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<html><head><title>[SONATA] 8 N-groups</title></head>
<body text="#000000" bgcolor="#ffffff">
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<h1>8 N-groups</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP008.htm#SECT001">Construction of N-groups</a>
<li> <A HREF="CHAP008.htm#SECT002">Operation tables of N-groups</a>
<li> <A HREF="CHAP008.htm#SECT003">Functions for N-groups</a>
<li> <A HREF="CHAP008.htm#SECT004">N-subgroups</a>
<li> <A HREF="CHAP008.htm#SECT005">N0-subgroups</a>
<li> <A HREF="CHAP008.htm#SECT006">Ideals of N-groups</a>
<li> <A HREF="CHAP008.htm#SECT007">Special properties of N-groups</a>
<li> <A HREF="CHAP008.htm#SECT008">Noetherian quotients</a>
<li> <A HREF="CHAP008.htm#SECT009">Nearring radicals</a>
</ol>

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 <code>*</code>, not <code>+</code>.

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


<h2><a name="SECT001">8.1 Construction of N-groups</a></h2>

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 <code>NGroup</code> 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
<code>NGroupByNearRingMultiplication</code>.

<a name = "SSEC001.1"></a>
<li><code>NGroup( </code><var>G</var><code>, </code><var>nr</var><code>, </code><var>action</var><code> )</code>

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

<pre>
 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 );
</pre>

<a name = "SSEC001.2"></a>
<li><code>NGroupByNearRingMultiplication( </code><var>nr</var><code> )</code>

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 <code>NGroupByNearRingMultiplication</code> returns this
N-group of the nearring <var>nr</var>.

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

<a name = "SSEC001.3"></a>
<li><code>NGroupByApplication( </code><var>tfmnr</var><code> )</code>

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 <code>NGroupByApplication</code> returns this <code>N-group</code> of the nearring
<var>tfmnr</var>.

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

<a name = "SSEC001.4"></a>
<li><code>NGroupByRightIdealFactor( </code><var>nr</var><code>, </code><var>R</var><code> )</code>

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

<pre>
 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

</pre>


<h2><a name="SECT002">8.2 Operation tables of N-groups</a></h2>

<a name = "SSEC002.1"></a>
<li><code>PrintTable( </code><var>G</var><code> )</code>

The function <code>PrintTable</code> prints out the operation table of the
action of a nearring on its N-group <var>G</var>

<pre>
 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

</pre>


<h2><a name="SECT003">8.3 Functions for N-groups</a></h2>

<a name = "SSEC003.1"></a>
<li><code>IsNGroup( </code><var>G</var><code> )</code>

For any group <var>G</var> the function <code>IsNGroup</code> tests whether there is a nearring
defined that acts on <var>G</var>.

<a name = "SSEC003.2"></a>
<li><code>NearRingActingOnNGroup( </code><var>G</var><code> )</code>

The function <code>NearRingActingOnNGroup</code> returns the nearring that acts on the
N-group <var>G</var>.

<pre>
 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)
</pre>

<a name = "SSEC003.3"></a>
<li><code>ActionOfNearRingOnNGroup( </code><var>G</var><code> )</code>

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

<pre>
 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
</pre>


<h2><a name="SECT004">8.4 N-subgroups</a></h2>

<a name = "SSEC004.1"></a>
<li><code>NSubgroup( </code><var>G</var><code>, </code><var>gens</var><code> )</code>

The function <code>NSubgroup</code> returns the N-subgroup of the N-group <var>G</var>
generated by <var>gens</var>.

<a name = "SSEC004.2"></a>
<li><code>NSubgroups( </code><var>G</var><code> )</code>

The function <code>NSubgroups</code> returns a list of all N-subgroups of the
N-group <var>G</var>.

<a name = "SSEC004.3"></a>
<li><code>IsNSubgroup( </code><var>G</var><code>, </code><var>S</var><code> )</code>

The function <code>IsNSubgroup</code> returns <code>true</code> iff <var>S</var> is an N-subgroup of
<var>G</var>.


