From sal at dcs.st-and.ac.uk Mon Jan 2 16:08:18 2006
From: sal at dcs.st-and.ac.uk (Steve Linton)
Date: Thu Jan 5 13:04:32 2006
Subject: [GAP Forum] Fw: [GAP Support] Itanium
Message-ID: <20060102160818.577e85ea@localhost.localdomain>
Dear GAP Forum,
Dima Pasechnik reported problems compiling GAP for the Itanium processor. This
is a known problem, see
http://www.gap-system.org/Faq/Hardware-OS/hardware-os8.html
If anyone wants to work on this, I can give you some more clues,
but that page summarises the issue. A non-optimising compile usually works (but
very slowly).
Steve
--
Steve Linton School of Computer Science &
Centre for Interdisciplinary Research in Computational Algebra
University of St Andrews Tel +44 (1334) 463269
http://www.dcs.st-and.ac.uk/~sal Fax +44 (1334) 463278
From welcometn at yahoo.fr Tue Jan 3 12:57:37 2006
From: welcometn at yahoo.fr (Saber)
Date: Thu Jan 5 13:04:32 2006
Subject: [GAP Forum] colored tetrahedron
Message-ID: <20060103125737.23179.qmail@web26801.mail.ukl.yahoo.com>
Dear GAP-Forum,
i want to give all the possibilities to color the
nodes (up to 4 colors) of a regular tetrahedron:
1) related to the symmetric group $S_4$.
2) related to the alternating group $A_4$.
is there any idea to draw this in (x)gap?
thanks in advance,
Saber.
___________________________________________________________________________
Nouveau : t?l?phonez moins cher avec Yahoo! Messenger ! D?couvez les tarifs exceptionnels pour appeler la France et l'international.
T?l?chargez sur http://fr.messenger.yahoo.com
From wdj at usna.edu Wed Jan 4 12:53:06 2006
From: wdj at usna.edu (David Joyner)
Date: Thu Jan 5 13:04:32 2006
Subject: [GAP Forum] guava 2.5 beta2
Message-ID: <43BBC532.2010500@usna.edu>
Hello GAP people:
I have posted a second beta version of GUAVA 2.5
on the GAP webpage
http://cadigweb.ew.usna.edu/~wdj/gap/GUAVA/
(it is the link guava2.5.tar.gz
near the
top).
Because of the function Peter Mayr and I wrote (see below),
one major difference is that SONATA must be loaded to install
GUAVA. This doesn't bother me but might bother others.
Opinions or comments on this?
Here are the changes:
Version 2.5: (1-2006)
o Fixed undesired feature in Decodeword (spotten by Cayanne
McFarlane).
o Added MinimumWeightWords to manual; modified GAP code
for MinimumWeightWords to speed it up.
o Modified CheckMatCode to "Set" *mutable* generator
and check matrices.
o Added BitFlipDecoder, a fast decoder for LDPC codes,
written with Gordon McDonald.
o Added GuavaVersion and guava_version
o Added FerrorDesignCode, written with Peter Mayr at Linz
(one of the SONATA authors). This requires SONATA.
o Miscellaneous additions to the GUAVA manual.
Again, if there are problems, please email me. Modulo
installation problems which might arise, this is the planned
next release.
- David Joyner
From kohl at mathematik.uni-stuttgart.de Mon Jan 9 14:56:50 2006
From: kohl at mathematik.uni-stuttgart.de (Stefan Kohl)
Date: Mon Jan 9 14:57:03 2006
Subject: [GAP Forum] RCWA 1.3
Message-ID: <43C279B2.8040304@mathematik.uni-stuttgart.de>
Dear Forum,
This is to announce the release of RCWA 1.3.
The RCWA package provides methods for computing in certain
infinite permutation groups acting on the integers.
Since the initial release of version 1.0 in spring last year,
significant functionality has been added.
As usual, the RCWA package is available at
http://www.gap-system.org/Packages/rcwa.html.
Wishing you fun and success using this package,
Stefan Kohl
From bob.heffernan at gmail.com Mon Jan 9 18:25:29 2006
From: bob.heffernan at gmail.com (Robert Heffernan)
Date: Mon Jan 9 18:26:38 2006
Subject: [GAP Forum] computing automorphism groups
Message-ID: <6d9a83e90601091025u1e22c2bcsf74a5f536ba50a42@mail.gmail.com>
Hi,
I'm computing the automorphism groups of finite groups (particularly,finite perfect groups) but GAP seems to be taking an awful long timeto do this.
For example, the following has been running for several hours now:
A:=AutomorphismGroup(PerfectGroup(168,1));
The machine I am running this on is quite decently specced. Also, Istarted gap with the following:gap -o 1000Mto give GAP plenty of memory to work with.
Should I be expecting such a delay?
thank you,Bob
From hulpke at math.colostate.edu Mon Jan 9 18:40:01 2006
From: hulpke at math.colostate.edu (Alexander Hulpke)
Date: Mon Jan 9 18:40:28 2006
Subject: [GAP Forum] computing automorphism groups
In-Reply-To: <6d9a83e90601091025u1e22c2bcsf74a5f536ba50a42@mail.gmail.com>
References: <6d9a83e90601091025u1e22c2bcsf74a5f536ba50a42@mail.gmail.com>
Message-ID: <2F415FC2-8945-4C10-84BF-F63DD6736EB1@math.colostate.edu>
Dear GAP Forum,
Rober Heffernan wrot:
> I'm computing the automorphism groups of finite groups
> (particularly,finite perfect groups) but GAP seems to be taking an
> awful long timeto do this.
> For example, the following has been running for several hours now:
> A:=AutomorphismGroup(PerfectGroup(168,1));
> The machine I am running this on is quite decently specced. Also,
> Istarted gap with the following:gap -o 1000Mto give GAP plenty of
> memory to work with.
PerfectGroup returns by default a finitely presented group. An
automorphism group calculation for these is likely to be very
inefficient. If you use a permutation representation instead you are
likely to get a much better performance:
A:=AutomorphismGroup(PerfectGroup(IsPermGroup,168,1));
Best wishes,
Alexander Hulpke
From vdabbagh at math.carleton.ca Tue Jan 10 05:21:47 2006
From: vdabbagh at math.carleton.ca (Vahid Dabbaghian-Abdoly)
Date: Tue Jan 10 05:23:54 2006
Subject: [GAP Forum] Repsn 2.0. package release
Message-ID:
Dear GAP Forum,
I am pleased to announce the release of Repsn 2.0. The modifications
include:
- computing constituents of reducible representations.
- computing a block diagonal representation equivalent
to a given reducible representation.
- improving the search method for finding character subgroups.
- a bugfix in the extending representation functions.
The Repsn package is available at the following pages;
http://www.gap-system.org/Packages/repsn.html
http://www.math.carleton.ca/~vdabbagh/gap/repsn.html
Please send bug reports, suggestions and other comments to
vdabbagh@math.carleton.ca.
Best wishes,
Vahid Dabbaghian-Abdoly
From alice at maths.uwa.edu.au Wed Jan 11 10:48:19 2006
From: alice at maths.uwa.edu.au (Alice C. Niemeyer)
Date: Wed Jan 11 10:48:52 2006
Subject: [GAP Forum] colored tetrahedron
In-Reply-To: <20060103125737.23179.qmail@web26801.mail.ukl.yahoo.com>
References: <20060103125737.23179.qmail@web26801.mail.ukl.yahoo.com>
Message-ID:
Dear Saber,
It is currently not possible to draw tetrahedra in different colours
using Xgap. You might have to find a different graphical package to do
this.
However, you can use GAP to count the number of different colourings
of a tetrahedron for the groups A_n or S_n. You can find an exposition
of the basic ideas on the GAP web site under
Documentation -> Teaching -> Lectures and Workshops by Alice Niemeyer ->
Lecture 3.
Accompanying these lectures are also some GAP functions.
Another reference is
N. G. de Bruijn, in {\it Selecta Mathematica, III}, 1--26. Heidelberger
Taschenb\"ucher, 86, Springer, Berlin, 1971;
All the best, Alice Niemeyer.
=*=*=*=*=*=*=*=*=*=*=*=M=*=*=*=*M*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*
Alice C. Niemeyer =@\___) =@\ School of Mathematics & Statistics
alice@maths.uwa.edu.au \_ ( ( \ University of Western Australia
+61-8-6488 3890 .| .| .|_(\_) Nedlands, WA 6009, Australia.
=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*
WWW: http://www.maths.uwa.edu.au/~alice/
On Tue, 3 Jan 2006, Saber wrote:
>Dear GAP-Forum,
>
>i want to give all the possibilities to color the
>nodes (up to 4 colors) of a regular tetrahedron:
>1) related to the symmetric group $S_4$.
>2) related to the alternating group $A_4$.
>
>is there any idea to draw this in (x)gap?
>
>
>thanks in advance,
>Saber.
>
>
>
>
>
>
>___________________________________________________________________________
>Nouveau : t?l?phonez moins cher avec Yahoo! Messenger ! D?couvez les tarifs exceptionnels pour appeler la France et l'international.
>T?l?chargez sur http://fr.messenger.yahoo.com
>
>_______________________________________________
>Forum mailing list
>Forum@mail.gap-system.org
>http://mail.gap-system.org/mailman/listinfo/forum
>
>
>
From raghu_juliet at rediffmail.com Thu Jan 12 07:56:03 2006
From: raghu_juliet at rediffmail.com (raghunathan)
Date: Thu Jan 12 10:43:12 2006
Subject: [GAP Forum] Class structure identification of symmetric groups
Message-ID: <20060112075603.19345.qmail@webmail30.rediffmail.com>
Hello GAP forum,
Is there a way to find the different classes of permutations corresponding to different columns of the character table of a symmetric group?
For ex.,
The character table of S3 is displayed by GAP as
2 1 1 .
3 1 . 1
1a 2a 3a
2P 1a 1a 3a
3P 1a 2a 1a
X.1 1 -1 1
X.2 2 . -1
X.3 1 1 1
Is there a way to relate the symbols displayed just above the table i.e., 1a 2a 3a etc., to the different classes (1)(2)(3),(1)(2,3) &(1,2,3) of SymmetricGroup(3)?
Thanks,
Raghunathan,R.
?
From welcometn at yahoo.fr Thu Jan 12 15:24:28 2006
From: welcometn at yahoo.fr (Saber)
Date: Thu Jan 12 15:24:58 2006
Subject: [GAP Forum] colored tetrahedron
In-Reply-To:
Message-ID: <20060112152428.82236.qmail@web26815.mail.ukl.yahoo.com>
Dear Alice Niemeyer,
thank you very much for your help.
i have added a link to your program.
Saber Mbarek
Algebra und Number Theory
University of Siegen, Germany
www.math.uni-siegen.de/~mbarek
--- "Alice C. Niemeyer" a
?crit?:
> Dear Saber,
>
> It is currently not possible to draw tetrahedra in
> different colours
> using Xgap. You might have to find a different
> graphical package to do
> this.
>
> However, you can use GAP to count the number of
> different colourings
> of a tetrahedron for the groups A_n or S_n. You can
> find an exposition
> of the basic ideas on the GAP web site under
> Documentation -> Teaching -> Lectures and Workshops
> by Alice Niemeyer ->
> Lecture 3.
> Accompanying these lectures are also some GAP
> functions.
>
> Another reference is
> N. G. de Bruijn, in {\it Selecta Mathematica, III},
> 1--26. Heidelberger
> Taschenb\"ucher, 86, Springer, Berlin, 1971;
>
> All the best, Alice Niemeyer.
>
>
>
=*=*=*=*=*=*=*=*=*=*=*=M=*=*=*=*M*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*
> Alice C. Niemeyer =@\___) =@\ School of
> Mathematics & Statistics
> alice@maths.uwa.edu.au \_ ( ( \ University of
> Western Australia
> +61-8-6488 3890 .| .| .|_(\_) Nedlands, WA
> 6009, Australia.
>
=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*
> WWW: http://www.maths.uwa.edu.au/~alice/
>
> On Tue, 3 Jan 2006, Saber wrote:
>
> >Dear GAP-Forum,
> >
> >i want to give all the possibilities to color the
> >nodes (up to 4 colors) of a regular tetrahedron:
> >1) related to the symmetric group $S_4$.
> >2) related to the alternating group $A_4$.
> >
> >is there any idea to draw this in (x)gap?
> >
> >
> >thanks in advance,
> >Saber.
> >
> >
> >
> >
> >
> >
>
>___________________________________________________________________________
> >Nouveau : t?l?phonez moins cher avec Yahoo!
> Messenger ! D?couvez les tarifs exceptionnels pour
> appeler la France et l'international.
> >T?l?chargez sur http://fr.messenger.yahoo.com
> >
> >_______________________________________________
> >Forum mailing list
> >Forum@mail.gap-system.org
> >http://mail.gap-system.org/mailman/listinfo/forum
> >
> >
> >
>
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
___________________________________________________________________________
Nouveau : t?l?phonez moins cher avec Yahoo! Messenger ! D?couvez les tarifs exceptionnels pour appeler la France et l'international.
T?l?chargez sur http://fr.messenger.yahoo.com
From laurent.bartholdi at gmail.com Thu Jan 12 18:24:28 2006
From: laurent.bartholdi at gmail.com (Laurent Bartholdi)
Date: Thu Jan 12 18:25:51 2006
Subject: [GAP Forum] weird bug with wreath products
Message-ID: <1ff637850601121024r3cff4ecbn9d56d784905bc951@mail.gmail.com>
hello world,
it seems to me that there's a problem with wreath products:
GAP4, Version: 4.4.5 of 13-May-05, i686-pc-linux-gnu-gcc
gap> G := WreathProduct(CyclicGroup(3),Group((1,2,3),(4,5,6)));
gap> Gc := Image(IsomorphismPcGroup(G));
Group([ f1, f2, f3, f4, f5 ])
gap> Size(Gc);
243
gap> G := WreathProduct(CyclicGroup(IsPermGroup,3),Group((1,2,3),(4,5,6)));
gap> Gc := Image(IsomorphismPcGroup(G));
Group([ f1, f2, f3, f4, f5, f6, f7, f8 ])
gap> Size(Gc);
6561
--
Laurent Bartholdi \ laurent.bartholdigmailcom
EPFL SB SMA IMB MAD \ T?l?phone: +41 21-6930380
CH-1015 Lausanne, Switzerland \ Fax: +41 21-6930385
From thoffman at coastal.edu Thu Jan 12 19:20:15 2006
From: thoffman at coastal.edu (Tom Hoffman)
Date: Thu Jan 12 19:23:31 2006
Subject: [GAP Forum] Special unitary groups
Message-ID: <1137093615.43c6abef6b711@mail.coastal.edu>
Does anyone know how to get a permutation representation of a special unitary
group in GAP? The manual says that the command SpecialUnitaryGroup will take a
filter as its first argument but when I tried this I got the following
GAP4, Version: 4.4.6 of 02-Sep-2005, i686-pc-linux-gnu-gcc
gap> SpecialUnitaryGroup(IsPermGroup, 4,2);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `SpecialUnitaryGroupCons' on 3 arguments\
called from
SpecialUnitaryGroupCons( arg[1], arg[2], arg[3] ) called from
( ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk>
Any help would be appreciated.
Tom
Dr. Thomas R. Hoffman
Department of Mathematics and Statistics
Coastal Carolina University
-----------------------------------------------------------------------
This message was sent from the Coastal Carolina University Mail System.
From wdjoyner at comcast.net Thu Jan 12 19:57:50 2006
From: wdjoyner at comcast.net (David Joyner)
Date: Thu Jan 12 19:54:57 2006
Subject: [GAP Forum] Special unitary groups
In-Reply-To: <1137093615.43c6abef6b711@mail.coastal.edu>
References: <1137093615.43c6abef6b711@mail.coastal.edu>
Message-ID: <43C6B4BE.2000609@comcast.net>
Try |IsomorphismPermGroup:|
gap> G:=SpecialUnitaryGroup(4,2);
SU(4,2)
gap> iso:=IsomorphismPermGroup(G);
gap> image:= Image( iso );;
For more details, see section 40.2 of the reference manual.
++++++++++++++++++++++
Tom Hoffman wrote:
> Does anyone know how to get a permutation representation of a special unitary
> group in GAP? The manual says that the command SpecialUnitaryGroup will take a
> filter as its first argument but when I tried this I got the following
>
> GAP4, Version: 4.4.6 of 02-Sep-2005, i686-pc-linux-gnu-gcc
> gap> SpecialUnitaryGroup(IsPermGroup, 4,2);
> Error, no method found! For debugging hints type ?Recovery from NoMethodFound
> Error, no 1st choice method found for `SpecialUnitaryGroupCons' on 3 arguments\
> called from
> SpecialUnitaryGroupCons( arg[1], arg[2], arg[3] ) called from
> ( ) called from read-eval-loop
> Entering break read-eval-print loop ...
> you can 'quit;' to quit to outer loop, or
> you can 'return;' to continue
> brk>
>
> Any help would be appreciated.
> Tom
>
>
> Dr. Thomas R. Hoffman
> Department of Mathematics and Statistics
> Coastal Carolina University
>
> -----------------------------------------------------------------------
> This message was sent from the Coastal Carolina University Mail System.
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
>
From holmespe at for.mat.bham.ac.uk Thu Jan 12 20:33:37 2006
From: holmespe at for.mat.bham.ac.uk (Petra Holmes)
Date: Thu Jan 12 20:34:01 2006
Subject: [GAP Forum] Special unitary groups
In-Reply-To: <1137093615.43c6abef6b711@mail.coastal.edu>
Message-ID:
When you know that SU(n,q) is isomorphic to PSU(n,q) (such as in your
example SU(4,2)) then you can use PSU instead, which always arrives as a
perm group.
On Thu, 12 Jan 2006, Tom Hoffman wrote:
>
> Does anyone know how to get a permutation representation of a special unitary
> group in GAP? The manual says that the command SpecialUnitaryGroup will take a
> filter as its first argument but when I tried this I got the following
>
> GAP4, Version: 4.4.6 of 02-Sep-2005, i686-pc-linux-gnu-gcc
> gap> SpecialUnitaryGroup(IsPermGroup, 4,2);
> Error, no method found! For debugging hints type ?Recovery from NoMethodFound
> Error, no 1st choice method found for `SpecialUnitaryGroupCons' on 3 arguments\
> called from
> SpecialUnitaryGroupCons( arg[1], arg[2], arg[3] ) called from
> ( ) called from read-eval-loop
> Entering break read-eval-print loop ...
> you can 'quit;' to quit to outer loop, or
> you can 'return;' to continue
> brk>
>
> Any help would be appreciated.
> Tom
>
>
> Dr. Thomas R. Hoffman
> Department of Mathematics and Statistics
> Coastal Carolina University
>
> -----------------------------------------------------------------------
> This message was sent from the Coastal Carolina University Mail System.
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
From Frank.Luebeck at math.rwth-aachen.de Thu Jan 12 23:17:02 2006
From: Frank.Luebeck at math.rwth-aachen.de (Frank =?iso-8859-1?Q?L=FCbeck?=)
Date: Thu Jan 12 23:18:05 2006
Subject: [GAP Forum] Class structure identification of symmetric groups
In-Reply-To: <20060112075603.19345.qmail@webmail30.rediffmail.com>
References: <20060112075603.19345.qmail@webmail30.rediffmail.com>
Message-ID: <20060112231702.GA4245@math.rwth-aachen.de>
On Thu, Jan 12, 2006 at 07:56:03AM +0000, raghunathan wrote:
> Hello GAP forum,
> Is there a way to find the different classes of permutations
> corresponding to different columns of the character table of a
> symmetric group?
> For ex.,
> The character table of S3 is displayed by GAP as
> 2 1 1 .
> 3 1 . 1
>
> 1a 2a 3a
> 2P 1a 1a 3a
> 3P 1a 2a 1a
>
> X.1 1 -1 1
> X.2 2 . -1
> X.3 1 1 1
> Is there a way to relate the symbols displayed just above the table
> i.e., 1a 2a 3a etc., to the different classes (1)(2)(3),(1)(2,3) &(1,2,3)
> of SymmetricGroup(3)?
Dear Raghunathan, dear Forum,
In general it can be difficult to identify the conjugacy classes of a given
group in GAP with the columns of its abstract character table.
But for symmetric groups GAP can compute the character table, using the
labeling of conjugacy classes and irreducible characters by partitions:
gap> t := CharacterTable("Symmetric", 7);
CharacterTable( "Sym(7)" )
gap> ClassParameters(t);
[ [ 1, [ 1, 1, 1, 1, 1, 1, 1 ] ], [ 1, [ 2, 1, 1, 1, 1, 1 ] ],
[ 1, [ 2, 2, 1, 1, 1 ] ], [ 1, [ 2, 2, 2, 1 ] ], [ 1, [ 3, 1, 1, 1, 1 ] ],
[ 1, [ 3, 2, 1, 1 ] ], [ 1, [ 3, 2, 2 ] ], [ 1, [ 3, 3, 1 ] ],
[ 1, [ 4, 1, 1, 1 ] ], [ 1, [ 4, 2, 1 ] ], [ 1, [ 4, 3 ] ],
[ 1, [ 5, 1, 1 ] ], [ 1, [ 5, 2 ] ], [ 1, [ 6, 1 ] ], [ 1, [ 7 ] ] ]
The i-th entry of ClassParameters(t) describes the class of the i-th column
of t: its second component gives the cycle type of the elements in this class.
There is a similar CharacterParameters(t).
Remark: If G is a group in GAP then 'CharacterTable(G);' returns a table
which can be asked for 'IdentificationOfConjugacyClasses', e.g.:
gap> G := SymmetricGroup(20);
Sym( [ 1 .. 20 ] )
gap> t := CharacterTable(G);
CharacterTable( Sym( [ 1 .. 20 ] ) )
gap> IdentificationOfConjugacyClasses(t);
[ 1 .. 627 ]
But for bigger G GAP may not be able to compute the character table.
Nevertheless, in this particular case of symmetric groups, GAP knows that
it can use the efficient function mentioned above to compute the table.
With best regards,
Frank Luebeck
--
/// Dr. Frank L?beck, Lehrstuhl D f?r Mathematik, Templergraben 64, ///
\\\ 52062 Aachen, Germany \\\
/// E-mail: Frank.Luebeck@Math.RWTH-Aachen.De ///
\\\ WWW: http://www.math.rwth-aachen.de/~Frank.Luebeck/ \\\
From raghu_juliet at rediffmail.com Fri Jan 13 10:18:36 2006
From: raghu_juliet at rediffmail.com (raghunathan)
Date: Fri Jan 13 10:19:09 2006
Subject: [GAP Forum] Class structure identification of symmetric groups
Message-ID: <20060113101836.26675.qmail@webmail50.rediffmail.com>
>Dear Raghunathan, dear Forum,
>
>In general it can be difficult to identify the conjugacy classes of a given
>group in GAP with the columns of its abstract character table.
>
>But for symmetric groups GAP can compute the character table, using the
>labeling of conjugacy classes and irreducible characters by partitions:
>
>gap> t := CharacterTable("Symmetric", 7);
>CharacterTable( "Sym(7)" )
>gap> ClassParameters(t);
>[ [ 1, [ 1, 1, 1, 1, 1, 1, 1 ] ], [ 1, [ 2, 1, 1, 1, 1, 1 ] ],
> [ 1, [ 2, 2, 1, 1, 1 ] ], [ 1, [ 2, 2, 2, 1 ] ], [ 1, [ 3, 1, 1, 1, 1 ] ],
> [ 1, [ 3, 2, 1, 1 ] ], [ 1, [ 3, 2, 2 ] ], [ 1, [ 3, 3, 1 ] ],
> [ 1, [ 4, 1, 1, 1 ] ], [ 1, [ 4, 2, 1 ] ], [ 1, [ 4, 3 ] ],
> [ 1, [ 5, 1, 1 ] ], [ 1, [ 5, 2 ] ], [ 1, [ 6, 1 ] ], [ 1, [ 7 ] ] ]
>
>The i-th entry of ClassParameters(t) describes the class of the i-th column
>of t: its second component gives the cycle type of the elements in this class.
>
>There is a similar CharacterParameters(t).
>
>Remark: If G is a group in GAP then 'CharacterTable(G);' returns a table
>which can be asked for 'IdentificationOfConjugacyClasses', e.g.:
>
>gap> G := SymmetricGroup(20);
>Sym( [ 1 .. 20 ] )
>gap> t := CharacterTable(G);
>CharacterTable( Sym( [ 1 .. 20 ] ) )
>gap> IdentificationOfConjugacyClasses(t);
>[ 1 .. 627 ]
>
>But for bigger G GAP may not be able to compute the character table.
>Nevertheless, in this particular case of symmetric groups, GAP knows that
>it can use the efficient function mentioned above to compute the table.
>
>With best regards,
>
> Frank Luebeck
Respected Dr.Frank Luebeck and GAP Forum,
Thank you very much for ur valuable suggestion regarding class structures of symmetric groups which can be correlated to numerical partitions of the degree of the symmetric groups.But, for a wreath product group as simple as S2[S2] whose conjugacy classes' structures as given by GAP is as
gap> > [ ()^G, (3,4)^G, (1,2)(3,4)^G, (1,3)(2,4)^G, (1,3,2,4)^G ]
has two classes of permutations each with two cycles of length two (1,2)(3,4) and (1,3)(2,4). Also, the sequential commands,
gap>s:=SymmetricGroup(2);;w:=WreathProduct(s,s);;t:=CharacterTable(w);;ClassParameters(t);
did not work.Is there a way to overcome this problem by relating the abstract symbols usually displayed above any character table given by GAP as 1a 2a 2b...etc., which in the case of S4 is as follows to the class structures?
2 3 2 3 . 2
3 1 . . 1 .
1a 2a 2b 3a 4a
2P 1a 1a 1a 3a 2b
3P 1a 2a 2b 1a 4a
X.1 1 -1 1 1 -1
X.2 3 -1 -1 . 1
X.3 2 . 2 -1 .
X.4 3 1 -1 . -1
X.5 1 1 1 1 1
Thanks,
Raghunathan,R.,
M.Sc.Chemistry,
Department of Chemistry,
Indian Institute of Technology Madras,
Chennai-36,
India.
From Goetz.Pfeiffer at NUIGalway.ie Fri Jan 13 12:44:50 2006
From: Goetz.Pfeiffer at NUIGalway.ie (Goetz Pfeiffer)
Date: Fri Jan 13 12:45:12 2006
Subject: [GAP Forum] Class structure identification of symmetric groups
In-Reply-To: <20060113101836.26675.qmail@webmail50.rediffmail.com>
References: <20060113101836.26675.qmail@webmail50.rediffmail.com>
Message-ID: <20060113124450.GA12957@schmidt.nuigalway.ie>
Dear Raghunathan, dear Forum,
the function 'CharacterTableWreathSymmetric' constructs the character
table of the wreath product of a group G with the symmetric group on n
points from the character table of G and supplies it with lists of
partitions as labels for the classes (and the characters). The
underlying algorithm is described in
Character Tables of Weyl Groups in GAP.
Bayreuther Math. Schr. 47 (1994), 165-222.
(http://schmidt.nuigalway.ie/~goetz/pub/ctweyl.html)
Goetz Pfeiffer
On Fri, Jan 13, 2006 at 10:18:36AM +0000, raghunathan wrote:
> Respected Dr.Frank Luebeck and GAP Forum,
> Thank you very much for ur valuable suggestion regarding class structures of symmetric groups which can be correlated to numerical partitions of the degree of the symmetric groups.But, for a wreath product group as simple as S2[S2] whose conjugacy classes' structures as given by GAP is as
>
> gap> > [ ()^G, (3,4)^G, (1,2)(3,4)^G, (1,3)(2,4)^G, (1,3,2,4)^G ]
>
> has two classes of permutations each with two cycles of length two (1,2)(3,4) and (1,3)(2,4). Also, the sequential commands,
>
> gap>s:=SymmetricGroup(2);;w:=WreathProduct(s,s);;t:=CharacterTable(w);;ClassParameters(t);
> did not work.Is there a way to overcome this problem by relating the abstract symbols usually displayed above any character table given by GAP as 1a 2a 2b...etc., which in the case of S4 is as follows to the class structures?
>
> 2 3 2 3 . 2
> 3 1 . . 1 .
>
> 1a 2a 2b 3a 4a
> 2P 1a 1a 1a 3a 2b
> 3P 1a 2a 2b 1a 4a
>
> X.1 1 -1 1 1 -1
> X.2 3 -1 -1 . 1
> X.3 2 . 2 -1 .
> X.4 3 1 -1 . -1
> X.5 1 1 1 1 1
>
> Thanks,
> Raghunathan,R.,
> M.Sc.Chemistry,
> Department of Chemistry,
> Indian Institute of Technology Madras,
> Chennai-36,
> India.
