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%A anupq.tex ANUPQ documentation - non-interactive f'ns Eamonn O'Brien
%A & Frank Celler
%A & Benedikt Rothe
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%A @(#)$Id: anupq.tex,v 1.54 2004/02/05 17:17:25 gap Exp $
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\Chapter{Non-interactive ANUPQ functions}
Here we describe all the non-interactive functions of the {\ANUPQ}
package; i.e.~``one-shot'' functions that invoke the `pq' program in such
a way that once {\GAP} has got what it needs, the `pq' program is allowed
to exit. It is expected that most of the time users will only need these
functions. The functions interface with three of the four algorithms (see
Chapter~"Introduction") provided by the ANU `pq' C program, and are
mainly grouped according to the algorithm of the `pq' program they relate
to.
In Section~"Computing p-Quotients", we describe the functions that give
access to the $p$-quotient algorithm.
Section~"Computing Standard Presentations" describe functions that give
access to the standard presentation algorithm.
Section~"Testing p-Groups for Isomorphism" describe functions that
implement an isomorphism test for $p$-groups using the standard
presentation algorithm.
In Section~"Computing Descendants of a p-Group", we describe functions
that give access to the $p$-group generation algorithm.
To use any of the functions one must have at some stage previously typed:
\begintt
gap> LoadPackage("anupq");
\endtt
(the response of which we have omitted; see~"Loading the ANUPQ Package").
It is strongly recommended that the user try the examples provided. To
save typing there is a `PqExample' equivalent for each manual example. We
also suggest that to start with you may find the examples more
instructive if you set the `InfoANUPQ' level to 2 (see~"InfoANUPQ").
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Computing p-Quotients}
\>Pq( <F> : <options> ) F
returns for the fp or pc group <F>, the $p$-quotient of <F> specified by
<options>, as a pc group. Following the colon, <options> is a selection
of the options from the following list, separated by commas like record
components (see Section~"ref:function call!with options" in the {\GAP}
Reference Manual). As a minimum the user *must* supply a value for the
`Prime' option. Below we list the options recognised by `Pq' (see
Chapter~"ANUPQ options" for detailed descriptions).
\beginlist%unordered
\atindex{option Prime}{@option \noexpand`Prime'}
\item{}`Prime := <p>'
\atindex{option ClassBound}{@option \noexpand`ClassBound'}
\item{}`ClassBound := <n>'
\atindex{option Exponent}{@option \noexpand`Exponent'}
\item{}`Exponent := <n>'
\atindex{option Relators}{@option \noexpand`Relators'}
\item{}`Relators := <rels>'
\atindex{option Metabelian}{@option \noexpand`Metabelian'}
\item{}`Metabelian'
\atindex{option Identities}{@option \noexpand`Identities'}
\item{}`Identities := <funcs>'
\atindex{option GroupName}{@option \noexpand`GroupName'}
\item{}`GroupName := <name>'
\atindex{option OutputLevel}{@option \noexpand`OutputLevel'}
\item{}`OutputLevel := <n>'
\atindex{option SetupFile}{@option \noexpand`SetupFile'}
\item{}`SetupFile := <filename>'
\atindex{option PqWorkspace}{@option \noexpand`PqWorkspace'}
\item{}`PqWorkspace := <workspace>'
\endlist
*Notes:* `Pq' may also be called with no arguments or one integer
argument, in which case it is being used interactively
(see~"Pq!interactive"); the same options may be used, except that
`SetupFile' and `PqWorkspace' are ignored by the interactive `Pq'
function.
See Section~"Attributes and a Property for fp and pc p-groups" for the
attributes and property `NuclearRank', `MultiplicatorRank' and
`IsCapable' which may be applied to the group returned by `Pq'.
See also `PqEpimorphism' ("PqEpimorphism").
We now give a few examples of the use of `Pq'. Except for the addition of
a few comments and the non-suppression of output (by not using duplicated
semicolons) the next 3 examples may be run by typing: `PqExample( "Pq"
);' (see~"PqExample").
