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<p <l>
<pdt<strong="Mark">-n&;k>strongdt>
<div class="">< ="chapA.html#X78A212947932A6D3">A <spanclass"Heading"The command lineinterface</span<ajava.lang.StringIndexOutOfBoundsException: Index 126 out of bounds for length 126
< classContSectspan="tocline"><span class=nocss>nbsp/><ahref.X7B495102781E821B.1 span=""> to theANUNQspan/
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA
<span
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<div class="ContSect"><span class
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X829490077E75F283">A.4 <span class="Heading">Some remarks about the algorithm</span></a>
</span>
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<h3>A <span class="Heading">The nq command line interface</span></h3>

<p><a id="X7B495102781E821B" name="X7B495102781E821B"></a></p>

<h4>A.1 <span class="Heading">How to use the ANU NQ</span></h4>

<p>If you start the ANU NQ by typing</p>

<div class="example"><pre>
     nq -X
</pre></div>

<p>you will get the following message:</p>

<div class="example"><pre>
    unknown option:
    usage: </d>
              [-dt><strong classMark;</strong/dt>
[y] -][-p -][<presentation>[&;class>]
</pre></div>

<p>All parameters in square brackets are optional. The

<dl>
<dt>< classMarklt&;</strong<dt
<dd><p>This option forces

</dd
<d>
<dd><p>This forces the first k generators <span class="SimpleMath">g_1,...,g_k</span> of the nilpotent quotient Q to be left n-Engel elements, i.e., they satisfy <span class="SimpleMath">[x,...,x,g_idt<strong="Mark">a<strong<dt

</dd>
<dt><strong class="Mark">-r <n></strong></dt>
<dd><p>This forces the first k generators <span class="SimpleMath">g_1,...,g_k</span> of the java.lang.StringIndexOutOfBoundsException: Range [0, 102) out of bounds for length 0

</dd>
<dt classMark>e&;n&;/><>
<dd><p>This enforces the n-th

</dd>
<dt><strongdd><p>This option causes to check semiwords in the generating set of the nilpotent quotient first and then all other
<<This  much timeprogram allowed .It  after;ngt econds CPUtime ltgtis (ithoutspace one   m h r ,&;n&specifies  in,hours days./>

</dd>
</dl>
<p>The other options have thejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<dl>
<>strongclassMark">-a/>

<dd>>strong=Mark-/></dt

</dd>
<dt><dd
<dd><p class-/><dt

</dd>
<dt><strong class="Mark">-s</strong></dt>
<><p>hisoptioncauses theprogram check onlysemigroup  in generating of nilpotent when  conditionis enforced   of  -,-ror are,  is./>

</dd>
<dt><strong class="Mark">-f</strong></dt>
<dd><p>This option causes to check semiwords in the generating set of the nilpotent quotient first and then all other words that need to be checked. It is ignored if the option -s is used or none of the options -l, -r or -e are present.</p>

</dd
<dt>dt< =Mark-/strong<dt
<ddp option checking  law   if thechecks a  weightdidnot    ofthe./>

</dd>
<dt><strong >< "">/></dt
<ddpSwitch mode perform during .  implemented/p>

</dd>
<dt><strong class="Mark">-o</strong></dt>
<dd><p>In checking Engel identities< class"">strong>

</dd>
<dtjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<dd><p>Enforce the identities <span class="SimpleMath">x^8</span> and <

</dd>
<dt><strong class="Mark">-v</strong></dt>
<  c3a2*c3  #a relation

</dda^(c =a          conjugaterelation
<dt><strong class="Mark">-g</strong></dt>
<>GAP.Presently GAP consistsonly asequenceof matrices   relations the ofthe central as .   as  handle groups/>

</dd>
<dt><strong class="Mark">-E</strong
<>p> l*n java.lang.StringIndexOutOfBoundsException: Range [34, 32) out of bounds for length 98

</dd>
<dt><strong class="Mark">-m</strong></dt>
<dd><p>output the relation matrix for each factor of the lower central series. The matrices are written to files with the names 'matrix.<cl>' where <cl> is replaced by the number of the factor in the lower central series. Each file contains first the number of columns of the matrix and then the rows of the matrix. The matrix is written as each relation is produced and is