>
>
>
>
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
-------------------------------------------------------------------------
Goetz.Pfeiffer@NUIGalway.ie http://schmidt.nuigalway.ie/~goetz/
National University of Ireland, Galway. phone +353-91-49-3591
From vdabbagh at math.carleton.ca Mon Jan 16 00:23:06 2006
From: vdabbagh at math.carleton.ca (Vahid Dabbaghian-Abdoly)
Date: Mon Jan 16 00:23:24 2006
Subject: [GAP Forum] substitution
Message-ID:
Dear GAP Forum,
I have a dense matrix of large dimensions with entries in the cyclotomic
field CF(24) and in this matrix I would like to replace the primitive
element E(24) by a prime number p. Do you know any method for this substitution?
Best regards, Vahid
From sh_fouladi at yahoo.com Mon Jan 16 08:31:16 2006
From: sh_fouladi at yahoo.com (shirin fouladi)
Date: Mon Jan 16 08:32:36 2006
Subject: [GAP Forum] find some information
Message-ID: <20060116083116.40318.qmail@web53303.mail.yahoo.com>
Dear Gap Forum.
I Read in mathscinet a summery of this paper:
" th.Exarchakos, LA-group.Math.Soc.Japan.33(1981)185-190"
but unfortunately this jornal is not available in my country
and I couldnot find the e-mail of writer.
I am very thankful if anyone tell me any information.
Best regards.
Shirin fouladi.
Faculty of Mathematical Sciences
and Computer Engineering
University For Teacher Education.
---------------------------------
Yahoo! Photos
Got holiday prints? See all the ways to get quality prints in your hands ASAP.
From laurent.bartholdi at gmail.com Tue Jan 17 14:07:51 2006
From: laurent.bartholdi at gmail.com (Laurent Bartholdi)
Date: Tue Jan 17 14:08:11 2006
Subject: [GAP Forum] CanComputeSize
Message-ID: <1ff637850601170607u72f6474l931939b2e91a9a43@mail.gmail.com>
Hi,
I try to understand Size(), HasSize() and CanComputeSize(). It seems very
strange to me that
gap> l := [];;
gap> HasSize(l);
false
gap> CanComputeSize(l);
false
isn't there a missing method/attribute here? I would expect that all dense
lists have these attributes set. However
InstallTrueMethod(CanComputeSize,IsDenseList);
doesn't change the results of the code above.
My interest is that I want to display an object either by its size (if
it exists or
is easy to compute) or as an object by itself (in other cases).
Thanks, Laurent
--
Laurent Bartholdi \ laurent.bartholdigmailcom
EPFL SB SMA IMB MAD \ T?l?phone: +41 21-6930380
CH-1015 Lausanne, Switzerland \ Fax: +41 21-6930385
From joachim.neubueser at math.rwth-aachen.de Tue Jan 17 16:23:30 2006
From: joachim.neubueser at math.rwth-aachen.de (Joachim Neubueser)
Date: Tue Jan 17 16:23:43 2006
Subject: [jneubues: Re: [GAP Forum] find some information]
Message-ID: <20060117162330.GB8114@math.rwth-aachen.de>
----- Forwarded message from jneubues -----
To: shirin fouladi
Subject: Re: [GAP Forum] find some information
Reply-To: Joachim Neubueser
Dear Shirin Fouladi,
On Mon, Jan 16, 2006 at 12:31:16AM -0800, you wrote to the GAP Forum:
> Dear Gap Forum.
> I Read in mathscinet a summery of this paper:
> " th.Exarchakos, LA-group.Math.Soc.Japan.33(1981)185-190"
> but unfortunately this jornal is not available in my country
> and I couldnot find the e-mail of writer.
The e-mail address of the writer is
texarcha@primedu.uoa.gr
The paper is probably available in our library, if you cannot get it
from the author, just send me your postal address and I will try to
send you a Xerox. But please do understand that the GAP Forum normally
is not the address for such requests.
With kind regards Joachim Neubueser
----------------------------------------------------------
Prof. em. J. Neubueser
LDFM, RWTH Aachen
Germany
------------------------------------------------------------
----- End forwarded message -----
From hulpke at math.colostate.edu Wed Jan 18 16:56:09 2006
From: hulpke at math.colostate.edu (Alexander Hulpke)
Date: Wed Jan 18 16:56:25 2006
Subject: [GAP Forum] weird bug with wreath products
In-Reply-To: <1ff637850601121024r3cff4ecbn9d56d784905bc951@mail.gmail.com>
References: <1ff637850601121024r3cff4ecbn9d56d784905bc951@mail.gmail.com>
Message-ID: <238A8206-0C6C-4707-8CDE-AC68382BC7BC@math.colostate.edu>
Dear GAP- Forum,
On Jan 12, 2006, at 11:24 , Laurent Bartholdi wrote:
>
> it seems to me that there's a problem with wreath products:
>
> GAP4, Version: 4.4.5 of 13-May-05, i686-pc-linux-gnu-gcc
> gap> G := WreathProduct(CyclicGroup(3),Group((1,2,3),(4,5,6)));
>
> gap> Gc := Image(IsomorphismPcGroup(G));
> Group([ f1, f2, f3, f4, f5 ])
> gap> Size(Gc);
> 243
Thank you for reporting this bug. The problem arises if the group H
in G\wr H is not acting transitively. In this case GAP might miss
some generators for the base of the product.
This will be corrected in the next bugfix, please write me privately
if you want a workaround now.
Best wishes and many thanks!
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke@math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
From thomas.breuer at math.rwth-aachen.de Wed Jan 18 18:07:52 2006
From: thomas.breuer at math.rwth-aachen.de (Thomas Breuer)
Date: Wed Jan 18 18:08:34 2006
Subject: [GAP Forum] CanComputeSize
Message-ID: <20060118180752.C9667777D4@altair.math.rwth-aachen.de>
Dear GAP Forum,
Laurent Bartholdi wrote
> I try to understand Size(), HasSize() and CanComputeSize(). It seems very
> strange to me that
>
> gap> l := [];;
> gap> HasSize(l);
> false
> gap> CanComputeSize(l);
> false
>
> isn't there a missing method/attribute here? I would expect that all dense
> lists have these attributes set. However
>
> InstallTrueMethod(CanComputeSize,IsDenseList);
>
> doesn't change the results of the code above.
>
> My interest is that I want to display an object either by its size (if
> it exists or
> is easy to compute) or as an object by itself (in other cases).
According to the GAP Reference Manual,
- the operation `Size' is applicable to domains and lists
(see Chapter "Collections"),
- the filter `HasSize' is `true' for an object if the `Size' value is
known (See Section "Setter and Tester for Attributes"), and
- the filter `CanComputeSize' indicates whether it is not too complicated
to compute the `Size' value of a domain
(see Section "Tests for the Availability of Methods").
The term ``known'' in the second item means that it is very cheap to
get the value when `Size' is called (see Section "Attributes");
except perhaps in special cases, this means that the object actually
stores its `Size' value, and that the object lies in the filter
`IsAttributeStoringRep'; then this value has been computed using `Size'
or has been set using `SetSize'.
In principle, it would be possible to set `HasSize' to `true' for
internally represented lists; currently this is not the case.
`CanComputeSize' was not designed as a filter for lists.
Again, in principle it could be set automatically also for internally
represented lists.
The implication from `IsDenseList' to `CanComputeSize' does not work
for internally represented lists because these lists do not carry
around their individual types:
Only a small number of types is supported for these lists,
and `CanComputeSize' is not involved in these types.
Thanks to Laurent for these suggestions how to improve the behaviour
of GAP.
(I am not sure whether these rather technical questions are of general
interest for the GAP Forum.
Perhaps the address support@gap-system.org would be more appropriate.)
All the best,
Thomas
From thomas.breuer at math.rwth-aachen.de Wed Jan 18 18:26:01 2006
From: thomas.breuer at math.rwth-aachen.de (Thomas Breuer)
Date: Wed Jan 18 18:26:27 2006
Subject: [GAP Forum] substitution
Message-ID: <20060118182601.6401E777D4@altair.math.rwth-aachen.de>
Dear GAP Forum,
Vahid Dabbaghian-Abdoly wrote
> I have a dense matrix of large dimensions with entries in the cyclotomic
> field CF(24) and in this matrix I would like to replace the primitive
> element E(24) by a prime number p.
> Do you know any method for this substitution?
If only one value shall be replaced by another value and all other values
in the matrix shall remain unchanged,
a short solution is the following.
mat:= ...; # the matrix, i.e., list of lists
old:= ...; # the value that shall be replaced
new:= ...; # the value that shall replace it
replace:= function( value )
if value = old then return new; fi; return value; end;
List( mat, row -> List( row, replace ) );
Another possible interpretation of replacement would be that
all entries of the given matrix over `CF(24)' shall be replaced
by the images under a linear map (or a ring homomorphism that is defined
by the image of `E(24)').
The replacement by a prime mentioned in the question is not of that kind,
but if one is interested in this then one can use the GAP function
`CoeffsCyc' to compute the (rational) coefficients of an element `x'
in `CF(n)' w.r.t. a basis of `CF(n)' that consists of certain powers of
`E(n)'.
`CoeffsCyc( x, n )' is a list of length `n' in which the entry at
position `i' is the coefficient of `E(n)^i'.
(See "CoeffsCyc" in the GAP Reference Manual,
more about the basis used can be found in the manual section
"Integral Bases of Abelian Number Fields".)
All the best,
Thomas
From dutour at liga.ens.fr Thu Jan 19 07:42:32 2006
From: dutour at liga.ens.fr (Mathieu Dutour)
Date: Thu Jan 19 09:38:15 2006
Subject: [GAP Forum] Function demand
Message-ID: <20060119074232.GA25528@orge.ens.fr>
Dear Gap forum,
there are two functions, which are reasonably basic, which would
perhaps be useful to have in GAP:
---Pfaffian of an antisymmetric matrix.
---sylvester inertia coefficients of a symmetric matrix.
I can contribute myself those functions, but the code may not be up
to the standard of GAP.
Mathieu
--
Mathieu Dutour Sikiric Researcher in Math
Tel. (+972)2 65 84 103 and Computer Science
Fax. (+972)2 56 30 702 Einstein Institute of Mathematics
E-mail: Mathieu.Dutour@ens.fr Hebrew University of Jerusalem
http://www.liga.ens.fr/~dutour Israel
From r_n_tsai at yahoo.com Fri Jan 20 21:22:28 2006
From: r_n_tsai at yahoo.com (R.N. Tsai)
Date: Fri Jan 20 21:22:46 2006
Subject: [GAP Forum] universal enveloping algebras
Message-ID: <20060120212228.53035.qmail@web33704.mail.mud.yahoo.com>
Dear gap forum,
I'm having difficulty in manipulating elements of the universal enveloping algebra
of a Lie algebra. This simple example brings out the problem :
Test:=function()local L,UL,g,x,d,t,e,f,h,a1,a2,a3,M,emb;
L:=SimpleLieAlgebra("A",1,Rationals);
UL:=UniversalEnvelopingAlgebra(L);
g:=GeneratorsOfAlgebraWithOne(UL);
d:=g[1];x:=g[2];t:=g[3];
a1:=d*x-x*d;Print(" [d,x] ",a1,"\n");
e:=x+d;f:=x-d;h:=f*e;
a2:=h*e-e*h;Print(" [h,f] ",a2,"\n");
a3:=h*f-f*h;Print(" [h,e] ",a3,"\n");
M:=FreeMagmaRing(Rationals,UL);
emb:=Embedding(UL,M);
end;
when executed this gives :
gap> Read("Test.g");
gap> Test();
[d,x] [(1)*x.3]
[h,f] [(-4)*x.1+(-2)*x.1*x.3+(4)*x.2+(-2)*x.2*x.3]
[h,e] [(-2)*x.1*x.3+(2)*x.2*x.3]
Error, this case will eventually be handled by the Vector Enumerator
which is not available yet called from
OperationAlgebraHomomorphism( A, [ [ Zero( A ) ] ], OnRight ) called from
IsomorphismMatrixFLMLOR( A ) called from
IsFiniteDimensional( V ) called from
IsFinite( V ) called from
GeneratorsOfMagma( M ) called from
...
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk>
If there's another way to map the generators of UL such that :
x -> x
d -> d
t -> 1
so that [d,x]=1 and not [d,x]=t, then this would suffice for what I'm doing.
I am running "GAP4, Version 4.4.6 of 02-Sep-2005, i686-pc-cygwin-gcc"
Thanks for you help.
R.N.
---------------------------------
Yahoo! Photos ? Showcase holiday pictures in hardcover
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From degraaf at science.unitn.it Mon Jan 23 09:52:24 2006
From: degraaf at science.unitn.it (Willem De Graaf)
Date: Mon Jan 23 09:52:49 2006
Subject: [GAP Forum] universal enveloping algebras
In-Reply-To: <20060120212228.53035.qmail@web33704.mail.mud.yahoo.com>
References: <20060120212228.53035.qmail@web33704.mail.mud.yahoo.com>
Message-ID: <43D4A758.3050109@science.unitn.it>
Dear R.N. Tsai,
You asked
>
> If there's another way to map the generators of UL such that :
> x -> x
> d -> d
> t -> 1
> so that [d,x]=1 and not [d,x]=t, then this would suffice for what I'm doing.
>I am running "GAP4, Version 4.4.6 of 02-Sep-2005, i686-pc-cygwin-gcc"
> Thanks for you help.
>
>
>
As far as I am aware, there is currently no way in GAP for doing this,
because there are no constructions of algebras with generators that satisfy
the ralations you indicate.
Also, I am not exactly sure what you want. There is no algebra homomorphism
that does the indicated thing: in the universal enveloping algebra we also
have the relation dt-td=-2d. So if you map t to 1 by an algebra
homomorphism,
then you have to map d to 0.
All the best,
Willem de Graaf
From r_n_tsai at yahoo.com Tue Jan 24 00:45:10 2006
From: r_n_tsai at yahoo.com (R.N. Tsai)
Date: Tue Jan 24 00:46:10 2006
Subject: [GAP Forum] universal enveloping algebras
In-Reply-To: <43D4A758.3050109@science.unitn.it>
Message-ID: <20060124004510.14425.qmail@web33701.mail.mud.yahoo.com>
Dear gap forum and Willem,
Thanks for your response. Maybe my example made things more complicated instead of clarifying them. I was hoping to find an equivalent to "CoefficientsAndMagmaElements" that would work with universal enveloping algebras; with that I could map the algebra elements to anything I want.
Fortunately I was pointed by email to gap code written by Jan Draisma that defines Weyl Algebras. This actually provides the ability to do the calculations I had in mind in a much better setting so I'll switch to using that.
Thanks again for your help and for Jan Draisma for very useful code.
R.N.
---------------------------------
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From ms-swissoft at arcor.de Thu Jan 26 16:22:33 2006
From: ms-swissoft at arcor.de (Michael Schweitzer)
Date: Thu Jan 26 16:22:45 2006
Subject: [GAP Forum] How to identify a group
Message-ID: <000601c62294$b772dc20$2101a8c0@SWISSOFT>
Dear forum members,
given a group G of order n (given by generators). Is GAP
able to identify the group by name or as the
group of symmetries of some geometric object?
For example: I define G such that G is isomorphic to A5. Can I ask GAP:
which group is G? And GAP answers: A5- which is, for example,
the symmetry group of the icosahedron.
That is, does GAP contain a database of finite groups of
small orders ( < several hundrets, say) which includes
information about the transformation group aspect: this
group, among other things, is the symmetry group of X or
operates in a natural manner on X (I know that GAP
does contain a database of small groups - but is this kind
of information stored there?).
The group in question is of order 432 with generators
g1 := (4,6,5)(7,8,9) and g2 := (1,7,2,4,6,9,5,3)
Regards,
Michael Schweitzer
Michael Schweitzer
Alt-Heiligensee 51 A
13503 Berlin
email: ms-swissoft@arcor.de
From klin at cs.bgu.ac.il Thu Jan 26 17:39:50 2006
From: klin at cs.bgu.ac.il (Mikhail Klin)
Date: Thu Jan 26 17:39:37 2006
Subject: [GAP Forum] How to identify a group
In-Reply-To: <000601c62294$b772dc20$2101a8c0@SWISSOFT>
Message-ID:
Dear Michael,
GAP may identify your group
or as a group of order 432,
or as a transitive group of degree 9,
however it does not have requested catalogues
of symmetries of geometrical
or combinatorial structures.
In fact, your group is the automorphism group
of the affine plane of order 3.
Moreover, this plane has is in a sense
most famous labelling.
Best regards,
Mikhail
********************************************************************
Dr. Mikhail Klin
Department of Mathematics
Ben-Gurion University of the Negev
P.O.Box 653, Beer Sheva 84105, Israel
Tel: (0)8/6477-802 (office)
(0)8/641-37-15 (home)
Fax: +972-(0)8/6477-648
e-mail: klin@cs.bgu.ac.il
On Thu, 26 Jan 2006, Michael Schweitzer wrote:
> Dear forum members,
>
> given a group G of order n (given by generators). Is GAP
> able to identify the group by name or as the
> group of symmetries of some geometric object?
>
> For example: I define G suchthat G is isomorphic to A5. Can I ask GAP:
> which group is G? And GAP answers: A5- which is,for example,
> the symmetry group of the icosahedron.
>
> That is, does GAP contain a database of finite groups of
> small orders ( < several hundrets, say) which includes
> information about the transformation group aspect: this
> group, among other things, is the symmetry group of X or
> operates in a natural manner on X (I know that GAP
> does contain a database of small groups - but is this kind
> of information stored there?).
>
> The group in question is of order 432 with generators
>
> g1 := (4,6,5)(7,8,9) andg2 := (1,7,2,4,6,9,5,3)
>
>
> Regards,
> Michael Schweitzer
>
>
> Michael Schweitzer
> Alt-Heiligensee 51 A
> 13503 Berlin
> email: ms-swissoft@arcor.de
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
From ndroock1 at gmail.com Fri Jan 27 09:46:02 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Fri Jan 27 09:46:18 2006
Subject: [GAP Forum] How to identify a group
In-Reply-To: <000601c62294$b772dc20$2101a8c0@SWISSOFT>
References: <000601c62294$b772dc20$2101a8c0@SWISSOFT>
Message-ID:
gap> G:=Group((4,6,5)(7,8,9),(1,7,2,4,6,9,5,3));
Group([ (4,6,5)(7,8,9), (1,7,2,4,6,9,5,3) ])
gap> Size(G);
432
gap> StructureDescription(G);
"(((C3 x C3) : Q8) : C3) : C2"
2006/1/26, Michael Schweitzer :
> Dear forum members,
>
> given a group G of order n (given by generators). Is GAP
> able to identify the group by name or as the
> group of symmetries of some geometric object?
>
> For example: I define G such that G is isomorphic to A5. Can I ask GAP:
> which group is G? And GAP answers: A5- which is, for example,
> the symmetry group of the icosahedron.
>
> That is, does GAP contain a database of finite groups of
> small orders ( < several hundrets, say) which includes
> information about the transformation group aspect: this
> group, among other things, is the symmetry group of X or
> operates in a natural manner on X (I know that GAP
> does contain a database of small groups - but is this kind
> of information stored there?).
>
> The group in question is of order 432 with generators
>
> g1 := (4,6,5)(7,8,9) and g2 := (1,7,2,4,6,9,5,3)
>
>
> Regards,
> Michael Schweitzer
>
>
> Michael Schweitzer
> Alt-Heiligensee 51 A
> 13503 Berlin
> email: ms-swissoft@arcor.de
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
From ndroock1 at gmail.com Fri Jan 27 09:48:59 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Fri Jan 27 09:49:05 2006
Subject: [GAP Forum] How to identify a group
In-Reply-To:
References: <000601c62294$b772dc20$2101a8c0@SWISSOFT>
Message-ID:
Sorry for the noise on the list, this reply was meant for my archive.
2006/1/27, Nilo de Roock :
> gap> G:=Group((4,6,5)(7,8,9),(1,7,2,4,6,9,5,3));
> Group([ (4,6,5)(7,8,9), (1,7,2,4,6,9,5,3) ])
> gap> Size(G);
> 432
> gap> StructureDescription(G);
> "(((C3 x C3) : Q8) : C3) : C2"
>
> 2006/1/26, Michael Schweitzer :
> > Dear forum members,
> >
> > given a group G of order n (given by generators). Is GAP
> > able to identify the group by name or as the
> > group of symmetries of some geometric object?
> >
> > For example: I define G such that G is isomorphic to A5. Can I ask GAP:
> > which group is G? And GAP answers: A5- which is, for example,
> > the symmetry group of the icosahedron.
> >
> > That is, does GAP contain a database of finite groups of
> > small orders ( < several hundrets, say) which includes
> > information about the transformation group aspect: this
> > group, among other things, is the symmetry group of X or
> > operates in a natural manner on X (I know that GAP
> > does contain a database of small groups - but is this kind
> > of information stored there?).
> >
> > The group in question is of order 432 with generators
> >
> > g1 := (4,6,5)(7,8,9) and g2 := (1,7,2,4,6,9,5,3)
> >
> >
> > Regards,
> > Michael Schweitzer
> >
> >
> > Michael Schweitzer
> > Alt-Heiligensee 51 A
> > 13503 Berlin
> > email: ms-swissoft@arcor.de
> >
> > _______________________________________________
> > Forum mailing list
> > Forum@mail.gap-system.org
> > http://mail.gap-system.org/mailman/listinfo/forum
> >
>
From ndroock1 at gmail.com Mon Jan 30 23:07:03 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Mon Jan 30 23:10:29 2006
Subject: [GAP Forum] Unexpected behaviour on StructureDescription()
Message-ID:
Hello GAP Forum,
The command
StructureDescription(DirectProduct(CyclicGroup(5),SymmetricGroup(4)));
generated the following response:
gap> StructureDescription(DirectProduct(CyclicGroup(5),SymmetricGroup(4)));
#I default `IsGeneratorsOfMagmaWithInverses' method returns `true' for
[ Tuple( [ f1, () ] ) ]
#I default `IsGeneratorsOfMagmaWithInverses' method returns `true' for
[ Tuple( [ of ..., (1,2,3,4) ] ) ]
#I default `IsGeneratorsOfMagmaWithInverses' method returns `true' for
[ Tuple( [ of ..., (1,2) ] ) ]
Error, lies not in group defined by called from
ExponentsOfPcElement( pcgs, elm ) called from
IN_LIST_DEFAULT( elm, list ) called from
Enumerator( D ) called from
called from
func( elm ) called from
...
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk>
I have never seen such a response from GAP. I use GAP mainly for
"educational algebra" i.e.: only very small, known groups - Is this
expected GAP behaviour?? If so how should I interpret the message?
Thanks in advance for an answer to my question.
Kind Regards,
nilo de roock
From greg.gamble at math.rwth-aachen.de Thu Feb 2 14:32:09 2006
From: greg.gamble at math.rwth-aachen.de (Greg Gamble)
Date: Thu Feb 2 14:32:35 2006
Subject: [GAP Forum] Package updates: ANUPQ 3.0, ACE 5.0
Message-ID: <20060202143208.GA16789@math.rwth-aachen.de>
Dear GAP Forum,
This is to announce the release of ANUPQ 3.0 and ACE 5.0.
Both packages are available from the GAP website:
http://www.gap-system.org
and my websites:
http://www.math.rwth-aachen.de/~Greg.Gamble/ (Aachen, Germany)
http://www.maths.uwa.edu.au/~gregg/ (Perth, Australia)
http://www.itee.uq.edu.au/~gregg/ (Brisbane, Australia)
There is no urgency for users to upgrade either package. The new
versions are *not* backward-compatible to versions of GAP prior to GAP
4.4. Thus, users of these packages must *not* upgrade if they are
still using GAP 4.3 (and must continue to use ANUPQ 2.2 and ACE 4.1),
though such users are strongly encouraged to upgrade their GAP to
version 4.4, in which case it makes sense to upgrade ACE and ANUPQ at
the same time.
Neither package contains new features nor fixes any outstanding bugs
(the previous versions of ACE and ANUPQ are not known to contain any
bugs). These new versions of the packages update them fully to GAP
4.4; all obsolete pre-GAP 4.4 code has been removed. While GAP
continues to support `ReadPkg' (and, in the case of ANUPQ,
`PrimeOfPGroup') users of ANUPQ and ACE may continue to use ANUPQ 2.2
and ACE 4.1, safely.
In the case of ANUPQ, the `IsPqIsomorphicPGroup' has been improved
thanks to suggestions from Marco Costantini and Jack Schmidt. Users
installing ANUPQ are urged to use testPq to check for correct
installation (and can be confident they do have a correct installation
if it responds as described in the README).
For ACE also, one of the internal functions has been simplified (by
removing GAP 4.2 compatibility). Users doing a ReadTest of the
tst/aceds.tst file of the package should be aware that if the only
difference is one of timing then their installation is quite ok (a
cputime of 0 just means the relevant instruction took less than 5ms).
Regards,
Greg Gamble
From greg.gamble at math.rwth-aachen.de Tue Feb 7 06:01:19 2006
From: greg.gamble at math.rwth-aachen.de (Greg Gamble)
Date: Wed Feb 8 14:29:07 2006
Subject: [GAP Forum] Example package updated
Message-ID: <20060207060119.GB32735@math.rwth-aachen.de>
Dear GAP Forum,
This is to announce the release of version 2.0 of the Example package.
The Example package is intended as a prototype/template of a GAP
package that hopefully new package writers will find useful. The
package complements the detailed instructions found in the Reference
Manual (chapter 74) and the Extending GAP Manual (chapter 4), links to
which, along with other useful information, can be found by following
the `Packages' then `For Authors' links at the GAP website:
http://www.gap-system.org
Also on the `Packages' page(s) is a link to the Example package.
This new version of the Example package is fully up-to-date with
respect to the new package loading mechanism of GAP 4.4, and removes
all obsolete code. New GAP package writers are urged to base their
package on this new version of the Example package and not older
versions of the package. Comments exist in the package files to
explain the changes from GAP 4.3 to GAP 4.4. The `Authorhints' link
from the `Packages' -> `For Authors' page described above, has a
section `How to adjust a package after the release of GAP 4.4?' that
provides details of these changes.
The new version of the Example package can be downloaded directly from
http://www.gap-system.org/Packages/example.html
or the package homepage:
http://www.math.rwth-aachen.de/~Greg.Gamble/Example/ (Aachen, Germany)
or its Australian mirrors:
http://www.maths.uwa.edu.au/~gregg/Example/ (Perth)
http://www.itee.uq.edu.au/~gregg/Example/ (Brisbane)
Regards,
Greg Gamble for the GAP Group
From kohl at mathematik.uni-stuttgart.de Wed Feb 8 15:54:29 2006
From: kohl at mathematik.uni-stuttgart.de (Stefan Kohl)
Date: Wed Feb 8 15:54:35 2006
Subject: [GAP Forum] Unexpected behaviour on StructureDescription()
In-Reply-To:
References:
Message-ID: <43EA1435.6020709@mathematik.uni-stuttgart.de>
Dear Forum,
Nilo de Roock wrote:
> The command
>
> StructureDescription(DirectProduct(CyclicGroup(5),SymmetricGroup(4)));
>
> generated the following response:
>
> gap> StructureDescription(DirectProduct(CyclicGroup(5),SymmetricGroup(4)));
> #I default `IsGeneratorsOfMagmaWithInverses' method returns `true' for
> [ Tuple( [ f1, () ] ) ]
This is due to a problem in a method in the package CRISP.
In the meantime the author of this package has fixed this bug.
Remedies are:
- upgrade to CRISP 1.3, or
- use
StructureDescription(DirectProduct(CyclicGroup(IsPermGroup,5),
SymmetricGroup(4))); or
StructureDescription(DirectProduct(CyclicGroup(5),
SymmetricGroup(IsPcGroup,4)));
instead or
- start GAP without autoloading of packages.
Hope this helps,
Stefan Kohl
---------------------------------------------------------------------------
Stefan Kohl
Institut f?r Geometrie und Topologie
Pfaffenwaldring 57
Universit?t Stuttgart
70550 Stuttgart / Germany
E-mail: kohl@mathematik.uni-stuttgart.de
Web: http://www.cip.mathematik.uni-stuttgart.de/~kohlsn/
---------------------------------------------------------------------------
From ndroock1 at gmail.com Fri Feb 10 09:11:34 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Fri Feb 10 09:12:13 2006
Subject: [GAP Forum] "(C4 x C2) : C2"
Message-ID:
Hello GAP Forum,
Could someone please explain why AllGroups(16)[3] and
AllGroups(16)[13] both return "(C4 x C2) : C2" on the function
StructureDescription?