\beginexample
gap> LoadPackage("anupq");; # does nothing if ANUPQ is already loaded
gap> # First we get a p-quotient of a free group of rank 2
gap> F := FreeGroup("a", "b");; a := F.1;; b := F.2;;
gap> Pq( F : Prime := 2, ClassBound := 3 );
<pc group of size 1024 with 10 generators>
gap> # Now let us get a p-quotient of an fp group
gap> G := F / [a^4, b^4];
<fp group on the generators [ a, b ]>
gap> Pq( G : Prime := 2, ClassBound := 3 );
<pc group of size 256 with 8 generators>
gap> # Now let's get a different p-quotient of the same group
gap> Pq( G : Prime := 2, ClassBound := 3, Exponent := 4 );
<pc group of size 128 with 7 generators>
gap> # Now we'll get a p-quotient of another fp group
gap> # which we will redo using the `Relators' option
gap> R := [ a^25, Comm(Comm(b, a), a), b^5 ];
[ a^25, a^-1*b^-1*a*b*a^-1*b^-1*a^-1*b*a^2, b^5 ]
gap> H := F / R;
<fp group on the generators [ a, b ]>
gap> Pq( H : Prime := 5, ClassBound := 5, Metabelian );
<pc group of size 78125 with 7 generators>
\endexample
\atindex{option Relators!example of usage}%
{@option \noexpand`Relators'!example of usage}
Now we redo the last example to show how one may use the `Relators'
option. Observe that `Comm(Comm(b, a), a)' is a left normed commutator
which must be written in square bracket notation for the `pq' program and
embedded in a pair of double quotes. The function `PqGAPRelators'
(see~"PqGAPRelators") can be used to translate a list of strings prepared
for the `Relators' option into {\GAP} format. Below we use it. Observe
that the value of `R' is the same as before.
\beginexample
gap> F := FreeGroup("a", "b");;
gap> # `F' was defined for `Relators'. We use the same strings that GAP uses
gap> # for printing the free group generators. It is *not* necessary to
gap> # predefine: a := F.1; etc. (as it was above).
gap> rels := [ "a^25", "[b, a, a]", "b^5" ];
[ "a^25", "[b, a, a]", "b^5" ]
gap> R := PqGAPRelators(F, rels);
[ a^25, a^-1*b^-1*a*b*a^-1*b^-1*a^-1*b*a^2, b^5 ]
gap> H := F / R;
<fp group on the generators [ a, b ]>
gap> Pq( H : Prime := 5, ClassBound := 5, Metabelian,
> Relators := rels );
<pc group of size 78125 with 7 generators>
\endexample
In fact, above we could have just passed `F' (rather than `H'), i.e.~we
could have done:
\beginexample
gap> F := FreeGroup("a", "b");;
gap> rels := [ "a^25", "[b, a, a]", "b^5" ];
[ "a^25", "[b, a, a]", "b^5" ]
gap> Pq( F : Prime := 5, ClassBound := 5, Metabelian,
> Relators := rels );
<pc group of size 78125 with 7 generators>
\endexample
The non-interactive `Pq' function also allows the options to be passed in
two other ways; these alternatives have been included for those familiar
with the {\GAP}~3 version of the {\ANUPQ} package; the preferred method
of passing options is the one already described. Firstly, they may be
passed in a record as a second argument; note that any boolean options
must be set explicitly e.g.
\beginexample
gap> Pq( H, rec( Prime := 5, ClassBound := 5, Metabelian := true ) );
<pc group of size 78125 with 7 generators>
\endexample
It is also possible to pass them as extra arguments, where each option
name appears as a string followed immediately by its value (if not a
boolean option) e.g.
\beginexample
gap> Pq( H, "Prime", 5, "ClassBound", 5, "Metabelian" );
<pc group of size 78125 with 7 generators>
\endexample
The preceding two examples can be run from {\GAP} via `PqExample( "Pq-ni"
);' (see~"PqExample").
This method of passing options permits abbreviation; the only restriction
is that the abbreviation must be unique. So `"Pr"' may be used for
`"Prime"', `"Class"' or even just `"C"' for `"ClassBound"', etc.
\atindex{option Identities!example of usage}%
{@option \noexpand`Identities'!example of usage}
The following example illustrates the use of the option `Identities'. We
compute the largest finite Burnside group of exponent $5$ that also
satisfies the $3$-Engel identity. Each identity is defined by a function
whose arguments correspond to the variables of the identity. The return
value of each of those functions is the identity evaluated on the
arguments of the function.
