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8 N-groups

Sections

  1. Construction of N-groups
  2. Operation tables of N-groups
  3. Functions for N-groups
  4. N-subgroups
  5. N0-subgroups
  6. Ideals of N-groups
  7. Special properties of N-groups
  8. Noetherian quotients
  9. Nearring radicals

In SONATA every N-group is a group, the only difference is, that there is a nearring that acts on the group. And since in SONATA all nearrings are left distributive, they act on the elements of an N-group from the right. Note, that the elements of an N-group are added via *, not +.

The functions described in this section can be found in the source files ngroups.g?.

8.1 Construction of N-groups

There is a natural way to construct an N-group. It is to take a group, a nearring and define an action of the nearring on the group. The function NGroup allows one to do this. The special case, where (the group reduct of) a nearring is viewed as an N-group over the nearring itself, can be constructed easily via NGroupByNearRingMultiplication.

  • NGroup( G, nr, action )

    The function NGroup has three arguments. G must be a group, nr the nearring that acts on the group and action a binary operation from the direct product of G and nr into G. It returns the N-group.

        gap> G := GTW4_2;                
        4/2
        gap> n := MapNearRing( G );
        TransformationNearRing(4/2)
        gap> action := function ( g, f )
        > return Image( f, g );
        > end;  
        function ( g, f ) ... end
        gap> gamma := NGroup( G, n, action );
        < N-group of TransformationNearRing(4/2) >
        gap> IsNGroup( gamma );
        true
        gap> NearRingActingOnNGroup( gamma );
        TransformationNearRing(4/2)
        gap> ActionOfNearRingOnNGroup( gamma );
        function ( g, f ) ... end
        gap> Print( ActionOfNearRingOnNGroup( gamma ) );
        function ( g, f )
            return Image( f, g );
    

  • NGroupByNearRingMultiplication( nr )

    For every (left) nearring (N,+,.) the group (N,+) is an N-group over N with respect to nearring multiplication from the right as the action. The function NGroupByNearRingMultiplication returns this N-group of the nearring nr.

        gap> n := LibraryNearRing( GTW8_2, 3 );
        LibraryNearRing(8/2, 3)
        gap> NGroupByNearRingMultiplication( n ) = GTW8_2;
        true
    

  • NGroupByApplication( tfmnr )

    For a nearring T of transformations on a group G, G is an N-group of T with the application of functions as the action. The function NGroupByApplication returns this N-group of the nearring tfmnr.

    Another way to construct an N-Group is to take a nearring N, a right ideal R and let N act on the factor N/R in the canonical way. This is accomplished by

  • NGroupByRightIdealFactor( nr, R )

    The function NGroupByRightIdealFactor has two arguments, a nearring nr and a right ideal R. It returns the N-group nr/R.

        gap> N := LibraryNearRing( GTW4_2, 11 );
        LibraryNearRing(4/2, 11)
        gap> R := NearRingRightIdeals( N )[ 3 ];
        < nearring right ideal >
        gap> ng := NGroupByRightIdealFactor( N, R );
        < N-group of LibraryNearRing(4/2, 11) >
        gap> PrintTable( ng );
        Let:
        (0,0) := (())
        (1,0) := ((3,4))
        (0,1) := ((1,2))
        (1,1) := ((1,2)(3,4))
        --------------------------------------------------------------------
        g0 := <identity> of ...
        g1 := f1
    
        N = LibraryNearRing(4/2, 11) acts on 
        G = Group( [ f1 ] )
        from the right by the following action: 
    
                 | g0  g1  
          ---------------
          (0,0)  | g0  g0  
          (1,0)  | g0  g0  
          (0,1)  | g0  g1  
          (1,1)  | g0  g1  
    
    

    8.2 Operation tables of N-groups

  • PrintTable( G )

