This package is written by Eddie Lo. The original program is available for anonymous ftp at math.rutgers.edu. The program is an implementation of the Baumslag-Cannonito-Miller polycyclic quotient algorithm and is written in C. For more details read BCMa,BCMb, Section 11.6 of Sims94and Lo.

This package contains functions to compute the polycyclic quotients which appear in the derived series of a finitely presented group.

- Currently, there are five functions implemented in this package:

`CallPCQA`

(see CallPCQA),

`ExtendPCQA`

(see ExtendPCQA),

`AbelianComponent`

(see AbelianComponent),

`HirschLength`

(see HirschLength),

`ModuleAction`

(see ModuleAction).

Eddie Lo

email: hlo@math.rutgers.edu

- Installing the PCQA Package
- Input format
- CallPCQA
- ExtendPCQA
- AbelianComponent
- HirschLength
- ModuleAction

The PCQA is written in C and the package can only be installed under UNIX. It has been tested on SUNs running SunOS and on IBM PCs running FreeBSD 2.1.0. It requires the GNU multiple precision arithmetic. Make sure that this library is installed before trying to install the PCQA.

If you got a complete binary and source distribution for your machine, nothing has to be done if you want to use the PCQA for a single architecture. If you want to use the PCQA for machines with different architectures skip the extraction and compilation part of this section and proceed with the creation of shell scripts described below.

If you got a complete source distribution, skip the extraction part of this section and proceed with the compilation part below.

In the example we will assume that you, as user `gap`

, are installing the
PCQA package for use by several users on a network of two SUNs, called
`bert`

and `tiffy`

, and a NeXTstation, called `bjerun`

. We assume that
**GAP3** is also installed on these machines following the instructions
given in Installation of GAP for UNIX.

Note that certain parts of the output in the examples should only be
taken as rough outline, especially file sizes and file dates are **not** to
be taken literally.

First of all you have to get the file `pcqa.zoo`

(see Getting GAP).
Then you must locate the **GAP3** directories containing `lib/`

and `doc/`

,
this is usually `gap3r4p?`

where `?`

is to be be replaced by the patch
level.

gap@tiffy:~ > ls -l drwxr-xr-x 11 gap gap 1024 Nov 8 15:16 gap3r4p3 -rw-r--r-- 1 gap gap 106307 Jan 24 15:16 pcqa.zoo

Unpack the package using `unzoo`

(see Installation of GAP for UNIX).
Note that you must be in the directory containing `gap3r4p?`

to unpack
the files. After you have unpacked the source you may remove the
**archive-file**.

gap@tiffy:~ > unzoo -x pcqa.zoo gap@tiffy:~ > ls -l gap3r4p3/pkg/pcqa -rw-r--r-- 1 gap gap 3697 Dec 14 15:58 Makefile drwxr-xr-x 2 gap gap 1024 Dec 14 15:57 bin/ drwxr-xr-x 2 gap gap 1024 Dec 14 16:12 doc/ drwxr-xr-x 2 gap gap 1024 Dec 15 18:28 examples/ -rw-r--r-- 1 gap gap 11819 Dec 14 13:31 init.g drwxr-xr-x 2 gap gap 3072 Dec 14 16:03 src/

Switch into the directory `src/`

and type `make`

to compile the PCQA. If
the header files for the GNU multiple precision arithmetic are **not** in
`/usr/local/include`

you must set `GNUINC`

to the correct directory. If
the library for the GNU multiple precision arithmetic is **not**
`/usr/local/lib/libgmp.a`

you must set `GNULIB`

. In our case we first
compile the SUN version.

gap@tiffy:~ > cd gap3r4p3/pkg/pcqa/src gap@tiffy:../src > make GNUINC=/usr/gnu/include \ GNULIB=/usr/gnu/lib/libmp.a#

` you will see a lot of messages `

If you want to use the PCQA on multiple architectures you have to move the executable to unique name.

` gap@tiffy:../pcqa > mv bin/pcqa bin/pcqa-sun-sparc-sunos `

Now repeat the compilation for the NeXTstation. **Do not** forget to clean
up.

gap@tiffy:../pcqa > rlogin bjerun gap@bjerun:~ > cd gap3r4p3/pkg/pcqa/src gap@bjerun:../src > make clean gap@bjerun:../src > make#

you will see a lot of messages gap@bjerun:../src > mv bin/pcqa ../bin/pcqa-next-m68k-mach gap@bjerun:../src > exit gap@tiffy:../src >

Switch into the subdirectory `bin/`

and create a script which will call
the correct binary for each machine. A skeleton shell script is provided
in `bin/pcqa.sh`

.

gap@tiffy:../src > cd .. gap@tiffy:../pcqa > cat bin/pcqa.sh#

!/bin/csh switch ( `hostname` ) case 'bert': case 'tiffy': exec $0-dec-mips-ultrix $* ; breaksw ; case 'bjerun': exec $0-next-m68k-mach $* ; breaksw ; default: echo "pcqa: sorry, no executable exists for this machine" ; breaksw ; endsw

