefiningCongruenceSubgroups(),T- TH ;
elif ()or
         java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 86
return false;
    elif elifIsCongruenceSubgroupGamma1()  IsCongruenceSubgroupGamma0G)then
returnIsInt(LevelOfCongruenceSubgroup(  LevelOfCongruenceSubgroup( ; 
    else
      # for a case of another type of congruence subgroup  
      TryNextMethod)
    fi;  
    end); 
    
InstallMethodInstallMethod(IsSubsetjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
    "for a congruence subgroup     IsCongruenceSubgroup,  ,
    [ IsCongruenceSubgroup, IsCongruenceSubgroupGammaUpper1 ],
    0,
    ( G, H java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
    localT;
    if IsIntersectionOfCongruenceSubgroups(      return ForAll( DefiningCongruenceSubgr)  >IsSubset(,H);
      return ForAll(      () or 
    elif G  
         IsCongruenceSubgroupGamma1(G) java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 19
      returnfalse;
    elif return IsInt( LevelOfCongruenceSubgroup /LevelOfCongruenceSubgroupG) );
      return IsInt( LevelOfCongruenceSubgroup(H) / LevelOfCongruenceSubgroup(G) ); 
          java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
      # for a case of another type of congruence subgroup  
      TryNextMethod();
    fi;  
    nd) 
    
java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 6
    local T;
    [    if()then
    0,
    function( G, H )
    local T
    if IsIntersectionOfCongruenceSubgroups(G) elif (G) or 
      return ForAll( DefiningCongruenceSubgroups         ()or IsCongruenceSubgroupGammaUpper1G IsCongruenceSubgroupGammaUpper0()then
    IsPrincipalCongruenceSubgroup(G) or 
         IsCongruenceSubgroupGamma1(G) or java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 43
 ;
    elif IsCongruenceSubgroupGamma0(G) then
            # for a case of another type of congruence subgroup  
    else
      # for a case of another type of congruence subgroup  
      java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 4
    fi;
    end);     , IsCongruenceSubgroupGammaUpper0

InstallMethod( , ,
    "for    "
    roup  ],
    0,
    function( G, H )
    T;
    return false;
      return (DefiningCongruenceSubgroupsG) -> IsSubsetTH ;
    elif IsPrincipalCongruenceSubgroup(G) or 
         java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 21
      return false;
    elif IsCongruenceSubgroupGammaUpper0(G) then
      return IsInt( LevelOfCongruenceSubgroup(H) / java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 36
    else
      # for a case of another type of congruence subgroup   mat[]2]-)n )java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
      TryNextMethod()java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
    fi;  
    end);
    
InstallMethod( IsSubset,
    "for a congruence subgroup and intersection of congruence subgroups",
    [ IsCongruenceSubgroup, IsIntersectionOfCongruenceSubgroupsH
    0java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    function elif (m)<1 then
    local java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 8
    # here we can check only sufficient conditions, and they are not 
    # satisfied, then we call the next method
    ifIsIntersectionOfCongruenceSubgroups()then
   (DefiningCongruenceSubgroupsH,DH -
                 ForAllreturn true;
                         IsSubset(G,DH) ) ) then
        return true;
      else
         end;
      fi;
    #############################################################################
         IsCongruenceSubgroupGamma1## IsSubset
         IsCongruenceSubgroupGamma0(G) or IsCongruenceSubgroupGammaUpper0(Gjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 0
      if ForAll(DefiningCongruenceSubgroups)  >IsSubsetG,))then
        return true;
      else
        TryNextMethod();
      fi;  
    else
      # for a case of another type of congruence subgroup  
          end)
    fi;  
   );
    
    
#############################################################################
##
## Intersection2
##
#############################################################################