\beginexample
gap> F := FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> Burnside5 := x->x^5;
function( x ) ... end
gap> Engel3 := function( x,y ) return PqLeftNormComm( [x,y,y,y] ); end;
function( x, y ) ... end
gap> Pq( F : Prime := 5, Identities := [ Burnside5, Engel3 ] );
#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 3125 with 5 generators>
\endexample
The above example can be run from {\GAP} via `PqExample( "B5-5-Engel3-Id"
);' (see~"PqExample").
\>PqEpimorphism( <F> : <options> ) F
returns for the fp or pc group <F> an epimorphism from <F> onto the
$p$-quotient of <F> specified by <options>; the possible options
<options> and *required* option (`"Prime"') are as for `Pq' (see~"Pq").
`PqEpimorphism' only differs from `Pq' in what it outputs; everything
about what must/may be passed as input to `PqEpimorphism' is the same as
for `Pq'. The same alternative methods of passing options to the
non-interactive `Pq' function are available to the non-interactive
version of `PqEpimorphism'.
*Notes:* `PqEpimorphism' may also be called with no arguments or one
integer argument, in which case it is being used interactively
(see~"PqEpimorphism!interactive"), and the options `SetupFile' and
`PqWorkspace' are ignored by the interactive `PqEpimorphism' function.
See Section~"Attributes and a Property for fp and pc p-groups" for the
attributes and property `NuclearRank', `MultiplicatorRank' and
`IsCapable' which may be applied to the image group of the epimorphism
returned by `PqEpimorphism'.
\beginexample
gap> F := FreeGroup (2, "F");
<free group on the generators [ F1, F2 ]>
gap> phi := PqEpimorphism( F : Prime := 5, ClassBound := 2 );
[ F1, F2 ] -> [ f1, f2 ]
gap> Image( phi );
<pc group of size 3125 with 5 generators>
\endexample
Typing: `PqExample( "PqEpimorphism" );' runs the above example in {\GAP}
(see~"PqExample").
\>PqPCover( <F> : <options> ) F
returns for the fp or pc group <F>, the $p$-covering group of the
$p$-quotient of <F> specified by <options>, as a pc group, i.e.~the
$p$-covering group of the $p$-quotient `Pq( <F> : <options> )'. Thus the
options that `PqPCover' accepts are exactly those expected for `Pq' (and
hence as a minimum the user *must* supply a value for the `Prime' option;
see~"Pq" for more details), except in the following special case.
If <F> is already a $p$-group, in the sense that `HasIsPGroup(<F>) and
IsPGroup(<F>)' is `true', then
\beginitems
`Prime'&
defaults to `PrimePGroup(<F>)', if not supplied and `HasPrimePGroup(<F>)
= true'; and
`ClassBound'&
defaults to `PClassPGroup(<F>)' if `HasPClassPGroup(<F>) = true' if not
supplied, or to the usual default of 63, otherwise.
\enditems
The same alternative methods of passing options to the non-interactive
`Pq' function are available to the non-interactive version of `PqPCover'.
We now give a few examples of the use of `PqPCover'. These examples are
just a subset of the ones we gave for `Pq' (see~"Pq"), except that in
each instance the command `Pq' has been replaced with `PqPCover'.
Essentially the same examples may be run by typing: `PqExample(
"PqPCover" );' (see~"PqExample").