`D`

gap@tiffy:../pcqa > chmod 755 bin/pcqa

Now it is time to test the package.

gap> RequirePackage("pcqa"); gap> f := FreeGroup(2); Group( f.1, f.2 ) gap> g := f/[f.1*f.2*f.1*f.2^-1*f.1^-1*f.2^-1];; gap> ds := CallPCQA( g, 2 ); rec( isDerivedSeries := true, DerivedLength := 2, QuotientStatus := 0, PolycyclicPresentation := rec( Generators := 3, ExponentList := [ 0, 0, 0 ], ppRelations := [ [ [ 0, 1, -1 ], [ 0, 1, 0 ] ], [ [ 0, 0, 1 ] ] ], pnRelations := [ [ [ 0, -1, 1 ], [ 0, -1, 0 ] ], [ [ 0, 0, -1 ] ] ], npRelations := [ [ [ 0, 0, 1 ], [ 0, -1, 1 ] ], [ [ 0, 0, 1 ] ] ], nnRelations := [ [ [ 0, 0, -1 ], [ 0, 1, -1 ] ], [ [ 0, 0, -1 ] ] ], PowerRelations := [ ] ), Homomorphisms := rec( Epimorphism := [ [ 1, 1, 0 ], [ 1, 0, 0 ] ], InverseMap := [ [ [ 2, 1 ] ], [ [ 3, -1 ], [ 1, 1 ] ], [ [ 1, 1 ], [ 3, -1 ] ] ] ), MembershipArray := [ 1, 3 ] ) gap> ExtendPCQA( g, ds.PolycyclicPresentation, ds.Homomorphisms ); rec( QuotientStatus := 5 )

This package uses the finitely presented group data structure defined in
GAP (see **Finitely Presented Groups**). It also defines and uses two types
of data structures. One data structure defines a consistent polycyclic
presentation of a polycyclic group and the other defines a homomorphism
and an inverse map between the finitely presented group and its quotient.

`CallPCQA( `

`G, n` )

This function attempts to compute the quotient of a finitely presented
group *G* by the *n+1*-st term of its derived series. A record made up
of four fields is returned. The fields are **DerivedLength**,
**QuotientStatus** , **PolycyclicPresentation** and **Homomorphisms** . If the
quotient is not polycyclic then the field **QuotientStatus** will return a
positive number. The group element represented by the module element
with that positive number generates normally a subgroup which cannot be
finitely generated. In this case the field **DerivedLength** will denote
the biggest integer *k* such that the quotient of *G* by the *k+1*-st
term in the derived series is polycyclic. The appropriate polycyclic
presentation and maps will be returned. If the field **QuotientStatus**
returns -1, then for some number *k < n*, the *k*-th term of the derived
series is the same as the *k+1*-st term of the derived series. In the
remaining case **QuotientStatus** returns 0.

The field **PolycyclicPresentation** is a record made up of seven fields.
The various conjugacy relations are stored in the fields **ppRelations** ,
**pnRelations**, **npRelations** and **nnRelations**. Each of these four fields
is an array of exponent sequences which correspond to the appropriate
left sides of the conjugacy relations . If *a _{1},a_{2},...,a_{n}* denotes the
polycyclic generators and

The field **Homomorphisms** consists of a homomorphism from the finitely
presented group to the polycyclic group and an inverse map backward. The
field **Epimorphism** stores the image of the generators of the finitely
presented group as exponent sequences of the polycyclic group . The field
**InverseMap** stores a preimage of the polycyclic generators as a word in
the finitely presented group.

gap> F := FreeGroup(2); Group( f.1, f.2 ) gap> G := F/[F.1*F.2*F.1*F.2^-1*F.1^-1*F.2^-1]; Group( f.1, f.2 ) gap> ans := CallPCQA(G,2); rec( DerivedLength := 2, QuotientStatus := 0, PolycyclicPresentation := rec( Generators := 3, ExponentList := [ 0, 0, 0 ], ppRelations := [ [ [ 0, 1, -1 ], [ 0, 1, 0 ] ], [ [ 0, 0, 1 ] ] ], pnRelations := [ [ [ 0, -1, 1 ], [ 0, -1, 0 ] ], [ [ 0, 0, -1 ] ] ], npRelations := [ [ [ 0, 0, 1 ], [ 0, -1, 1 ] ], [ [ 0, 0, 1 ] ] ], nnRelations := [ [ [ 0, 0, -1 ], [ 0, 1, -1 ] ], [ [ 0, 0, -1 ] ] ], PowerRelations := [ ] ), Homomorphisms := rec( Epimorphism := [ [ 1, 1, 0 ], [ 1, 0, 0 ] ], InverseMap := [ [ [ 2, 1 ] ], [ [ 3, -1 ], [ 1, 1 ] ], [ [ 1, 1 ], [ 3, -1 ] ] ] ), MembershipArray := [ 1, 3 ] )

`ExtendPCQA( `

`G, CPP, HOM, m, n` )

This function takes as input a finitely presented group *G*, a consistent
polycyclic presentation *CPP* (CallPCQA) of a polycyclic quotient *G/N*
of *G*, an epimorphism and an inverse map as in the field **Homomorphisms**
in CallPCQA. It determines whether the quotient *G/[N,N]* is polycyclic
and returns the flag **QuotientStatus** . It also returns the polycyclic
presentation and the appropriate homomorphism and map if the quotient is
polycyclic.

