examples.xml

<Appendix Label="Examples">
<Heading>Examples</Heading>

There are a large number of examples provided with the &ANUPQ; package. 
These may be executed or displayed via the function <C>PqExample</C>
(see&nbsp;<Ref Func="PqExample" Style="Text"/>). Each example resides in a file of the same name in the
directory <C>examples</C>. Most of the examples are translations to &GAP; of
examples provided for the <C>pq</C> standalone by Eamonn O'Brien; the
standalone examples are found in directories <C>standalone/examples</C>
(<M>p</M>-quotient and <M>p</M>-group generation examples) and <C>standalone/isom</C>
(standard presentation examples). The first line of each example
indicates its origin. All the examples seen in earlier chapters of this
manual are also available as examples, in a slightly modified form (the
example which one can run in order to see something very close to the
text example <Q>live</Q> is always indicated near -- usually immediately
after -- the text example). The format of the (<C>PqExample</C>) examples is
such that they can be read by the standard <C>Read</C> function of &GAP;, but
certain features and comments are interpreted by the function <C>PqExample</C>
to do somewhat more than <C>Read</C> does. In particular, any function without
a <C>-i</C>, <C>-ni</C> or <C>.g</C> suffix has both a non-interactive and interactive
form; in these cases, the default form is the non-interactive form, and
giving <C>PqStart</C> as second argument generates the interactive form.

<P/>

Running <C>PqExample</C> without an argument or with a non-existent example
<C>Info</C>s the available examples and some hints on usage:

<Example><![CDATA[
gap> PqExample();
#I                   PqExample Index (Table of Contents)
#I                   -----------------------------------
#I  This table of possible examples is displayed when calling `PqExample'
#I  with no arguments, or with the argument: "index" (meant in the  sense
#I  of ``list''), or with a non-existent example name.
#I  
#I  Examples that have a name ending in `-ni' are  non-interactive  only.
#I  Examples that have a  name  ending  in  `-i'  are  interactive  only.
#I  Examples with names ending in `.g' also have  only  one  form.  Other
#I  examples have both a non-interactive and an  interactive  form;  call
#I  `PqExample' with 2nd argument `PqStart' to get the  interactive  form
#I  of the example. The substring `PG' in an  example  name  indicates  a
#I  p-Group Generation example, `SP' indicates  a  Standard  Presentation
#I  example, `Rel' indicates it uses  the  `Relators'  option,  and  `Id'
#I  indicates it uses the `Identities' option.
#I  
#I  The following ANUPQ examples are available:
#I  
#I   p-Quotient examples:
#I    general:
#I     "Pq"                   "Pq-ni"                "PqEpimorphism"        
#I     "PqPCover"             "PqSupplementInnerAutomorphisms"
#I    2-groups:
#I     "2gp-Rel"              "2gp-Rel-i"            "2gp-a-Rel-i"
#I     "B2-4"                 "B2-4-Id"              "B2-8-i"
#I     "B4-4-i"               "B4-4-a-i"             "B5-4.g"
#I    3-groups:
#I     "3gp-Rel-i"            "3gp-a-Rel"            "3gp-a-Rel-i"
#I     "3gp-a-x-Rel-i"        "3gp-maxoccur-Rel-i"
#I    5-groups:
#I     "5gp-Rel-i"            "5gp-a-Rel-i"          "5gp-b-Rel-i"
#I     "5gp-c-Rel-i"          "5gp-metabelian-Rel-i" "5gp-maxoccur-Rel-i"
#I     "F2-5-i"               "B2-5-i"               "R2-5-i"
#I     "R2-5-x-i"             "B5-5-Engel3-Id"
#I    7-groups:
#I     "7gp-Rel-i"
#I    11-groups:
#I     "11gp-i"               "11gp-Rel-i"           "11gp-a-Rel-i"
#I     "11gp-3-Engel-Id"      "11gp-3-Engel-Id-i"
#I  
#I   p-Group Generation examples:
#I    general:
#I     "PqDescendants-1"      "PqDescendants-2"      "PqDescendants-3"
#I     "PqDescendants-1-i"
#I    2-groups:
#I     "2gp-PG-i"             "2gp-PG-2-i"           "2gp-PG-3-i"
#I     "2gp-PG-4-i"           "2gp-PG-e4-i"
#I     "PqDescendantsTreeCoclassOne-16-i"
#I    3-groups:
#I     "3gp-PG-i"             "3gp-PG-4-i"           "3gp-PG-x-i"
#I     "3gp-PG-x-1-i"         "PqDescendants-treetraverse-i"
#I     "PqDescendantsTreeCoclassOne-9-i"
#I    5-groups:
#I     "5gp-PG-i"             "Nott-PG-Rel-i"        "Nott-APG-Rel-i"
#I     "PqDescendantsTreeCoclassOne-25-i"
#I    7,11-groups:
#I     "7gp-PG-i"             "11gp-PG-i"
#I  
#I   Standard Presentation examples:
#I    general:
#I     "StandardPresentation" "StandardPresentation-i"
#I     "EpimorphismStandardPresentation"
#I     "EpimorphismStandardPresentation-i"           "IsIsomorphicPGroup-ni"
#I    2-groups:
#I     "2gp-SP-Rel-i"         "2gp-SP-1-Rel-i"       "2gp-SP-2-Rel-i"
#I     "2gp-SP-3-Rel-i"       "2gp-SP-4-Rel-i"       "2gp-SP-d-Rel-i"
#I     "gp-256-SP-Rel-i"      "B2-4-SP-i"            "G2-SP-Rel-i"
#I    3-groups:
#I     "3gp-SP-Rel-i"         "3gp-SP-1-Rel-i"       "3gp-SP-2-Rel-i"
#I     "3gp-SP-3-Rel-i"       "3gp-SP-4-Rel-i"       "G3-SP-Rel-i"
#I    5-groups:
#I     "5gp-SP-Rel-i"         "5gp-SP-a-Rel-i"       "5gp-SP-b-Rel-i"
#I     "5gp-SP-big-Rel-i"     "5gp-SP-d-Rel-i"       "G5-SP-Rel-i"
#I     "G5-SP-a-Rel-i"        "Nott-SP-Rel-i"
#I    7-groups:
#I     "7gp-SP-Rel-i"         "7gp-SP-a-Rel-i"       "7gp-SP-b-Rel-i"
#I    11-groups:
#I     "11gp-SP-a-i"          "11gp-SP-a-Rel-i"      "11gp-SP-a-Rel-1-i"
#I     "11gp-SP-b-i"          "11gp-SP-b-Rel-i"      "11gp-SP-c-Rel-i"
#I  
#I  Notes
#I  -----
#I  1. The example (first) argument of  `PqExample'  is  a  string;  each
#I     example above is in double quotes to remind you to include them.
#I  2. Some examples accept options. To find  out  whether  a  particular
#I     example accepts options, display it first (by including  `Display'
#I     as  last  argument)  which  will  also  indicate  how  `PqExample'
#I     interprets the options, e.g. `PqExample("11gp-SP-a-i", Display);'.
#I  3. Try `SetInfoLevel(InfoANUPQ, <n>);' for  some  <n>  in  [2  ..  4]
#I     before calling PqExample, to see what's going on behind the scenes.
#I  
]]></Example>

If on your terminal you are unable to scroll back, an alternative to
typing <C>PqExample();</C> to see the displayed examples is to use on-line
help, i.e.&nbsp; you may type:

<Log><![CDATA[
gap> ?anupq:examples
]]></Log>

which will display this appendix in a &GAP; session. If you are not
fussed about the order in which the examples are organised,
<C>AllPqExamples();</C> lists the available examples relatively compactly
(see&nbsp;<Ref Func="AllPqExamples" Style="Text"/>).
<P/>

In the remainder of this appendix we will discuss particular aspects
related to the <C>Relators</C> (see&nbsp;<Ref Label="option Relators" Style="Text"/>) and <C>Identities</C>
(see&nbsp;<Ref Label="option Identities" Style="Text"/>) options, and the construction of the Burnside
group <M>B(5, 4)</M>.


