<html><head><title>[NQL] 4 The underlying functions</title></head>
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<h1>4 The underlying functions</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP004.htm#SECT001">Nilpotent Quotient Systems for invariant L-presentations</a>
<li> <A HREF="CHAP004.htm#SECT002">Attributes of L-presented groups related with the nilpotent quotient algorithm</a>
<li> <A HREF="CHAP004.htm#SECT003">The Info-Class InfoNQL</a>
</ol><p>
<p>
<p>
<h2><a name="SECT001">4.1 Nilpotent Quotient Systems for invariant L-presentations</a></h2>
<p><p>
For an invariantly <var>L</var>-presented group <var>G</var>, our algorithm computes a
nilpotent presentation for <var>G/gamma<sub>c+1</sub>(G)</var> by computing a <var>weighted
nilpotent quotient system</var> for <var>G/G'</var> and extending it inductively to
a weighted nilpotent quotient system for <var>G/gamma<sub>c+1</sub>(G)</var>.
<p>
In the <font face="Gill Sans,Helvetica,Arial">NQL</font> package, a weighted nilpotent quotient system is a record
containing the following entries:
<p>
<p>
<dl compact>
<dt><var>Lpres</var> <dd> the invariantly <var>L</var>-presented group <var>G</var>.
<p>
<dt><var>Pccol</var> <dd> <code>FromTheLeftCollector</code> of the nilpotent quotient represented by this
quotient system.
<p>
<dt><var>Imgs</var> <dd> the images of the generators of the <var>L</var>-presented group <var>G</var> under
the epimorphism onto the nilpotent quotient <var>Pccol</var>. For
each generator of <var>G</var> there is an integer or a generator
exponent list. If the image is an integer <var>int</var>, the image
is a definition of the <var>int</var>-th generator of the nilpotent
presentation <var>Pccol</var>.
<p>
<dt><var>Epimorphism</var><dd> an epimorphism from the <var>L</var>-presented group <var>G</var> onto its
nilpotent quotient <var>Pccol</var> with the images of the generators
given by <var>Imgs</var>.
<p>
<dt><var>Weights</var> <dd> a list of the weight of each generator of the nilpotent
presentation <var>Pccol</var>.
<p>
<var>Definitions</var>
<dt> <dd> the definition of each generator of <var>Pccol</var>. Each generator in the
quotient system has a definition as an image or as a commutator
of the form <var>[a<sub>j</sub>,a<sub>i</sub>]</var> where <var>a<sub>j</sub></var> and <var>a<sub>i</sub></var> are generators of
a certain weight. If the <var>i</var>-th entry is an integer, the <var>i</var>-th
generator of <var>Pccol</var> has a definition as an image. Otherwise,
the <var>i</var>-th entry is a <var>2</var>-tuple <var>[k,l]</var> and the <var>i</var>-th generator
has a definition as commutator <var>[a<sub>k</sub>,a<sub>l</sub>]</var>.
</dl>
<p>
A weighted nilpotent quotient system of an invariantly <var>L</var>-presented group
can be computed with the following functions.
<p>
<a name = "SSEC001.1"></a>
<li><code>InitQuotientSystem( </code><var>LpGroup</var><code> ) O</code>
<p>
computes a weighted nilpotent quotient system for the abelian quotient
of the <var>L</var>-presented group <var>LpGroup</var>.
<p>
<a name = "SSEC001.2"></a>
<li><code>ExtendQuotientSystem( </code><var>QS</var><code> ) F</code>
<p>
extends the weighted nilpotent quotient system <var>QS</var> for a class-<var>c</var>
quotient of an invariantly <var>L</var>-presented group to a weighted nilpotent
quotient system of its class-<var>c+1</var> quotient.
<p>
<pre>
gap> G := ExamplesOfLPresentations( 1 );
<L-presented group on the generators [ a, b, c, d ]>
gap> Q := InitQuotientSystem( G );
rec( Lpres := <L-presented group on the generators [ a, b, c, d ]>,
Pccol := <<from the left collector with 3 generators>>,
Imgs := [ 1, [ 2, 1, 3, 1 ], 2, 3 ], Epimorphism := [ a, b, c, d ] ->
[ g1, g2*g3, g2, g3 ], Weights := [ 1, 1, 1 ], Definitions := [ 1, 3, 4 ]
)
gap> ExtendQuotientSystem( Q );
rec( Lpres := <L-presented group on the generators [ a, b, c, d ]>,
Pccol := <<from the left collector with 5 generators>>,
Imgs := [ 1, [ 2, 1, 3, 1 ], 2, 3 ],
Definitions := [ 1, 3, 4, [ 2, 1 ], [ 3, 1 ] ],
Weights := [ 1, 1, 1, 2, 2 ], Epimorphism := [ a, b, c, d ] ->
[ g1, g2*g3, g2, g3 ] )
</pre>
<p>
<p>
<h2><a name="SECT002">4.2 Attributes of L-presented groups related with the nilpotent quotient algorithm</a></h2>
<p><p>
To avoid repeated extensions of a weighted nilpotent quotient system
the largest known quotient system is stored as an attribute of the
invariantly <var>L</var>-presented group. For non-invariantly <var>L</var>-presented groups
(or groups which are not known to be invariantly <var>L</var>-presented) the known
epimorphisms onto the nilpotent quotients are stored as an attribute.