Thanks in advance,
nilo
From ken.w.smith at cmich.edu Fri Feb 10 16:41:54 2006
From: ken.w.smith at cmich.edu (Ken W Smith)
Date: Fri Feb 10 16:42:08 2006
Subject: [GAP Forum] "(C4 x C2) : C2"
In-Reply-To:
References:
Message-ID: <2d4ac61a0e3c30dbdae57cb083631ad4@cmich.edu>
Hi Nilo,
Both groups are semidirect products of a normal subgroup isomorphic to
C4 x C2 with a subgroup of order 2.
(More explicitly, according to some notes of mine, group [16,3] is
generated by elements x, y, z where x has order 4, y and z have order
2, x and y commute (thus = C4 x C2), y and z commute and zxz=xy.
Group [16,13] is generated by x, y, z with orders 4, 2, 2, respectively
where xy=yx, xz=zx, zyz=x^2y.)
ken
---
On Feb 10, 2006, at 4:11 AM, Nilo de Roock wrote:
> Hello GAP Forum,
>
> Could someone please explain why AllGroups(16)[3] and
> AllGroups(16)[13] both return "(C4 x C2) : C2" on the function
> StructureDescription?
>
> Thanks in advance,
> nilo
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
>
---
Ken W. Smith, Professor of Mathematics, Central Michigan University
989-854-0185 (Cell)
http://www.cst.cmich.edu/users/smith1kw
Address for 2005-06:
22 Chase Gayton Terrace, Apt 1518
Richmond, VA 23238-6526
From ndroock1 at gmail.com Sat Feb 11 12:56:35 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Sat Feb 11 12:56:47 2006
Subject: [GAP Forum] StructureDescription Revisited
Message-ID:
Hello GAP forum,
I have posted some questions regarding StructureDescription()
recently, thank you for all answers. ( It came a bit as a shock to me
but I now understand that non-isomorphic groups can have the same
structure description. So yes the answers have been -very- helpful. I
have also updated to the latest GAP releases, both kernel and
packages. )
I am however still a bit uncertain about how I should interpret the
answers of StructureDescription() and what is the best (simplest)
method of finding the structure of a group. Let me give an example.
I am doing some experiments regarding generating sets of matrices and
the structure of the group they generate. I use for example the
following function:
testG:=function(a,b)
local M1;
M1:=[[ [ 0, -E(a)^-1 ], [ -E(a), 0 ] ],[ [ 0, -1 ], [ 1, 0 ] ], [ [
E(4*b), 0 ], [ 0, -E(4*b) ] ],[ [ -1, 0 ], [ 0, -1 ] ]];
return (Group(M1));
end;
I noticed the (for me...) interesting result that
StructureDescription(testG(8,1)) = QD16
StructureDescription(testG(8,3)) =C3 X QD16
StructureDescription(testG(8,5))= C5 X QD16
For other numbers however...
StructureDescription(testG(8,2))= GAP Error*
StructureDescription(testG(8,4))= GAP Error*
StructureDescription(testG(8,7))= GAP Error*
*="... Error, no method found! For debugging hints type ?Recovery from
NoMethodFound
Error, no 2nd choice method found for `IsNaturalGL' on 1 arguments called fro\
m..."
( I interpreted the errors as "does not generate a group", or a bug in
StructureDescription() for which a fix is due...)
Size(testG(8,1))= 16
Size(testG(8,2))= 64
Size(testG(8,3))= 48
Size(testG(8,4))= 128
Size(testG(8,5))= 80
Size(testG(8,6))= 192
Size(testG(8,7))= 112
So there -are- groups generated. But which ones?
( Starting from Size(testG(4,7*11))= 176 computation time increased noticably. )
In this particular example I would very much like to know which group
is generated in testG(8,2). Can GAP give an answer to that? Will the
forthcoming update in StructureDescription() address this issue(if an
issue at all)?
More in general, am I perhaps using StructureDescription() in a wrong
way or am I expecting too much from the command?
Thanks in advance for any advice.
nilo
From rm43 at evansville.edu Sat Feb 11 13:18:52 2006
From: rm43 at evansville.edu (Robert F. Morse)
Date: Sat Feb 11 13:19:09 2006
Subject: [GAP Forum] StructureDescription Revisited
In-Reply-To:
References:
Message-ID: <43EDE43C.8050605@evansville.edu>
Nilo de Roock wrote:
> Hello GAP forum,
>
> I have posted some questions regarding StructureDescription()
> recently, thank you for all answers. ( It came a bit as a shock to me
> but I now understand that non-isomorphic groups can have the same
> structure description. So yes the answers have been -very- helpful. I
> have also updated to the latest GAP releases, both kernel and
> packages. )
>
> I am however still a bit uncertain about how I should interpret the
> answers of StructureDescription() and what is the best (simplest)
> method of finding the structure of a group.
Dear Nilo,
One safe approach is to find an isomorphic permutation group for the
group in question and then attempt to find the structural description.
Hence for your examples:
gap> g8_2 := Image(IsomorphismPermGroup(testG(8,2)));
gap> StructureDescription(g8_2);
"(C8 x C4) : C2"
gap> g8_4 := Image(IsomorphismPermGroup(testG(8,4)));
gap> StructureDescription(g8_4);
"(C16 x C4) : C2"
gap> g8_7 := Image(IsomorphismPermGroup(testG(8,7)));
gap> StructureDescription(g8_7);
"C7 x QD16"
Regards, Robert F. Morse
From max at quendi.de Sat Feb 11 14:21:53 2006
From: max at quendi.de (Max Horn)
Date: Sat Feb 11 14:22:01 2006
Subject: [GAP Forum] StructureDescription Revisited
In-Reply-To:
References:
Message-ID:
Am 11.02.2006 um 13:56 schrieb Nilo de Roock:
> Hello GAP forum,
>
[...]
>
> *="... Error, no method found! For debugging hints type ?Recovery from
> NoMethodFound
> Error, no 2nd choice method found for `IsNaturalGL' on 1 arguments
> called fro\
> m..."
> ( I interpreted the errors as "does not generate a group", or a bug in
> StructureDescription() for which a fix is due...)
To me this seems like a bug...
>
> Size(testG(8,1))= 16
> Size(testG(8,2))= 64
> Size(testG(8,3))= 48
> Size(testG(8,4))= 128
> Size(testG(8,5))= 80
> Size(testG(8,6))= 192
> Size(testG(8,7))= 112
> So there -are- groups generated. But which ones?
You can work around the problem by first converting to e.g. a
permutation group.
gap> G:=testG(8,2);;
gap> phi:=IsomorphismPermGroup(G);
gap> StructureDescription(Image(phi));
"(C8 x C4) : C2"
Hope that helps!
Max
From R.W.Barraclough at qmul.ac.uk Fri Feb 10 18:08:23 2006
From: R.W.Barraclough at qmul.ac.uk (Richard Barraclough)
Date: Mon Feb 13 10:38:24 2006
Subject: [GAP Forum] "(C4 x C2) : C2"
In-Reply-To: <2d4ac61a0e3c30dbdae57cb083631ad4@cmich.edu>
Message-ID:
Hi Nilo,
As you have noticed, the 'shape' of a group, i.e., what you get from
StructureDescription(), does not determine the isomorphism type of the
group.
There are two split extensions of (4 x 2) by 2, the action of the outer 2 on
the normal 4x2 is different.
With Ken's notation we have ( x ) : .
Now, 4x2 has two cyclic subgroups of order 4, one generated by x that we can
see clearly, the other generated by xy. In case [16,3] z acts to swap these.
4x2 has three cyclic subgroups of order 2, generators are x^2, y and x^2y.
In case [16,3] z acts to swap with . Notice that this fixes the
two subgroups of order 4
You can't swap any other pair of order 2 subgroups. For example,
y -> x^2 -> xyxy = x^2 -> x^2 -> ...
which is nonsense.
It is also impossible to do both of these swaps at once: The first requires
you to swap x with xy, the second forces you to swap x with x^3y.
Therefore these are the only two groups of shape (4x2):2.
I seem to remember that "Groups for Undergraduates" by J. Moody determines
all of the groups of order up . I expect there are many other
references.
Richard.
> Hi Nilo,
> Both groups are semidirect products of a normal subgroup isomorphic to
> C4 x C2 with a subgroup of order 2.
>
> (More explicitly, according to some notes of mine, group [16,3] is
> generated by elements x, y, z where x has order 4, y and z have order
> 2, x and y commute (thus = C4 x C2), y and z commute and zxz=xy.
> Group [16,13] is generated by x, y, z with orders 4, 2, 2, respectively
> where xy=yx, xz=zx, zyz=x^2y.)
>
> ken
> ---
> On Feb 10, 2006, at 4:11 AM, Nilo de Roock wrote:
>
>> Hello GAP Forum,
>>
>> Could someone please explain why AllGroups(16)[3] and
>> AllGroups(16)[13] both return "(C4 x C2) : C2" on the function
>> StructureDescription?
>>
>> Thanks in advance,
>> nilo
>>
>> _______________________________________________
>> Forum mailing list
>> Forum@mail.gap-system.org
>> http://mail.gap-system.org/mailman/listinfo/forum
>>
>>
> ---
> Ken W. Smith, Professor of Mathematics, Central Michigan University
> 989-854-0185 (Cell)
> http://www.cst.cmich.edu/users/smith1kw
> Address for 2005-06:
> 22 Chase Gayton Terrace, Apt 1518
> Richmond, VA 23238-6526
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
From ally at dcs.gla.ac.uk Mon Feb 13 15:12:08 2006
From: ally at dcs.gla.ac.uk (Alastair Donaldson)
Date: Mon Feb 13 15:12:18 2006
Subject: [GAP Forum] Wreath Product Decomposition
Message-ID:
Dear Forum members
I am trying to write an algorithm which, given a permutation group G
acting (not necessarily transitively) on {1,2,.....,n} for some n>0, will
determine whether or not G is a wreath product of subgroups H and K.
According to a previous posting by Burkhard Hoefling, I think this should
be fairly easy:
"Wreath products of permutation groups can easily be recognized by looking
at their block structure, see `Blocks' in the GAP reference manual. In
general, a transitive permutation group G embeds in the wreath
product (action of block stabilizer on block) wr (action of G on
orbit of block), and you can easily check equality by comparing orders."
However, the groups which crop up in my application domain do not tend to
act transitively. As an example, consider the group
G := Group( [ (1,2), (2,3), (1,4)(2,5)(3,6)(19,20), (26,27), (28,29),
(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(19,21)(20,22)(25,30)(26,31)(27,32)(28,33)(29,34)(40,41),
(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(21,23)(22,24)(30,35)(31,36)(32,37)(33,38)(34,39)(41,42)
]);
I have worked out that this decomposes in several possible ways as H wr K,
one of which is
H := Group([ (33,34), (31,32), (11,12), (8,9), (10,11,12), (7,8,9),
(7,10)(8,11)(9,12)(21,22) ]);
and
K := Group([ (7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(21,23)(22,24)(30,35)(31,36)(32,37)(33,38)(34,39)(41,42),
(1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,21,23)(20,22,24)(25,30,35)(26,31,36)(27,32,37)(28,33,38)(29,34,39)(40,41,42) ]
(the group H can then be decomposed further as a direct product, one
factor of which is itself a wreath product).
I have written a rather complex algorithm which does this decomposition
correctly for my example. However, I'm having trouble proving the
correctness of my algorithm, which is not nearly as simple as Buckhard's approach of looking at blocks
and comparing orders. However, since blocks are not immediately
applicable when the group does not act transitively, I'm not sure how to
extend his suggested approach.
Any ideas would be greatly appreciated
-Alastair
From barracrw at for.mat.bham.ac.uk Mon Feb 13 18:51:23 2006
From: barracrw at for.mat.bham.ac.uk (Richard Barraclough)
Date: Mon Feb 13 22:01:48 2006
Subject: [GAP Forum] Wreath Product Decomposition
In-Reply-To:
Message-ID:
Dear Alastair,
Transitive imprimitive permutation groups can be decomposed as wreath
products. All intransitive permutation groups are imprimitive: The orbits
are the blocks. I guess that's not what you had in mind though.
The usual way of dealing with permutation groups is to write them as a
direct product of transitive permutation groups, then deal with the
transitive groups separately.
You probably then want at least one transitive constituent in order to be
imprimitive.
Here are some calculations.
gens := [ (1,2), (2,3), (1,4)(2,5)(3,6)(19,20), (26,27), (28,29),
(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(19,21)(20,22)(25,30)(26,31)(27,32)(28,33)(
29,34)(40,41),
(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(21,23)(22,24)(30,35)(31,36)(32,37)(3
3,38)(34,39)(41,42)
];
gp := Group(gens);
orbs := Orbits(gp);
gpsGens := []; gps := [];
for i in [1..Length(orbs)] do
gpsGens[i] := List([1..Length(gens)], j -> RestrictedPerm( gens[j],
orbs[i]));
gps[i] := Group(gpsGens[i]);
od;
gp is the direct product of the gps
You can now investigate them using GAP's built in functions, which have
already been proved correct.
looks relatively interesting.
gap> Order(gps[1]);
2239488
gap> # looks relatively interesting
gap> IsTransitive(gps[1]);
true
gap> IsPrimitive(gps[1]);
false
So that may be all you need.
You can write gps[1] as a wreath product as follows:
gap> bl := MaximalBlocks(gps[1],MovedPoints(gps[1]));
[ [ 1, 2, 3, 4, 5, 6 ], [ 7, 8, 9, 10, 11, 12 ], [ 13, 14, 15, 16, 17, 18 ]
]
# (Explained in Chapter 39 of the Ref. Manual.)
gap> H := Stabilizer(gps[1], bl[1], OnSets);
gap> HH := Group(List(GeneratorsOfGroup(H), g -> RestrictedPerm(g,bl[1])));
gap> K := Action(gps[1],bl,OnSets);
Group([ (1,2,3), (2,3) ])
gap> wr := WreathProduct(HH,K);
By construction, wr embeds in gps[1]. (I've assumed that gps[1] is acting
the same way on each block.) Burkhard Hoefling suggests comparing orders:
gap> Order(gps[1]);
2239488
gap> Order(wr);
2239488
They're the same, therefore the groups are isomorphic.
gap> # This will take a while and is probably not a good idea unless you're
gap> # _really_ sceptical
gap> iso := IsomorphismGroups(wr,gps[1]);
Once you've done this for all the gps you have gp as a direct product of
wreath products, say
(H1 wr K1) x (H2 wr K2) x ... x (Hn wr Kn)
You can then write this as a subgroup of the wreath product
(H1 x H2 x ... x Hn) wr (K1 x K2 x ... x Kn)
but that seems a bit silly because it needs many more points to support its
action.
If you're interested in the actual algorithms, I seem to remember Greg
Butler's 'Fundamental algorithms for permutation groups' being easy reading
(if your library has it).
Richard.
-------------
Richard Barraclough
School of Mathematical Sciences
Queen Mary College web: www.maths.qmul.ac.uk/~rwb
Mile End Road email: R.W.Barraclough@qmul.ac.uk
London E1 4NS
On 13/2/06 3:12 pm, "Alastair Donaldson" wrote:
> Dear Forum members
>
> I am trying to write an algorithm which, given a permutation group G
> acting (not necessarily transitively) on {1,2,.....,n} for some n>0, will
> determine whether or not G is a wreath product of subgroups H and K.
>
> According to a previous posting by Burkhard Hoefling, I think this should
> be fairly easy:
>
> "Wreath products of permutation groups can easily be recognized by looking
> at their block structure, see `Blocks' in the GAP reference manual. In
> general, a transitive permutation group G embeds in the wreath
> product (action of block stabilizer on block) wr (action of G on
> orbit of block), and you can easily check equality by comparing orders."
>
> However, the groups which crop up in my application domain do not tend to
> act transitively. As an example, consider the group
>
> G := Group( [ (1,2), (2,3), (1,4)(2,5)(3,6)(19,20), (26,27), (28,29),
> (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(19,21)(20,22)(25,30)(26,31)(27,32)(28,33)(29
> ,34)(40,41),
> (7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(21,23)(22,24)(30,35)(31,36)(32,37)(33,
> 38)(34,39)(41,42)
> ]);
>
> I have worked out that this decomposes in several possible ways as H wr K,
> one of which is
>
> H := Group([ (33,34), (31,32), (11,12), (8,9), (10,11,12), (7,8,9),
> (7,10)(8,11)(9,12)(21,22) ]);
>
> and
>
> K := Group([
> (7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(21,23)(22,24)(30,35)(31,36)(32,37)(33,
> 38)(34,39)(41,42),
> (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,21,23)(20,22,24)(25,30,
> 35)(26,31,36)(27,32,37)(28,33,38)(29,34,39)(40,41,42) ]
>
> (the group H can then be decomposed further as a direct product, one
> factor of which is itself a wreath product).
>
> I have written a rather complex algorithm which does this decomposition
> correctly for my example. However, I'm having trouble proving the
> correctness of my algorithm, which is not nearly as simple as Buckhard's
> approach of looking at blocks
> and comparing orders. However, since blocks are not immediately
> applicable when the group does not act transitively, I'm not sure how to
> extend his suggested approach.
>
> Any ideas would be greatly appreciated
>
> -Alastair
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
From ndroock1 at gmail.com Wed Feb 15 00:13:41 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Wed Feb 15 00:13:56 2006
Subject: [GAP Forum] StructureDescription Revisited
In-Reply-To: <60206.12.211.103.182.1139683037.squirrel@webmail.ms.uky.edu>
References:
<60206.12.211.103.182.1139683037.squirrel@webmail.ms.uky.edu>
Message-ID:
Thanks,
Your reply was very helpful. Since I can also report that while using
...SmallGroup(IdGroup... calculation was measurably faster. In my
case: calculations that took >> 10 minutes ( I don't know the actual
time, I cancelled after 10 min) are now done in seconds.
Kind regards,
nilo
2006/2/11, Jack Schmidt :
> Howdy, as a quick fix you convert the matrix group to a
> representation that is easier for GAP to use:
>
> for i in [1..10] do
> Print(i,": ",
> StructureDescription(SmallGroup(IdGroup(testG(8,i)))),
> "\n");
> od;
>
> This should print:
>
> 1: QD16
> 2: (C8 x C4) : C2
> 3: C3 x QD16
> 4: (C16 x C4) : C2
> 5: C5 x QD16
> 6: C3 x ((C8 x C4) : C2)
> 7: C7 x QD16
> 8: (C32 x C4) : C2
> 9: C9 x QD16
> 10: C5 x ((C8 x C4) : C2)
>
>
> The (silly) reason this works is because IdGroup is an older more
> mature function which has been tested and fixed for many years to
> handle all sorts of groups given to it. StructureDescription is
> much newer and does not yet protect itself against hard questions.
>
>
>
> On Sat, February 11, 2006 07:56, Nilo de Roock wrote:
> > Hello GAP forum,
> >
> > I have posted some questions regarding StructureDescription()
> > recently, thank you for all answers. ( It came a bit as a shock to me
> > but I now understand that non-isomorphic groups can have the same
> > structure description. So yes the answers have been -very- helpful. I
> > have also updated to the latest GAP releases, both kernel and
> > packages. )
> >
> > I am however still a bit uncertain about how I should interpret the
> > answers of StructureDescription() and what is the best (simplest)
> > method of finding the structure of a group. Let me give an example.
> >
> >
> > I am doing some experiments regarding generating sets of matrices and
> > the structure of the group they generate. I use for example the
> > following function:
> >
> > testG:=function(a,b)
> > local M1;
> > M1:=[[ [ 0, -E(a)^-1 ], [ -E(a), 0 ] ],[ [ 0, -1 ], [ 1, 0 ] ], [ [
> > E(4*b), 0 ], [ 0, -E(4*b) ] ],[ [ -1, 0 ], [ 0, -1 ] ]];
> > return (Group(M1));
> > end;
> >
> > I noticed the (for me...) interesting result that
> > StructureDescription(testG(8,1)) = QD16
> > StructureDescription(testG(8,3)) =C3 X QD16
> > StructureDescription(testG(8,5))= C5 X QD16
> > For other numbers however...
> > StructureDescription(testG(8,2))= GAP Error*
> > StructureDescription(testG(8,4))= GAP Error*
> > StructureDescription(testG(8,7))= GAP Error*
> >
> > *="... Error, no method found! For debugging hints type ?Recovery from
> > NoMethodFound
> > Error, no 2nd choice method found for `IsNaturalGL' on 1 arguments called
> > fro\
> > m..."
> > ( I interpreted the errors as "does not generate a group", or a bug in
> > StructureDescription() for which a fix is due...)
> >
> > Size(testG(8,1))= 16
> > Size(testG(8,2))= 64
> > Size(testG(8,3))= 48
> > Size(testG(8,4))= 128
> > Size(testG(8,5))= 80
> > Size(testG(8,6))= 192
> > Size(testG(8,7))= 112
> > So there -are- groups generated. But which ones?
> >
> > ( Starting from Size(testG(4,7*11))= 176 computation time increased
> > noticably. )
> >
> >
> > In this particular example I would very much like to know which group
> > is generated in testG(8,2). Can GAP give an answer to that? Will the
> > forthcoming update in StructureDescription() address this issue(if an
> > issue at all)?
> >
> > More in general, am I perhaps using StructureDescription() in a wrong
> > way or am I expecting too much from the command?
> >
> > Thanks in advance for any advice.
> > nilo
> >
> > _______________________________________________
> > Forum mailing list
> > Forum@mail.gap-system.org
> > http://mail.gap-system.org/mailman/listinfo/forum
> >
>
>
>
From ndroock1 at gmail.com Wed Feb 15 00:22:50 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Wed Feb 15 00:22:56 2006
Subject: [GAP Forum] StructureDescription Revisited
In-Reply-To:
References:
<60206.12.211.103.182.1139683037.squirrel@webmail.ms.uky.edu>
Message-ID:
Hello Forum,
This is a correction on my previous post. I wrote: "...are now done in
seconds.", this should be: "are now done in a few minutes, some even
in a few seconds."
nilo
2006/2/15, Nilo de Roock :
> Thanks,
>
> Your reply was very helpful. Since I can also report that while using
> ...SmallGroup(IdGroup... calculation was measurably faster. In my
> case: calculations that took >> 10 minutes ( I don't know the actual
> time, I cancelled after 10 min) are now done in seconds.
>
> Kind regards,
> nilo
>
>
> 2006/2/11, Jack Schmidt :
> > Howdy, as a quick fix you convert the matrix group to a
> > representation that is easier for GAP to use:
> >
> > for i in [1..10] do
> > Print(i,": ",
> > StructureDescription(SmallGroup(IdGroup(testG(8,i)))),
> > "\n");
> > od;
> >
> > This should print:
> >
> > 1: QD16
> > 2: (C8 x C4) : C2
> > 3: C3 x QD16
> > 4: (C16 x C4) : C2
> > 5: C5 x QD16
> > 6: C3 x ((C8 x C4) : C2)
> > 7: C7 x QD16
> > 8: (C32 x C4) : C2
> > 9: C9 x QD16
> > 10: C5 x ((C8 x C4) : C2)
> >
> >
> > The (silly) reason this works is because IdGroup is an older more
> > mature function which has been tested and fixed for many years to
> > handle all sorts of groups given to it. StructureDescription is
> > much newer and does not yet protect itself against hard questions.
> >
> >
> >
> > On Sat, February 11, 2006 07:56, Nilo de Roock wrote:
> > > Hello GAP forum,
> > >
> > > I have posted some questions regarding StructureDescription()
> > > recently, thank you for all answers. ( It came a bit as a shock to me
> > > but I now understand that non-isomorphic groups can have the same
> > > structure description. So yes the answers have been -very- helpful. I
> > > have also updated to the latest GAP releases, both kernel and
> > > packages. )
> > >
> > > I am however still a bit uncertain about how I should interpret the
> > > answers of StructureDescription() and what is the best (simplest)
> > > method of finding the structure of a group. Let me give an example.
> > >
> > >
> > > I am doing some experiments regarding generating sets of matrices and
> > > the structure of the group they generate. I use for example the
> > > following function:
> > >
> > > testG:=function(a,b)
> > > local M1;
> > > M1:=[[ [ 0, -E(a)^-1 ], [ -E(a), 0 ] ],[ [ 0, -1 ], [ 1, 0 ] ], [ [
> > > E(4*b), 0 ], [ 0, -E(4*b) ] ],[ [ -1, 0 ], [ 0, -1 ] ]];
> > > return (Group(M1));
> > > end;
> > >
> > > I noticed the (for me...) interesting result that
> > > StructureDescription(testG(8,1)) = QD16
> > > StructureDescription(testG(8,3)) =C3 X QD16
> > > StructureDescription(testG(8,5))= C5 X QD16
> > > For other numbers however...
> > > StructureDescription(testG(8,2))= GAP Error*
> > > StructureDescription(testG(8,4))= GAP Error*
> > > StructureDescription(testG(8,7))= GAP Error*
> > >
> > > *="... Error, no method found! For debugging hints type ?Recovery from
> > > NoMethodFound
> > > Error, no 2nd choice method found for `IsNaturalGL' on 1 arguments called
> > > fro\
> > > m..."
> > > ( I interpreted the errors as "does not generate a group", or a bug in
> > > StructureDescription() for which a fix is due...)
> > >
> > > Size(testG(8,1))= 16
> > > Size(testG(8,2))= 64
> > > Size(testG(8,3))= 48
> > > Size(testG(8,4))= 128
> > > Size(testG(8,5))= 80
> > > Size(testG(8,6))= 192
> > > Size(testG(8,7))= 112
> > > So there -are- groups generated. But which ones?
> > >
> > > ( Starting from Size(testG(4,7*11))= 176 computation time increased
> > > noticably. )
> > >
> > >
> > > In this particular example I would very much like to know which group
> > > is generated in testG(8,2). Can GAP give an answer to that? Will the
> > > forthcoming update in StructureDescription() address this issue(if an
> > > issue at all)?
> > >
> > > More in general, am I perhaps using StructureDescription() in a wrong
> > > way or am I expecting too much from the command?
> > >
> > > Thanks in advance for any advice.
> > > nilo
> > >
> > > _______________________________________________
> > > Forum mailing list
> > > Forum@mail.gap-system.org
> > > http://mail.gap-system.org/mailman/listinfo/forum
> > >
> >
> >
> >
>
From kohl at mathematik.uni-stuttgart.de Wed Feb 15 10:35:52 2006
From: kohl at mathematik.uni-stuttgart.de (Stefan Kohl)
Date: Wed Feb 15 10:35:59 2006
Subject: [GAP Forum] StructureDescription Revisited
In-Reply-To:
References:
Message-ID: <43F30408.7050604@mathematik.uni-stuttgart.de>
Dear Forum,
Nilo de Roock wrote:
> I am doing some experiments regarding generating sets of matrices and
> the structure of the group they generate. I use for example the
> following function:
>
> testG:=function(a,b)
> local M1;
> M1:=[[ [ 0, -E(a)^-1 ], [ -E(a), 0 ] ],[ [ 0, -1 ], [ 1, 0 ] ], [ [
> E(4*b), 0 ], [ 0, -E(4*b) ] ],[ [ -1, 0 ], [ 0, -1 ] ]];
> return (Group(M1));
> end;
>
> I noticed the (for me...) interesting result that
> StructureDescription(testG(8,1)) = QD16
> StructureDescription(testG(8,3)) =C3 X QD16
> StructureDescription(testG(8,5))= C5 X QD16
> For other numbers however...
> StructureDescription(testG(8,2))= GAP Error*
> StructureDescription(testG(8,4))= GAP Error*
> StructureDescription(testG(8,7))= GAP Error*
>
> *="... Error, no method found! For debugging hints type ?Recovery from
> NoMethodFound
> Error, no 2nd choice method found for `IsNaturalGL' on 1 arguments called fro\
> m..."
> ( I interpreted the errors as "does not generate a group", or a bug in
> StructureDescription() for which a fix is due...)
Thanks for reporting this!
This problem will be fixed in the next update.
In the meantime -- as already several people have suggested --
you can use
StructureDescription( Image ( IsomorphismPermGroup( ) ) )
instead.
Technically, the reason for the error message is that a trivial method for
`IsGeneralLinearGroup' for matrix groups in lib/grpmat.gi which is ranked
higher than the nontrivial method for generic groups in lib/grpnames.gi
calls the operation `IsNaturalGL', for which currently no nontrivial method
is available.