    The function PrintTable prints out the operation table of the action of a nearring on its N-group G

        gap> n := LibraryNearRing( TWGroup( 8, 2 ), 3 );
        LibraryNearRing(8/2, 3)
        gap> gamma := NGroupByNearRingMultiplication( n );
        < N-group of LibraryNearRing(8/2, 3) >
        gap> PrintTable( gamma );
        Let:
        n0 := (())
        n1 := ((3,4,5,6))
        n2 := ((3,5)(4,6))
        n3 := ((3,6,5,4))
        n4 := ((1,2))
        n5 := ((1,2)(3,4,5,6))
        n6 := ((1,2)(3,5)(4,6))
        n7 := ((1,2)(3,6,5,4))
        --------------------------------------------------------------------
        g0 := ()
        g1 := (3,4,5,6)
        g2 := (3,5)(4,6)
        g3 := (3,6,5,4)
        g4 := (1,2)
        g5 := (1,2)(3,4,5,6)
        g6 := (1,2)(3,5)(4,6)
        g7 := (1,2)(3,6,5,4)
    
        N = LibraryNearRing(8/2, 3) acts on 
        G = Group( [ (1,2), (3,4,5,6) ] )
        from the right by the following action: 
    
              | g0  g1  g2  g3  g4  g5  g6  g7  
          ------------------------------------
          n0  | g0  g0  g0  g0  g0  g0  g0  g0  
          n1  | g0  g0  g0  g0  g0  g0  g0  g2  
          n2  | g0  g0  g0  g0  g0  g0  g0  g0  
          n3  | g0  g0  g0  g0  g0  g0  g0  g2  
          n4  | g0  g0  g0  g0  g0  g0  g0  g0  
          n5  | g0  g0  g0  g0  g0  g0  g0  g2  
          n6  | g0  g0  g0  g0  g0  g0  g0  g0  
          n7  | g0  g0  g0  g0  g0  g0  g0  g2  
    
    

    8.3 Functions for N-groups

  • IsNGroup( G )

    For any group G the function IsNGroup tests whether there is a nearring defined that acts on G.

  • NearRingActingOnNGroup( G )

    The function NearRingActingOnNGroup returns the nearring that acts on the N-group G.

        gap> n := LibraryNearRing( TWGroup( 8, 2 ), 3 );
        LibraryNearRing(8/2, 3)
        gap> gamma := NGroupByNearRingMultiplication( n );
        < N-group of LibraryNearRing(8/2, 3) >
        gap> NearRingActingOnNGroup( gamma );                   
        LibraryNearRing(8/2, 3)
    

  • ActionOfNearRingOnNGroup( G )

    The function ActionOfNearRingOnNGroup returns the action of the nearring that acts on the N-group G as a 2-ary GAP-function.

        gap> n := LibraryNearRing( TWGroup( 8, 2 ), 3 );
        LibraryNearRing(8/2, 3)
        gap> gamma := NGroupByNearRingMultiplication( n );
        < N-group of LibraryNearRing(8/2, 3) >
        gap> ActionOfNearRingOnNGroup( gamma );
        function ( g, n ) ... end
    

    8.4 N-subgroups

  • NSubgroup( G, gens )

    The function NSubgroup returns the N-subgroup of the N-group G generated by gens.

  • NSubgroups( G )

    The function NSubgroups returns a list of all N-subgroups of the N-group G.

  • IsNSubgroup( G, S )

    The function IsNSubgroup returns true iff S is an N-subgroup of G.

    8.5 N0-subgroups

  • N0Subgroups( G )

    The function N0Subgroups returns a list of all N0-subgroups of the N-group G.

        gap> n := LibraryNearRing(GTW12_3,20465);
        LibraryNearRing(12/3, 20465)
        gap> ng := NGroupByNearRingMultiplication( n );
        < N-group of LibraryNearRing(12/3, 20465) >
        gap> Length( N0Subgroups( ng ) );
        9
    

    8.6 Ideals of N-groups

  • NIdeal( G, gens )

    The function NIdeal returns the N-ideal of the N-group G generated by gens.

  • NIdeals( G )

    The function NGroupIdeals returns a list of all ideals of the N-group G.

        gap> n:=LibraryNearRing(GTW12_3,20465);
        LibraryNearRing(12/3, 20465)
        gap> ng := NGroupByNearRingMultiplication( n );
        < N-group of LibraryNearRing(12/3, 20465) >
        gap> NIdeals( ng );
        [ < N-group of LibraryNearRing(12/3, 20465) >, 
          < N-group of LibraryNearRing(12/3, 20465) >, 
          < N-group of LibraryNearRing(12/3, 20465) > ]
    

  • IsNIdeal( G, I )

    The function IsNIdeal returns true iff I is an N-ideal of the N-group G.