<p>
<a name = "SSEC002.1"></a>
<li><code>NilpotentQuotientSystem( </code><var>LpGroup</var><code> ) A</code>
<p>
stores the largest known weighted nilpotent quotient system of an
invariantly <var>L</var>-presented group.
<p>
<pre>
gap> G := ExamplesOfLPresentations( 1 );;
gap> NilpotentQuotient( G, 5 );
Pcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
gap> NilpotentQuotientSystem( G );
rec( Lpres := <L-presented group on the generators [ a, b, c, d ]>,
Pccol := <<from the left collector with 10 generators>>,
Imgs := [ 1, [ 2, 1, 3, 1 ], 2, 3 ],
Definitions := [ 1, 3, 4, [ 2, 1 ], [ 3, 1 ], [ 4, 2 ], [ 4, 3 ], [ 7, 1 ],
[ 8, 2 ], [ 8, 3 ] ], Weights := [ 1, 1, 1, 2, 2, 3, 3, 4, 5, 5 ],
Epimorphism := [ a, b, c, d ] -> [ g1, g2*g3, g2, g3 ] )
gap> NilpotencyClassOfGroup( PcpGroupByCollectorNC( last.Pccol ) );
5
</pre>
<p>
<a name = "SSEC002.2"></a>
<li><code>NilpotentQuotients( </code><var>LpGroup</var><code> ) A</code>
<p>
stores all known epimorphisms onto the nilpotent quotients of <var>LpGroup</var>.
The nilpotent quotients are accessible by the operation <code>Range</code>.
<p>
<pre>
gap> G:=ExamplesOfLPresentations( 3 );;
gap> HasIsInvariantLPresentation( G );
false
gap> NilpotentQuotient( G, 3 );
Pcp-group with orders [ 0, 2, 2, 2 ]
gap> NilpotentQuotients( G );
[ [ a, t, u ] -> [ g2, g1, g2 ], [ a, t, u ] -> [ g2, g1, g2 ],
[ a, t, u ] -> [ g2, g1, g2 ] ]
gap> Range( last[2] );
Pcp-group with orders [ 0, 2, 2 ]
</pre>
The underlying invariant <var>L</var>-presentation has stored its largest
weighted nilpotent quotient system as an attribute.
<pre>
gap> NilpotentQuotientSystem( UnderlyingInvariantLPresentation( G ) );
rec( Lpres := <L-presented group on the generators [ a, t, u ]>,
Pccol := <<from the left collector with 9 generators>>, Imgs := [ 1, 2, 3 ],
Definitions := [ 1, 2, 3, [ 2, 1 ], [ 3, 2 ], [ 4, 1 ], [ 4, 2 ], [ 5, 2 ],
[ 5, 3 ] ], Weights := [ 1, 1, 1, 2, 2, 3, 3, 3, 3 ],
Epimorphism := [ a, t, u ] -> [ g1, g2, g3 ] )
</pre>
<p>
<p>
<h2><a name="SECT003">4.3 The Info-Class InfoNQL</a></h2>
<p><p>
To get some information about the progress of the algorithm,
one can use the info class <code>InfoNQL</code>.
<p>
<a name = "SSEC003.1"></a>
<li><code>InfoNQL</code>
<p>
is the info class of the <font face="Gill Sans,Helvetica,Arial">NQL</font>-package. If the info-level is <var>1</var>, the
info-class gives further information on the progress of the nilpotent
quotient algorithm for <var>L</var>-presented groups. The info-level <var>2</var> also
includes some information on the runtime of our algorithm while the
info-level <var>3</var> is mainly used for debugging-purposes. An example of such
a session for the Grigorchuk group is shown below:
<p>
<pre>
gap> SetInfoLevel( InfoNQL, 1 );;
gap> G:=ExamplesOfLPresentations( 1 );
#I The Grigorchuk group on 4 generators
<L-presented group on the generators [ a, b, c, d ]>
gap> NilpotentQuotient( G, 3 );
#I Class 1: 3 generators with relative orders: [ 2, 2, 2 ]
#I Class 2: 2 generators with relative orders: [ 2, 2 ]
#I Class 3: 2 generators with relative orders: [ 2, 2 ]
Pcp-group with orders [ 2, 2, 2, 2, 2, 2, 2 ]
gap> SetInfoLevel( InfoNQL, 2 );
gap> NilpotentQuotient( G, 5 );
#I Time spent for spinning algo: 0:00:00.004
#I Class 4: 1 generators with relative orders: [ 2 ]
#I Runtime for this step 0:00:00.028
#I Time spent for spinning algo: 0:00:00.008
#I Class 5: 2 generators with relative orders: [ 2, 2 ]
#I Runtime for this step 0:00:00.036
Pcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
</pre>
<p>
<a name = "SSEC003.2"></a>
<li><code>InfoNQL_MAX_GENS </code>
<p>
this global variable sets the limit of generators whose relative order
will be shown on each step of the nilpotent quotient algorithm, if the
info-level of <code>InfoNQL</code> is positive.
<p>
<p>
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<address>NQL manual<br>November 2008
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