Thanks again and best wishes,
Stefan Kohl
---------------------------------------------------------------------------
Stefan Kohl
Institut f?r Geometrie und Topologie
Pfaffenwaldring 57
Universit?t Stuttgart
70550 Stuttgart / Germany
E-mail: kohl@mathematik.uni-stuttgart.de
Web: http://www.cip.mathematik.uni-stuttgart.de/~kohlsn/
---------------------------------------------------------------------------
From wdjoyner at comcast.net Thu Feb 16 03:24:39 2006
From: wdjoyner at comcast.net (David Joyner)
Date: Thu Feb 16 18:16:47 2006
Subject: [GAP Forum] guava 2.5
Message-ID: <43F3F077.4000801@comcast.net>
Hello GAP people:
GUAVA 2.5 is now officially released. You can find
it on the GAP website or the URL
http://cadigweb.ew.usna.edu/~wdj/gap/GUAVA/
New features are described in
http://cadigweb.ew.usna.edu/~wdj/gap/GUAVA/CHANGES
The main difference is that GUAVA 2.5 requires
SONATA 2.3.
Recently, Cary Huffman has discovered a number of
codes for which AutomorphismGroup fails. This
program calls Leon's C code. The error seems to be
so serious that Leon's code will have to eventually
abandoned (and hopefully replaced by someone's
else's some day).
- David Joyner
From ndroock1 at gmail.com Sat Feb 18 20:19:05 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Sat Feb 18 20:20:07 2006
Subject: [GAP Forum] Bug Report
Message-ID:
Hello GAP Forum,
Below you'll find
- GAP output
- listing of the GAP program I was running
- contents of stackdumpfile
I hope this helps.
Kind regards,
nilo
I got the following output...
gap> str(testH(3));
"(C12 x C2) : C2"
gap> str(testH(9));
"(C36 x C2) : C2"
gap> str(testH(27));
4 [main] gapw95 101740 handle_exceptions: Exception: STATUS_ACCESS_VIOLATI
ON
18102 [main] gapw95 101740 open_stackdumpfile: Dumping stack trace to gapw95.e
xe.stackdump
testH:=function(p)
local M, M1, T;
# QD16 (8)
# M1:=[ [[ 1, 0, 0 ], [0, 0, E(8)^-1 ], [0, E(8), 0 ] ],
# [[ 1, 0, 0 ], [0, E(4), 0 ], [0, 0, -E(4) ] ],
# [[ 1, 0, 0 ], [0, 0, -1 ], [0, 1, 0 ] ],
# [[ 1, 0, 0 ], [0, 1, 0 ], [0, 0, 1 ] ] ];
M1:=[ [[ 1, 0, 0, 0 ], [0, 0, E(p)^-1, 0 ], [0, E(p), 0, 0 ], [0,
0, 0, 1 ] ],
[[ 1, 0, 0, 0 ], [0, E(4), 0, 0 ], [0, 0, -E(4), 0 ], [0, 0, 0, 1 ] ],
[[ 1, 0, 0, 0 ], [0, 0, -1, 0 ], [0, 1, 0, 0 ], [0, 0, 0, 1 ] ],
[[ 1, 0, 0, 0 ], [0, 1, 0, 0 ], [0, 0, 1, 0 ], [0, 0, 0, 1 ] ]];
return (SmallGroup(IdGroup(Group(M1))));
end;
Exception: STATUS_ACCESS_VIOLATION at eip=00483218
eax=00000001 ebx=0AAE6098 ecx=00000000 edx=0A4A8FF8 esi=10F71F04 edi=0EA3D19C
ebp=0022DE48 esp=0022DE00 program=C:\GAP4R4\bin\gapw95.exe, pid
101740, thread main
cs=001B ds=0023 es=0023 fs=003B gs=0000 ss=0023
Stack trace:
Frame Function Args
0022DE48 00483218 (00000010, 00000000, 00000001, 0DA3434F)
0022DE68 00482B14 (00000002, 00000010, 0E990EBC, 00000000)
0022DEA8 00487552 (368D0D3D, 0A9AED74, 00000002, 0A9AECEC)
0022DED8 004D88B0 (0A66CE54, 0A9ADF38, 0022DF18, 0A070A4C)
0022DF08 0049F6E0 (0A06D150, 0AC644D4, 0A658648, 0A9ADF38)
0022DF98 004BCCC3 (0A093200, 0AC644D4, 0A658648, 0A9ADF38)
0022DFD8 004778AB (000004CC, 0A52A354, 00000504, 000005EC)
0022DFF8 004EDE54 (0000053C, 0B80DA28, 0022E028, 004ECA66)
0022E008 004EC714 (000005EC, 0AAE8D90, 0022E048, 0B80DA28)
0022E028 004ECA66 (00000600, 00000638, 0022E048, 004A4100)
0022E048 004EC9E6 (00000614, 0AAD14A4, 0022E088, 000003B8)
0022E068 004EC9E6 (00000620, 00000005, 0022E098, 00000007)
0022E098 004ED698 (0000062C, 00000714, 0022E0C8, 00000240)
0022E0B8 004EC5D5 (0000063C, 00000009, 0022E0D8, 00000005)
0022E0D8 004EDBC9 (00000664, 0A529E7C, 0022E0F8, 00000004)
0022E0F8 004EC5D5 (00000718, 0000016C, 0022E128, 0047975D)
End of stack trace (more stack frames may be present)
From ndroock1 at gmail.com Sat Feb 18 20:36:48 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Sat Feb 18 20:38:03 2006
Subject: [GAP Forum] Re: Bug Report
In-Reply-To:
References:
Message-ID:
Hello GAP Forum,
Two remarks about the bug report I just emailed.
1. Please note that although my GAP program is in
C:\GAP4R4\bin\gapw95.exe I only recently updated to the latest
releases.
2. str is an alias for StructureDescription (if that wasn't clear from
the context already).
Kind regards,
nilo
From andrew_johnson at uk.ibm.com Sun Feb 19 15:43:48 2006
From: andrew_johnson at uk.ibm.com (Andrew Johnson)
Date: Sun Feb 19 15:43:57 2006
Subject: [GAP Forum] RepresentativeAction gives error
Message-ID:
I'm using GAP (gap4r4p6-win.zip) to investigate some permutation groups.
group1 := Group([ (1,3)(2,5)(4,7)(6,8), (1,4)(2,6)(3,7)(5,8),
(1,5)(2,3)(4,8)(6,7),
(2,3,4,5,7,8,6), (3,4,7)(5,6,8) ]);
group2 := Group([ (1,3,4,7,2,6,8), (1,8,7,5,3,6,2) ]);
group3 := SymmetricGroup([1..8]);
RepresentativeAction(group3,group1,group2);
gives:
Error, no method found! For debugging hints type ?Recovery from
NoMethodFound
Error, no 1st choice method found for `ONE' on 1 arguments called from
OneOp( elm ) called from
One( F ) called from
CallFuncList( Refinements.(Rf.func), Concatenation( [ rbase, image ],
Rf.args
) ) called from
RRefine( rbase, image, false ) called from
PBEnumerate( 1, not repr ) called from
...
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
Is this expected?
My work around is to use
First(AsList(group3),e->group1^e=group2);
If this gets too slow, then I might use
IsomorphismGroups(group1, group2)
to see first of all whether the groups are isomorphic
or compare cycle structures of the elements.
signature2:= function(g)
local cg,c2;
cg := ConjugacyClasses(g);
c2 := List(cg, c->[CycleStructurePerm(Representative(c)),Size(c)]);
return SortedList(c2);
end;
to weed out groups which are clearly not conjugate.
[The reference manual for IsConjugate says 'This command is only a shortcut
to RepresentativeOperation.' I presume this should be updated to the
prefered name of RepresentativeAction.]
Andrew Johnson
From teron at udm.ru Wed Feb 22 18:57:54 2006
From: teron at udm.ru (Serge)
Date: Wed Feb 22 18:59:26 2006
Subject: [GAP Forum] Question about semigroup and finite-state machine
Message-ID:
Hello.
My problem is simple, but it is very urgent and
unfortunately I cannot solve it myself.
I have a task: There is a jump table of finite-state
machine. I must calculate
the semigroup of this FSM.
Following information is known:
1. Alphabet of FSM, set of initial and final states are the
same
2. Size of jump table is limited to 5*5
As far as I know, semigroup of finite-state machine is the
set of congruence classes of its elements.
In my case, FSM is representation from A*A to A (f: AA ->
A), where A is alphabet, set of initial and final states.
I thought that AA means Cartesian product, or in this case
second Cartesian power of set A, but I was wrong.
So, the first question is:
Can anyone explain, how should I treat record AA?
According to the definition of congruence relation, x is
congruent to y, if they are equivalent and
for any t xt is equivalent to yt and tx is equivalent to
ty. According to the definition of semigroup of
machine, t1 is congruent to t2 if for all a and b from AA
f(at1b)=f(at2b), where f is our machine.
The second question is:
How should I apply machine to the string? My teacher said
that at1b is concatenation of strings, but
I am still unclear, how to calculate f(at1b).
And the last question:
My teacher recommends me to use GAP in order to solve this
task. But I didn't use GAP earlier.
Can you tell me, which advantages I receive, if I will use
GAP for this task?
Any help is appreciated. Thank you in advance.
P.S. Any advices about algorithm are greatly appreciated.
_____
Best Regards, Serge.
mailto:teron@udm.ru
ICQ 315293596
----------------------------------------------------
????? ???????? ????? ?? - ??????????? ???????? ???
????? ?????? - ?????? ?????
http://www.mark-itt.ru/MARK-ITT/Contract/current/price.htm
From hulpke at math.colostate.edu Fri Feb 24 17:16:34 2006
From: hulpke at math.colostate.edu (Alexander Hulpke)
Date: Fri Feb 24 17:16:52 2006
Subject: [GAP Forum] Fwd: [GAP Support] Suggested response:
RepresentativeAction gives error
References: <7CA77904-F8E4-4095-96FC-EB3E2BAE122A@mac.com>
Message-ID:
Dear GAP-Forum,
Andrew Johnson reported a bug in `RepresentativeAction' for
subgroups. Thank you very much.
This error will be corrected in the next bugfix.
Before going into details of the bug in question, let me take this
opportunity to remind everyone of the existence of the email address
support@gap-system.org, which is intended for bug reports or to
request help with installation problems. The forum email list goes to
several hundred people worldwide, most of whom likely are not
interested in lengthy error descriptions or stack dumps.
Now to the bug in question. What happens is that the conjugacy test
in permutation groups contains a special treatment for groups with an
elementary abelian regular subgroup (EARNS), as such groups are
important in the classification of primitive groups. What is missing
is the (easy) check that not just one, but both groups have an EARNS.
In the example this was not the case, the error was triggered when
trying to use the EARNS of group2. (You can verify this by swapping
group2 and group1.)
As mentioned this will be corrected in the next bugfix.
In case you are testing a larger set of subgroups of S_n (for n<=31)
for conjugacy, you might want to look as well as the command
`TransitiveIdentification', which uses the classification of
transitive subgroups of the symmetric group (known up to degree 31)
and which is likely to be much faster.
Best wishes,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke@math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
From ndroock1 at gmail.com Sun Feb 26 10:44:56 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Sun Feb 26 10:45:23 2006
Subject: [GAP Forum] StructureDescription
Message-ID:
Hello GAP forum,
Forgive for yet another question (...) on this command, perhaps I am using
the wrong command for my purpose. When I execute the following command.
gap> List(AllGroups(20),StructureDescription);
GAP responds with
[ "C5 : C4", "C20", "C5 : C4", "D20", "C10 x C2" ]
The issue I have here is that "C5:C4" occurs twice in the list. What I am
looking for is a command that would recognize that the Frobenius group
~~ is in there somewhere.
Even if I do...
gap> F:=FreeGroup(2);
gap> s:=F.1;
f1
gap> t:=F.2;
f2
gap> G:=F/[s^4,t^5,t*s*t^-2*s^-1];
gap> Size(G);
20
gap> str(G);
"C5 : C4"
gap>
the answer is "C5:C4".
I am getting lost on this command, and thus on GAP. Please help.
--
met vriendelijke groet,
Nilo de Roock
From bob.heffernan at gmail.com Sun Feb 26 13:46:10 2006
From: bob.heffernan at gmail.com (Robert Heffernan)
Date: Sun Feb 26 13:46:17 2006
Subject: [GAP Forum] StructureDescription
In-Reply-To:
References:
Message-ID: <6d9a83e90602260546l300e247boebb2d1d449125949@mail.gmail.com>
On 2/26/06, Nilo de Roock wrote:> The issue I have here is that "C5:C4" occurs twice in the list. What I am> looking for is a command that would recognize that the Frobenius group> ~~~~ is in there somewhere.
The semidirect product of two groups is not unique, this is why C5:C4occurs twice. You can ask GAP to construct an explicit isomorphismbetween F and one of the two C5:C4 groups:
gap> F:=FreeGroup(2);;gap> s:=F.1;; t:=F.2;;gap> G:=F/[s^4,t^5,t*s*t^-2*s^-1];;gap> StructureDescription(G);"C5 : C4"gap> IsomorphismGroups(G,SmallGroup(20,1));failgap> IsomorphismGroups(G,SmallGroup(20,3));[ f1, f2 ] -> [ f1, f3 ]
So, as you can see, F is isomorphic to SmallGroup(20,3).
I hope this helps.
From ndroock1 at gmail.com Sun Feb 26 16:39:12 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Sun Feb 26 16:39:29 2006
Subject: [GAP Forum] Unexpected Behaviour in Size()
Message-ID:
Hello GAP forum,
I created the free group ( i got this spec. for the group from a
textbook-exercise ):
gap> F:=FreeGroup(3);
gap> x:=F.1;
f1
gap> y:=F.2;
f2
gap> z:=F.3;
f3
gap> G:=F/[y^3*z^15,x^4*y^7*z^3,x^8,y^14,z^18];
Then when I wanted to now the size of the group, GAP became a bit of
erratic.
gap> Size(G);
#I Coset table calculation failed -- trying with bigger table limit
#I Coset table calculation failed -- trying with bigger table limit
#I Coset table calculation failed -- trying with bigger table limit
user interrupt at
firstFree := app[6];
called from
TCENUM.CosetTableFromGensAndRels( fgens, grels, fsgens ) called from
CosetTableFromGensAndRels( fgens, grels, List( trial, UnderlyingElement )
) called from
Attempt( gens ) called from
FinIndexCyclicSubgroupGenerator( G, infinity ) called from
( ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
#I Options stack has been reset
gap> G:=F/[y^3*z^15,x^4*y^7*z^3,x^8*y^14*z^18];
gap> Size(G);
#I Coset table calculation failed -- trying with bigger table limit
#I Coset table calculation failed -- trying with bigger table limit
#I Coset table calculation failed -- trying with bigger table limit
#I Coset table calculation failed -- trying with bigger table limit
#I Coset table calculation failed -- trying with bigger table limit
exceeded the permitted memory (`-o' command line option) at
g[2 * limit] := 0;
called from
TCENUM.CosetTableFromGensAndRels( fgens, grels, fsgens ) called from
CosetTableFromGensAndRels( fgens, grels, List( trial, UnderlyingElement )
) called from
Attempt( trial ) called from
Attempt( gens ) called from
FinIndexCyclicSubgroupGenerator( G, infinity ) called from
...
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk>
Sorry but, any idea what's wrong in this case? Thanks on beforehand for
any advice.
--
met vriendelijke groet,
Nilo de Roock
From dima at ntu.edu.sg Sun Feb 26 18:08:25 2006
From: dima at ntu.edu.sg (Dima Pasechnik)
Date: Sun Feb 26 18:09:50 2006
Subject: [GAP Forum] Unexpected Behaviour in Size()
In-Reply-To:
Message-ID:
Dear Forum,
On 2/27/06 12:39 AM, "Nilo de Roock" wrote:
> Hello GAP forum,
>
> I created the free group ( i got this spec. for the group from a
> textbook-exercise ):
> gap> F:=FreeGroup(3);
>
> gap> x:=F.1;
> f1
> gap> y:=F.2;
> f2
> gap> z:=F.3;
> f3
> gap> G:=F/[y^3*z^15,x^4*y^7*z^3,x^8,y^14,z^18];
>
>
> Then when I wanted to now the size of the group, GAP became a bit of
> erratic.
>
That's a huge group for sure (as you can gather, using PQuotient, that it's
at least 2^97)
Can it be ininite?
Anyhow, enumerating conjugacy classes modulo the trivial subgroup (that's
what Size will try to do) is out of the question.
--
Dima Pasechnik
http://www.ntu.edu.sg/home/dima/
From hulpke at mac.com Sun Feb 26 18:21:56 2006
From: hulpke at mac.com (Alexander Hulpke)
Date: Sun Feb 26 18:22:29 2006
Subject: [GAP Forum] Unexpected Behaviour in Size()
In-Reply-To:
References:
Message-ID: <15FD609B-88E8-45F2-81C5-BC8B74E31162@mac.com>
Dear GAP-Forum,
Nilo de Roock wrote:
On Feb 26, 2006, at 9:39 AM, Nilo de Roock wrote:
> Hello GAP forum,
>
> I created the free group ( i got this spec. for the group from a
> textbook-exercise ):
> gap> F:=FreeGroup(3);
>
> gap> x:=F.1;
> f1
> gap> y:=F.2;
> f2
> gap> z:=F.3;
> f3
> gap> G:=F/[y^3*z^15,x^4*y^7*z^3,x^8,y^14,z^18];
>
>
> Then when I wanted to now the size of the group, GAP became a bit
> of erratic.
I would not call this erratic.
This is the expected behaviour, as there are fundamental difficulties
on algorithmic methods for finitely presented groups. (The so-called
``word problem''.)
You might want to read the AMS notices article by 'Akos Seress
(notices.ps on the page http://www.math.ohio-state.edu/~akos/ )
or the recent ``Handbook of Computational Group Theory'' by Holt et.
al. for methods used and some of the fundamental problems arising.
GAP issues warnings that it is performing a lot (probably more than
you expected) work, and still does not have a result. This is an
indication that you might needs lots of memory or would be on the way
of overloading your computer without ever getting a result -- GAP is
trying to stop you doing something you don't really want.
In your case, GAP tries to compute the size of an fp group by
calculating the index of a cyclic subgroup and rewriting the
presentation. However an easy calculation (abelian invariants of G')
shows that your group is infinite and cannot have any cyclic subgroup
of finite index.
Best wishes,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke@math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
From dfh at maths.warwick.ac.uk Sun Feb 26 20:49:06 2006
From: dfh at maths.warwick.ac.uk (Derek Holt)
Date: Sun Feb 26 20:50:54 2006
Subject: [GAP Forum] Unexpected Behaviour in Size()
In-Reply-To: <15FD609B-88E8-45F2-81C5-BC8B74E31162@mac.com>
References:
<15FD609B-88E8-45F2-81C5-BC8B74E31162@mac.com>
Message-ID: <20060226204906.GA4036@maths.warwick.ac.uk>
Dear GAP-Forum,
On Sun, Feb 26, 2006 at 11:21:56AM -0700, Alexander Hulpke wrote:
> Dear GAP-Forum,
>
> Nilo de Roock wrote:
>
> On Feb 26, 2006, at 9:39 AM, Nilo de Roock wrote:
>
> >Hello GAP forum,
> >
> >I created the free group ( i got this spec. for the group from a
> >textbook-exercise ):
> >gap> F:=FreeGroup(3);
> >
> >gap> x:=F.1;
> >f1
> >gap> y:=F.2;
> >f2
> >gap> z:=F.3;
> >f3
> >gap> G:=F/[y^3*z^15,x^4*y^7*z^3,x^8,y^14,z^18];
> >
> >
> >Then when I wanted to now the size of the group, GAP became a bit
> >of erratic.
>
> I would not call this erratic.
>
> This is the expected behaviour, as there are fundamental difficulties
> on algorithmic methods for finitely presented groups. (The so-called
> ``word problem''.)
> You might want to read the AMS notices article by 'Akos Seress
> (notices.ps on the page http://www.math.ohio-state.edu/~akos/ )
> or the recent ``Handbook of Computational Group Theory'' by Holt et.
> al. for methods used and some of the fundamental problems arising.
>
> GAP issues warnings that it is performing a lot (probably more than
> you expected) work, and still does not have a result. This is an
> indication that you might needs lots of memory or would be on the way
> of overloading your computer without ever getting a result -- GAP is
> trying to stop you doing something you don't really want.
>
> In your case, GAP tries to compute the size of an fp group by
> calculating the index of a cyclic subgroup and rewriting the
> presentation. However an easy calculation (abelian invariants of G')
> shows that your group is infinite and cannot have any cyclic subgroup
> of finite index.
Another way to show that the group is infinite is to use rewriting systems.
The MakeConfluent command succeeds quickly, and by inspecting the rules
in the confluent system, we find that
G = < x,y,z | z^3=y, y^2=1, x^4=1, yz=zy >,
so G is the free product of the cyclic groups and or orders 6 and 4.
In fact it is not difficult to establish that with a hand calculation,
which was probably what was intended in the textbook exercise!
I guess in a perfect world, asking whether a finitely presented group was
finite or infinite would automatically trigger these types of calculations
(coset enumeration, abelian invariants of finite index subgroups, rewriting
systems) each being tried in succession (or perhaps in parallel) for
progressively longer times.
Derek Holt.
From dutour at liga.ens.fr Thu Feb 23 05:17:19 2006
From: dutour at liga.ens.fr (Mathieu Dutour)
Date: Mon Feb 27 10:16:54 2006
Subject: [GAP Forum] A strange error
Message-ID: <20060223051719.GA24588@orge.ens.fr>
Hi all,
I got the following error with gap 4.4.6:
gap> RepresentativeAction(Group(()), [1], [2], OnSets);
Error, no method found! For debugging hints type ?Recovery from
NoMethodFound
Error, no 1st choice method found for epresentativeActionOp' on 3 arguments called from
RepresentativeActionOp( G, d[1], e[1] ) called from
RepresentativeActionOp( G, d, e, act ) called from
( ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk>
The following commands work as expected.
RepresentativeAction(Group(()), [1], [1], OnSets);
RepresentativeAction(Group(()), [1], [1,3], OnSets);
RepresentativeAction(Group(()), [1,3], [2,3], OnSets);
Mathieu
--
Mathieu Dutour Sikiric Researcher in Math
Tel. (+972)2 65 84 103 and Computer Science
Fax. (+972)2 56 30 702 Einstein Institute of Mathematics
E-mail: Mathieu.Dutour@ens.fr Hebrew University of Jerusalem
http://www.liga.ens.fr/~dutour Israel
From hulpke at math.colostate.edu Mon Feb 27 20:05:31 2006
From: hulpke at math.colostate.edu (Alexander Hulpke)
Date: Mon Feb 27 20:06:21 2006
Subject: [GAP Forum] A strange error
In-Reply-To: <20060223051719.GA24588@orge.ens.fr>
References: <20060223051719.GA24588@orge.ens.fr>
Message-ID:
Dear GAP-Forum,
On Feb 22, 2006, at 22:17 , Mathieu Dutour wrote:
> I got the following error with gap 4.4.6:
> gap> RepresentativeAction(Group(()), [1], [2], OnSets);
Thank you for the error report. This will be corrected in the next
bugfix. (Let me know in private, if you want a temporary patch.)
Best wishes,
Alexander Hulpke
PS: I would like to take the opportunity to once more remind everyone
of the `support@gap-system.org' email address for reporting errors
which likely are of little interest to the whole list.
From ndroock1 at gmail.com Mon Feb 27 21:05:51 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Mon Feb 27 21:07:01 2006
Subject: [GAP Forum] StructureDescription
In-Reply-To: <200602261339.19621.costanti@science.unitn.it>
References:
<200602261339.19621.costanti@science.unitn.it>
Message-ID:
Hello Marco,
Thank you very much for your help. And, yes I will use the support
address in the future for this sort of questions. Since GAP is Open
Source anyway, is it possible to subscribe to the support-list? These
lists usually become the best knowledgebase around, at least for the
other software I work with. ( The only thing that's preventing me from
reading the GAP source code is the language, C instead Java. Well,
Java wasn't around when it all started. )
Although I am still fairly new to GAP I think it's fair to say that
the feedback and answers on this list are both excellent and friendly.
Isn't that' a rather unique combination in the field?
kind regards,
nilo de roock
2006/2/26, Marco Costantini :
> Dear Nilo de Roock,
>
> (Warning:
> this answer is a draft, and has not yet been discussed with other people
> of the GAP Support Group. More detailed information may (or may not)
> arrive later. Feel free to write again if you need more help. For any
> remark concerning this mail, please do not reply to me, but write to
> support@gap-system.org .)
>
> We offer to react as well as we can to questions, requests for help with
> problems, or complaints that you may have. However, for the sake of the whole
> user community of GAP, we ask you to separate these into two different
> categories.
>
> GAP Support. We would like to deal with those topics that are more or less
> local to you, that is, are likely not of interest to most of the other GAP
> users by direct correspondence with you. Please send letters about such local
> problems and questions also to the address support@gap-system.org.
>
> GAP Forum. On the other hand, the GAP Forum should be reserved for
> discussions about problems that are likely to interest many of the GAP users.
> It would also be welcome if you could occasionally tell other users in the
> GAP Forum about interesting applications you have made of GAP.
>
>
> On Sunday 26 February 2006 11:44, Nilo de Roock wrote:
> > Hello GAP forum,
> >
> > Forgive for yet another question (...) on this command, perhaps I am using
> > the wrong command for my purpose. When I execute the following command.
> >
> > gap> List(AllGroups(20),StructureDescription);
> >
> > GAP responds with
> > [ "C5 : C4", "C20", "C5 : C4", "D20", "C10 x C2" ]
> >
> > The issue I have here is that "C5:C4" occurs twice in the list.
>
> Yes, in fact StructureDescription is not injective, an two of the groups with
> 20 elements have a similar structure.
>
> > What I am
> > looking for is a command that would recognize that the Frobenius group
> > ~~~~ is in there somewhere.
> >
> >
> > Even if I do...
> >
> > gap> F:=FreeGroup(2);
> >
> > gap> s:=F.1;
> > f1
> > gap> t:=F.2;
> > f2
> > gap> G:=F/[s^4,t^5,t*s*t^-2*s^-1];
> >
> > gap> Size(G);
> > 20
> > gap> str(G);
> > "C5 : C4"
> > gap>
> >
> > the answer is "C5:C4".
> >
> >
> > I am getting lost on this command, and thus on GAP. Please help.
>
> A possibility is simply
>
> gap> IdGroup(G);
> [ 20, 3 ]
>
> that is, G is the 3rd group in AllGroups(20) .
>
> Another possibility is to use StructureDescription only to restrict to some of
> the groups, and then to proceed with something else. For instance
>
> gap> G_20_1 := SmallGroup(20,1);
>
> gap> G_20_3 := SmallGroup(20,3);
>
>
> After using StructureDescription you know that the other groups with 20
> elements are not isomorphic to your G.
>
> After that you can use:
>
> gap> IsomorphismGroups( G_20_1, G );
> fail
> gap> IsomorphismGroups( G_20_3, G );
> [ f1, f2, f3 ] -> [ f1^5, f1^2, f2 ]
>
> you can also use:
>
> gap> Length( ConjugacyClasses( G ) );
> 5
> gap> Length( ConjugacyClasses( G_20_1 ) );
> 8
> gap> Length( ConjugacyClasses( G_20_3 ) );
> 5
>
> you can also use anything else analogous.
>
> All the best,
> Marco Costantini
>
From wdjoyner at comcast.net Mon Feb 27 12:08:22 2006
From: wdjoyner at comcast.net (David Joyner)
Date: Tue Feb 28 02:57:38 2006
Subject: [GAP Forum] GAP's impact on mathematics
Message-ID: <4402EBB6.50406@comcast.net>
Hello:
Although the GAP website has a bibliography,
and an examples page
http://www.gap-system.org/Doc/Examples/examples.html ,
I was looking for a page summarizing some of the
problems GAP has helped resolve.
What are some interesting problems in mathematics
that GAP was instrumental in resolving?
Has GAP resolved any famous conjectures?
Work on Riemann surfaces using "braid", based
on Thomas Breuer's ideas, comes to mind as an example.
Unfortunately, that isn't even mentioned, as far as I
could see.
Anyone else have any favorite applications of GAP?
- David Joyner
From aodabas at ogu.edu.tr Tue Feb 28 16:47:35 2006
From: aodabas at ogu.edu.tr (=?iso-8859-9?Q?Alper_Odaba=FE?=)
Date: Tue Feb 28 16:47:22 2006
Subject: [GAP Forum] elements of algebra
Message-ID: <001101c63c86$ad5a0540$a8838cc1@ogu209>
Good day.
I have some (maybe stupid) questions:
How can I get all elements of given algebra?
For elements a,b in group G the command a^b in GAP computes (b^-1ab). How can I compute it in algebra case.
is GAP best way in Commutative Algebra??? or CoCoA, Magma
Thank you.