<h2><a name="SECT005">8.5 N0-subgroups</a></h2>

<a name = "SSEC005.1"></a>
<li><code>N0Subgroups( </code><var>G</var><code> )</code>

The function <code>N0Subgroups</code> returns a list of all N0-subgroups
of the N-group <var>G</var>.

<pre>
 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
</pre>


<h2><a name="SECT006">8.6 Ideals of N-groups</a></h2>

<a name = "SSEC006.1"></a>
<li><code>NIdeal( </code><var>G</var><code>, </code><var>gens</var><code> )</code>

The function <code>NIdeal</code> returns the N-ideal of the N-group <var>G</var> generated
by <var>gens</var>.

<a name = "SSEC006.2"></a>
<li><code>NIdeals( </code><var>G</var><code> )</code>

The function <code>NGroupIdeals</code> returns a list of all ideals of the N-group <var>G</var>.

<pre>
 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) > ]
</pre>

<a name = "SSEC006.3"></a>
<li><code>IsNIdeal( </code><var>G</var><code>, </code><var>I</var><code> )</code>

The function <code>IsNIdeal</code> returns <code>true</code> iff <var>I</var> is an N-ideal of the
N-group <var>G</var>.

<a name = "SSEC006.4"></a>
<li><code>IsSimpleNGroup( </code><var>G</var><code> )</code>

The function <code>IsSimpleNGroup</code> returns <code>true</code> if <var>G</var> is a simple N-group
and <code>false</code> otherwise.

<a name = "SSEC006.5"></a>
<li><code>IsN0SimpleNGroup( </code><var>G</var><code> )</code>

The function <code>IsN0SimpleNGroup</code> returns <code>true</code> if the N-group <var>G</var> is
N0-simple and <code>false</code> otherwise.


<h2><a name="SECT007">8.7 Special properties of N-groups</a></h2>

<a name = "SSEC007.1"></a>
<li><code>IsCompatible( </code><var>G</var><code> )</code>

The function <code>IsCompatible</code> returns <code>true</code> if the N-group <var>G</var> is compatible
and <code>false</code> otherwise.

<a name = "SSEC007.2"></a>
<li><code>IsTameNGroup( </code><var>G</var><code> )</code>

The function <code>IsTameNGroup</code> returns <code>true</code> if <var>G</var> is a tame N-group
and <code>false</code> otherwise.

<a name = "SSEC007.3"></a>
<li><code>Is2TameNGroup( </code><var>G</var><code> )</code>

The function <code>Is2TameNGroup</code> returns <code>true</code> if the N-group <var>G</var> is 2-tame
and <code>false</code> otherwise.

<a name = "SSEC007.4"></a>
<li><code>Is3TameNGroup( </code><var>G</var><code> )</code>

The function <code>Is3TameNGroup</code> returns <code>true</code> if the N-group <var>G</var> is 3-tame
and <code>false</code> otherwise.

<a name = "SSEC007.5"></a>
<li><code>IsMonogenic( </code><var>G</var><code> )</code>

The function <code>IsMonogenic</code> returns <code>true</code> if the N-group <var>G</var> is monogenic
and <code>false</code> otherwise.

<a name = "SSEC007.6"></a>
<li><code>IsStronglyMonogenic( </code><var>G</var><code> )</code>

The function <code>IsStronglyMonogenic</code> returns <code>true</code> if the N-group <var>G</var> is
strongly monogenic and <code>false</code> otherwise.

<a name = "SSEC007.7"></a>
<li><code>TypeOfNGroup( </code><var>G</var><code> )</code>

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

<pre>
 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
</pre>


<h2><a name="SECT008">8.8 Noetherian quotients</a></h2>

<a name = "SSEC008.1"></a>
<li><code>NoetherianQuotient( </code><var>nr</var><code>, </code><var>ngrp</var><code>, </code><var>target</var><code>, </code><var>source</var><code> )</code>

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

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.

<pre>
 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
</pre>


<h2><a name="SECT009">8.9 Nearring radicals</a></h2>

<a name = "SSEC009.1"></a>
<li><code>NuRadical( </code><var>nr</var><code>, </code><var>nu</var><code> )</code>

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

<a name = "SSEC009.2"></a>
<li><code>NuRadicals( </code><var>nr</var><code> )</code>

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

<pre>
 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
</pre>


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