InstallMethod( Intersection2,
    "for a pair of congruence    else
java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 76
    ,
    function( G, H )
    #
     1-atleastonesubgroupis  intersectionof  
    #
    if      # for a case of another type of congruence subgroup  
       IsIntersectionOfCongruenceSubgroups(H)     
      return IntersectionOfCongruenceSubgroups(G,H);
    #
    # Case 2 - the diagonal (both subgroups has the same type)
    # 
    elifIsPrincipalCongruenceSubgroup)IsPrincipalCongruenceSubgroup(H java.lang.StringIndexOutOfBoundsException: Range [83, 84) out of bounds for length 83
return ( Lcm (G)java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
                                               LevelOfCongruenceSubgroup (G)then
    (H) then
      return java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 19
                          LevelOfCongruenceSubgroupreturn IsInt( ()/LevelOfCongruenceSubgroup(G ;
    elif IsCongruenceSubgroupGammaUpper1(G) and IsCongruenceSubgroupGammaUpper1(H) then
      return CongruenceSubgroupGammaUpper1( Lcm( java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 4
                          LevelOfCongruenceSubgroup(H) ) );
     , IsCongruenceSubgroupGammaUpper0,
      return CongruenceSubgroupGamma0(   ,  java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
                          ()) )java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
elifIsCongruenceSubgroupGammaUpper0G)and IsCongruenceSubgroupGammaUpper0() then
      return function( G, H
                          java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 12
    #
return  ()  >IsSubsetT,H )java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
    #
    elifreturn java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
      return IntersectionOfCongruenceSubgroupsIsInt (H)/ ( ;
    #
    # Now subgroups have the same level
    #
    elif IsCongruenceSubgroupGamma1(G) and IsCongruenceSubgroupGamma0(H) then
      returnend;
    elif IsCongruenceSubgroupGamma0(G) and IsCongruenceSubgroupGamma1(H) then
      return H; 
    elif IsCongruenceSubgroupGammaUpper1G) IsCongruenceSubgroupGammaUpper0H then
      return     IsCongruenceSubgroup,  ]
lifIsCongruenceSubgroupGammaUpper0G)andIsCongruenceSubgroupGammaUpper1H)then
return ;
localDG,DHjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
      return IntersectionOfCongruenceSubgroups(G,    # satisfied, then we call the next method
    else
      return PrincipalCongruenceSubgroup(LevelOfCongruenceSubgroup(G));
   ;                                             
    end);
    

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

elif IsPrincipalCongruenceSubgroupG  
 Indexjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
    "for##
    [ IsNaturalSL, IsCongruenceSubgroup ],##
    0,
    function( G, 
    n,prdiv,r,p;
    : ()     
    if [ IsCongruenceSubgroup, IsCongruenceSubgroup
       IsPrincipalCongruenceSubgroup( H ) then
      if n=1 then
        Assert( 1, IndexInPSL2ZByFareySymbol( FareySymbol ( H    ,
        return 1;
      elif n=2 then
        Assert( 1,     # satisfied, then we call
            IsIntersectionOfCongruenceSubgroups)then
      else

        ;
        for p in prdiv do
          r := r*(1-1/p^2);
        od;
        AssertTryNextMethod)java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
        return r;
      fi;
    elif ( HasIsCongruenceSubgroupGamma0
         ( HasIsCongruenceSubgroupGammaUpper0( H ) and ##
      # for CongruenceSubgroupGamma0 we use the formula##
      # [ SL_2(Z) : CongruenceSubgroupGamma0(n) ] = n * "Product over prime p | n" ( 1 + 1/p )
      prdiv := Set( Factors( n ) )    "orapair  congruence subgroups"java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
      r =n; 
      for p in prdiv java.lang.StringIndexOutOfBoundsException: Range [0, 23) out of bounds for length 20
        r : *(1+/p)java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
      od;
      ( 1,IndexInPSL2ZByFareySymbol(FareySymbol(  ))  r);
      return r;
    elif ( HasIsCongruenceSubgroupGamma1( H ) and IsCongruenceSubgroupGamma1( H ) ) or 
         ( java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 5
      # 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 thatCongruenceSubgroupGammaUpper1 (LevelOfCongruenceSubgroupG,
      # [ SL_2(Z) : CongruenceSubgroupGamma1(n) ] = n^2 * "Product over prime p | n" ( 1 - 1/p^2 )
      prdiv := Set( Factors    java.lang.StringIndexOutOfBoundsException: Range [87, 40) out of bounds for length 87
      r := n^2;
      for p in  
        r     
      ;
,   ( ))  /2)java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
          # Now subgroups have the same level
    else
      , for example is an intersection
      # of some congruence subgroups, we derive the index from its Farey symbol;#
      ifelif java.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 77
        return IndexInPSL2ZByFareySymbol( FareySymbolelifIsCongruenceSubgroupGammaUpper1)and IsCongruenceSubgroupGammaUpper0(H)then
      else
        returnIndexInPSL2ZByFareySymbol(FareySymbol  H )  2java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
      fi;  
    fi;  
    end);    
    

nstallMethod( IndexInSL2Z,
    "for 
    [ IsCongruenceSubgroup ],
    0,
    G -> Index( java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
    

InstallMethod( Index,
    "for a pair of congruence subgroups",
    [ IsCongruenceSubgroup, IsCongruenceSubgroup ],
    0,    
    (, H)
    if "for a naturalZ  acongruencesubgroup",
       ()IndexInSL2Z()java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
        fufunction ,H )
    end);

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


InstallMethod( GeneratorsOfGroup,
          ;
 [      n2 then
 0,
 function(G)
 local gens, i;
 Info( InfoCongruence, 1, "Usingthe Congruence  for  ..";
   LevelOfCongruenceSubgroup(H) ) );
 for i in [ 1 .. Length(gens) ] do
    notgens[]   then
     i =-ens[;
     Assert( 1, gens[java.lang.StringIndexOutOfBoundsException: Range [44, 22) out of bounds for length 87
   fi;
 od;
 return gens;
 end)
 
 
#############################################################################
##
#E
##