Alper
From kohl at mathematik.uni-stuttgart.de Thu Mar 2 12:14:33 2006
From: kohl at mathematik.uni-stuttgart.de (Stefan Kohl)
Date: Thu Mar 2 12:14:37 2006
Subject: [GAP Forum] elements of algebra
In-Reply-To: <001101c63c86$ad5a0540$a8838cc1@ogu209>
References: <001101c63c86$ad5a0540$a8838cc1@ogu209>
Message-ID: <4406E1A9.5000000@mathematik.uni-stuttgart.de>
Dear Forum,
Alper Odaba? wrote:
> Good day.
> I have some (maybe stupid) questions:
> How can I get all elements of given algebra?
If your algebra is finite, then you can use `AsList'. E.g.:
gap> A := FullMatrixAlgebra(GF(2),2);
( GF(2)^[ 2, 2 ] )
gap> AsList(A);
[ [ , ],
[ ... (rest of output omitted for saving space) ... ]
> For elements a,b in group G the command a^b in GAP computes (b^-1ab). How can I compute it in algebra case.
In general you cannot.
The reason for this is simply that in general not all elements
of an algebra are invertible.
> is GAP best way in Commutative Algebra??? or CoCoA, Magma
I feel that I am not a neutral person in this respect, thus prefer to leave
commenting on this to other people.
Hope this helps,
Stefan Kohl
From hulpke at math.colostate.edu Thu Mar 2 18:15:55 2006
From: hulpke at math.colostate.edu (Alexander Hulpke)
Date: Thu Mar 2 18:17:04 2006
Subject: [GAP Forum] Serious bug in `PolynomialReduction'
Message-ID: <9B0852C4-F263-4392-AB2A-42222D3D7237@math.colostate.edu>
Dear GAP Forum,
A user has reported a serious bug in the code for polynomial
reduction (thus also affecting Groebner bases), which may return
wrong results without warning. (See below for an example)
This bug will be corrected in the forthcoming next bugfix.
If you are using Groebner bases and need a temporary workaround
already now, you can
download the file
http://www.math.colostate.edu/~hulpke/workaround.gi
Read this file in with *Reread* (not Read, as it overwrites some
library functions)
We apologize for this problem!
Alexander Hulpke
Example with correct output:
x:=X(Rationals,"x");;
y:=X(Rationals,"y");;
a:=X(Rationals,"a");;
c:=X(Rationals,"c");;
s:=X(Rationals,"s");;
L:=[(a+c)*s-x,(a+c)*c-y,s^2+c^2-1];;
ReducedGroebnerBasis(L,MonomialLexOrdering([x,y,a,c,s]));
[ c^2+s^2-1, -a*c+s^2+y-1, -a*s-c*s+x ]
(2nd polynomial was wrong!)
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke@math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
From anvita21 at usa.com Fri Mar 3 05:11:01 2006
From: anvita21 at usa.com (Anvita)
Date: Fri Mar 3 05:14:16 2006
Subject: [GAP Forum] Coefficients of constant polynomials
Message-ID: <20060303051101.4C0A0BA44C@ws3-2.us4.outblaze.com>
Dear Forum,
When I apply the function "CoefficientsOfUnivariatePolynomial" to the
unit polynomial I get the result [ 1 ], as expected:
-----------------------------------------------
gap> R:=PolynomialRing(Integers,["x"]);
PolynomialRing(..., [ x ])
gap> i:=One(R);
1
gap> CoefficientsOfUnivariatePolynomial(i);
[ 1 ]
-----------------------------------------------
For the zero polynomial, however, it returns an empty set:
-----------------------------------------------
gap> o:=Zero(R);
0
gap> CoefficientsOfUnivariatePolynomial(o);
[ ]
-----------------------------------------------
Why isn't the result [ 0 ] ?
Could this be a bug?
Thank you
Anvita
--
___________________________________________________
Play 100s of games for FREE! http://games.mail.com/
From alexander.konovalov at gmail.com Fri Mar 3 10:55:43 2006
From: alexander.konovalov at gmail.com (Alexander Konovalov)
Date: Fri Mar 3 10:57:17 2006
Subject: [GAP Forum] LAGUNA 3.3.2
Message-ID: <5b6b8f6c0603030255u47926866iafa7cdb4ef2bd20@mail.gmail.com>
Dear GAP Forum,
this is to announce the availability of the new version of the LAGUNA package.
The LAGUNA package provides functionality for calculation of the
normalized unit group of the modular group algebra of the finite
p-group and for investigations of Lie algebras associated with group
algebras.
LAGUNA 3.3.2 was released on March 01, 2006 and it is available from
the following pages:
- http://ukrgap.exponenta.ru/laguna.htm
- http://www.gap-system.org/Packages/laguna.html
and also in a merged archive of all currently redistributed GAP packages
available from http://www.gap-system.org/Download/index.html
The new version resolves a problem in compatibility with GAP methods
for Lie algebras and essentially improves performance for computation
of the normalized unit group in abelian case.
Sincerely yours,
Alexander Konovalov
From hulpke at mac.com Fri Mar 3 16:07:39 2006
From: hulpke at mac.com (Alexander Hulpke)
Date: Fri Mar 3 16:09:22 2006
Subject: [GAP Forum] Coefficients of constant polynomials
In-Reply-To: <20060303051101.4C0A0BA44C@ws3-2.us4.outblaze.com>
References: <20060303051101.4C0A0BA44C@ws3-2.us4.outblaze.com>
Message-ID: <571DE562-15CE-4DA9-A6B3-BD626AE7B173@mac.com>
Dear Forum,
`Anvita21' wrote:
> When I apply the function "CoefficientsOfUnivariatePolynomial" to the
>
> unit polynomial I get the result [ 1 ], as expected:
>
>
>
> For the zero polynomial, however, it returns an empty set:
Yes. the zero polynomial is stored by an empty coefficient list, as
there are no nonzero coefficients.
> Why isn't the result [ 0 ] ?
Because storing the zero polynomial that way would require extra
treatment in some arithmetic routines (e.g. multiplication).
>
> Could this be a bug?
No. This is intended.
Best wishes,
Alexander Hulpke
> ___________________________________________________
> Play 100s of games for FREE! http://crappy.advertising.dump/
From alexk at mcs.st-and.ac.uk Sat Mar 4 10:21:39 2006
From: alexk at mcs.st-and.ac.uk (Alexander Konovalov)
Date: Sat Mar 4 10:08:17 2006
Subject: [GAP Forum] LAGUNA 3.3.2
Message-ID:
Dear GAP Forum,
this is to announce the availability of the new version of the LAGUNA
package.
The LAGUNA package provides functionality for calculation of the
normalized unit group of the modular group algebra of the finite
p-group and for investigations of Lie algebras associated with group
algebras.
LAGUNA 3.3.2 was released on March 01, 2006 and it is available from
the following pages:
- http://ukrgap.exponenta.ru/laguna.htm
- http://www.gap-system.org/Packages/laguna.html
and also in a merged archive of all currently redistributed GAP packages
available from http://www.gap-system.org/Download/index.html
The new version resolves a problem in compatibility with GAP methods
for Lie algebras and essentially improves performance for computation
of the normalized unit group in abelian case.
Sincerely yours,
Alexander Konovalov
From dn2447 at yahoo.com Sat Mar 4 21:19:04 2006
From: dn2447 at yahoo.com (D N)
Date: Sat Mar 4 21:22:58 2006
Subject: [GAP Forum] Character group and semi-direct product
Message-ID: <20060304211904.41058.qmail@web37406.mail.mud.yahoo.com>
Hello All,
Let G be a finite group and H be a finite left G-module.
Let H^ := Hom(H, C*) denote the character group of H.
Then, H^ is a right G-module: (\rho \dot g)(h) := \rho(g \dot h)
for \rho \in H^, g \in G and h \in H.
Let G' := H^ : G (semi-direct product of H^ and G).
My question is: how to construct the group G' in GAP?
Any help is greatly appreciated.
Thanks,
DN
---------------------------------
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From anvita21 at usa.com Mon Mar 6 02:03:43 2006
From: anvita21 at usa.com (Anvita)
Date: Mon Mar 6 02:05:17 2006
Subject: [GAP Forum] Re: Coefficients of constant polynomials
Message-ID: <20060306020343.3221A21B32F@ws3-5.us4.outblaze.com>
>> If the function "CoefficientsOfUnivariatePolynomial" is applied to the
>> unit polynomial, the result is [ 1 ], as expected:
>> ...
>> For the zero polynomial, however, it returns an empty set:
>
>Yes. the zero polynomial is stored by an empty coefficient list, as
>there are no nonzero coefficients.
>
If the zero polynomial is stored as an empty list then why does the function
"UnivariatePolynomial" return an error with an empty list as the second
argument?
---------------------------------------------------------------------------------------
gap> p:=UnivariatePolynomial(Integers,[]);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `UnivariatePolynomial' on 2 arguments called from
( ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk>
---------------------------------------------------------------------------------------
Anvita
--
___________________________________________________
Play 100s of games for FREE! http://games.mail.com/
From hulpke at mac.com Mon Mar 6 02:33:14 2006
From: hulpke at mac.com (Alexander Hulpke)
Date: Mon Mar 6 02:36:27 2006
Subject: [GAP Forum] Re: Coefficients of constant polynomials
In-Reply-To: <20060306020343.3221A21B32F@ws3-5.us4.outblaze.com>
References: <20060306020343.3221A21B32F@ws3-5.us4.outblaze.com>
Message-ID: <8DF3C528-F950-4FDE-9938-547FF57F411B@mac.com>
Dear Gap Forum,
Someone called `Anvita' wrote:
>
>>> If the function "CoefficientsOfUnivariatePolynomial" is applied
>>> to the
>>> unit polynomial, the result is [ 1 ], as expected:
>>> ...
>>> For the zero polynomial, however, it returns an empty set:
>>
>> Yes. the zero polynomial is stored by an empty coefficient list, as
>> there are no nonzero coefficients.
>>
>
> If the zero polynomial is stored as an empty list then why does the
> function
> "UnivariatePolynomial" return an error with an empty list as the
> second
> argument?
That is an oversight. (The conditions on coefficients is to be a list
of ring elements, the empty list does not qualify and needs to be
treated specially.) Thank you for spotting this. It will be corrected
in a future release.
As a workaround you can issue the command:
InstallOtherMethod( UnivariatePolynomial, "ring,empty cof",true,
[ IsRing, IsEmpty ], 0,
function( ring, cofs )
return LaurentPolynomialByCoefficients( ElementsFamily(FamilyObj
(ring)),cofs, 0, 1 );
end );
Best wishes,
Alexander Hulpke
From ndroock1 at gmail.com Mon Mar 6 07:04:57 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Mon Mar 6 07:07:39 2006
Subject: [GAP Forum] Re: GAP issue
In-Reply-To:
References:
Message-ID:
Thanks Ignat,
This idea about IdGroup() was also in most of the other replies and
yes, it helps. I suppose that if StructureDescription() was perfect
we wouldn't be doing Group Theory in the first place. ;-)
Kind regards,
nilo
2006/3/5, ignat soroko :
> Hello, Nilo,
>
> I read some of your questions in GAP forum and I'd like to suggest a solution.
>
> You like the function StructureDescription. This is a wonderful
> function, but it gives only a rough idea what the group is. It does
> not determine the group up to isomorphism. What determines the group
> up to isomorphism, is IdGroup() function. Type IdGroup(G) and you will
> get a pair [ord, nr]. That means that G is isomorphic to the group
> number nr among all groups of order ord, that is to
> SmallGroup(ord,nr);.
>
> Thus to determine a group up to isomorphism one should use IdGroup().
> And to get some idea what is the structure of the group, one can use
> StructureDescription().
>
> Hope this helps.
>
> Ignat
>
From dn2447 at yahoo.com Mon Mar 6 18:31:46 2006
From: dn2447 at yahoo.com (D N)
Date: Mon Mar 6 18:35:58 2006
Subject: [GAP Forum] Character group and semi-direct product
Message-ID: <20060306183146.86690.qmail@web37411.mail.mud.yahoo.com>
Dear GAP Forum,
Actually, I found (quite inefficient though) a way to do this.
My apologies if the question and the follow-up is too trivial
to post on this forum.
Let H be any Abelian group. The irreducible characters of H
can be obtained by typing "Display(Irr(CharacterTable(H)));"
Form diagonal matrices with rows of the above output. The
group generated by these diagonal matrices is of course
isomorphic to the character group H^. Putting the required
action on H^ and forming the semidirect product is quite
straight-forward.
Thanks,
DN
D N wrote: Hello All,
Let G be a finite group and H be a finite left G-module.
Let H^ := Hom(H, C*) denote the character group of H.
Then, H^ is a right G-module: (\rho \dot g)(h) := \rho(g \dot h)
for \rho \in H^, g \in G and h \in H.
Let G' := H^ : G (semi-direct product of H^ and G).
My question is: how to construct the group G' in GAP?
Any help is greatly appreciated.
Thanks,
DN
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From dn2447 at yahoo.com Wed Mar 8 20:11:53 2006
From: dn2447 at yahoo.com (D N)
Date: Wed Mar 8 20:12:43 2006
Subject: [GAP Forum] Having trouble with cohomolo package
Message-ID: <20060308201153.4761.qmail@web37406.mail.mud.yahoo.com>
Dear GAP Forum,
I get the following message:
gap> Read("cohomolo.tst");
Function: number of arguments must be 2 (not 5) at
Cohomology( chr, true, false, false, TmpName( ) );
called from
( ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can replace the argument list via 'return ;' to continue
brk>
What is going on here?
DN
---------------------------------
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From l.h.soicher at qmul.ac.uk Thu Mar 9 14:26:13 2006
From: l.h.soicher at qmul.ac.uk (Leonard Soicher)
Date: Thu Mar 9 14:26:49 2006
Subject: [GAP Forum] Groups and Computation 2006 - A Leedham-Green Fest
Message-ID: <20060309142613.GA6441@mrcpc02.maths.qmul.ac.uk>
Groups and Computation 2006
A Leedham-Green Fest
We are pleased to announce a one-day conference to celebrate Charles
Leedham-Green's contributions to mathematics on the occasion of his
retirement.
10th July 2006, starting at 10.30
School of Mathematical Sciences
Queen Mary, University of London
The main speakers are:
Bettina Eick, Eamonn O'Brien, and Aner Shalev (tbc)
There is a registration fee of 10 pounds to cover refreshments and lunch.
There will also be a celebratory dinner on the evening of July 10th at
L'Oasis, costing 20 pounds.
Places are limited, and are available on a first-come first-served basis.
Should you wish to attend, please mail a cheque or postal order in GBP,
payable to "Queen Mary, University of London", covering the registration
fee and dinner (if applicable), to:
Dr S. McKay
Groups and Computation 2006
School of Mathematical Sciences
Queen Mary, University of London
Mile End Road
London E1 4NS
U.K.
Please include your name and email address. We shall contact you by
email with more information in due course, including menu choices for
the dinner. If you have any queries, please email S.McKay@qmul.ac.uk
Some accommodation will be available in the new Queen Mary student village.
If you wish to stay for one or more nights, it is important to book as
soon as possible. You can book online at www.qmulholidays.co.uk
If you experience any problems with this, then phone +44(0)20-7882-3642.
We look forward to seeing you at Queen Mary!
Sue McKay, Leonard Soicher, Peter Cameron
From aodabas at ogu.edu.tr Thu Mar 9 15:10:17 2006
From: aodabas at ogu.edu.tr (=?iso-8859-9?Q?Alper_Odaba=FE?=)
Date: Thu Mar 9 15:09:48 2006
Subject: [GAP Forum] Automorphism of Algebra
Message-ID: <004201c6438b$93383f90$be838cc1@ogu209>
Hi all,
I have a question for algebra ,
Let A and B algebras. Suppose that B acts on A, is there a algebra homomorphism B --> Aut(A) ??
has GAP any function Automorphism of commutative algebras??
gap> G:=Group((1,2,3,4));;
gap> A:=GroupRing(GF(3),G);;
gap> Automorphism(A);
Variable: 'Automorphism' must have a value
gap> AutomorphismAlgebra(A);
Variable: 'AutomorphismAlgebra' must have a value
gap> Automorphisms(A);
Variable: 'Automorphisms' must have a value
gap> Automorphism(A);
Variable: 'Automorphism' must have a value
gap> AutomorphismOfAlgebra(A);
Variable: 'AutomorphismOfAlgebra' must have a value
gap> AutomorphismRing(A);
Variable: 'AutomorphismRing' must have a value
gap> AutomorphismOfRing(A);
Variable: 'AutomorphismOfRing' must have a value
thanks.
Alper
From sal at dcs.st-and.ac.uk Fri Mar 10 10:07:19 2006
From: sal at dcs.st-and.ac.uk (Steve Linton)
Date: Fri Mar 10 10:07:33 2006
Subject: [GAP Forum] Cohomolo
Message-ID: <20060310100719.6730df52@localhost.localdomain>
"DN" wrote:
I get the following message:
gap> Read("cohomolo.tst");
Function: number of arguments must be 2 (not 5) at
Cohomology( chr, true, false, false, TmpName( ) );
called from
( ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can replace the argument list via 'return ;' to continue
brk>
What is going on here?
DN
This problem is an incompatibility which we recently
discovered between the recently deposited Hap package and the
long-esablished cohomolo package, which define the global variable Cohomology
differently. Unfortunately, Hap version 1.1 was set to load automatically
when GAP starts, so anyone trying to use cohomolo would see this problem.
Version 1.2 released a few days ago fixes this problem, so that Hap is only
loaded if you load it explicitly. Thus, unless you need to use Hap and
cohomolo in the same session, you can avoid the problem by updating to this new
version.
For the moment it is not possible to use both of these packages in the same
GAP session. We will work with the authors to resolve this.
Steve Linton
A couple of remarks: firstly, specific problems of this kind are probably best
sent to support@gap-system.org rather than to the forum. If we find issues of
general interest arising in response to pronblems, we will inform the forum;
secondly, although we don't insist,it would be nice to know the names of
people we are dealing with, rather than initials or nicknames.
--
Steve Linton School of Computer Science &
Centre for Interdisciplinary Research in Computational Algebra
University of St Andrews Tel +44 (1334) 463269
http://www.dcs.st-and.ac.uk/~sal Fax +44 (1334) 463278
From degraaf at science.unitn.it Fri Mar 10 14:51:37 2006
From: degraaf at science.unitn.it (Willem De Graaf)
Date: Fri Mar 10 14:51:55 2006
Subject: [GAP Forum] Automorphism of Algebra
In-Reply-To: <004201c6438b$93383f90$be838cc1@ogu209>
References: <004201c6438b$93383f90$be838cc1@ogu209>
Message-ID: <44119279.1050807@science.unitn.it>
Dear Alper Odaba?,
Currently GAP does not have much functionality for working
with automorphisms of algebras. However, there are functions
for constructing homomorphisms (which, as a special case,
can be automorphisms). This is done by using the function
"AlgebraHomomorphismByImages".
The GAP manual contains a description of this function,
it is available online:
http://www.gap-system.org/Manuals/doc/htm/ref/CHAP060.htm#SECT009
(Note that in order to use it, one must specify set of source generators,
and a set of image generators.)
Best wishes,
Willem de Graaf
Alper Odaba? wrote:
>Hi all,
>
>I have a question for algebra ,
>Let A and B algebras. Suppose that B acts on A, is there a algebra homomorphism B --> Aut(A) ??
>
>has GAP any function Automorphism of commutative algebras??
>
>gap> G:=Group((1,2,3,4));;
>gap> A:=GroupRing(GF(3),G);;
>gap> Automorphism(A);
>Variable: 'Automorphism' must have a value
>
>gap> AutomorphismAlgebra(A);
>Variable: 'AutomorphismAlgebra' must have a value
>
>gap> Automorphisms(A);
>Variable: 'Automorphisms' must have a value
>
>gap> Automorphism(A);
>Variable: 'Automorphism' must have a value
>
>gap> AutomorphismOfAlgebra(A);
>Variable: 'AutomorphismOfAlgebra' must have a value
>
>gap> AutomorphismRing(A);
>Variable: 'AutomorphismRing' must have a value
>
>gap> AutomorphismOfRing(A);
>Variable: 'AutomorphismOfRing' must have a value
>
>
>thanks.
>
>Alper
>
>_______________________________________________
>Forum mailing list
>Forum@mail.gap-system.org
>http://mail.gap-system.org/mailman/listinfo/forum
>
>
>
From dn2447 at yahoo.com Sun Mar 12 03:46:24 2006
From: dn2447 at yahoo.com (D N)
Date: Sun Mar 12 03:46:37 2006
Subject: [GAP Forum] Action of a group on the Schur multiplier of a subgroup
Message-ID: <20060312034624.1774.qmail@web37408.mail.mud.yahoo.com>
Dear GAP Forum,
Let H be a normal subgroup of a finite group G. The action of G on H, by conjugation,
induces an action of G on the Schur multiplier H^2(H, C*) of H.
Is there any way I can use GAP to find the number of orbits of H^2(H, C*) under the action of G?
Thanks,
Deepak Naidu
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From Graham.Ellis at NUIGALWAY.IE Mon Mar 13 12:34:54 2006
From: Graham.Ellis at NUIGALWAY.IE (0002319s)
Date: Mon Mar 13 12:39:18 2006
Subject: [GAP Forum] action on Schur multipliers
Message-ID: <44155CA7@bodkin.nuigalway.ie>
Dear GAP Forum,
Deepak Naidu wrote:
> Let H be a normal subgroup of a finite group G. The action of G on H, by
conjugation,
> induces an action of G on the Schur multiplier H^2(H, C*) of H.
> Is there any way I can use GAP to find the number of orbits of H^2(H, C*)
under the action of G?
The following commands (which use the HAP package) show that the there are 4
orbits in the Schur multiplier of N:=AltenratingGroup(6) under the conjugation
action of G:=SymmetricGroup(6).
Note that the Schur multiplier of N is isomorphic to the Second integral
homology of N, and that N acts trivial on its Schur multiplier/second homology
(for any group N). So we are really interested in orbits under an action of
the quotient group G/N.
############## CUT ######################
gap> G:=SymmetricGroup(6);;
gap> N:=AlternatingGroup(6);;
gap> R:=ResolutionFiniteGroup(N,3);;
gap> SchurMultiplier:=Homology(TensorWithIntegers(R),2);
[6]
gap> #So the Schur multiplier of N is cyclic of order 6.
gap> ConjugationHomomorphism:=function(g);
> return GroupHomomorphismByFunction(N,N,x->g*x*g^-1);
end;;
gap> HomologyMapInducedByConjugation:=function(g)
> local f;
> f:=EquivariantChainMap(R,R,ConjugationHomomorphism(g));
> f:=TensorWithIntegers(f);
> f:=Homology(f,2);
> return f;
> end;;
gap> #Note that (1,2) represents a nontrivial element in G/N.
gap> HM:=HomologyMapInducedByConjugation((1,2));
[ f1, f2, f3, f4, f5, f6 ] -> [ f1^4*f2^-3, f1^2*f2^-1, f3, f3^-2*f4^2*f5,
f3^-2*f4^2*f5, f6 ]
gap> IsomorphismPermGroup(Source(HM));
[ f1, f2, f3, f4, f5, f6 ] -> [ (), (1,2,3)(4,5,6), (1,4)(2,5)(3,6), (), (),
((1,4)((2,5)(3,6) ]
##################### CUT ##################
Here Source(HM) is a finitely presented group isomorphic to the cyclic group
H_2(N) of order 6.We see that conjugation by (1,2) in G induces an
endomorphism on H_2(N)=C_2 x C_3 which fixes the generator f3 of order two,
and inverts the generator f2 of order three.
Graham
From aodabas at ogu.edu.tr Wed Mar 15 14:01:01 2006
From: aodabas at ogu.edu.tr (=?iso-8859-9?Q?Alper_Odaba=FE?=)
Date: Wed Mar 15 14:00:27 2006
Subject: [GAP Forum] bug with group homomorphism
Message-ID: <001201c64838$e463c4f0$af838cc1@ogu209>
hi,
i think i found a bug with the group homomorphisms.
AB is not a group in spite of f: AB --> B is a group homomorphism.
Alper
gap> A:=Group((1,2,3,4));
Group([ (1,2,3,4) ])
gap> elA:=Elements(A);
[ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2) ]
gap> B:=Subgroup(A,[elA[3]]);
Group([ (1,3)(2,4) ])
gap> A=B;
false
gap> IsSubgroup(A,B);
true
gap> AB:=Cartesian(A,B);
[ [ (), () ], [ (), (1,3)(2,4) ], [ (1,2,3,4), () ],
[ (1,2,3,4), (1,3)(2,4) ], [ (1,3)(2,4), () ], [ (1,3)(2,4), (1,3)(2,4) ],
[ (1,4,3,2), () ], [ (1,4,3,2), (1,3)(2,4) ] ]
gap> f:=GroupHomomorphismByFunction(AB,B,x->x^2);
MappingByFunction( [ [ (), () ], [ (), (1,3)(2,4) ], [ (1,2,3,4), () ],
[ (1,2,3,4), (1,3)(2,4) ], [ (1,3)(2,4), () ], [ (1,3)(2,4), (1,3)(2,4) ],
[ (1,4,3,2), () ], [ (1,4,3,2), (1,3)(2,4) ] ], Group(
[ (1,3)(2,4) ]), function( x ) ... end )
gap> IsGroupHomomorphism(f);
true
gap> IsGroup(AB);
false
gap> Image(f);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `GeneratorsOfMagmaWithInverses' on 1 arg\
uments called from
GeneratorsOfGroup( PreImagesRange( map ) ) called from
MappingGeneratorsImages( hom ) called from
ImagesSource( arg[1] ) called from
( ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk>
From bob.heffernan at gmail.com Wed Mar 15 17:26:45 2006
From: bob.heffernan at gmail.com (Robert Heffernan)
Date: Wed Mar 15 17:26:56 2006
Subject: [GAP Forum] finding automorphisms of finitely presented groups
Message-ID: <6d9a83e90603150926q23165364pe112f83eee0c8701@mail.gmail.com>
I am dealing with groups constructed in this manner:
F:=FreeGroup("x","y","z");G:=F/rels;
where rels is some list of relations in terms of x,y and z.
If the group is large GAP seems to have trouble constructing the groupof automorphisms of G.
I understand that doing this:G:=Image(IsomorphismPermGroup(G));would give me a representation of the group that GAP can deal witheasily (and easily compute Aut G, etc.), but I want to look at theautomorphisms in terms of the generators x,y and z above.
Specifially, I am trying to find all the automorphisms G->G of a given order.
Is there an efficient way to do this?
thank you,
Bob
From hulpke at math.colostate.edu Wed Mar 15 18:16:43 2006
From: hulpke at math.colostate.edu (Alexander Hulpke)
Date: Wed Mar 15 18:17:20 2006
Subject: [GAP Forum] finding automorphisms of finitely presented groups
In-Reply-To: <6d9a83e90603150926q23165364pe112f83eee0c8701@mail.gmail.com>
References: <6d9a83e90603150926q23165364pe112f83eee0c8701@mail.gmail.com>
Message-ID: <5D7AA287-DF9E-4C67-A87F-1E34E22032C6@math.colostate.edu>
Dear Robert Heffernan,
> I am dealing with groups constructed in this manner:
> F:=FreeGroup("x","y","z");G:=F/rels;
> where rels is some list of relations in terms of x,y and z.
> If the group is large GAP seems to have trouble constructing the
> groupof automorphisms of G.
> I understand that doing this:G:=Image(IsomorphismPermGroup
> (G));would give me a representation of the group that GAP can deal
> witheasily (and easily compute Aut G, etc.), but I want to look at
> theautomorphisms in terms of the generators x,y and z above.
That is not a contradiction. Compute the automorphism group for the
permutation representation and then pull the generators back to the
finitely presented group.
Concretely, if
phi:=IsomorphismPermGroup(G);
P:=Image(phi);
A:=AutomorphismGroup(P);
you can do
List(GeneratorsOfGroup(A),a->GroupHomomorphismByImagesNC
(G,G,GeneratorsOfGroup(G),
List(GeneratorsOfGroup(G),x->PreImagesRepresentative(phi,Image
(a,Image(phi,x)))));
to get generators of the automorphism group in terms of x,y and z.
Best,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke@math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
From bob.heffernan at gmail.com Thu Mar 16 12:58:23 2006
From: bob.heffernan at gmail.com (Robert Heffernan)
Date: Thu Mar 16 12:58:32 2006
Subject: [GAP Forum] finding automorphisms of finitely presented groups
In-Reply-To: <5D7AA287-DF9E-4C67-A87F-1E34E22032C6@math.colostate.edu>
References: <6d9a83e90603150926q23165364pe112f83eee0c8701@mail.gmail.com>
<5D7AA287-DF9E-4C67-A87F-1E34E22032C6@math.colostate.edu>
Message-ID: <6d9a83e90603160458o2cff5f49h1c3ee6f0748d4b0f@mail.gmail.com>
Thank you for your help Alexander.
