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?.
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
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
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
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.
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
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.
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
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
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
ν-radical for ν = 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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