<Section Label="The Relators Option">
<Heading>The Relators Option</Heading>

The <C>Relators</C> option was included because computations involving words
containing commutators that are pre-expanded by &GAP; before being
passed to the <C>pq</C> program may run considerably more slowly, than the
same computations being run with &GAP; pre-expansions avoided. The
following examples demonstrate a case where the performance hit due to
pre-expansion of commutators by &GAP; is a factor of order 100 (in order
to see timing information from the <C>pq</C> program, we set the <C>InfoANUPQ</C>
level to 2).

<P/>

Firstly, we run the example that allows pre-expansion of commutators (the
function <C>PqLeftNormComm</C> is provided by the &ANUPQ; package;
see&nbsp;<Ref Func="PqLeftNormComm" Style="Text"/>). Note that since the two commutators of this
example are <E>very</E> long (taking more than an page to print), we have
edited the output at this point.

<Log><![CDATA[
gap> SetInfoLevel(InfoANUPQ, 2); #to see timing information
gap> PqExample("11gp-i");
#I  #Example: "11gp-i" . . . based on: examples/11gp
#I  F, a, b, c, R, procId are local to `PqExample'
gap> F := FreeGroup("a", "b", "c"); a := F.1; b := F.2; c := F.3;
<free group on the generators [ a, b, c ]>
a
b
c
gap> R := [PqLeftNormComm([b, a, a, b, c])^11, 
>          PqLeftNormComm([a, b, b, a, b, c])^11, (a * b)^11];;
gap> procId := PqStart(F/R : Prime := 11);
1
gap> PqPcPresentation(procId : ClassBound := 7, 
>                              OutputLevel := 1);
#I  Lower exponent-11 central series for [grp]
#I  Group: [grp] to lower exponent-11 central class 1 has order 11^3
#I  Group: [grp] to lower exponent-11 central class 2 has order 11^8
#I  Group: [grp] to lower exponent-11 central class 3 has order 11^19
#I  Group: [grp] to lower exponent-11 central class 4 has order 11^42
#I  Group: [grp] to lower exponent-11 central class 5 has order 11^98
#I  Group: [grp] to lower exponent-11 central class 6 has order 11^228
#I  Group: [grp] to lower exponent-11 central class 7 has order 11^563
#I  Computation of presentation took 27.04 seconds
gap> PqSavePcPresentation(procId, ANUPQData.outfile);
#I  Variables used in `PqExample' are saved in `ANUPQData.example.vars'.
]]></Log>

Now we do the same calculation using the <C>Relators</C> option. In this way,
the commutators are passed directly as strings to the <C>pq</C> program, so
that &GAP; does not <Q>see</Q> them and pre-expand them.

<Log><![CDATA[
gap> PqExample("11gp-Rel-i");
#I  #Example: "11gp-Rel-i" . . . based on: examples/11gp
#I  #(equivalent to "11gp-i" example but uses `Relators' option)
#I  F, rels, procId are local to `PqExample'
gap> F := FreeGroup("a", "b", "c");
<free group on the generators [ a, b, c ]>
gap> rels := ["[b, a, a, b, c]^11", "[a, b, b, a, b, c]^11", "(a * b)^11"];
[ "[b, a, a, b, c]^11", "[a, b, b, a, b, c]^11", "(a * b)^11" ]
gap> procId := PqStart(F : Prime := 11, Relators := rels);
2
gap> PqPcPresentation(procId : ClassBound := 7, 
>                              OutputLevel := 1);
#I  Relators parsed ok.
#I  Lower exponent-11 central series for [grp]
#I  Group: [grp] to lower exponent-11 central class 1 has order 11^3
#I  Group: [grp] to lower exponent-11 central class 2 has order 11^8
#I  Group: [grp] to lower exponent-11 central class 3 has order 11^19
#I  Group: [grp] to lower exponent-11 central class 4 has order 11^42
#I  Group: [grp] to lower exponent-11 central class 5 has order 11^98
#I  Group: [grp] to lower exponent-11 central class 6 has order 11^228
#I  Group: [grp] to lower exponent-11 central class 7 has order 11^563
#I  Computation of presentation took 0.27 seconds
gap> PqSavePcPresentation(procId, ANUPQData.outfile);
#I  Variables used in `PqExample' are saved in `ANUPQData.example.vars'.
]]></Log>
</Section>


<Section Label="The Identities Option and PqEvaluateIdentities Function">
<Heading>The Identities Option and PqEvaluateIdentities Function</Heading>