I do have a followup question, if anybody can help.
I have created a finitely presented group G
F:=FreeGroup("a","b","c","d");;a:=F.1;;b:=F.2;;c:=F.3;;d:=F.4;;rels:= G:=F/rels;a:=G.1;;b:=G.2;;c:=G.3;;d:=G.4;;
I have then created a new set of words/relations, rels2 say, in termsof a,b,c and d (by calculating with things in G).
Now I want to create a new finitely presented group in a manner such as this:H:=F/rels2;
However, I can't do this directly as a,b,c and d are now elements of G, not F.
I can't find a simple way to relate a,b,c and d back to the generatorsof the free group F, even though I'm sure such a thing must exist.
Any help would be wonderful.
thank you,Bob
From hulpke at mac.com Thu Mar 16 17:03:14 2006
From: hulpke at mac.com (Alexander Hulpke)
Date: Thu Mar 16 17:04:07 2006
Subject: [GAP Forum] finding automorphisms of finitely presented groups
In-Reply-To: <6d9a83e90603160458o2cff5f49h1c3ee6f0748d4b0f@mail.gmail.com>
References: <6d9a83e90603150926q23165364pe112f83eee0c8701@mail.gmail.com>
<5D7AA287-DF9E-4C67-A87F-1E34E22032C6@math.colostate.edu>
<6d9a83e90603160458o2cff5f49h1c3ee6f0748d4b0f@mail.gmail.com>
Message-ID: <9780E964-2DC5-433C-A6FF-CDEFB03FD94D@mac.com>
Dear GAP-Forum,
On Mar 16, 2006, at 5:58 AM, Robert Heffernan wrote:
> I do have a followup question, if anybody can help.
> I have created a finitely presented group G
> F:=FreeGroup("a","b","c","d");;a:=F.1;;b:=F.2;;c:=F.3;;d:=F.
> 4;;rels:= G:=F/rels;a:=G.
> 1;;b:=G.2;;c:=G.3;;d:=G.4;;
> I have then created a new set of words/relations, rels2 say, in
> termsof a,b,c and d (by calculating with things in G).
> Now I want to create a new finitely presented group in a manner
> such as this:H:=F/rels2;
> However, I can't do this directly as a,b,c and d are now elements
> of G, not F.
> I can't find a simple way to relate a,b,c and d back to the
> generatorsof the free group F,
You have three options. In order from ``cleanest'' to ``most
technical'':
a) From a `GroupHomomorphismByImages' from F to G, mapping F.i to
G.i. Then take the `PreImagesRepresentative' of your words.
b) Use `MappedWord' to map a word in the G.i to a word in the F.i
c) For any element of G, `UnderlyingElement' returns the
corresponding element of F, i.e. the relators you want.
Best wishes,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke@math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
From laurent.bartholdi at gmail.com Fri Mar 17 20:40:45 2006
From: laurent.bartholdi at gmail.com (Laurent Bartholdi)
Date: Fri Mar 17 20:40:57 2006
Subject: [GAP Forum] restricted lie algebras
Message-ID: <1ff637850603171240x3e471dd8h5dcf06ee82591bea@mail.gmail.com>
dear forum,
i see in gap the command "FreeLieAlgebra", but no command
"FreeRestrictedLieAlgebra". Has anybody implemented this? is it
difficult?
I would very much like, then, to have
"NilpotentQuotientRestrictedFpLieAlgebra"; though this may pose
a problem in the current implementation because the p-mapping
is always given as a function, and not as an intrinsic operation.
any ideas?
thanks in advance, laurent
--
Laurent Bartholdi \ laurent.bartholdigmailcom
EPFL SB SMA IMB MAD \ T?l?phone: +41 21-6935458
Station 8 \ Secr?taire: +41 21-6935501
CH-1015 Lausanne, Switzerland \ Fax: +41 21-6930339
From laurent.bartholdi at gmail.com Fri Mar 17 22:37:21 2006
From: laurent.bartholdi at gmail.com (Laurent Bartholdi)
Date: Fri Mar 17 22:37:29 2006
Subject: [GAP Forum] three remarks
Message-ID: <1ff637850603171437t3b0b5578x515ca352e591f5bd@mail.gmail.com>
hi again, gap world,
1) it would be very nice, i think, if LowerCentralSeries() accepted a second
argument, which is the length of the series to be computed. There are
sometimes groups for which the whole computation is very long, but only
the first few terms are needed.
2) It would also be nice (in terms of shortening code) that if L is a list, then
L[infinity] returns the last element. I tried
InstallOtherMethod(\[\],[IsList,IsInfinity],100,function(l,i)return
l[Length(l)];end);
but it doesn't seem to work.
3) AlgebraHomomorphismByImages is very slow, and I don't see why. I have
finite-dimensional algebras, and it seems, to me, that to check whether a
mapping (on generators) is a homomorphism amounts to finding a basis, and
checking the homomorphism on the basis. Therefore I would use
f := AlgebraHomomorphismByImagesNC(source,range,gens,images);
CheckIsAlgebraHomorphism(f);
using the code
CheckIsAlgebraHomomorphism := function(f)
local S,R,V,i,added;
S := Source(f);
R := Range(f);
B := ShallowCopy(MappingGeneratorsImages(f)[1]);
C := ShallowCopy(MappingGeneratorsImages(f)[2]);
V := VectorSpace(LeftActingDomain(S),B);
while V <> S do
added := false;
for i in Tuples([1..Length(B)],2) do
if not B[i[1]]*B[i[2]] in V then
Add(B,B[i[1]]*B[i[2]]);
Add(C,C[i[1]]*C[i[2]]);
V := VectorSpace(LeftActingDomain(S),B);
added := true;
fi;
od;
if not added then return fail; fi; # does not generate
od;
B := Basis(S,B);
return ForAll(Tuples([1..Length(B)],2),
i->LinearCombination(C,Coefficients(B,B[i[1]]*B[i[2]]))=C[i[1]]*C[i[2]]);
end;
--
Laurent Bartholdi \ laurent.bartholdigmailcom
EPFL SB SMA IMB MAD \ T?l?phone: +41 21-6935458
Station 8 \ Secr?taire: +41 21-6935501
CH-1015 Lausanne, Switzerland \ Fax: +41 21-6930339
From marta31 at gmail.com Sat Mar 18 05:33:48 2006
From: marta31 at gmail.com (marta asaeda)
Date: Sat Mar 18 05:33:58 2006
Subject: [GAP Forum] 1/sqrt(polynomial) not recognized
Message-ID: <4dd3929a0603172133m1a8b90dme33e215ce9c13d74@mail.gmail.com>
Hello,
I am having a problem: I would like to find galois group for each
polynomial in a sequence of polynomials parametrized by n. It is given
by a recursive formula, so it does involve non-polynomials for
expression in terms of n. I have this ( I'm mixing mathematica and
gap notations just for this message):
A[x]:=(1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x - Sqrt(x^2
- 4*x))/2))
B[x]:=(-1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x + Sqrt(x^2
- 4*x))/2))
a := (2 - x + Sqrt[x^2 - 4x])/2
b := (2 - x - Sqrt[x^2 - 4x])/2
P[n_, x_] := A[x]a^(n - 1) + B[x]b^(n - 1)
R[k_, x_] :=
(-2+x)^2(3-5x+x^2) P[2(k - 1), x] - (8-14x+7x^2-x^3) P[2(k - 1) - 1, x]
R[k,x] is a polynomial for any positive integer k. I would like to
give the list of galois groups for each k, say, 5 x:=Indeterminate(Rationals);
gap> Ax:=(1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x -
Sqrt(x^2 - 4*x))/2));
then I get error message like
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `Sqrt' on 1 arguments called from
Error( no_method_found ); called from
( ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
Could anyone please tell me what I should do ?
Thank you ~
marta
From dima at ntu.edu.sg Sat Mar 18 07:09:35 2006
From: dima at ntu.edu.sg (Dima Pasechnik)
Date: Sat Mar 18 07:11:24 2006
Subject: [GAP Forum] 1/sqrt(polynomial) not recognized
In-Reply-To: <4dd3929a0603172133m1a8b90dme33e215ce9c13d74@mail.gmail.com>
Message-ID:
GAP cannot deal with non-rational expressions involving indeterminants.
Realistically, you most probably would not be able to go much further than
k=15, say. So you can just use another CAS to write these polynomials out
explicitly and feed them into GAP.
HTH,
Dmitrii
On 3/18/06 1:33 PM, "marta asaeda" wrote:
> Hello,
>
> I am having a problem: I would like to find galois group for each
> polynomial in a sequence of polynomials parametrized by n. It is given
> by a recursive formula, so it does involve non-polynomials for
> expression in terms of n. I have this ( I'm mixing mathematica and
> gap notations just for this message):
>
> A[x]:=(1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x - Sqrt(x^2
> - 4*x))/2))
> B[x]:=(-1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x + Sqrt(x^2
> - 4*x))/2))
>
> a := (2 - x + Sqrt[x^2 - 4x])/2
> b := (2 - x - Sqrt[x^2 - 4x])/2
>
> P[n_, x_] := A[x]a^(n - 1) + B[x]b^(n - 1)
>
> R[k_, x_] :=
> (-2+x)^2(3-5x+x^2) P[2(k - 1), x] - (8-14x+7x^2-x^3) P[2(k - 1) - 1, x]
>
> R[k,x] is a polynomial for any positive integer k. I would like to
> give the list of galois groups for each k, say, 5 as much as it is doable by gap in a few days. If I just set it at
> night, go to bed, and see 1000 of galois groups spitted out, that will
> be wonderful. However, it seems gap is having problem dealing with
> 1/Sqrt(x^2 - 4*x). I just tried to teach A, B, a, b one by one, so I
> did like
>
> gap> x:=Indeterminate(Rationals);
> gap> Ax:=(1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x -
> Sqrt(x^2 - 4*x))/2));
>
> then I get error message like
> Error, no method found! For debugging hints type ?Recovery from NoMethodFound
> Error, no 1st choice method found for `Sqrt' on 1 arguments called from
> Error( no_method_found ); called from
> ( ) called from read-eval-loop
> Entering break read-eval-print loop ...
> you can 'quit;' to quit to outer loop, or
> you can 'return;' to continue
>
>
> Could anyone please tell me what I should do ?
>
> Thank you ~
>
> marta
>
From laurent.bartholdi at gmail.com Sat Mar 18 10:13:43 2006
From: laurent.bartholdi at gmail.com (Laurent Bartholdi)
Date: Sat Mar 18 10:13:49 2006
Subject: [GAP Forum] 1/sqrt(polynomial) not recognized
In-Reply-To: <4dd3929a0603172133m1a8b90dme33e215ce9c13d74@mail.gmail.com>
References: <4dd3929a0603172133m1a8b90dme33e215ce9c13d74@mail.gmail.com>
Message-ID: <1ff637850603180213l6af55c5fkfd88bc834b8289c9@mail.gmail.com>
Hi Marta,
1) most systems, like Maple, can compute Galois groups, in some
standard format (a set of generating permutations, e.g.)
2) like Dima wrote, it's hopeless to compute for degree above a hundred;
and usually the limit is much lower
3) the expressions you gave don't produce irreducible polynomials. most
computer command require an irreducible polynomial.
Here's my sample code, in Maple:
A:=(1/sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x - sqrt(x^2-4*x))/2)):
B:=(-1/sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x + sqrt(x^2-4*x))/2)):
a := (2 - x + sqrt(x^2 - 4*x))/2:
b := (2 - x - sqrt(x^2 - 4*x))/2:
P := n->A*a^(n-1)+B*b^(n-1):
R := k->(-2+x)^2*(3-5*x+x^2)*P(2*k-2)-(8-14*x+7*x^2-x^3)*P(2*k-3):
L := 'factor(convert(series(R(n),x,2*n+3),polynom))'$n=0..5;
2 2 2 3 2 2
(-2 + x) , (3 - 5 x + x ) (-2 + x) , (x - 1) (x - 8 x + 17 x - 5) (-2 + x) ,
6 5 4 3 2 2
(x - 13 x + 63 x - 140 x + 142 x - 59 x + 7) (-2 + x) ,
8 7 6 5 4 3 2
2
(x - 17 x + 117 x - 418 x + 827 x - 898 x + 502 x - 124 x +
9) (-2 + x) ,
9 8 7 6 5 4
3 2 2
(x - 1) (x - 20 x + 167 x - 753 x + 1979 x - 3050 x + 2635 x
- 1153 x + 214 x - 11) (-2 + x)
so i assume you're interested in the "big" factor:
galois(L[2]/(x-2)^2);
"2T1", {"S(2)"}, "-", 2, {"(1 2)"}
galois(L[3]/(x-2)^2/(x-1));
"3T1", {"A(3)"}, "+", 3, {"(1 2 3)"}
galois(L[4]/(x-2)^2);
"6T16", {"S(6)"}, "-", 720, {"(3 6)", "(1 6)",
"(2 6)", "(4 6)", "(5 6)"}
galois(L[5]/(x-2)^2);
"8T50", {"S(8)"}, "-", 40320, {"(4 8)", "(1 8)", "(7 8)",
"(2 8)", "(5 8)", "(6 8)", "(3 8)"}
galois(L[6]/(x-2)^2/(x-1));
"9T34", {"S(9)"}, "-", 362880, {"(8 9)", "(5 9)", "(6 9)", "(7
9)", "(3 9)", "(4 9)", "(1 9)", "(2 9)"}
this is the limit of Maple's implementation.
Best, Laurent
On 3/18/06, marta asaeda wrote:
> Hello,
>
> I am having a problem: I would like to find galois group for each
> polynomial in a sequence of polynomials parametrized by n. It is given
> by a recursive formula, so it does involve non-polynomials for
> expression in terms of n. I have this ( I'm mixing mathematica and
> gap notations just for this message):
>
> A[x]:=(1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x - Sqrt(x^2
> - 4*x))/2))
> B[x]:=(-1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x + Sqrt(x^2
> - 4*x))/2))
>
> a := (2 - x + Sqrt[x^2 - 4x])/2
> b := (2 - x - Sqrt[x^2 - 4x])/2
>
> P[n_, x_] := A[x]a^(n - 1) + B[x]b^(n - 1)
>
> R[k_, x_] :=
> (-2+x)^2(3-5x+x^2) P[2(k - 1), x] - (8-14x+7x^2-x^3) P[2(k - 1) - 1, x]
>
> R[k,x] is a polynomial for any positive integer k. I would like to
> give the list of galois groups for each k, say, 5 as much as it is doable by gap in a few days. If I just set it at
> night, go to bed, and see 1000 of galois groups spitted out, that will
> be wonderful. However, it seems gap is having problem dealing with
> 1/Sqrt(x^2 - 4*x). I just tried to teach A, B, a, b one by one, so I
> did like
>
> gap> x:=Indeterminate(Rationals);
> gap> Ax:=(1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x -
> Sqrt(x^2 - 4*x))/2));
>
> then I get error message like
> Error, no method found! For debugging hints type ?Recovery from NoMethodFound
> Error, no 1st choice method found for `Sqrt' on 1 arguments called from
> Error( no_method_found ); called from
> ( ) called from read-eval-loop
> Entering break read-eval-print loop ...
> you can 'quit;' to quit to outer loop, or
> you can 'return;' to continue
>
>
> Could anyone please tell me what I should do ?
>
> Thank you ~
>
> marta
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
--
Laurent Bartholdi \ laurent.bartholdigmailcom
EPFL SB SMA IMB MAD \ T?l?phone: +41 21-6935458
Station 8 \ Secr?taire: +41 21-6935501
CH-1015 Lausanne, Switzerland \ Fax: +41 21-6930339
From marta31 at gmail.com Sun Mar 19 00:47:08 2006
From: marta31 at gmail.com (marta asaeda)
Date: Sun Mar 19 00:47:17 2006
Subject: [GAP Forum] 1/sqrt(polynomial) not recognized
In-Reply-To: <1ff637850603180213l6af55c5fkfd88bc834b8289c9@mail.gmail.com>
References: <4dd3929a0603172133m1a8b90dme33e215ce9c13d74@mail.gmail.com>
<1ff637850603180213l6af55c5fkfd88bc834b8289c9@mail.gmail.com>
Message-ID: <4dd3929a0603181647r465a2081pd553f59620b06c8a@mail.gmail.com>
Hi Laurent, Dima, Ju'rgen,
Thank you very much for all the suggestions. I needed the minimal
polynomial of the largest eigen value. I will try as you suggest,
whatever I have an access and can handle with my very limited
experience in math softwares;;
Best
marta
From degraaf at science.unitn.it Mon Mar 20 08:35:05 2006
From: degraaf at science.unitn.it (Willem De Graaf)
Date: Mon Mar 20 08:35:22 2006
Subject: [GAP Forum] restricted lie algebras
In-Reply-To: <1ff637850603171240x3e471dd8h5dcf06ee82591bea@mail.gmail.com>
References: <1ff637850603171240x3e471dd8h5dcf06ee82591bea@mail.gmail.com>
Message-ID: <441E6939.2010304@science.unitn.it>
Dear Laurent,
>dear forum,
>i see in gap the command "FreeLieAlgebra", but no command
>"FreeRestrictedLieAlgebra". Has anybody implemented this? is it
>difficult?
>
>
>I would very much like, then, to have
>"NilpotentQuotientRestrictedFpLieAlgebra"; though this may pose
>a problem in the current implementation because the p-mapping
>is always given as a function, and not as an intrinsic operation.
>any ideas?
>
>
As far as I know this problem has not been considered before (but I might
be wrong of course). It would involve adding a basis element b^p for
every basis element of the free Lie algebra (and then (b^p)^p and so on).
At the moment I don't see any theoretical restrictions for making a function
"NilpotentQuotientRestrictedFpLieAlgebra". (But I might be wrong there too.)
In any case, the way the p-mapping is represented should not make a
difference, I think.
Best wishes,
Willem
From Rudolf.Zlabinger at chello.at Thu Mar 23 17:55:15 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Thu Mar 23 17:49:02 2006
Subject: [GAP Forum] Fw: Tabulatoren in Editoren
Message-ID: <00b901c64ea2$f0aee4e0$eec57254@chello.at>
----- Original Message -----
From: Rudolf Zlabinger
To: Forum@mail.gap-system.org
Sent: Thursday, March 23, 2006 3:43 PM
Subject: Tabulatoren in Editoren
Falls nicht ohnehin bekannt: Read und ReadTest ignorieren Tabulatoren in Editoren wie zB. im Windows Text Editor.
In case of that is unknown: Read and ReadTest ignore tabulator tags in editors such as for example Windows Test Editor.
regards, Rudolf Zlabinger
From sal at dcs.st-and.ac.uk Fri Mar 24 09:03:29 2006
From: sal at dcs.st-and.ac.uk (Steve Linton)
Date: Fri Mar 24 09:06:15 2006
Subject: [GAP Forum] New update of GAP released
Message-ID: <20060324090329.6eb6943a@localhost.localdomain>
Dear GAP Forum,
We are delighted to announce the release of GAP 4 release 4 update 7 (GAP 4.4.7
for short), which is available now from the GAP Web pages and FTP site. The
priority of this upgrade is very high, since it contains fixes for bugs which
can return wrong results without warnings. All users should update to this
release as soon as possible.
The upgrade also fixes many less dangerous bugs and adds new functionality
including improvements to the display of character tables, support for larger
finite fields, improved semigroup functionality, better support for free
products and more. Full details can be found on the web pages at
http://www.gap-system.org/Download/Updates/gap4r4p7.html.
Steve Linton for the GAP group
--
Steve Linton School of Computer Science &
Centre for Interdisciplinary Research in Computational Algebra
University of St Andrews Tel +44 (1334) 463269
http://www.dcs.st-and.ac.uk/~sal Fax +44 (1334) 463278
From joachim.s at web.de Fri Mar 24 14:18:43 2006
From: joachim.s at web.de (Joachim Schittenhelm)
Date: Fri Mar 24 14:19:06 2006
Subject: [GAP Forum] large finite fields
Message-ID: <4423FFC3.5060603@web.de>
Hi Members,
I am quite new to GAP and have to work with GaloisFields with a
large number of elements. Unfortunately the field is limited to
approximately 65k elements (~2^16).
By searching i discovered a 12 year old article that states the same
problem. Is this somehow solved in any extra package, or is it still
unpossible to use/create greater fields?
Regards
Joachim Schittenhelm
From raghu_juliet at rediffmail.com Sat Mar 25 17:35:48 2006
From: raghu_juliet at rediffmail.com (Raghunathan,R.)
Date: Sat Mar 25 17:36:35 2006
Subject: [GAP Forum] product groups
Message-ID: <20060325173548.22929.qmail@webmail6.rediffmail.com>
Dear GAP forum,
Thank you for the previous responses.
I have two questions.
Question 1:
Considering two groups 's' and 't' such that
s = ((),(3,4)) and t = ((1,3)(2,4)(5,6)), the sequential GAP commands
s:=Group((3,4));;
t:=Group((1,3)(2,4)(5,6))l;;
u:=DirectProduct(s,t); returns
Group([(1,2),(3,5)(4,6)(7,8)]) as the answer.
The command
Elements(u); returns
((),(3,5)(4,6)(7,8),(1,2),(1,2)(3,5)(4,6)(7,8)) as the answer which doesn't contain, the generators of both the groups 's' and 't' as elements. Is there an interpretation for the result?
Question 2:
If there exist two groups 'p' and 'q' such that
p=(generator1) and q=(generator2), are the groups
r =(generator1,generator2) and s = {p} U {p}*{q} always equal?
Thank you,
Raghunathan.
?
From wdjoyner at comcast.net Sat Mar 25 21:44:51 2006
From: wdjoyner at comcast.net (David Joyner)
Date: Sat Mar 25 21:41:47 2006
Subject: [GAP Forum] GAP permutation question
Message-ID: <4425B9D3.6040302@comcast.net>
Hello all:
Let L1, L2 be lists of objects (lists of vectors or lists of strings).
Is there a GAP function which returns a permutation which
send L1 to L2, if one exists, and "false" otherwise?
In case L1 = L2 = columns of a matrix M then the
function could be paired with the MatrixAutomorphism group of
M to return an "isomorphism" between two matrices.
- David Joyner
From dima at ntu.edu.sg Sun Mar 26 04:31:28 2006
From: dima at ntu.edu.sg (Dima Pasechnik)
Date: Sun Mar 26 04:32:07 2006
Subject: [GAP Forum] GAP permutation question
In-Reply-To: <4425B9D3.6040302@comcast.net>
Message-ID:
Dear all, dear David,
When L1, L2 are 0-1 matrices, one can use the package GRAPE to construct
(bipartite) graphs from them, and test them for isomorphism (such a test, if
successful, returns an isomorphism, too)
As GRAPE does not work with edge-coloured graphs, for more general matrices
this won't work, at least not out of the box.
One can think of constructing edge-coloured bipartite graphs, then their
vertex-coloured line graphs, and test the latter for the isomorphism. (GRAPE
can deal with vertex colourings).
One can probably use the general GAP backtracking on permutation groups, but
I doubt that this will be feasible for big matrices...
regards,
Dima
On 3/26/06 5:44 AM, "David Joyner" wrote:
> Hello all:
>
> Let L1, L2 be lists of objects (lists of vectors or lists of strings).
> Is there a GAP function which returns a permutation which
> send L1 to L2, if one exists, and "false" otherwise?
>
> In case L1 = L2 = columns of a matrix M then the
> function could be paired with the MatrixAutomorphism group of
> M to return an "isomorphism" between two matrices.
>
> - David Joyner
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
--
Dima Pasechnik
http://www.ntu.edu.sg/home/dima/
From wdjoyner at comcast.net Sun Mar 26 04:36:55 2006
From: wdjoyner at comcast.net (David Joyner)
Date: Sun Mar 26 04:33:33 2006
Subject: [GAP Forum] GAP permutation question
In-Reply-To: <4425B9D3.6040302@comcast.net>
References: <4425B9D3.6040302@comcast.net>
Message-ID: <44260C57.1020904@comcast.net>
This is simpler than I thought. Here is
an answer to my own question:
ListIsomorphism:=function(L1,L2)
local i,x,p,L,n1,n2;
L:=[];
n1:=Length(L1);
n2:=Length(L2);
if n1<>n2 then return fail; fi;
if n1<>Length(Set(L1)) then Print("Contains duplicates.\n"); return
fail; fi;
if n2<>Length(Set(L2)) then Print("Contains duplicates.\n"); return
fail; fi;
for x in L1 do
i:=Position(L2,x);
if i=fail then return fail; fi;
Append(L,[i]);
od;
return PermList(L);
end;
gap> L1:=[1,2,"a",[1,1]]; L2:=[[1,1],"a",2,1];
[ 1, 2, "a", [ 1, 1 ] ]
[ [ 1, 1 ], "a", 2, 1 ]
gap> ListIsomorphism(L1,L2);
(1,4)(2,3)
+++++++++++++++++++++++++
David Joyner wrote:
> Hello all:
>
> Let L1, L2 be lists of objects (lists of vectors or lists of strings).
> Is there a GAP function which returns a permutation which
> send L1 to L2, if one exists, and "false" otherwise?
>
> In case L1 = L2 = columns of a matrix M then the
> function could be paired with the MatrixAutomorphism group of
> M to return an "isomorphism" between two matrices.
>
> - David Joyner
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
From Dima at ntu.edu.sg Sun Mar 26 06:09:33 2006
From: Dima at ntu.edu.sg (Dmitrii V Pasechnik (Asst Prof))
Date: Sun Mar 26 06:10:00 2006
Subject: [GAP Forum] GAP permutation question
Message-ID: <48BABA4E8A54DF4AAF9CCF36D614027302214D3A@EXCHANGE23.staff.main.ntu.edu.sg>
Sorry, I thought that a permutation on the coordinates of the vectors
that makes L1=L2 equal as sets was sought. Otherwise it is indeed next
to trivial.
Regards,
Dima
-----Original Message-----
From: forum-bounces@gap-system.org [mailto:forum-bounces@gap-system.org]
On Behalf Of Dima Pasechnik
Sent: Sunday, March 26, 2006 11:31 AM
To: David Joyner; GAP forum
Subject: Re: [GAP Forum] GAP permutation question
Dear all, dear David,
When L1, L2 are 0-1 matrices, one can use the package GRAPE to construct
(bipartite) graphs from them, and test them for isomorphism (such a
test, if successful, returns an isomorphism, too)
As GRAPE does not work with edge-coloured graphs, for more general
matrices this won't work, at least not out of the box.
One can think of constructing edge-coloured bipartite graphs, then their
vertex-coloured line graphs, and test the latter for the isomorphism.
(GRAPE can deal with vertex colourings).
One can probably use the general GAP backtracking on permutation groups,
but I doubt that this will be feasible for big matrices...
regards,
Dima
On 3/26/06 5:44 AM, "David Joyner" wrote:
> Hello all:
>
> Let L1, L2 be lists of objects (lists of vectors or lists of strings).
> Is there a GAP function which returns a permutation which send L1 to
> L2, if one exists, and "false" otherwise?
>
> In case L1 = L2 = columns of a matrix M then the function could be
> paired with the MatrixAutomorphism group of M to return an
> "isomorphism" between two matrices.
>
> - David Joyner
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
--
Dima Pasechnik
http://www.ntu.edu.sg/home/dima/
_______________________________________________
Forum mailing list
Forum@mail.gap-system.org
http://mail.gap-system.org/mailman/listinfo/forum
From helge.ruddat at math.uni-freiburg.de Fri Mar 24 16:23:10 2006
From: helge.ruddat at math.uni-freiburg.de (Helge Ruddat)
Date: Mon Mar 27 10:40:18 2006
Subject: [GAP Forum] AppendTo problem with large strings
Message-ID: <200603241623.11427.helge.ruddat@math.uni-freiburg.de>
Dear GAP supporters!
I am trying to write a large matrix into a text file in such a way
that scilab is able to read it as input. For this I need to write
long strings (about 600 characters) into the file and I am doing this
via AppendTo( filename, string ).
I realized that this function inserts backslashes and newlines to cut my
strings into smaller pieces, although I do not intend this. It also is not
documented in the "AppendTo - function - documentation".
Funny is, that the length of the dissected pieces correllates with
the width of my gap terminal window. When I tried to increase
the width of my gap window significantly, gap quit with a
memory access error.
Can someone help me?
I have version 4.4.6.
Thanks in advance,
Helge Ruddat
--
Helge Ruddat / Tivolistra?e 16 / 79104 Freiburg i.Br.