<
<>strong=MarkM<></java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
<ddpoutput relation beforeand relations been. This in groups of  with names<&;nilp;cl;' and '<presgtmult;&;  &;pres;  the  of input nd;cl;is classThematrices in  form<pjava.lang.StringIndexOutOfBoundsException: Index 319 out of bounds for length 319

</dd>
</dl>
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0

< classjava.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 20

<

<pre=normal

    ;ab &;                        group  2

    <      abelian .
                ,^#another  topower
                 =c-3a^2c3  a relation
                a
                (a*[b,(a*c)])^6      # something :  2  java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    >

</pre>

<p>A presentation starts with: 0 0

<p><a id="X7B5623E3821CC0D0" name="X7B5623E3821CC0D0"></a></p>

<h4>A.3 <span class="Heading">An example

<p>Let be  free  two   y  input ( free2 here) containsfollowing:/>

<pre class="normal">

        &t x y|&;

</

<p>Computing nilpotentquotient

<pre

        nq free2.fp 3

</pre>

<p>produces the following output:</p>

<pre class="normal">

#
#    The ANU NilpotentQuotientProgramVersion.)
#    Calculating a nilpotent quotient
#    Input: free2.fp
#    Nilpotency class: 3
#ram:java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
#  java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
#    Calculating the= B,Ajava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
#    total  1 java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
#        with the following exponents: 0 0
#
#    Calculating the class
#s 23
#    Layer 2 of the lower central series has 1 generators
#           the  :java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
#
#    Calculating the class  of    straight. a     definitions representa  .Calculations the   done     the withoutcombinatorial .Generators thespan="">c<spanth  factorare  commutators  form classSimpleMathyx]span  span="SimpleMath>"y/pan is generatorofweight< class"">c-1/pan> Then the tails all definitions, throughthecheck the original the polycyclic . a list ofwords be trivial inordertoobtainaconsistent polycyclicpresentationrepresenting quotient of givenfinitelypresented .This is ainteger ,whichisintouppertriangularformusingthe Kannan-Bachem .The GNU multiple package usedforthis
##  Sizes:  2  3  5
#    Layer 3 of the lower central java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 0
#with followingexponents  0
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

#    The epimorphism :
#    x |---> A
#    y |->

#    The nilpotent quotient :
    <A,B,C,D,E
      |
        B^A           =: B*C,
        B^(A^-1)      =  B*C^-1*D,
        C^A           =: C*D,
        C^(A^-1)      =  C*D^-1,
        C^B           =: C*E,
        C^(B^-1)      =  C*E^-1 >

#    Class : 3
#    Nr of generators of each class : 2 1 2

#    The definitions:
#    C := [ B, A ]
#    D := [ B, A, A ]
#    E := [ B, A, B ]
#    total runtime : 1 msec
##  Total time spent on integer matrices: 0

</pre>

<p>Most of the comments are fairly self-explanatory. One note of caution is necessary: The number of generators for each factor of the lower central series is not the minimal number possible but is the number of generators that the ANU NQ chose to use. This will be improved in one of the future version of the program. The epimorphism from the original group onto the nilpotent quotient is printed in a somewhat confusing way. The generators on the left hand side of the arrows correspond to the generators in the original presentation but are printed with different names. This will be fixed in one of the next version.</p>

<p><a id="X829490077E75F283" name="X829490077E75F283"></a></p>

<h4>A.4 <span class="Heading">Some remarks about the algorithm</span></h4>

<p>The implementation of the algorithm is fairly straight forward. The program uses a weighted nilpotent presentation with definitions to represent a nilpotent group. Calculations in the nilpotent group are done using a collector from the left without combinatorial collection. Generators for the <span class="SimpleMath">c</span>-th lower central factor are defined as commutators of the form <span class="SimpleMath">[y,x]</span>, where <span class="SimpleMath">x</span> is a generator of weight 1 and <span class="SimpleMath">y</span> is a generator of weight <span class="SimpleMath">c-1</span>. Then the program calculates the necessary changes (tails) for all relations which are not definitions, runs through the consistency check and evaluates the original relations on the polycyclic presentation. This gives a list of words, which have to be made trivial in order to obtain a consistent polycyclic presentation representing a nilpotent quotient of the given finitely presented group. This list is converted into a integer matrix, which is transformed into upper triangular form using the Kannan-Bachem algorithm. The GNU multiple precision package is used for this.</p>

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