Please pay heed to the warnings given for the <C>Identities</C> option
(see&nbsp;<Ref Label="option Identities" Style="Text"/>); it is written mainly at the &GAP; level and
is not particularly optimised. The <C>Identities</C> option allows one to
compute <M>p</M>-quotients that satisfy an identity. A trivial example better
done using the <C>Exponent</C> option, but which nevertheless demonstrates the
usage of the <C>Identities</C> option, is as follows:

<Log><![CDATA[
gap> SetInfoLevel(InfoANUPQ, 1);
gap> PqExample("B2-4-Id");
#I  #Example: "B2-4-Id" . . . alternative way to generate B(2, 4)
#I  #Generates B(2, 4) by using the `Identities' option
#I  #... this is not as efficient as using `Exponent' but
#I  #demonstrates the usage of the `Identities' option.
#I  F, f, procId are local to `PqExample'
gap> F := FreeGroup("a", "b");
<free group on the generators [ a, b ]>
gap> # All words w in the pc generators of B(2, 4) satisfy f(w) = 1 
gap> f := w -> w^4;
function( w ) ... end
gap> Pq( F : Prime := 2, Identities := [ f ] );
#I  Class 1 with 2 generators.
#I  Class 2 with 5 generators.
#I  Class 3 with 7 generators.
#I  Class 4 with 10 generators.
#I  Class 5 with 12 generators.
#I  Class 5 with 12 generators.
<pc group of size 4096 with 12 generators>
#I  Variables used in `PqExample' are saved in `ANUPQData.example.vars'.
gap> time; 
1400
]]></Log>

Note that the <C>time</C> statement gives the time in milliseconds spent by
&GAP; in executing the <C>PqExample("B2-4-Id");</C> command (i.e.&nbsp;everything
up to the <C>Info</C>-ing of the variables used), but over 90% of that time
is spent in the final <C>Pq</C> statement. The time spent by the <C>pq</C> program,
which is negligible anyway (you can check this by running the example
while the <C>InfoANUPQ</C> level is set to 2), is not counted by <C>time</C>.

<P/>

Since the identity used in the above construction of <M>B(2, 4)</M> is just an
exponent law, the <Q>right</Q> way to compute it is via the <C>Exponent</C>
option (see&nbsp;<Ref Label="option Exponent" Style="Text"/>), which is implemented at the C level and
<E>is</E> highly optimised. Consequently, the <C>Exponent</C> option is
significantly faster, generally by several orders of magnitude:

<Log><![CDATA[
gap> SetInfoLevel(InfoANUPQ, 2); # to see time spent by the `pq' program
gap> PqExample("B2-4");
#I  #Example: "B2-4" . . . the ``right'' way to generate B(2, 4)
#I  #Generates B(2, 4) by using the `Exponent' option
#I  F, procId are local to `PqExample'
gap> F := FreeGroup("a", "b");
<free group on the generators [ a, b ]>
gap> Pq( F : Prime := 2, Exponent := 4 );
#I  Computation of presentation took 0.00 seconds
<pc group of size 4096 with 12 generators>
#I  Variables used in `PqExample' are saved in `ANUPQData.example.vars'.
gap> time; # time spent by GAP in executing `PqExample("B2-4");' 
50
]]></Log>

The following example uses the <C>Identities</C> option to compute a 3-Engel
group for the prime 11. As is the case for the example <C>"B2-4-Id"</C>, the
example has both a non-interactive and an interactive form; below, we
demonstrate the interactive form.