Tel.: +49 761 2925203 Uni.: +49 761 2035620
mobil: +49 176 20155350 Heimzone: +49 761 1528137
From maasiru at yahoo.com Mon Mar 27 11:58:22 2006
From: maasiru at yahoo.com (muniru asiru)
Date: Mon Mar 27 11:58:35 2006
Subject: [GAP Forum] GAP permutation question
Message-ID: <20060327105822.52447.qmail@web53314.mail.yahoo.com>
Form, David,
I think you should use Loops package for Gap which
contains a function by which two loops (and groups)
can be tested for isomorphism.
For example: Given L1 and L2 you can test
IsIsomorphism(LoopByCayleyTable(L1),LoopByCayleyTable(L2));
sincerely,
Muniru
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From holmespe at for.mat.bham.ac.uk Mon Mar 27 15:09:08 2006
From: holmespe at for.mat.bham.ac.uk (Petra Holmes)
Date: Mon Mar 27 15:09:39 2006
Subject: [GAP Forum] AppendTo problem with large strings
In-Reply-To: <200603241623.11427.helge.ruddat@math.uni-freiburg.de>
Message-ID:
Helge,
I'm not sure if there's a proper way of doing it, but this way works for
me.
If you print the string out one character at a time then it doesn't insert
newlines, like this:
for s in string do
AppendTo(file,s);
od;
This gives an output file that is what you want except that it's full of
the character ' but you can get rid of those at the command line using
sed. The command is
sed "s/'//g" file > newfile
using a bash shell, and I suppose it's almost the same in other shells.
Hope this helps,
Beth
On Fri, 24 Mar 2006, Helge Ruddat wrote:
> Dear GAP supporters!
>
> I am trying to write a large matrix into a text file in such a way
> that scilab is able to read it as input. For this I need to write
> long strings (about 600 characters) into the file and I am doing this
> via AppendTo( filename, string ).
> I realized that this function inserts backslashes and newlines to cut my
> strings into smaller pieces, although I do not intend this. It also is not
> documented in the "AppendTo - function - documentation".
> Funny is, that the length of the dissected pieces correllates with
> the width of my gap terminal window. When I tried to increase
> the width of my gap window significantly, gap quit with a
> memory access error.
> Can someone help me?
>
> I have version 4.4.6.
>
> Thanks in advance,
> Helge Ruddat
>
> --
> Helge Ruddat / Tivolistraße 16 / 79104 Freiburg i.Br.
> Tel.: +49 761 2925203 Uni.: +49 761 2035620
> mobil: +49 176 20155350 Heimzone: +49 761 1528137
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
From pjd at maths.gla.ac.uk Mon Mar 27 16:48:33 2006
From: pjd at maths.gla.ac.uk (pjd@maths.gla.ac.uk)
Date: Mon Mar 27 16:48:36 2006
Subject: [GAP Forum] Cyclically Reduced Words
Message-ID: <1143474513.4428095158316@mail.maths.gla.ac.uk>
Dear GAP Forum,
Given the alphabet {a,a^-1,b,b^-1}, how can I get GAP to produce a list of
cyclically reduced words of length at most 3, 4, etc?
Thanks
Peter
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From l.h.soicher at qmul.ac.uk Mon Mar 27 20:30:51 2006
From: l.h.soicher at qmul.ac.uk (Leonard Soicher)
Date: Mon Mar 27 20:31:23 2006
Subject: [GAP Forum] AppendTo problem with large strings
In-Reply-To: <200603241623.11427.helge.ruddat@math.uni-freiburg.de>
References: <200603241623.11427.helge.ruddat@math.uni-freiburg.de>
Message-ID: <20060327193051.GA16460@mrcpc02.maths.qmul.ac.uk>
Dear Helge Ruddat, Dear GAP-Forum,
In GRAPE 4.2 I use text-file streams to communicate with
external programs, with print formatting switched off
(via the GAP function SetPrintFormattingStatus ).
For an example of this, look at the function AutGroupGraph
in pkg/grape/lib/grape.g
Best wishes,
Leonard
On Fri, Mar 24, 2006 at 04:23:10PM +0000, Helge Ruddat wrote:
> Dear GAP supporters!
>
> I am trying to write a large matrix into a text file in such a way
> that scilab is able to read it as input. For this I need to write
> long strings (about 600 characters) into the file and I am doing this
> via AppendTo( filename, string ).
> I realized that this function inserts backslashes and newlines to cut my
> strings into smaller pieces, although I do not intend this. It also is not
> documented in the "AppendTo - function - documentation".
> Funny is, that the length of the dissected pieces correllates with
> the width of my gap terminal window. When I tried to increase
> the width of my gap window significantly, gap quit with a
> memory access error.
> Can someone help me?
>
> I have version 4.4.6.
>
> Thanks in advance,
> Helge Ruddat
>
> --
> Helge Ruddat / Tivolistra?e 16 / 79104 Freiburg i.Br.
> Tel.: +49 761 2925203 Uni.: +49 761 2035620
> mobil: +49 176 20155350 Heimzone: +49 761 1528137
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
From alexk at mcs.st-and.ac.uk Tue Mar 28 09:15:15 2006
From: alexk at mcs.st-and.ac.uk (Alexander Konovalov)
Date: Tue Mar 28 09:15:54 2006
Subject: [GAP Forum] AppendTo problem with large strings
In-Reply-To: <200603241623.11427.helge.ruddat@math.uni-freiburg.de>
References: <200603241623.11427.helge.ruddat@math.uni-freiburg.de>
Message-ID: <69EED167-AEF3-4353-B856-87C44C2DEA01@mcs.st-and.ac.uk>
Dear Helge Ruddat,
You can use the 'SetPrintFormattingStatus' operation, which sets
whether the output will be formatted with line breaks and indentation,
or not.
There is also an operation 'PrintFormattingStatus' that can be used
to get information about the current setup for this option.
Please see their description and examples in the GAP Manual in your
GAP installation or online at the page
http://www.gap-system.org/Manuals/doc/htm/ref/CHAP010.htm#SSEC004.8
Also, for writing huge amounts of data efficiently with GAP the
utility functions 'PrintTo1', 'AppendTo1' and 'FileString' from the
GAPDoc package by Frank L?beck and Max Neunh?ffer
(http://www.gap-system.org/Packages/gapdoc.html) may also be
interesting for you.
Best wishes,
Alexander Konovalov
On 24 Mar 2006, at 17:23, Helge Ruddat wrote:
> Dear GAP supporters!
>
> I am trying to write a large matrix into a text file in such a way
> that scilab is able to read it as input. For this I need to write
> long strings (about 600 characters) into the file and I am doing this
> via AppendTo( filename, string ).
> I realized that this function inserts backslashes and newlines to
> cut my
> strings into smaller pieces, although I do not intend this. It also
> is not
> documented in the "AppendTo - function - documentation".
> Funny is, that the length of the dissected pieces correllates with
> the width of my gap terminal window. When I tried to increase
> the width of my gap window significantly, gap quit with a
> memory access error.
> Can someone help me?
>
> I have version 4.4.6.
>
> Thanks in advance,
> Helge Ruddat
>
> --
> Helge Ruddat / Tivolistra?e 16 / 79104 Freiburg i.Br.
> Tel.: +49 761 2925203 Uni.: +49 761 2035620
> mobil: +49 176 20155350 Heimzone: +49 761 1528137
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
From aodabas at ogu.edu.tr Tue Mar 28 10:59:15 2006
From: aodabas at ogu.edu.tr (=?iso-8859-9?Q?Alper_Odaba=FE?=)
Date: Tue Mar 28 10:58:12 2006
Subject: [GAP Forum] algebra autmorphism
Message-ID: <003401c6524e$4571cb10$be838cc1@ogu209>
hi all
1) has gap any function for autmorphism algebra like AutomorphismGroup()
2) A is a GroupRing (Algebra) . How we can determine Aut(A)???
gap> A:=GroupRing(GF(3),CyclicGroup(2));
gap> IsAlgebra(A);
true
gap> AutA:=GroupRing(GF(3),AutomorphismGroup(CyclicGroup(2)));
????????????
thanks
Alper
From thomas.breuer at math.rwth-aachen.de Wed Mar 29 09:37:50 2006
From: thomas.breuer at math.rwth-aachen.de (Thomas Breuer)
Date: Wed Mar 29 09:39:26 2006
Subject: [GAP Forum] product groups
Message-ID: <20060329083750.GB2523@math.rwth-aachen.de>
Dear GAP Forum,
R. Raghunathan wrote
> I have two questions.
> Question 1:
> Considering two groups 's' and 't' such that
> s = ((),(3,4)) and t = ((1,3)(2,4)(5,6)), the sequential GAP commands
> s:=Group((3,4));;
> t:=Group((1,3)(2,4)(5,6))l;;
> u:=DirectProduct(s,t); returns
> Group([(1,2),(3,5)(4,6)(7,8)]) as the answer.
> The command
> Elements(u); returns
> ((),(3,5)(4,6)(7,8),(1,2),(1,2)(3,5)(4,6)(7,8)) as the answer which doesn't
> +contain, the generators of both the groups 's' and 't' as elements. Is there
+an
> +interpretation for the result?
>
> Question 2:
> If there exist two groups 'p' and 'q' such that
> p=(generator1) and q=(generator2), are the groups
> r =(generator1,generator2) and s = {p} U {p}*{q} always equal?
Question 1 addresses the question which generators GAP chooses for
a direct product of groups.
Of course one can represent the elements of a direct product
of two groups $G$, $H$ by pairs $(g,h)$ with $g \in G$, $h \in H$.
However, such pairs are not very suitable for GAP computations.
In the case that $G$ and $H$ are permutation groups,
GAP decides to represent the direct product as a permutation group;
for that, first the groups are conjugated in a bigger symmetric group
in order to achieve that the sets of moved points are disjoint.
The connection between the generators of the original groups $G$ and $H$
and the direct product is given by embeddings and projections.
In the above example, this looks as follows.
gap> s:= Group( (3,4) );;
gap> t:= Group( (1,3)(2,4)(5,6) );;
gap> u:= DirectProduct( s, t );
Group([ (1,2), (3,5)(4,6)(7,8) ])
gap> emb1:= Embedding( u, 1 );
1st embedding into Group([ (1,2), (3,5)(4,6)(7,8) ])
gap> emb2:= Embedding( u, 2 );
2nd embedding into Group([ (1,2), (3,5)(4,6)(7,8) ])
gap> Source( emb1 ); # this is s
Group([ (3,4) ])
gap> Image( emb1, s ); # this is the first factor of the direct product
Group([ (1,2) ])
gap> Image( emb2, t ); # this is the second factor
Group([ (3,5)(4,6)(7,8) ])
If one wants to construct the group generated by two groups in GAP,
which is in general *not* a direct product, one can use the function
`ClosureGroup'.
The result is a group (in fact the smallest group) that contains
the two given groups as subsets.
In the above example, the closure is a group of order eight.
gap> c:= ClosureGroup( s, t );
Group([ (3,4), (1,3)(2,4)(5,6) ])
gap> Size( c );
8
I am not sure whether I understand Question 2.
If ``U'' denotes the union of sets and ``*'' denotes the complex product
of sets then the number of elements in ``{p} U {p}*{q}'' is at most
the order of `p' plus the product of the orders of `p' and `q'.
The group generated by `p' and `q' can be much bigger.
For example, the groups < (1,2) > and < (1,3)(2,4) > have order two,
and the group < (1,2), (1,3)(2,4) > is a dihedral group of order eight.
Or is the question whether the group generated by `p' and `q' equals the
group *generated by* `p' and all products of elements in `p' and `q'?
Since the elements of `q' are a subset of these products,
the answer to this question is yes.
All the best,
Thomas Breuer
From kohl at mathematik.uni-stuttgart.de Wed Mar 29 12:41:44 2006
From: kohl at mathematik.uni-stuttgart.de (Stefan Kohl)
Date: Wed Mar 29 12:41:52 2006
Subject: [GAP Forum] RCWA 1.5
Message-ID: <442A7278.2030306@mathematik.uni-stuttgart.de>
Dear Forum,
This is to announce the release of RCWA 1.5.
The RCWA package provides methods for computing in
certain infinite permutation groups acting on the integers.
Due to a change of maintainership of the GAP website it may take
some time until the new version is available at the usual location
http://www.gap-system.org/Packages/rcwa.html. In any case, you can
get RCWA 1.5 from
http://www.cip.mathematik.uni-stuttgart.de/~kohlsn/rcwa.html .
Let me take the opportunity to point to a related recent preprint of mine:
----------------------------------------------------------------------------
Title: Three Countable Highly Transitive Simple Groups
Containing Collatz' Permutation
Abstract: We construct three countable simple permutation groups which act
highly transitively on N_0 resp. Z. These groups form an ascending
chain w.r.t. inclusion. The smallest of them is generated by
involutions which interchange two disjoint residue classes, each.
The free group of rank 2 and the modular group PSL(2,Z) embed in
this group. We show by means of computation that it further
contains a certain permutation of the integers which has already
been investigated by Lothar Collatz in 1932, and whose cycle
structure is unknown so far.
http://www.cip.mathematik.uni-stuttgart.de/~kohlsn/preprints/simplegp.pdf
----------------------------------------------------------------------------
Wishing you fun and success using RCWA,
Stefan Kohl
From alexander.konovalov at gmail.com Wed Mar 29 16:59:27 2006
From: alexander.konovalov at gmail.com (Alexander Konovalov)
Date: Wed Mar 29 16:59:41 2006
Subject: [GAP Forum] Experimental GAP Installer for Windows
Message-ID:
Dear GAP Forum members,
Following the announcement of GAP 4.4.7, I would like to inform
Windows users that I had updated the experimental GAP Installer for
Windows.
Now GAP 4.4.7 is available also in exe-format from the page http://
ukrgap.exponenta.ru/wininst/wininst.htm.
To install the GAP system and GAP packages, you need to download the
following files:
- ftp://ftp.gap-system.org/pub/gap/windowsinstaller/gap4r4p7.exe (52
MB):
The core GAP system with optional components tools, htmie and xtom
- the most recent merged archive of all currently redistributed GAP
packages packages-.exe (about 40 MB)
from the directory ftp://ftp.gap-system.org/pub/gap/
windowsinstaller/.
If you already used the experimental GAP installer for Windows to
install GAP 4.4.6,
you can upgrade your system to GAP 4.4.7 installing the following
update/bugfix:
ftp://ftp.gap-system.org/pub/gap/windowsinstaller/fix4r4p7.exe (16,6 MB)
The procedure of installing this upgrade/bugfix is similar to the
procedure of GAP installation from the exe-file.
Best wishes,
Alexander Konovalov
From joachim.neubueser at math.rwth-aachen.de Thu Mar 30 11:50:59 2006
From: joachim.neubueser at math.rwth-aachen.de (Joachim Neubueser)
Date: Thu Mar 30 11:51:43 2006
Subject: [degraaf@science.unitn.it: Re: [GAP Forum] Automorphism of Algebra]
Message-ID: <20060330105059.GA8567@math.rwth-aachen.de>
Dear Alper Odaba?
You are asking again in the GAP Forum about automorphisms of algebras,
after you had asked the same question before on March 9 and your
letter had been answered by Willem De Graaf on March 10.
Just in case for some reason Willem's letter has not reached you, I
append a copy.
However please understand that we want to keep the GAP Forum, that is
read by several hundred colleagues worldwide, free from unnecessary
mail. So certainly repeating a problem is unnecessary, but moreover we
make clear on the GAP website, that we ask all users to send questions
that are rather particular to them but likely not of interest to many
of the Forum members to the address
support@gap-system.org
rather than to the GAP Forum.
Kind regards Joachim Neubueser
----- Forwarded message from Willem De Graaf -----
X-Original-To: joachim.neubueser@math.rwth-aachen.de
From: Willem De Graaf
Subject: Re: [GAP Forum] Automorphism of Algebra
To: Alper Odaba? , forum@gap-system.org
Dear Alper Odaba?,
Currently GAP does not have much functionality for working
with automorphisms of algebras. However, there are functions
for constructing homomorphisms (which, as a special case,
can be automorphisms). This is done by using the function
"AlgebraHomomorphismByImages".
The GAP manual contains a description of this function,
it is available online:
http://www.gap-system.org/Manuals/doc/htm/ref/CHAP060.htm#SECT009
(Note that in order to use it, one must specify set of source generators,
and a set of image generators.)
Best wishes,
Willem de Graaf
Alper Odaba? wrote:
>Hi all,
>
>I have a question for algebra ,
>Let A and B algebras. Suppose that B acts on A, is there a algebra
>homomorphism B --> Aut(A) ??
>has GAP any function Automorphism of commutative algebras??
>
>gap> G:=Group((1,2,3,4));;
>gap> A:=GroupRing(GF(3),G);;
>gap> Automorphism(A);
>Variable: 'Automorphism' must have a value
>
>gap> AutomorphismAlgebra(A);
>Variable: 'AutomorphismAlgebra' must have a value
>
>gap> Automorphisms(A);
>Variable: 'Automorphisms' must have a value
>
>gap> Automorphism(A);
>Variable: 'Automorphism' must have a value
>
>gap> AutomorphismOfAlgebra(A);
>Variable: 'AutomorphismOfAlgebra' must have a value
>
>gap> AutomorphismRing(A);
>Variable: 'AutomorphismRing' must have a value
>
>gap> AutomorphismOfRing(A);
>Variable: 'AutomorphismOfRing' must have a value
>
>
>thanks.
>
>Alper
>
>_______________________________________________
>Forum mailing list
>Forum@mail.gap-system.org
>http://mail.gap-system.org/mailman/listinfo/forum
>
>
>
_______________________________________________
Forum mailing list
Forum@mail.gap-system.org
http://mail.gap-system.org/mailman/listinfo/forum
----- End forwarded message -----
From Rudolf.Zlabinger at chello.at Sat Apr 1 14:49:36 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Sat Apr 1 14:43:05 2006
Subject: [GAP Forum] Rubiks cube and Fp groups
Message-ID: <006801c65593$1d1affe0$eec57254@chello.at>
I refer to http://www.gap-system.org/Doc/Examples/rubik.html as given by
Martin Sch?nert.
I tested the presentation of a random element of the cubes permutations in a
finitely presented group.
I took the free group in the sample and formed a Fp group giving the
relators f ^ 4 for all generators of the free group, as this is also given
by the original generators. Of course, its a infinite group again, as
unnecessarily also tested by the Newman Infinity Criterion. The Preimage for
the Free group gave a solution of length 120, the same performed with the Fp
group, same permutation, resulted in a chain of 84 moves. To be secure I
tested the image also in reverse order successfully, as given by the sample.
The random permutation was in both cases
(1,22,33,17,19,40,30,35,16,3,6,25,46,24,9,41,27,11,8,14,43)(2,13,31,29,15,42
,
28)(4,5,39,18,37)(7,12,10,26,47)(20,45,36,44,23,21,34)(32,38,48)
.
The algorithm mentioned in the sample was using stabilizer chains. Is it the
same for the Fp group I used for my test? Or is there help for a "better"
algorithm caused by the relators? Or is it pure random behaviour?
By the way, another question to the same sample. There are given " wreath
products of a 3 cycle (2 cycle) with S(8)". Is it right to interprete them
as wreath products of the cycles ^ 8 and S(8)?
best regards, Rudolf Zlabinger
From hulpke at mac.com Sat Apr 1 17:36:35 2006
From: hulpke at mac.com (Alexander Hulpke)
Date: Sat Apr 1 17:37:00 2006
Subject: [GAP Forum] Rubiks cube and Fp groups
In-Reply-To: <006801c65593$1d1affe0$eec57254@chello.at>
References: <006801c65593$1d1affe0$eec57254@chello.at>
Message-ID:
Dear GAP Forum,
On Apr 1, 2006, at 6:49 AM, Rudolf Zlabinger wrote:
> I tested the presentation of a random element of the cubes
> permutations in a
> finitely presented group.
>
> The Preimage for
> the Free group gave a solution of length 120, the same performed
> with the Fp
> group, same permutation, resulted in a chain of 84 moves.
>
> The algorithm mentioned in the sample was using stabilizer chains.
> Is it the
> same for the Fp group I used for my test? Or is there help for a
> "better"
> algorithm caused by the relators? Or is it pure random behaviour?
The algorithm is the same. I uses random words, thus the different
length will be due to random behavior. If you construct the same
homomorphism anew and try you will see some length discrepancies.
>
> By the way, another question to the same sample. There are given "
> wreath
> products of a 3 cycle (2 cycle) with S(8)". Is it right to
> interprete them
> as wreath products of the cycles ^ 8 and S(8)?
I suppose you mean ``semidirect'' in the last line. Then yes.
Best wishes,
Alexander Hulpke
From Rudolf.Zlabinger at chello.at Sat Apr 1 19:23:50 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Sat Apr 1 19:17:14 2006
Subject: [GAP Forum] Rubiks cube and Fp groups
References: <006801c65593$1d1affe0$eec57254@chello.at>
Message-ID: <00c201c655b9$6cff01c0$eec57254@chello.at>
Dear Mr. Hulpke,
thank you for your quick answer. I was not surprised to see, that there are
random algorithms in use, i would have used also for similar purposes. But
the idea, that the relators could contribute, was too attractive for me.
Semidirect product is correct in this case, where S(8) doesn't p e r m u t
e the components of cycle^8, as defined for wreath products in general, so
the only permutation on cycle^8 components would be the identity. My
question was not formulated correctly and therefore misleading. But in our
context there is no ambiguity at all, as I can see now.
As you can see, I had (erroneously?) some more general definition of "wreath
product" in mind, where the power of first factor is somewhat independent
from the cardinality of workingset of the second factor, and semidirect
product would be a special case. The action on the components of first
factor by second factor had to be defined separately in this case.
In the meantime i found, that "wreath product" has a more strict meaning at
least in the context of GAP and from there my question was already answered.
thank you and best wishes, Rudolf Zlabinger
----- Original Message -----
From: "Alexander Hulpke"
To: "Rudolf Zlabinger"
Cc:
Sent: Saturday, April 01, 2006 6:36 PM
Subject: Re: [GAP Forum] Rubiks cube and Fp groups
Dear GAP Forum,
On Apr 1, 2006, at 6:49 AM, Rudolf Zlabinger wrote:
> I tested the presentation of a random element of the cubes
> permutations in a
> finitely presented group.
>
> The Preimage for
> the Free group gave a solution of length 120, the same performed
> with the Fp
> group, same permutation, resulted in a chain of 84 moves.
>
> The algorithm mentioned in the sample was using stabilizer chains.
> Is it the
> same for the Fp group I used for my test? Or is there help for a
> "better"
> algorithm caused by the relators? Or is it pure random behaviour?
The algorithm is the same. I uses random words, thus the different
length will be due to random behavior. If you construct the same
homomorphism anew and try you will see some length discrepancies.
>
> By the way, another question to the same sample. There are given "
> wreath
> products of a 3 cycle (2 cycle) with S(8)". Is it right to
> interprete them
> as wreath products of the cycles ^ 8 and S(8)?
I suppose you mean ``semidirect'' in the last line. Then yes.
Best wishes,
Alexander Hulpke
From hulpke at mac.com Sun Apr 2 23:00:51 2006
From: hulpke at mac.com (Alexander Hulpke)
Date: Sun Apr 2 23:02:22 2006
Subject: [GAP Forum] Cyclically Reduced Words
In-Reply-To: <1143474513.4428095158316@mail.maths.gla.ac.uk>
References: <1143474513.4428095158316@mail.maths.gla.ac.uk>
Message-ID: <798CB985-C101-4F80-A0C2-969E83A29019@mac.com>
Dear GAP-Forum,
On Mar 27, 2006, at 8:48 AM, pjd@maths.gla.ac.uk wrote:
>
> Given the alphabet {a,a^-1,b,b^-1}, how can I get GAP to produce a
> list of
> cyclically reduced words of length at most 3, 4, etc?
There is no predefined function which does this. What I would do if I
had to construct these words is to take the code for `Tuples' (in lib/
combinat.gi) to construct combinations of [-n,-(n-1),-
(n-2)..-2,-1,1,2,3,..n] and modify it to not permit i following -i
and vice versa or have a word starting in i and ending in -i and vice
versa.
If your application is not time/memory critical the following naive
approach is likely the easiest:
f:=FreeGroup("a","b");
IsCycRed:=l->l[1]<>-l[Length(l)] and ForAll([1..Length(l)-1],j->l[j]
<>-l[j+1]);
len:=4; # or whatever
w:=Filtered(Tuples([-2,-1,1,2],len),IsCycRed);
fam:=FamilyObj(One(f));;
List(w,i->AssocWordByLetterRep(fam,i));
Best,
Alexander Hulpke
From tmeletn at gmail.com Wed Apr 5 00:18:28 2006
From: tmeletn at gmail.com (Tom Meletn)
Date: Wed Apr 5 00:18:39 2006
Subject: [GAP Forum] Character table of D4(q)
Message-ID: <9ee207f00604041618j553aaa9k7130154d5c684f49@mail.gmail.com>
Dear GAP Froum,
Do you know how I can get the whole character table of the Chevalley group
D4(3), and maybe D4(5)?
Thanks, Tom
From alexander.konovalov at gmail.com Thu Apr 6 07:54:22 2006
From: alexander.konovalov at gmail.com (Alexander Konovalov)
Date: Thu Apr 6 07:54:23 2006
Subject: [GAP Forum] LAGUNA 3.3.3
Message-ID: <519BD48C-AF8F-45CB-94AA-5E00B228469E@gmail.com>
Dear GAP Forum,
this is to announce the availability of the new version of the LAGUNA
package.
The LAGUNA package provides functionality for calculation of the
normalized unit group of the modular group algebra of the finite
p-group and for investigations of Lie algebras associated with group
algebras.
LAGUNA 3.3.3 was released on April 03, 2006. It is available from
the following pages:
- http://ukrgap.exponenta.ru/laguna.htm
- http://www.gap-system.org/Packages/laguna.html
and also in a merged archive of all currently redistributed GAP packages
available from http://www.gap-system.org/Download/index.html
The new version fixes a rare no-method-found error that appear during
the computation of an inverse of a group ring element in characteristic
zero, in the case when the support subgroup is a p-group.
Also it contains more explicit statement about the licensing policy
and distributing LAGUNA under the GNU General Public License.
Sincerely yours,
Alexander Konovalov
From Rudolf.Zlabinger at chello.at Thu Apr 6 08:15:05 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Thu Apr 6 08:08:16 2006
Subject: [GAP Forum] Answer to Tom Meletn; Chevalley groups
Message-ID: <002901c65949$d4603f80$eec57254@chello.at>
I found the character table of D4(3), but not D4(5), under O+8(3) in the ATLAS.
(see GAPs Bibliography).
Yours sincerely, Rudolf Zlabinger
From Rudolf.Zlabinger at chello.at Thu Apr 6 16:29:11 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Thu Apr 6 16:22:24 2006
Subject: [GAP Forum] Conway Co2 group
Message-ID: <011501c6598e$dac96e60$eec57254@chello.at>
I try to handle the Conway Co2 group in GAP. I formed a Fpgroup with the relators as given in the ATLAS as follows:
gap> START_TEST("The Conway Group Co2.txt");
gap> fgco2:=FreeGroup("a","b","c","d","e","f","g");;
gap> a:=fgco2.1;;
gap> b:=fgco2.2;;
gap> c:=fgco2.3;;
gap> d:=fgco2.4;;
gap> e:=fgco2.5;;
gap> f:=fgco2.6;;
gap> g:=fgco2.7;;
gap> relators:=[a/(c*f)^2,b/(e*f)^3,e/(b*g)^2,(a*e*c*d)^4,(c*e*f)^7,(b*a*e*f*g)^3];;
gap> Append(relators,[a^2,b^2,c^2,d^2,e^2,f^2,g^2]);;
gap> Append(relators,[(a*b)^3,(a*c)^2,(a*d)^2,(a*e)^4,(a*f)^2,(a*g)^2]);;
gap> Append(relators,[(b*a)^3,(b*c)^5,(b*d)^2,(b*e)^2,(b*f)^2,(b*g)^4]);;
gap> Append(relators,[(c*b)^5,(c*d)^3,(c*e)^3,(c*f)^4,(c*g)^2]);;
gap> Append(relators,[(d*c)^3,(d*e)^2,(d*f)^3,(d*g)^2]);;
gap> Append(relators,[(e*a)^4,(e*c)^3,(e*f)^6,(e*g)^2]);;
gap> Append(relators,[(f*c)^4,(f*d)^3,(f*e)^6,(f*g)^4]);;
gap> Append(relators,[(g*b)^4,(g*f)^4]);;
gap> co2:=fgco2/relators;;
gap> STOP_TEST( "The Conway Group Co2",1000);
The size of the group is 2^18*3^6*5^3*7*11*23 = 42305421312000;
The standard operations as IsFinite, ConjugacyClasses, Size and others, run out of time > 1 hour on my small machine of 500 Mhz, 128MB.
