[SONATA] 8 N-groups

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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.

- `IsTame(` G`)`

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

- `Is2Tame(` G`)`

  The function `Is2Tame` returns `true` if the N-group G is 2-tame and `false` otherwise.

- `Is3Tame(` G`)`

  The function `Is3Tame` 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 n-radical for n = 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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