  • IsSimpleNGroup( G )

    The function IsSimpleNGroup returns true if G is a simple N-group and false otherwise.

  • IsN0SimpleNGroup( G )

    The function IsN0SimpleNGroup returns true if the N-group G is N0-simple and false otherwise.

    8.7 Special properties of N-groups

  • IsCompatible( G )

    The function IsCompatible returns true if the N-group G is compatible and false otherwise.

  • IsTameNGroup( G )

    The function IsTameNGroup returns true if G is a tame N-group and false otherwise.

  • Is2TameNGroup( G )

    The function Is2TameNGroup returns true if the N-group G is 2-tame and false otherwise.

  • Is3TameNGroup( G )

    The function Is3TameNGroup returns true if the N-group G is 3-tame and false otherwise.

  • IsMonogenic( G )

    The function IsMonogenic returns true if the N-group G is monogenic and false otherwise.

  • IsStronglyMonogenic( G )

    The function IsStronglyMonogenic returns true if the N-group G is strongly monogenic and false otherwise.

  • TypeOfNGroup( G )

    The function TypeOfNGroup returns the type of a monogenic N-group G. If N is not monogenic or not of type 0, 1 or 2 it returns fail.

        gap> n:=LibraryNearRing(GTW12_3,20465);
        LibraryNearRing(12/3, 20465)
        gap> ng := NGroupByNearRingMultiplication( n );
        < N-group of LibraryNearRing(12/3, 20465) >
        gap> TypeOfNGroup( ng );
        fail
    

    8.8 Noetherian quotients

  • NoetherianQuotient( nr, ngrp, target, source )

    It is assumed that source and target are subsets of the nr-group ngrp. The function NoetherianQuotient computes the set of all elements f of nr such that source*f is a subset of target. If target is an nr-ideal of ngrp, the Noetherian quotient is returned as a near ring ideal, if target is an nr-subgroup of ngrp, a left ideal of nr is returned. Otherwise the result is a subset of nr.

    In the following example we let a nearring act on its group reduct and compute the noetherian quotient (I,I)N for an ideal I of N.

        gap> N := LibraryNearRing( GTW12_3, 100 );
        LibraryNearRing(12/3, 100)
        gap> I := NearRingIdeals( N );            
        [ < nearring ideal >, < nearring ideal >, < nearring ideal > ]
        gap> List(I,Size);
        [ 1, 6, 12 ]
        gap> NN := NGroupByNearRingMultiplication( N );
        < N-group of LibraryNearRing(12/3, 100) >
        gap> NoetherianQuotient( N, NN, GroupReduct(I[2]), GroupReduct(I[2]) );
        < nearring ideal >
        gap> Size(last);
        12
    

    8.9 Nearring radicals

  • NuRadical( nr, nu )

    The function NuRadical has two arguments, a nearring nr and a number nu which must be one of 0, 1/2, 1 and 2. It returns the nu-radical for nu= 0, 1/2, 1, 2 respectively.

  • NuRadicals( nr )

    the function NuRadicals returns a record with the components J_0, J1_2, J1 and J2 with the corresponding radicals.

        gap> f := LibraryNearRing( GTW8_4, 3 );
        LibraryNearRing(8/4, 3)
        gap> NuRadicals( f );
        rec( J2 := < nearring ideal >, J1 := < nearring ideal >, 
          J1_2 := < nearring right ideal >, J0 := < nearring ideal > )
        gap> NuRadical( f, 1/2 );
        < nearring right ideal >
        gap> Size( NuRadical( f, 0 ) );  
        8
        gap> AsList( NuRadical( f, 1 ) );
        [ (()), ((2,4)), ((1,2)(3,4)), ((1,2,3,4)), ((1,3)), ((1,3)(2,4)), 
          ((1,4,3,2)), ((1,4)(2,3)) ]
        gap> NuRadical( f, 1/2 ) = NuRadical( f, 2 );
        true
    

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    SONATA manual
    November 2012