\beginexample
gap> F := FreeGroup("a", "b");; a := F.1;; b := F.2;;
gap> PqPCover( F : Prime := 2, ClassBound := 3 );
<pc group of size 262144 with 18 generators>
gap>
gap> # Now let's get a p-cover of a p-quotient of an fp group
gap> G := F / [a^4, b^4];
<fp group on the generators [ a, b ]>
gap> PqPCover( G : Prime := 2, ClassBound := 3 );
<pc group of size 16384 with 14 generators>
gap>
gap> # Now let's get a p-cover of a different p-quotient of the same group
gap> PqPCover( G : Prime := 2, ClassBound := 3, Exponent := 4 );
<pc group of size 8192 with 13 generators>
gap>
gap> # Now we'll get a p-cover of a p-quotient of another fp group
gap> # which we will redo using the `Relators' option
gap> R := [ a^25, Comm(Comm(b, a), a), b^5 ];
[ a^25, a^-1*b^-1*a*b*a^-1*b^-1*a^-1*b*a^2, b^5 ]
gap> H := F / R;
<fp group on the generators [ a, b ]>
gap> PqPCover( H : Prime := 5, ClassBound := 5, Metabelian );
<pc group of size 48828125 with 11 generators>
gap>
gap> # Now we redo the previous example using the `Relators' option
gap> F := FreeGroup("a", "b");;
gap> rels := [ "a^25", "[b, a, a]", "b^5" ];
[ "a^25", "[b, a, a]", "b^5" ]
gap> PqPCover( F : Prime := 5, ClassBound := 5, Metabelian,
> Relators := rels );
<pc group of size 48828125 with 11 generators>
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Computing Standard Presentations}
\index{automorphisms!of $p$-groups}
\>PqStandardPresentation( <F> : <options> ) F
\>StandardPresentation( <F> : <options> ) M
return the <p>-quotient specified by <options> of the fp or pc $p$-group
<F>, as an *fp group* which has a standard presentation. Here <options>
is a selection of the options from the following list (see Chapter~"ANUPQ
options" for detailed descriptions). Section~"Hints and Warnings
regarding the use of Options" gives some important hints and warnings
regarding option usage, and Section~"ref:function call!with options" in
the {\GAP} Reference Manual describes their ``record''-like syntax.
\beginlist%unordered
\atindex{option Prime}{@option \noexpand`Prime'}
\item{}`Prime := <p>'
\atindex{option pQuotient}{@option \noexpand`pQuotient'}
\item{}`pQuotient := <Q>'
\atindex{option ClassBound}{@option \noexpand`ClassBound'}
\item{}`ClassBound := <n>'
\atindex{option Exponent}{@option \noexpand`Exponent'}
\item{}`Exponent := <n>'
\atindex{option Metabelian}{@option \noexpand`Metabelian'}
\item{}`Metabelian'
\atindex{option GroupName}{@option \noexpand`GroupName'}
\item{}`GroupName := <name>'
\atindex{option OutputLevel}{@option \noexpand`OutputLevel'}
\item{}`OutputLevel := <n>'
\atindex{option StandardPresentationFile}%
{@option \noexpand`StandardPresentationFile'}
\item{}`StandardPresentationFile := <filename>'
\atindex{option SetupFile}{@option \noexpand`SetupFile'}
\item{}`SetupFile := <filename>'
\atindex{option PqWorkspace}{@option \noexpand`PqWorkspace'}
\item{}`PqWorkspace := <workspace>'
\endlist
Unless <F> is a pc <p>-group, the user *must* supply either the option
`Prime' or the option `pQuotient' (if both `Prime' and `pQuotient' are
supplied, the prime <p> is determined by applying `PrimePGroup'
(see~"ref:PrimePGroup" in the Reference Manual) to the value of
`pQuotient').
The options for `PqStandardPresentation' may also be passed in the two
alternative ways described for `Pq' (see~"Pq"). `StandardPresentation'
does not provide these alternative ways of passing options.
*Notes:*
In contrast to the function `Pq' (see~"Pq") which returns a pc group,
`PqStandardPresentation' or `StandardPresentation' returns an fp group.
This is because the output is mainly used for isomorphism testing for
which an fp group is enough. However, the presentation is a polycyclic
presentation and if you need to do any further computation with this
group (e.g.~to find the order) you can use the function `PcGroupFpGroup'
(see~"ref:PcGroupFpGroup" in the {\GAP} Reference Manual) to form a pc
group.
If the user does not supply a <p>-quotient <Q> via the `pQuotient' option
and the prime <p> is either supplied or <F> is a pc <p>-group, then a
<p>-quotient <Q> is computed. If the user does supply a <p>-quotient <Q>
via the `pQuotient' option, the package {\AutPGrp} is called to compute
the automorphism group of <Q>; an error will occur that asks the user to
install the package {\AutPGrp} if the automorphism group cannot be
computed.
The attributes and property `NuclearRank', `MultiplicatorRank' and
`IsCapable' are set for the group returned by `PqStandardPresentation' or
`StandardPresentation' (see Section~"Attributes and a Property for fp and
pc p-groups").