<Log><![CDATA[
gap> SetInfoLevel(InfoANUPQ, 1); # reset InfoANUPQ to default level
gap> PqExample("11gp-3-Engel-Id", PqStart);
#I  #Example: "11gp-3-Engel-Id" . . . 3-Engel group for prime 11
#I  #Non-trivial example of using the `Identities' option
#I  F, a, b, G, f, procId, Q are local to `PqExample'
gap> F := FreeGroup("a", "b"); a := F.1; b := F.2;
<free group on the generators [ a, b ]>
a
b
gap> G := F/[ a^11, b^11 ];
<fp group on the generators [ a, b ]>
gap> # All word pairs u, v in the pc generators of the 11-quotient Q of G 
gap> # must satisfy the Engel identity: [u, v, v, v] = 1.
gap> f := function(u, v) return PqLeftNormComm( [u, v, v, v] ); end;
function( u, v ) ... end
gap> procId := PqStart( G );
3
gap> Q := Pq( procId : Prime := 11, Identities := [ f ] );
#I  Class 1 with 2 generators.
#I  Class 2 with 3 generators.
#I  Class 3 with 5 generators.
#I  Class 3 with 5 generators.
<pc group of size 161051 with 5 generators>
gap> # We do a ``sample'' check that pairs of elements of Q do satisfy
gap> # the given identity:
gap> f( Random(Q), Random(Q) );
<identity> of ...
gap> f( Q.1, Q.2 );
<identity> of ...
#I  Variables used in `PqExample' are saved in `ANUPQData.example.vars'.
]]></Log>

The (interactive) call to <C>Pq</C> above is essentially equivalent to a call
to <C>PqPcPresentation</C> with the same arguments and options followed by a
call to <C>PqCurrentGroup</C>. Moreover, the call to <C>PqPcPresentation</C> (as
described in&nbsp;<Ref Func="PqPcPresentation" Style="Text"/>) is equivalent to a <Q>class 1</Q> call to
<C>PqPcPresentation</C> followed by the requisite number of calls to
<C>PqNextClass</C>, and with the <C>Identities</C> option set, both
<C>PqPcPresentation</C> and <C>PqNextClass</C> <Q>quietly</Q> perform the equivalent
of a <C>PqEvaluateIdentities</C> call. In the following example we break down
the <C>Pq</C> call into its low-level equivalents, and set and unset the
<C>Identities</C> option to show where <C>PqEvaluateIdentities</C> fits into this
scheme.

<Log><![CDATA[
gap> PqExample("11gp-3-Engel-Id-i");
#I  #Example: "11gp-3-Engel-Id-i" . . . 3-Engel grp for prime 11
#I  #Variation of "11gp-3-Engel-Id" broken down into its lower-level component
#I  #command parts.
#I  F, a, b, G, f, procId, Q are local to `PqExample'
gap> F := FreeGroup("a", "b"); a := F.1; b := F.2;
<free group on the generators [ a, b ]>
a
b
gap> G := F/[ a^11, b^11 ];
<fp group on the generators [ a, b ]>
gap> # All word pairs u, v in the pc generators of the 11-quotient Q of G 
gap> # must satisfy the Engel identity: [u, v, v, v] = 1.
gap> f := function(u, v) return PqLeftNormComm( [u, v, v, v] ); end;
function( u, v ) ... end
gap> procId := PqStart( G : Prime := 11 );
4
gap> PqPcPresentation( procId : ClassBound := 1);
gap> PqEvaluateIdentities( procId : Identities := [f] );
#I  Class 1 with 2 generators.
gap> for c in [2 .. 4] do
>      PqNextClass( procId : Identities := [] ); #reset `Identities' option
>      PqEvaluateIdentities( procId : Identities := [f] );
>    od;
#I  Class 2 with 3 generators.
#I  Class 3 with 5 generators.
#I  Class 3 with 5 generators.
gap> Q := PqCurrentGroup( procId );
<pc group of size 161051 with 5 generators>
gap> # We do a ``sample'' check that pairs of elements of Q do satisfy
gap> # the given identity:
gap> f( Random(Q), Random(Q) );
<identity> of ...
gap> f( Q.1, Q.2 );
<identity> of ...
#I  Variables used in `PqExample' are saved in `ANUPQData.example.vars'.
]]></Log>

</Section>


<Section Label="A Large Example">
<Heading>A Large Example</Heading>

<Index Key="B(5,4)"><M>B(5,4)</M></Index>
An example demonstrating how a large computation can be organised with the
&ANUPQ; package is the computation of the Burnside group <M>B(5, 4)</M>, the
largest group of exponent 4 generated by 5 elements. It has order
<M>2^{2728}</M> and lower exponent-<M>p</M> central class 13. The example
<C>"B5-4.g"</C> computes <M>B(5, 4)</M>; it is based on a <C>pq</C> standalone input
file written by M.&nbsp;F.&nbsp;Newman.
<P/>

To be able to do examples like this was part of the motivation to provide
access to the low-level functions of the standalone program from within
&GAP;.
<P/>

Please note that the construction uses the knowledge gained by Newman and
O'Brien in their initial construction of <M>B(5, 4)</M>, in particular,
insight into the commutator structure of the group and the knowledge of
the <M>p</M>-central class and the order of <M>B(5, 4)</M>. Therefore, the
construction cannot be used to prove that <M>B(5, 4)</M> has the order and
class mentioned above. It is merely a reconstruction of the group. More
information is contained in the header of the file <C>examples/B5-4.g</C>.

