[ANUPQ] A Examples

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# A Examples

### Sections

1. The Relators Option
2. The Identities Option and PqEvaluateIdentities Function
3. A Large Example
4. Developing descendants trees

There are a large number of examples provided with the ANUPQ package. These may be executed or displayed via the function `PqExample` (see PqExample). Each example resides in a file of the same name in the directory `examples`. Most of the examples are translations to GAP of examples provided for the `pq` standalone by Eamonn O'Brien; the standalone examples are found in directories `standalone/examples` (p-quotient and p-group generation examples) and `standalone/isom` (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 ``live'' is always indicated near -- usually immediately after -- the text example). The format of the (`PqExample`) examples is such that they can be read by the standard `Read` function of GAP, but certain features and comments are interpreted by the function `PqExample` to do somewhat more than `Read` does. In particular, any function without a `-i`, `-ni` or `.g` suffix has both a non-interactive and interactive form; in these cases, the default form is the non-interactive form, and giving `PqStart` as second argument generates the interactive form.

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

    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

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

    gap> ?anupq:examples

which will display this appendix in a GAP session. If you are not fussed about the order in which the examples are organised, `AllPqExamples();` lists the available examples relatively compactly (see AllPqExamples).

In the remainder of this appendix we will discuss particular aspects related to the `Relators` (see option Relators) and `Identities` (see option Identities) options, and the construction of the Burnside group B(5, 4).

## A.1 The Relators Option

The `Relators` option was included because computations involving words containing commutators that are pre-expanded by GAP before being passed to the `pq` 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 `pq` program, we set the `InfoANUPQ` level to 2).

Firstly, we run the example that allows pre-expansion of commutators (the function `PqLeftNormComm` is provided by the ANUPQ package; see PqLeftNormComm). Note that since the two commutators of this example are very long (taking more than an page to print), we have edited the output at this point.

    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];
    [ b^-1*a^-2*b^-1*a*b*a*b^-1*a^-1*b*a*b*a^-1*b^-1*a*b*a^-1*b^-1*a^-1*b*a^2*c^
        ... 22 lines deleted here ...
        -1*a*b*a^-1*b^-1*a^-1*b*a^2*b*c, b^-1*a^-1*b^-2*a^-1*b*a*b*a^-1*b^
        ... 43 lines deleted here ...
        -1*b^-1*a^-1*b*a*b^-1*a^-1*b^-1*a*b^2*a*b*c,
      a*b*a*b*a*b*a*b*a*b*a*b*a*b*a*b*a*b*a*b*a*b ]
    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'.

Now we do the same calculation using the `Relators` option. In this way, the commutators are passed directly as strings to the `pq` program, so that GAP does not ``see'' them and pre-expand them.

    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'.

## A.2 The Identities Option and PqEvaluateIdentities Function

Please pay heed to the warnings given for the `Identities` option (see option Identities); it is written mainly at the GAP level and is not particularly optimised. The `Identities` option allows one to compute p-quotients that satisfy an identity. A trivial example better done using the `Exponent` option, but which nevertheless demonstrates the usage of the `Identities` option, is as follows:

    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

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

Since the identity used in the above construction of B(2, 4) is just an exponent law, the ``right'' way to compute it is via the `Exponent` option (see option Exponent), which is implemented at the C level and is highly optimised. Consequently, the `Exponent` option is significantly faster, generally by several orders of magnitude:

    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

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

    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'.

The (interactive) call to `Pq` above is essentially equivalent to a call to `PqPcPresentation` with the same arguments and options followed by a call to `PqCurrentGroup`. Moreover, the call to `PqPcPresentation` (as described in PqPcPresentation) is equivalent to a ``class 1'' call to `PqPcPresentation` followed by the requisite number of calls to `PqNextClass`, and with the `Identities` option set, both `PqPcPresentation` and `PqNextClass` ``quietly'' perform the equivalent of a `PqEvaluateIdentities` call. In the following example we break down the `Pq` call into its low-level equivalents, and set and unset the `Identities` option to show where `PqEvaluateIdentities` fits into this scheme.

    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'.

## A.3 A Large Example

An example demonstrating how a large computation can be organised with the ANUPQ package is the computation of the Burnside group B(5, 4), the largest group of exponent 4 generated by 5 elements. It has order 22728 and lower exponent-p central class 13. The example `"B5-4.g"` computes B(5, 4); it is based on a `pq` standalone input file written by M. F. Newman.

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.

Please note that the construction uses the knowledge gained by Newman and O'Brien in their initial construction of B(5, 4), in particular, insight into the commutator structure of the group and the knowledge of the p-central class and the order of B(5, 4). Therefore, the construction cannot be used to prove that B(5, 4) 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 `examples/B5-4.g`.

    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;

## A.4 Developing descendants trees

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 p-group generation machinery of the package. We start with the elementary abelian 3-group of rank 2. From within GAP, run the example `"PqDescendants-treetraverse-i"` via `PqExample` (see PqExample).

    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

To automate the above procedure to some extent we provide:

- `PqDescendantsTreeCoclassOne(` i`) F`
- `PqDescendantsTreeCoclassOne() F`

  for the ith or default interactive ANUPQ process, generate a descendant tree for the group of the process (which must be a pc p-group) consisting of descendants of p-coclass 1 and extending to the class determined by the option `TreeDepth` (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 `PqDescendantsTreeCoclassOne` for the same process may be used to extend the descendant tree from the last descendant computed that itself has more than one descendant. `PqDescendantsTreeCoclassOne` also accepts the options `CapableDescendants` (or `AllDescendants`) and any options accepted by the interactive `PqDescendants` function (see PqDescendants!interactive).

  Notes

  1. `PqDescendantsTreeCoclassOne` first calls `PqDescendants`. If `PqDescendants` has already been called for the process, the previous value computed is used and a warning is `Info`-ed at `InfoANUPQ` level 1.

  2. As each descendant is processed its unique label defined by the `pq` program and number of descendants is `Info`-ed at `InfoANUPQ` level 1.

  3. `PqDescendantsTreeCoclassOne` is an ``experimental'' function that is included to demonstrate the sort of things that are possible with the p-group generation machinery.

  Ignoring the extra functionality provided in an XGAP session, `PqDescendantsTreeCoclassOne`, with one argument that is the index of an interactive ANUPQ process, is approximately equivalent to:

      PqDescendantsTreeCoclassOne := function( procId )
          local   des,  i;

          des := PqDescendants( procId : StepSize := 1 );
          RecurseDescendants( procId, 2, Length(des) );
      end;

  where `RecurseDescendants` is (approximately) defined as follows:

      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;

  The following examples (executed via `PqExample`; see PqExample), demonstrate the use of `PqDescendantsTreeCoclassOne`:

  `"PqDescendantsTreeCoclassOne-9-i"`
    approximately does example `"PqDescendants-treetraverse-i"` again using `PqDescendantsTreeCoclassOne`;

  `"PqDescendantsTreeCoclassOne-16-i"`
    uses the option `CapableDescendants`; and

  `"PqDescendantsTreeCoclassOne-25-i"`
    calculates all descendants by omitting the `CapableDescendants` option.

  The numbers `9`, `16` and `25` respectively, indicate the order of the elementary abelian group to which `PqDescendantsTreeCoclassOne` is applied for these examples.

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