We illustrate the method with the following examples.
\beginexample
gap> F := FreeGroup( "a", "b" );; a := F.1;; b := F.2;;
gap> G := F / [a^25, Comm(Comm(b, a), a), b^5];
<fp group on the generators [ a, b ]>
gap> S := StandardPresentation( G : Prime := 5, ClassBound := 10 );
<fp group on the generators [ f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11,
f12, f13, f14, f15, f16, f17, f18, f19, f20, f21, f22, f23, f24, f25, f26 ]>
gap> IsPcGroup( S );
false
gap> # if we need to compute with S we should convert it to a pc group
gap> Spc := PcGroupFpGroup( S );
<pc group of size 1490116119384765625 with 26 generators>
gap>
gap> H := F / [ a^625, Comm(Comm(Comm(Comm(b, a), a), a), a)/Comm(b, a)^5,
> Comm(Comm(b, a), b), b^625 ];;
gap> StandardPresentation( H : Prime := 5, ClassBound := 15, Metabelian );
<fp group on the generators [ f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11,
f12, f13, f14, f15, f16, f17, f18, f19, f20 ]>
gap>
gap> F4 := FreeGroup( "a", "b", "c", "d" );;
gap> a := F4.1;; b := F4.2;; c := F4.3;; d := F4.4;;
gap> G4 := F4 / [ b^4, b^2 / Comm(Comm (b, a), a), d^16,
> a^16 / (c * d), b^8 / (d * c^4) ];
<fp group on the generators [ a, b, c, d ]>
gap> K := Pq( G4 : Prime := 2, ClassBound := 1 );
<pc group of size 4 with 2 generators>
gap> StandardPresentation( G4 : pQuotient := K, ClassBound := 14 );
<fp group with 53 generators>
\endexample
Typing: `PqExample( "StandardPresentation" );' runs the above example in
{\GAP} (see~"PqExample").
\>EpimorphismPqStandardPresentation( <F> : <options> ) F
\>EpimorphismStandardPresentation( <F> : <options> ) M
Each of the above functions accepts the same arguments and options as the
function `StandardPresentation' (see~"StandardPresentation") and returns
an epimorphism from the fp or pc group <F> onto the finitely presented
group given by a standard presentation, i.e.~if <S> is the standard
presentation computed for the $p$-quotient of <F> by
`StandardPresentation' then `EpimorphismStandardPresentation' returns the
epimorphism from <F> to the group with presentation <S>.
*Note:*
The attributes and property `NuclearRank', `MultiplicatorRank' and
`IsCapable' are set for the image group of the epimorphism returned by
`EpimorphismPqStandardPresentation' or `EpimorphismStandardPresentation'
(see Section~"Attributes and a Property for fp and pc p-groups").
We illustrate the function with the following example.
\beginexample
gap> F := FreeGroup(6, "F");
<free group on the generators [ F1, F2, F3, F4, F5, F6 ]>
gap> # For printing GAP uses the symbols F1, ... for the generators of F
gap> x := F.1;; y := F.2;; z := F.3;; w := F.4;; a := F.5;; b := F.6;;
gap> R := [x^3 / w, y^3 / w * a^2 * b^2, w^3 / b,
> Comm (y, x) / z, Comm (z, x), Comm (z, y) / a, z^3 ];;
gap> Q := F / R;
<fp group on the generators [ F1, F2, F3, F4, F5, F6 ]>
gap> # For printing GAP also uses the symbols F1, ... for the generators of Q
gap> # (the same as used for F) ... but the gen'rs of Q and F are different:
gap> GeneratorsOfGroup(F) = GeneratorsOfGroup(Q);
false
gap> G := Pq( Q : Prime := 3, ClassBound := 3 );