<Log><![CDATA[
procId := PqStart( FreeGroup(5) : Exponent := 4, Prime := 2 );
Pq( procId : ClassBound := 2 );
PqSupplyAutomorphisms( procId,
      [
        [ [ 1, 1, 0, 0, 0],      # first automorphism
          [ 0, 1, 0, 0, 0],
          [ 0, 0, 1, 0, 0],
          [ 0, 0, 0, 1, 0],
          [ 0, 0, 0, 0, 1] ],

        [ [ 0, 0, 0, 0, 1],      # second automorphism
          [ 1, 0, 0, 0, 0],
          [ 0, 1, 0, 0, 0],
          [ 0, 0, 1, 0, 0],
          [ 0, 0, 0, 1, 0] ]
                             ] );;

Relations :=
  [ [],          ## class 1
    [],          ## class 2
    [],          ## class 3
    [],          ## class 4
    [],          ## class 5
    [],          ## class 6
    ## class 7     
    [ [ "x2","x1","x1","x3","x4","x4","x4" ] ],
    ## class 8
    [ [ "x2","x1","x1","x3","x4","x5","x5","x5" ] ],
    ## class 9
    [ [ "x2","x1","x1","x3","x4","x4","x5","x5","x5" ],
      [ "x2","x1","x1","x2","x3","x4","x5","x5","x5" ],
      [ "x2","x1","x1","x3","x3","x4","x5","x5","x5" ] ],
    ## class 10
    [ [ "x2","x1","x1","x2","x3","x3","x4","x5","x5","x5" ],
      [ "x2","x1","x1","x3","x3","x4","x4","x5","x5","x5" ] ],
    ## class 11
    [ [ "x2","x1","x1","x2","x3","x3","x4","x4","x5","x5","x5" ],
      [ "x2","x1","x1","x2","x3","x1","x3","x4","x2","x4","x3" ] ],
    ## class 12
    [ [ "x2","x1","x1","x2","x3","x1","x3","x4","x2","x5","x5","x5" ],
      [ "x2","x1","x1","x3","x2","x4","x3","x5","x4","x5","x5","x5" ] ],
    ## class 13
    [ [ "x2","x1","x1","x2","x3","x1","x3","x4","x2","x4","x5","x5","x5" 
        ] ]
];

for class in [ 3 .. 13 ] do
    Print( "Computing class ", class, "\n" );
    PqSetupTablesForNextClass( procId );

    for w in [ class, class-1 .. 7 ] do

        PqAddTails( procId, w );   
        PqDisplayPcPresentation( procId );

        if Relations[ w ] <> [] then
            # recalculate automorphisms
            PqExtendAutomorphisms( procId );

            for r in Relations[ w ] do
                Print( "Collecting ", r, "\n" );
                PqCommutator( procId, r, 1 );
                PqEchelonise( procId );
                PqApplyAutomorphisms( procId, 15 ); #queue factor = 15
            od;

            PqEliminateRedundantGenerators( procId );
        fi;   
        PqComputeTails( procId, w );
    od;
    PqDisplayPcPresentation( procId );

    smallclass := Minimum( class, 6 );
    for w in [ smallclass, smallclass-1 .. 2 ] do
        PqTails( procId, w );
    od;
    # recalculate automorphisms
    PqExtendAutomorphisms( procId );
    PqCollect( procId, "x5^4" );
    PqEchelonise( procId );
    PqApplyAutomorphisms( procId, 15 ); #queue factor = 15
    PqEliminateRedundantGenerators( procId );
    PqDisplayPcPresentation( procId );
od;
]]></Log>

</Section>


<Section Label="Developing descendants trees">
<Heading>Developing descendants trees</Heading>

In the following example we will explore the 3-groups of rank 2 and
3-coclass 1 up to 3-class 5. This will be done using the <M>p</M>-group
generation machinery of the package. We start with the elementary abelian
3-group of rank 2. From within &GAP;, run the example
<C>"PqDescendants-treetraverse-i"</C> via <C>PqExample</C> (see&nbsp;<Ref Func="PqExample" Style="Text"/>).

