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# 8 N-groups

### Sections

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), GroupReduct(I) );
< 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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SONATA manual
October 2018