<pc group of size 729 with 6 generators>
gap> phi := EpimorphismStandardPresentation( Q : Prime := 3,
> ClassBound := 3 );
[ F1, F2, F3, F4, F5, F6 ] -> [ f1*f2^2*f3*f4^2*f5^2, f1*f2*f3*f5, f3^2,
f4*f6^2, f5, f6 ]
gap> NamesOfComponents(phi);
[ "Source", "Range", "MappingGeneratorsImages" ]
gap> phi!.Source; # This is the group Q (GAP uses F1, ... for gen'r symbols)
<fp group of size infinity on the generators [ F1, F2, F3, F4, F5, F6 ]>
gap> phi!.Range; # This is the group G (GAP uses f1, ... for gen'r symbols)
<fp group on the generators [ f1, f2, f3, f4, f5, f6 ]>
gap> AssignGeneratorVariables(G);
#I Assigned the global variables [ f1, f2, f3, f4, f5, f6 ]
gap> # Just to see that the images of [F1, ..., F6] do generate G
gap> Group([ f1*f2^2*f3, f1*f2*f3*f4*f5^2*f6^2, f3^2, f4, f5, f6 ]) = G;
true
gap> Size( Image(phi) );
729
\endexample
Typing: `PqExample( "EpimorphismStandardPresentation" );' runs the above
example in {\GAP} (see~"PqExample"). Note that `AssignGeneratorVariables'
(see~"ref:AssignGeneratorVariables") has only been available since
{\GAP}~4.3.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Testing p-Groups for Isomorphism}
\>IsPqIsomorphicPGroup( <G>, <H> ) F
\>IsIsomorphicPGroup( <G>, <H> ) M
each return true if <G> is isomorphic to <H>, where both <G> and <H> must
be pc groups of prime power order. These functions compute and compare
in {\GAP} the fp groups given by standard presentations for <G> and <H>
(see "StandardPresentation").
\beginexample
gap> G := Group( (1,2,3,4), (1,3) );
Group([ (1,2,3,4), (1,3) ])
gap> P1 := Image( IsomorphismPcGroup( G ) );
Group([ f1, f2, f3 ])
gap> P2 := SmallGroup( 8, 5 );
<pc group of size 8 with 3 generators>
gap> IsIsomorphicPGroup( P1, P2 );
false
gap> P3 := SmallGroup( 8, 4 );
<pc group of size 8 with 3 generators>
gap> IsIsomorphicPGroup( P1, P3 );
false
gap> P4 := SmallGroup( 8, 3 );
<pc group of size 8 with 3 generators>
gap> IsIsomorphicPGroup( P1, P4 );
true
\endexample
Typing: `PqExample( "IsIsomorphicPGroup" );' runs the above example in
{\GAP} (see~"PqExample").
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Computing Descendants of a p-Group}
\>PqDescendants( <G> : <options> ) F
returns, for the pc group <G> which must be of prime power order with a
confluent pc presentation (see~"ref:IsConfluent!for pc groups" in the
{\GAP} Reference Manual), a list of descendants (pc groups) of <G>.
Following the colon <options> a selection of the options listed below
should be given, separated by commas like record components (see
Section~"ref:function call!with options" in the {\GAP} Reference Manual).
See Chapter~"ANUPQ options" for detailed descriptions of the options.
The automorphism group of each descendant <D> is also computed via a call
to the `AutomorphismGroupPGroup' function of the {\AutPGrp} package.
%For each descendant <D>, a subgroup of the automorphism group of <D> is
%computed which is a supplement to the inner automorphisms of <D> in the
%whole automorphism group of <D>. This subgroup can be accessed via the
%function `PqSupplementInnerAutomorphisms'
%(see~"PqSupplementInnerAutomorphisms").