<Example><![CDATA[
gap> G := ElementaryAbelianGroup( 9 );
<pc group of size 9 with 2 generators>
gap> procId := PqStart( G );
5
gap> #
gap> #  Below, we use the option StepSize in order to construct descendants
gap> #  of coclass 1. This is equivalent to setting the StepSize to 1 in
gap> #  each descendant calculation.
gap> #
gap> #  The elementary abelian group of order 9 has 3 descendants of
gap> #  3-class 2 and 3-coclass 1, as the result of the next command
gap> #  shows. 
gap> #
gap> PqDescendants( procId : StepSize := 1 );
[ <pc group of size 27 with 3 generators>, 
  <pc group of size 27 with 3 generators>, 
  <pc group of size 27 with 3 generators> ]
gap> #
gap> #  Now we will compute the descendants of coclass 1 for each of the
gap> #  groups above. Then we will compute the descendants  of coclass 1
gap> #  of each descendant and so  on.  Note  that the  pq program keeps
gap> #  one file for each class at a time.  For example, the descendants
gap> #  calculation for  the  second  group  of class  2  overwrites the
gap> #  descendant file  obtained  from  the  first  group  of  class 2.
gap> #  Hence,  we have to traverse the descendants tree  in depth first
gap> #  order.
gap> #
gap> PqPGSetDescendantToPcp( procId, 2, 1 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
2
gap> PqPGSetDescendantToPcp( procId, 3, 1 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
2
gap> PqPGSetDescendantToPcp( procId, 4, 1 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
2
gap> #
gap> #  At this point we stop traversing the ``left most'' branch of the
gap> #  descendants tree and move upwards.
gap> #
gap> PqPGSetDescendantToPcp( procId, 4, 2 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
#I  group restored from file is incapable
0
gap> PqPGSetDescendantToPcp( procId, 3, 2 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
#I  group restored from file is incapable
0
gap> #  
gap> #  The computations above indicate that the descendants subtree under
gap> #  the first descendant of the elementary abelian group of order 9
gap> #  will have only one path of infinite length.
gap> #
gap> PqPGSetDescendantToPcp( procId, 2, 2 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
4
gap> #
gap> #  We get four descendants here, three of which will turn out to be
gap> #  incapable, i.e., they have no descendants and are terminal nodes
gap> #  in the descendants tree.
gap> #
gap> PqPGSetDescendantToPcp( procId, 2, 3 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
#I  group restored from file is incapable
0
gap> #
gap> #  The third descendant of class three is incapable.  Let us return
gap> #  to the second descendant of class 2.
gap> #
gap> PqPGSetDescendantToPcp( procId, 2, 2 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
4
gap> PqPGSetDescendantToPcp( procId, 3, 1 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
#I  group restored from file is incapable
0
gap> PqPGSetDescendantToPcp( procId, 3, 2 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
#I  group restored from file is incapable
0
gap> #
gap> #  We skip the third descendant for the moment ... 
gap> #
gap> PqPGSetDescendantToPcp( procId, 3, 4 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
#I  group restored from file is incapable
0
gap> #
gap> #  ... and look at it now.
gap> #
gap> PqPGSetDescendantToPcp( procId, 3, 3 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
6
gap> #
gap> #  In this branch of the descendant tree we get 6 descendants of class
gap> #  three.  Of those 5 will turn out to be incapable and one will have
gap> #  7 descendants.
gap> #
gap> PqPGSetDescendantToPcp( procId, 4, 1 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
#I  group restored from file is incapable
0
gap> PqPGSetDescendantToPcp( procId, 4, 2 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
7
gap> PqPGSetDescendantToPcp( procId, 4, 3 );
gap> PqPGExtendAutomorphisms( procId );
gap> PqPGConstructDescendants( procId : StepSize := 1 );
#I  group restored from file is incapable
0
]]></Example>