When the parameter *m* is a positive number the quotient *G/[N,N]N ^{m}* is
computed. When it is a negative number, and if

When the parameter *n* is a nonzero number, the quotient *G/[N,G]* is
computed instead. Otherwise the quotient *G/[N,N]* is computed. If this
argument is not assigned by the user, then *n* is set to zero. Different
combinations of *m* and *n* give different quotients. For example, when
**ExtendPCQA** is called with *m = 6* and *n = 1*,the quotient *G/[N,G]N ^{6}*
is computed.

gap> ExtendPCQA(G,ans.PolycyclicPresentation,ans.Homomorphisms); rec( QuotientStatus := 5 ) gap> ExtendPCQA(G,ans.PolycyclicPresentation,ans.Homomorphisms,6,1); rec( QuotientStatus := 0, PolycyclicPresentation := rec( Generators := 4, ExponentList := [ 0, 0, 0, 6 ], ppRelations := [ [[ 0, 1, -1, 0 ],[ 0, 1, 0, 0 ],[ 0, 0, 0, 1 ]], [[ 0, 0, 1, 1 ],[ 0, 0, 0, 1 ]], [[ 0, 0, 0, 1 ]] ], pnRelations := [ [[ 0, -1, 1, 5 ],[ 0, -1, 0, 0 ],[ 0, 0, 0, 5]], [[ 0, 0, -1, 5 ],[ 0, 0, 0, 5 ]], [[ 0, 0, 0, 5 ]] ], npRelations := [ [[ 0, 0, 1, 0 ],[ 0, -1, 1, 0 ],[ 0, 0, 0, 1 ]], [[ 0, 0, 1, 5 ],[ 0, 0, 0, 1 ]], [[ 0, 0, 0, 1 ]] ], nnRelations := [ [[ 0, 0, -1, 0 ],[ 0, 1, -1, 5 ],[ 0, 0, 0, 5]], [[ 0, 0, -1, 1 ],[ 0, 0, 0, 5 ]], [[ 0, 0, 0, 5 ]] ], PowerRelations := [ ,,,,,, [ 0, 0, 0, 0 ], [ 0, 0, 0, 5 ] ] ), Homomorphisms := rec( Epimorphism := [ [ 1, 1, 0, 0 ], [ 1, 0, 0, 0 ] ], InverseMap := [ [[ 2, 1 ]], [[ 3, -1 ],[ 1, 1 ]], [[ 1, 1 ],[ 3, -1 ]], [[ 5, -1 ],[ 4, -1 ],[ 5, 1 ],[ 4, 1 ]] ] ), Next := 4 )

`AbelianComponent( `

`QUOT` )

This function takes as input the output of a **CallPCQA** function call
(see CallPCQA) or an **ExtendPCQA** function call (see ExtendPCQA) and
returns the structure of the abelian groups which appear as quotients in
the derived series. The structure of each of these quotients is given by
an array of nonnegative integers.Read the section on **ElementaryDivisors**
for details.

gap> F := FreeGroup(3); Group( f.1, f.2, f.3 ) gap> G := F/[F.1*F.2*F.1*F.2,F.2*F.3^2*F.2*F.3,F.3^6]; Group( f.1, f.2, f.3 ) gap> quot := CallPCQA(G,2);; gap> AbelianComponent(quot); [ [ 1, 2, 12 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ]

`HirschLength( `

`CPP` )

This function takes as input a consistent polycyclic presentation (see CallPCQA) and returns the Hirsch length of the group presented.

gap> HirschLength(quot.PolycyclicPresentation); 11

`ModuleAction( `

`QUOT` )

This function takes as input the output of a **CallPCQA** function call
(see CallPCQA) or an **ExtendPCQA** function call (see ExtendPCQA). If
the quotient *G/[N,N]* returned by the function call is polycyclic then
**ModuleAction** computes the action of the polycyclic generators
corresponding to *G/N* on the polycyclic generators of *N/[N,N]*. The
result is returned as an array of matrices. Notice that the Smith normal
form of *G/[N,N]* is returned by the function **CallPCQA** as part of the
polycyclic presentation.

gap> ModuleAction(quot); [ [ [ 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1 ], [ 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0 ], [ 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0 ] ], [ [ -1, 0, -1, -1, -1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1 ], [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] ], [ [ 1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1 ], [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ] ] ]

gap3-jm

11 Mar 2019