\beginlist%unordered
\atindex{option ClassBound}{@option \noexpand`ClassBound'}
\item{}`ClassBound := <n>'
\atindex{option Relators}{@option \noexpand`Relators'}
\item{}`Relators := <rels>'
\atindex{option OrderBound}{@option \noexpand`OrderBound'}
\item{}`OrderBound := <n>'
\atindex{option StepSize}{@option \noexpand`StepSize'}
\item{}`StepSize := <n>', `StepSize := <list>'
\atindex{option RankInitialSegmentSubgroups}%
{@option \noexpand`RankInitialSegmentSubgroups'}
\item{}`RankInitialSegmentSubgroups := <n>'
\atindex{option SpaceEfficient}{@option \noexpand`SpaceEfficient'}
\item{}`SpaceEfficient'
\atindex{option CapableDescendants}{@option \noexpand`CapableDescendants'}
\item{}`CapableDescendants'
\atindex{option AllDescendants}{@option \noexpand`AllDescendants'}
\item{}`AllDescendants := false'
\atindex{option Exponent}{@option \noexpand`Exponent'}
\item{}`Exponent := <n>'
\atindex{option Metabelian}{@option \noexpand`Metabelian'}
\item{}`Metabelian'
\atindex{option GroupName}{@option \noexpand`GroupName'}
\item{}`GroupName := <name>'
\atindex{option SubList}{@option \noexpand`SubList'}
\item{}`SubList := <sub>'
\atindex{option BasicAlgorithm}{@option \noexpand`BasicAlgorithm'}
\item{}`BasicAlgorithm'
\atindex{option CustomiseOutput}{@option \noexpand`CustomiseOutput'}
\item{}`CustomiseOutput := <rec>'
\atindex{option SetupFile}{@option \noexpand`SetupFile'}
\item{}`SetupFile := <filename>'
\atindex{option PqWorkspace}{@option \noexpand`PqWorkspace'}
\item{}`PqWorkspace := <workspace>'
\endlist
*Notes:*
The function `PqDescendants' uses the automorphism group of <G> which it
computes via the package {\AutPGrp}. If this package is not installed an
error may be raised. If the automorphism group of <G> is insoluble, the
`pq' program will call {\GAP} together with the {\AutPGrp} package for
certain orbit-stabilizer calculations. (So, in any case, one should
ensure the {\AutPGrp} package is installed.)
The attributes and property `NuclearRank', `MultiplicatorRank' and
`IsCapable' are set for each group of the list returned by
`PqDescendants' (see Section~"Attributes and a Property for fp and pc
p-groups").
The options <options> for `PqDescendants' may be passed in an alternative
manner to that already described, namely you can pass `PqDescendants' a
record as an argument, which contains as entries some (or all) of the
above mentioned. Those parameters which do not occur in the record are
set to their default values.
Note that you cannot set both `OrderBound' and `StepSize'.
In the first example we compute all descendants of the Klein four group
which have exponent-2 class at most 5 and order at most $2^6$.
\beginexample
gap> F := FreeGroup( "a", "b" );; a := F.1;; b := F.2;;
gap> G := PcGroupFpGroup( F / [ a^2, b^2, Comm(b, a) ] );
<pc group of size 4 with 2 generators>
gap> des := PqDescendants( G : OrderBound := 6, ClassBound := 5 );;
gap> Length(des);
83
gap> List(des, Size);
[ 8, 8, 8, 16, 16, 16, 32, 16, 16, 16, 16, 16, 32, 32, 64, 64, 32, 32, 32,
32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 32, 32, 32,
64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 32, 32, 32, 32, 64, 64, 64,
64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,
64, 64, 64, 64, 64, 64, 64 ]
gap> List(des, d -> Length( PCentralSeries( d, 2 ) ) - 1 );
[ 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4,
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
4, 4, 4, 5, 5, 5, 5, 5 ]
\endexample
Below, we compute all capable descendants of order 27 of the elementary
abelian group of order 9.
\beginexample
gap> F := FreeGroup( 2, "g" );
<free group on the generators [ g1, g2 ]>
gap> G := PcGroupFpGroup( F / [ F.1^3, F.2^3, Comm(F.1, F.2) ] );
<pc group of size 9 with 2 generators>
gap> des := PqDescendants( G : OrderBound := 3, ClassBound := 2,
> CapableDescendants );
[ <pc group of size 27 with 3 generators>,
<pc group of size 27 with 3 generators> ]
gap> List(des, d -> Length( PCentralSeries( d, 3 ) ) - 1 );
[ 2, 2 ]
gap> # For comparison let us now compute all descendants
gap> PqDescendants( G : OrderBound := 3, ClassBound := 2);
[ <pc group of size 27 with 3 generators>,
<pc group of size 27 with 3 generators>,
<pc group of size 27 with 3 generators> ]
\endexample
In the third example, we compute all capable descendants of the
elementary abelian group of order $5^2$ which have exponent-$5$ class at
most $3$, exponent $5$, and are metabelian.