To automate the above procedure to some extent we provide:

<ManSection>
<Func Name="PqDescendantsTreeCoclassOne" Arg="i"/>
<Func Name="PqDescendantsTreeCoclassOne" Arg="" Label="for default process"/>

<Description>
for the <A>i</A>th or default interactive &ANUPQ; process, generate a
descendant tree for the group of the process (which must be a pc
<M>p</M>-group) consisting of descendants of <M>p</M>-coclass 1 and extending to
the class determined by the option <C>TreeDepth</C> (or 6 if the option is
omitted). In an &XGAP; session, a graphical representation of the
descendants tree appears in a separate window. Subsequent calls to
<C>PqDescendantsTreeCoclassOne</C> for the same process may be used to extend
the descendant tree from the last descendant computed that itself has
more than one descendant. <C>PqDescendantsTreeCoclassOne</C> also accepts the
options <C>CapableDescendants</C> (or <C>AllDescendants</C>) and any options
accepted by the interactive <C>PqDescendants</C> function
(see&nbsp;<Ref Func="PqDescendants" Label="interactive" Style="Text"/>).
<P/>

<E>Notes</E>
<Enum>
    <Item>
    <C>PqDescendantsTreeCoclassOne</C> first calls <C>PqDescendants</C>. If
    <C>PqDescendants</C> has already been called for the process, the previous
    value computed is used and a warning is <C>Info</C>-ed at <C>InfoANUPQ</C> level 1.
    </Item>

    <Item>
    As each descendant is processed its unique label defined by the <C>pq</C>
    program and number of descendants is <C>Info</C>-ed at <C>InfoANUPQ</C> level 1.
    </Item>

    <Item>
    <C>PqDescendantsTreeCoclassOne</C> is an <Q>experimental</Q> function that is
    included to demonstrate the sort of things that are possible with the
    <M>p</M>-group generation machinery.
    </Item>
</Enum>

Ignoring the extra functionality provided in an &XGAP; session,
<C>PqDescendantsTreeCoclassOne</C>, with one argument that is the index of an
interactive &ANUPQ; process, is approximately equivalent to:

<Listing><![CDATA[
PqDescendantsTreeCoclassOne := function( procId )
    local des, i;

    des := PqDescendants( procId : StepSize := 1 );
    RecurseDescendants( procId, 2, Length(des) );
end;
]]></Listing>

where <C>RecurseDescendants</C> is (approximately) defined as follows:

<Listing><![CDATA[
RecurseDescendants := function( procId, class, n )
    local i, nr;

    if class > ValueOption("TreeDepth") then return; fi;

    for i in [1..n] do
        PqPGSetDescendantToPcp( procId, class, i );
        PqPGExtendAutomorphisms( procId );
        nr := PqPGConstructDescendants( procId : StepSize := 1 );
        Print( "Number of descendants of group ", i,
               " at class ", class, ": ", nr, "\n" );
        RecurseDescendants( procId, class+1, nr );
    od;
    return;
end;
]]></Listing>

The following examples (executed via <C>PqExample</C>; see&nbsp;<Ref Func="PqExample" Style="Text"/>),
demonstrate the use of <C>PqDescendantsTreeCoclassOne</C>:

<List>
<Mark><C>"PqDescendantsTreeCoclassOne-9-i"</C></Mark>
<Item>
approximately does example <C>"PqDescendants-treetraverse-i"</C> again using
<C>PqDescendantsTreeCoclassOne</C>;
</Item>
<Mark><C>"PqDescendantsTreeCoclassOne-16-i"</C></Mark>
<Item>
uses the option <C>CapableDescendants</C>; and
</Item>
<Mark><C>"PqDescendantsTreeCoclassOne-25-i"</C></Mark>
<Item>
calculates all descendants by omitting the <C>CapableDescendants</C> option.
</Item>
</List>

The numbers <C>9</C>, <C>16</C> and <C>25</C> respectively, indicate the order of the
elementary abelian group to which <C>PqDescendantsTreeCoclassOne</C> is
applied for these examples.
</Description>
</ManSection>

</Section>
</Appendix>