\beginexample
gap> F := FreeGroup( 2, "g" );;
gap> G := PcGroupFpGroup( F / [ F.1^5, F.2^5, Comm(F.2, F.1) ] );
<pc group of size 25 with 2 generators>
gap> des := PqDescendants( G : Metabelian, ClassBound := 3,
> Exponent := 5, CapableDescendants );
[ <pc group of size 125 with 3 generators>,
<pc group of size 625 with 4 generators>,
<pc group of size 3125 with 5 generators> ]
gap> List(des, d -> Length( PCentralSeries( d, 5 ) ) - 1 );
[ 2, 3, 3 ]
gap> List(des, d -> Length( DerivedSeries( d ) ) );
[ 3, 3, 3 ]
gap> List(des, d -> Maximum( List( Elements(d), Order ) ) );
[ 5, 5, 5 ]
\endexample
The examples `"PqDescendants-1"', `"PqDescendants-2"' and
`"PqDescendants-3"' (in order) are essentially the same as the above
three examples (see~"PqExample").
\>PqSupplementInnerAutomorphisms( <D> ) F
returns a generating set for a supplement to the inner automorphisms of
<D>, in the form of a record with fields `agAutos', `agOrder' and
`glAutos', as provided by the `pq' program. One should be very careful in
using these automorphisms for a descendant calculation.
%The automorphisms generate a subgroup of the automorphism group of the pc
%group <D> that supplements the inner automorphism group of <D> in the
%whole automorphism group of <D>. The group of automorphisms returned may
%be a proper subgroup of the full automorphism group. The descendant <D>
%must have been computed by the function `PqDescendants'
%(see~"PqDescendants").
*Note:*
In principle there must be a way to use those automorphisms in order to
compute descendants but there does not seem to be a way to hand back
these automorphisms properly to the `pq' program.
\beginexample
gap> Q := Pq( FreeGroup(2) : Prime := 3, ClassBound := 1 );
<pc group of size 9 with 2 generators>
gap> des := PqDescendants( Q : 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> S := PqSupplementInnerAutomorphisms( des[3] );
rec( agAutos := [ ], agOrder := [ 3, 2, 2, 2 ],
glAutos := [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2^2, f2, f3 ],
Pcgs([ f1, f2, f3 ]) -> [ f1^2, f2, f3^2 ],
Pcgs([ f1, f2, f3 ]) -> [ f1^2, f2, f3^2 ] ] )
gap> A := AutomorphismGroupPGroup( des[3] );
rec( glAutos := [ ], glOrder := 1,
agAutos := [ Pcgs([ f1, f2, f3 ]) -> [ f1^2, f2, f3^2 ],
Pcgs([ f1, f2, f3 ]) -> [ f1*f2^2, f2, f3 ],
Pcgs([ f1, f2, f3 ]) -> [ f1*f3, f2, f3 ],
Pcgs([ f1, f2, f3 ]) -> [ f1, f2*f3, f3 ] ], agOrder := [ 2, 3, 3, 3 ],
one := IdentityMapping( <pc group of size 27 with 3 generators> ),
group := <pc group of size 27 with 3 generators>, size := 54,
glOper := [ ]
\endexample
The above example may be executed on-line via `PqExample(
"PqSupplementInnerAutomorphisms" );' (see~"PqExample").
Note that by also including `PqStart' as a second argument to `PqExample'
one can see how it is possible, with the aid of `PqSetPQuotientToGroup'
(see~"PqSetPQuotientToGroup"), to do the equivalent computations with the
interactive versions of `Pq' and `PqDescendants' and a single `pq'
process (recall `pq' is the name of the external C program).
\>PqList( <filename> [: SubList := <sub> ]) F
reads a file with name <filename> (a string) and returns the list <L> of
pc groups (or with option `SubList' a sublist of <L> or a single pc group
in <L>) defined in that file. If the option `SubList' is passed and has
the value <sub>, then it has the same meaning as for `PqDescendants',
i.e.~if <sub> is an integer then `PqList' returns `<L>[<sub>]';
otherwise, if <sub> is a list of integers `PqList' returns `Sublist(<L>,
<sub> )'.
Both `PqList' and `SavePqList' (see "SavePqList") can be used to save and
restore a list of descendants (see "PqDescendants").
\>SavePqList( <filename>, <list> ) F
writes a list of descendants <list> to a file with name <filename> (a
string).
`SavePqList' and `PqList' (see "PqList") can be used to save and restore,
respectively, the results of `PqDescendants' (see "PqDescendants").
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%E