Is there, as you know, any chance to handle this Fpgroup smoothly on a machine of 3,5Ghz, 500MB ( I mean times <= 15 minutes)? I currently try to find a representation in a permutation group, would that help?
Thank you for hints, Rudolf Zlabinger
From reichard at maths.uwa.edu.au Fri Apr 7 02:45:11 2006
From: reichard at maths.uwa.edu.au (Sven Reichard)
Date: Fri Apr 7 02:45:41 2006
Subject: [GAP Forum] Conway Co2 group
In-Reply-To: <011501c6598e$dac96e60$eec57254@chello.at>
References: <011501c6598e$dac96e60$eec57254@chello.at>
Message-ID: <4435C427.7040603@maths.uwa.edu.au>
Rudolf,
you can get a permutation representation of Co2 on 2300 points from the
AtlasRep package. This representation can be dealt with quite efficiently.
gap> LoadPackage("atlasrep");
-----------------------------------------------------------------------------
Loading AtlasRep 1.2.1 (An Atlas of Group Representations)
by Robert A. Wilson (http://for.mat.bham.ac.uk/R.A.Wilson),
Richard A. Parker (http://www.ukonline.co.uk/richard),
John Bray (http://web.mat.bham.ac.uk/J.N.Bray), and
Thomas Breuer (http://www.math.rwth-aachen.de/~Thomas.Breuer).
-----------------------------------------------------------------------------
true
gap> DisplayAtlasInfo("Co2", IsPermGroup);
Representations for G = Co2: (all refer to std. generators 1)
----------------------------
1: G <= Sym(2300)
2: G <= Sym(4600)
gap> gens := AtlasGenerators("Co2", 1).generators;;
gap> G := Group(gens);
gap> Size(G);
42305421312000
If you want an FP group, you may also try a shorter set of generators,
found on the Atlas Homepage:
< a, b | a^2 = b^5 = (ab^2)^9 = [a, b]^4 = [a, b^2]^4 = [a, bab]^3 = [a,
bab^2ab]^2 = [a, bab^-2]^3 = [a, b^-2abab^-2]^2 = (abab^2ab^-1ab^-2)^7 =
1 >.
Hope this helps,
Sven Reichard.
From dima at ntu.edu.sg Fri Apr 7 06:03:21 2006
From: dima at ntu.edu.sg (Dmitrii Pasechnik)
Date: Fri Apr 7 06:03:52 2006
Subject: [GAP Forum] Conway Co2 group
In-Reply-To: <4435C427.7040603@maths.uwa.edu.au>
References: <011501c6598e$dac96e60$eec57254@chello.at>
<4435C427.7040603@maths.uwa.edu.au>
Message-ID: <1144386201.27084.13.camel@spms-michele.staff.main.ntu.edu.sg>
Dear Rudolf,
a presentation that you can work with should have the subgroup U_6(2)
visible, then you can enumerate cosets of it in Co_2 without much
trouble. I don't know whether such an explicit presentation was ever
published, but it is implicit in characterisations of Co_2 via
a geometry called extended dual polar space, see e.g.
A. A. Ivanov, Exceptional Extended Dual Polar Spaces, Eur. J. Comb. 18
(1997), 859-885, or
Hans Cuypers, Extended near hexagons and line systems, Advances in
Geometry 4 (2004), 181-214.
Hope this helps,
Dmitrii
From jjm at dcs.st-and.ac.uk Fri Apr 7 11:46:14 2006
From: jjm at dcs.st-and.ac.uk (John McDermott)
Date: Fri Apr 7 11:47:34 2006
Subject: [GAP Forum] Matrix Operations and saving cpu cycles
Message-ID:
Dear Forum,
the following message was sent by ariel@zarzamora.com.mx but held for
moderation - please direct any replies to that address (cc to
forum@gap-system.org).
>
> Im looking for a way to do simple matrix operations,
> specifically,
> swap one row with another &
> swap one column with another,
> also sums and product.
>
> I do not want to create identity matrices and make de
> product as this will involve much more CPU resources (i think),
> im doing this for a bigger proyect and want to save cpu cycles.
>
> Googling the gap system site didn't help much, i cant find an
> easy way to do this. Shall i code my own rutine?
>
> Thanks very much
>
> Ariel
>
>
From Rudolf.Zlabinger at chello.at Fri Apr 7 14:04:28 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Fri Apr 7 13:57:38 2006
Subject: [GAP Forum] Matrix Operations and saving cpu cycles
Message-ID: <000d01c65a43$cda09640$eec57254@chello.at>
The most operations could be done by the statistical application R
free downlod at
The R Project for Statistical Computing
Swapping rows and columns could be done at least by multiplying with permutation matrices.
best wishes, Rudolf Zlabinger
From Rudolf.Zlabinger at chello.at Fri Apr 7 19:20:50 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Fri Apr 7 19:13:58 2006
Subject: [GAP Forum] Fw: Matrix Operations and saving cpu cycles
Message-ID: <001e01c65a6f$fff5b400$eec57254@chello.at>
This should be a option only, GAP provides a rich set of matrix functionality too, you should compare for performance and usability. R may be somewhath simpler to get working with, the functionality is also significantly smaller than that of GAP.
best wishes, Rudolf Zlabinger
----- Original Message -----
From: Rudolf Zlabinger
To: ariel@zarzamora.com.mx
Cc: forum@gap-system.org
Sent: Friday, April 07, 2006 3:04 PM
Subject: Matrix Operations and saving cpu cycles
The most operations could be done by the statistical application R
free downlod at
The R Project for Statistical Computing
Swapping rows and columns could be done at least by multiplying with permutation matrices.
best wishes, Rudolf Zlabinger
From igor at txc.com Fri Apr 7 19:53:28 2006
From: igor at txc.com (Igor Schein)
Date: Fri Apr 7 19:53:41 2006
Subject: [GAP Forum] RPM package
Message-ID: <20060407185328.GY19808@txc.com>
Dear Forum,
does anyone (un)officially maintain gap.spec for building RPMs of
latest GAP releases?
Thanks
Igor
From hulpke at frii.com Mon Apr 10 03:45:54 2006
From: hulpke at frii.com (Alexander Hulpke)
Date: Mon Apr 10 03:46:18 2006
Subject: [GAP Forum] Conway Co2 group
References:
Message-ID: <2B1690C3-7795-4B0E-BC44-63F3AF35F3B9@frii.com>
Dear GAP Forum,
On Apr 6, 2006, at 9:29 , Rudolf Zlabinger wrote:
> I try to handle the Conway Co2 group in GAP. I formed a Fpgroup
> with the relators as given in the ATLAS as follows:
As this question concerns two aspects that come up again and again,
allow me a more general answer:
The first aspect is on how to get a particular simple (or close to
simple) group in GAP.
The most ``universal'' source is -- as already mentioned by several
contributors -- probably the ATLAS web pages maintained by Rob Wilson
and collaborators which provides a plethora of representations for
such groups. In particular, there is the AtlasRep package which
provides a very nice interface.
By searching through the forum archive you will find several mails
describing this package in more detail.
A second way to get these groups is by the fact that they often have
a faithful primitive representation of small degree. (For Example Co2
has a representation of degree 2300.)
Thus many of these groups can be found in the primitive groups
library. Even if you don't know the degree, but the order you can
search for it:
AllPrimitiveGroups(IsSimple,true Size,42305421312000);
(This will run a while as it has to read in the whole library to
check. However you only need to do this once.)
The second aspect is working with finitely presented groups.
Often the first example of defining a group in an abstract algebra
course is de facto as a finitely presented group. Unfortunately this
belies a serious problem:
There can not be an algorithm (i.e. there is a proof that no such
algorithm can exist)
which would be able to test for an arbitrary finitely presented
group whether
it is trivial. In particular just testing the equality of words
in a finitely presented
group cannot be solved in general.
There are heuristics which can work in certain cases, but in other
situations these heuristics fail and end up using huge amounts of
memory and runtime. With this in mind, the following rules of thumb
are worth remembering:
- If you have a finite group for which you know the order or even the
isomorphism type don't represent it as a finitely presented group or
-- if you have no other possibility -- convert it into another
representation (say a permutation group as soon as possible.
- If you run any ``generic'' command for finitely presented groups,
GAP internally has to find a way to compare elements. This is either
a faithful permutation representation or a confluent rewriting
system. Both can be potentially very large.
- Even with such an element comparison in place almost all algorithms
used will be very naive, for example enumerating elements. Even just
testing whether the group is finite can be unsurmountable.
- GAP does NOT automatically translate all complicated commands to
permutation groups, because such a permutation representation could
potentially be large and doing so would interfere with some
calculations for huge finitely presented groups. You have to form the
permutation representation by hand.
- If you want to get results as words in generators, still work in a
permutation group, but use the permutation representation to pull the
end result back into a finitely presented (or a free) group.
- It can sometimes be worth to preprocess a finitely presented group
using `SimplifiedFpGroup' or `IsomorphismSimplifiedFpGroup' to
eliminate redundant generators.
- If you know a priori the order of a group (of whatever form) you
can often help GAP by telling it the order:
SetSize(G,known_size);
So how could we go about finding a faithful permutation
representation. The most naive way is `IsomorphismPermGroup' but this
will by default for FP groups use via the representation on the
cosets of a cyclic subgroup. (The reason for this is that in general
such groups are not simple and we need to ensure that we get a
faithful representation.)
This is out of the question for a group as large as Co2.
The next possibility is to use the permutation representation on the
cosets of a subgroup (or a set of subgroups). Here we are in luck, as
CO2 is simple and any nontrivial representation will be faithful. So
we only need to find a subgroup.
To find subgroups one could use quotient algorithms and take
preimages of subgroups of quotients, or one could use the low index
algorithm to find subgroups of small (say 20-50 at most) index.
In our case alas the group is perfect and the smallest subgroup index
is 2300. So neither method will succeed. Thus we are left with a
slightly desperate attempt:
Construct subgroups from generators. (In fact, given the particular
presentation used in the example this is not as desperate as it
sounds. For these Coxeter-type presentations often subgroups
generated by some generators are a good try.
Indeed for the group co2 you defined in your mail, the following
yields a subgroup of reasonable finite index:
gap> s:=Subgroup(co2,GeneratorsOfGroup(co2){[1,2,3,4,5,6]});
Group([ a, b, c, d, e, f ])
gap> Index(co2,s);
47104
Now we can take the permutation action on the cosets to get a
permutation representation:
gap> act:=FactorCosetAction(co2,s);;
gap> p:=Image(act);
gap> IsPrimitive(p);
true
Alas this is primitive, so we don't have an easy way of reducing the
degree of the permutation action via blocks. (In fact it is rather
hard to construct the representation of degree 2300 from this, which
makes the first approach much more promising.)
(As Dima Pasechnik wrote, there are other representations in which
one can easily find generators for the subgroup of index 2300, which
would give us the degree 2300 representation.)
Best wishes,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke@math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
From l.h.soicher at qmul.ac.uk Mon Apr 10 11:47:40 2006
From: l.h.soicher at qmul.ac.uk (Leonard Soicher)
Date: Mon Apr 10 11:48:14 2006
Subject: [GAP Forum] Conway Co2 group
In-Reply-To: <2B1690C3-7795-4B0E-BC44-63F3AF35F3B9@frii.com>
References:
<2B1690C3-7795-4B0E-BC44-63F3AF35F3B9@frii.com>
Message-ID: <20060410104740.GA24163@mrcpc02.maths.qmul.ac.uk>
Dear GAP-Forum,
On Sun, Apr 09, 2006 at 08:45:54PM -0600, Alexander Hulpke wrote:
[...]
> Alas this is primitive, so we don't have an easy way of reducing the
> degree of the permutation action via blocks. (In fact it is rather
> hard to construct the representation of degree 2300 from this, which
> makes the first approach much more promising.)
>
It is worth pointing out that Praeger and Soicher's book, "Low Rank
Representations and Graphs for Sporadic Groups" (CUP, 1997), contains
presentations for many sporadic groups (often presentations from
the ATLAS), together with sets of words generating subgroups of low
(permutation) rank. The given presentations and subgroup generators
seem to be "coset enumeration friendly" and can be used to construct many
low-rank representations of sporadic groups. In particular, on page 106,
the ATLAS presentation for Co2 is given, together with sets of words (in
the generators of the given presentation) generating various subgroups,
including U6(2):2 of index 2300.
Best wishes,
Leonard Soicher
From alexander.konovalov at gmail.com Mon Apr 10 16:57:07 2006
From: alexander.konovalov at gmail.com (Alexander Konovalov)
Date: Mon Apr 10 16:57:11 2006
Subject: [GAP Forum] Matrix Operations and saving cpu cycles
In-Reply-To:
References:
Message-ID: <91E7580E-390A-4561-8C79-F4610CCEFF4A@gmail.com>
Dear Ariel,
You can find information about matrices in the GAP Documentation
available online, in particular, in the GAP Tutorial :
http://www.gap-system.org/Manuals/doc/htm/tut/CHAP003.htm#SECT008
and in the GAP Reference Manual :
http://www.gap-system.org/Manuals/doc/htm/ref/CHAP024.htm
Sum and product of matrices are standard operations in GAP.
If you have matrices m1 and m2, type m1+m2, m1*m2 to get the result:
gap> m1:=[[1,2],[3,-1]];
[ [ 1, 2 ], [ 3, -1 ] ]
gap> m2:=[[-1,2],[4,0]];
[ [ -1, 2 ], [ 4, 0 ] ]
gap> m1+m2;
[ [ 0, 4 ], [ 7, -1 ] ]
gap> m1*m2;
[ [ 7, 2 ], [ -7, 6 ] ]
GAP does not have standard functions to swap rows or columns,
but these operations can be easily described in GAP. Please
let me know if you need more help to implement them.
Let me also add the GAP homepage has also its own search tools
available on the page http://www.gap-system.org/search.html.
Besides this, after you installed GAP, the documentation is
available locally on your computer, and you can use the help system
to find necessary information, in particular, invoking the help from
the GAP session as it is explained on the page
http://www.gap-system.org/Manuals/doc/htm/ref/CHAP002.htm
or view it with HTML browser (starting pages is gap4r4/doc/htm/
index.htm).
Best wishes,
Alexander Konovalov
>> Im looking for a way to do simple matrix operations,
>> specifically,
>> swap one row with another &
>> swap one column with another,
>> also sums and product.
>>
>> I do not want to create identity matrices and make de
>> product as this will involve much more CPU resources (i think),
>> im doing this for a bigger proyect and want to save cpu cycles.
>>
>> Googling the gap system site didn't help much, i cant find an
>> easy way to do this. Shall i code my own rutine?
>>
>> Thanks very much
>>
>> Ariel
From alexk at mcs.st-and.ac.uk Mon Apr 10 16:58:11 2006
From: alexk at mcs.st-and.ac.uk (Alexander Konovalov)
Date: Mon Apr 10 16:58:02 2006
Subject: [GAP Forum] RPM package
In-Reply-To: <20060407185328.GY19808@txc.com>
References: <20060407185328.GY19808@txc.com>
Message-ID: <6AFB82F0-39B0-4547-8BDC-DF4292129BAA@mcs.st-and.ac.uk>
Dear Igor,
to my best knowledge, very long time ago Rex Dieter produced gap.spec
for Red Hat, and it seems no longer supported.
But then it was used in AltLinux distribution (http://www.altlinux.com/)
that contains a package (in a Linux sense) gap-4.4-alt2 by Andrey
Brindeew
(dated 23 May 2004): see
http://www.altlinux.com/index.php?module=sisyphus&package=gap
You can find gap.spec in gap-4.4-alt2.src.rpm therefore:
ftp://ftp.altlinux.ru/pub/distributions/ALTLinux/Sisyphus/
SRPMS.classic/gap-4.4-alt2.src.rpm
I have somewhere the previous version of gap.spec from this RPM,
so I can send it to you if you prefer not to download the full
src.rpm package
Please note the following:
1) AltLinux have some specific requirements for spec-files, described
somewhere in the
documentation for AltLinux developers on the AltLinux homepage
2) Looking on the spec file, it is easy to see some possible
improvements and necessary updates
It would be interesting to know, if you will improve this spec file
and will use it.
Sincerely yours,
Alexander Konovalov
On 07 Apr 2006, at 20:53, Igor Schein wrote:
> Dear Forum,
>
> does anyone (un)officially maintain gap.spec for building RPMs of
> latest GAP releases?
>
> Thanks
>
> Igor
From laurent.bartholdi at gmail.com Mon Apr 10 17:24:24 2006
From: laurent.bartholdi at gmail.com (Laurent Bartholdi)
Date: Mon Apr 10 17:24:35 2006
Subject: [GAP Forum] Matrix Operations and saving cpu cycles
In-Reply-To: <91E7580E-390A-4561-8C79-F4610CCEFF4A@gmail.com>
References:
<91E7580E-390A-4561-8C79-F4610CCEFF4A@gmail.com>
Message-ID: <1ff637850604100924i7b08f388l1a20cba16c0fb03a@mail.gmail.com>
Dear Alexander, Dear Forum,
> GAP does not have standard functions to swap rows or columns,
> but these operations can be easily described in GAP. Please
> let me know if you need more help to implement them.
In fact, GAP does have such commands -- even though they're not
documented as such. See Permuted() and PermutedCols()
--
Laurent Bartholdi \ laurent.bartholdigmailcom
EPFL SB SMA IMB MAD \ T?l?phone: +41 21-6935458
Station 8 \ Secr?taire: +41 21-6935501
CH-1015 Lausanne, Switzerland \ Fax: +41 21-6930339
From Rudolf.Zlabinger at chello.at Mon Apr 10 18:53:55 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Mon Apr 10 18:46:56 2006
Subject: [GAP Forum] this is a technical test only
References:
<91E7580E-390A-4561-8C79-F4610CCEFF4A@gmail.com>
Message-ID: <004f01c65cc7$bc56f2e0$eec57254@chello.at>
please ignore this message, as it is a technical test for forum software and
Outlook Express, ys, Rudolf Zlabinger
From Rudolf.Zlabinger at chello.at Mon Apr 10 20:09:24 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Mon Apr 10 20:02:26 2006
Subject: [GAP Forum] Co2 group; thank you for your valuable input
Message-ID: <00df01c65cd2$48171e40$eec57254@chello.at>
Thank you all for the numerous contributions concerning the Conway 2 group,
i have now stuff to evaluate for about 1 month. First i have to get work
Atlasrep, as this seams not to work under Windows without according effort,
but I hope, this should not stop me in going forward in my studies in group
theory.
Thank you all, best wishes, Rudolf Zlabinger
From Rudolf.Zlabinger at chello.at Tue Apr 11 09:34:52 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Tue Apr 11 09:27:54 2006
Subject: [GAP Forum] ATLASrep does work under Windows
Message-ID: <000f01c65d42$ce314d00$eec57254@chello.at>
Dear Thomas Breuer,
after investigating in various sources, i succeeded to setup ATLASrep using
wget under Windows.
As this information may be useful for other Windows users, i think, it would
be a good idea to publish that procedure
at a suiting central place.
Following was to do:
1) Get the wget.exe from http://www.interlog.com/~tcharron/wgetwin.html
2) Investigate the Directories for GAP system programs:
gap> DirectoriesSystemPrograms( )
giving im my case: cygdrive/c/windows/command for example, for
Windows users that means:
3) put the executable wget.exe in the folder c:/Windows/command (for
example). After that wget.exe is known
to ATLASrep.
4) after loading the package "atlasrep" set the component
AtlasOfGroupRepresentationsInfo.wget to true in order to
disable the default to pearl.
After doing these steps Atlasrep works, as I tested getting
AtlasGenerators("A5",1), for example
Thank you for your advice in getting the wget exe, yours sincerely, Rudolf
Zlabinger
From Rudolf.Zlabinger at chello.at Tue Apr 11 13:18:06 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Tue Apr 11 13:11:05 2006
Subject: Fw: [GAP Forum] this is a technical test only
Message-ID: <001101c65d61$fd7434a0$eec57254@chello.at>
----- Original Message -----
From: "Rudolf Zlabinger"
To: "John McDermott"
Cc: "Alexander Konovalov" ; "Thomas Breuer"
Sent: Tuesday, April 11, 2006 2:15 PM
Subject: Re: [GAP Forum] this is a technical test only
Dear John,
thank you for feedback. I seem presently to be the only active Windows user
in the forum, because nobody seems to know, what happens with forum against
Windows (only Alexander Konovalov told me, that he used Outlook Express a
time ago), and also other things, as for example ATLASrep.
Outlook Express has to be settled down to a minimal option set, causing a
format without anything except plaintext. It was a piece of hard work to
guess that option set, I only came to an end as Alexander adviced me to
compare the output to a message, that was an answer to an forum message,
thus having the required format. That was this test message. But to relate
that output format to the needed optionset was a merely puzzling thing.
ATLASrep on the other hand has nothing explicit in his docs as to install
for example even wget.exe, and it was some effort to do this myself. So I
collect experience over the time in dealing with GAP and forum under
Windows.
If you want, I will be available in the future to share my experience to
other Windows users, as needed, and also for some tests under Windows for
other parts of GAP, should the need arise (to execute some testfiles under
Windows for example).
thank you, yours sincerely, Rudolf Zlabinger
----- Original Message -----
From: "John McDermott"
To: "Rudolf Zlabinger"
Cc: "Alexander Hulpke"
Sent: Tuesday, April 11, 2006 11:26 AM
Subject: Fwd: [GAP Forum] this is a technical test only
Dear Rudolf,
Your message appeared - and looked normal - on the Forum. If you
receive another message saying that your mail was empty (after
filtering) or any other error, please let us know.
John.
Begin forwarded message:
> From: "Rudolf Zlabinger"
> Date: 10 April 2006 18:53:55 BDT
> To: "Alexander Konovalov"
> Cc: forum@gap-system.org
> Subject: Re: [GAP Forum] this is a technical test only
>
> please ignore this message, as it is a technical test for forum
> software and
> Outlook Express, ys, Rudolf Zlabinger
>
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
From gzabl at yahoo.com Tue Apr 11 14:21:03 2006
From: gzabl at yahoo.com (Gary Zablackis)
Date: Tue Apr 11 14:21:14 2006
Subject: Fw: [GAP Forum] this is a technical test only
In-Reply-To: <001101c65d61$fd7434a0$eec57254@chello.at>
Message-ID: <20060411132103.57807.qmail@web53009.mail.yahoo.com>
--- Rudolf Zlabinger
wrote:
> Dear John,
>
> thank you for feedback. I seem presently to be the
> only active Windows user
> in the forum, because nobody seems to know, what
> happens with forum against
> Windows (only Alexander Konovalov told me, that he
> used Outlook Express a
> time ago), and also other things, as for example
> ATLASrep.
>
Rudolf,
I also am an active Windows user of GAP. I have been
able to run all of the GAP packages except for Carat
and Edim. If you have any questions that I can help
you with, please feel free to ask.
Gary Zablackis
__________________________________________________
Do You Yahoo!?
Tired of spam? Yahoo! Mail has the best spam protection around
http://mail.yahoo.com
From joerongen at sprint.ca Tue Apr 11 16:19:39 2006
From: joerongen at sprint.ca (Joe Rongen)
Date: Wed Apr 12 09:55:40 2006
Subject: [GAP Forum] this is a technical test only
References: <001101c65d61$fd7434a0$eec57254@chello.at>
Message-ID: <001901c65d7b$7c7d8800$4f58fea9@joerongen>
----- Original Message -----
From: "Rudolf Zlabinger"
Sent: Tuesday, April 11, 2006 8:18 AM
...
> Outlook Express has to be settled down to a minimal option set,
> causing a format without anything except plaintext. It was a piece
> of hard work to guess that option set, I only came to an end as
> Alexander adviced me to compare the output to a message, that
> was an answer to an forum message, thus having the required format.
> That was this test message. But to relate that output format to the
> needed optionset was a merely puzzling thing.
In Outlook Express one simply clicks on "Format" or use
the keyboard shortcut: Alt + O. Next, select "Plain Text"
and that's it! Please note, Windows has lots of information-
help files, available by clicking: "Start" and "Help."
Regards Joe
-------
"Life is a funny school, where the tests are
given before the material can be studied."
From alexander.konovalov at gmail.com Wed Apr 12 12:56:17 2006
From: alexander.konovalov at gmail.com (Alexander Konovalov)
Date: Wed Apr 12 12:56:03 2006
Subject: [GAP Forum] Matrix Operations and saving cpu cycles
In-Reply-To: <1ff637850604100924i7b08f388l1a20cba16c0fb03a@mail.gmail.com>
References:
<91E7580E-390A-4561-8C79-F4610CCEFF4A@gmail.com>
<1ff637850604100924i7b08f388l1a20cba16c0fb03a@mail.gmail.com>
Message-ID: <65E7AB71-9F56-49D6-B167-3D6AC6E44851@gmail.com>
Dear Laurent,
thanks for your remark. Let me just add a few comments on it:
1) Dependently on the size of matrices and maybe other factors,
another implementation could be more CPU time saving. For
example, the following function swaps i-th and j-th row of a
matrix mat and returns the resulting matrix m:
Swap:=function(mat,i,j)
local m, t;
m := ShallowCopy(mat);
t := m[i];
m[i] := m[j];
m[j] := t;
return m;
end;
gap> e := RandomMat(1000,1000);;
gap> for i in [1..10000] do x:=Permuted(e,(1,2)); od; time;
4534
gap> for i in [1..10000] do x:=Swap(e,1,2); od; time;
211
And you can save even more time, if you can use the destructive
counterpart of this Swap function, changing the argument mat:
SwapDestructive:=function(m,i,j)
local t;
t := m[i];
m[i] := m[j];
m[j] := t;
end;
gap> for i in [1..10000] do SwapDestructive(e,1,2); od; time;
20
2) The function PermutedCols actually belongs to the GUAVA
package, see http://www.gap-system.org/Manuals/pkg/guava/htm/
chap7.html#s3ss8.
So the GUAVA package should be loaded first before calling PermutedCols
Best wishes,
Alexander
On 10 Apr 2006, at 18:24, Laurent Bartholdi wrote:
> Dear Alexander, Dear Forum,
>
>> GAP does not have standard functions to swap rows or columns,
>> but these operations can be easily described in GAP. Please
>> let me know if you need more help to implement them.
>
> In fact, GAP does have such commands -- even though they're not
> documented as such. See Permuted() and PermutedCols()
>
> --
> Laurent Bartholdi \ laurent.bartholdigmailcom
> EPFL SB SMA IMB MAD \ T?l?phone: +41 21-6935458
> Station 8 \ Secr?taire: +41 21-6935501
> CH-1015 Lausanne, Switzerland \ Fax: +41 21-6930339
From akshriv at gmail.com Thu Apr 13 12:27:03 2006
From: akshriv at gmail.com (Abhishek)
Date: Thu Apr 13 12:27:25 2006
Subject: [GAP Forum] orbits of a set under induced group action
Message-ID:
Dear GAP forum,
I need to solve the following problem.
Let X be a set and G a group action on X. Let Y={0,1}.
Consider the collection Y^X of all functions f:X --> Y.
I need to find the orbits of Y^X induced by G. (The image of any function f
in Y^X under a \in G is given by (a*f)(x)=f(a*x), for each x in X)
I don't know if this can be solved using GAP since I have little background
in Group Theory. I'll be glad if someone could point to some specific
topic(s) in the GAP Manuals or beyond.
Thank you,
Abhishek
From laurent.bartholdi at gmail.com Thu Apr 13 13:43:55 2006
From: laurent.bartholdi at gmail.com (Laurent Bartholdi)
Date: Thu Apr 13 13:44:10 2006
Subject: [GAP Forum] orbits of a set under induced group action
In-Reply-To:
References:
Message-ID: <1ff637850604130543y5393f63dhc00f6a6dd5dfeefe@mail.gmail.com>
Hi Abhishek,
Use the command Orbits().
Functions X -> Y are (characteristic functions of) subsets of X. The orbits
split according to size.
For example, if G=Alt(4) acting on 4 points, try
gap> Orbits(AlternatingGroup(4),Combinations([1..4]),OnSets);
[ [ [ ] ], [ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ],
[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 4 ], [ 1, 4 ], [ 2, 4 ] ],
[ [ 1, 2, 3 ], [ 1, 3, 4 ], [ 1, 2, 4 ], [ 2, 3, 4 ] ], [ [ 1, 2, 3, 4 ] ] ]
which tells you that there are five orbits, of sizes 1,4,6,4,1; for D_8, you get
gap> Orbits(DihedralGroup(IsPermGroup,8),Combinations([1..4]),OnSets);
[ [ [ ] ], [ [ 1 ], [ 2 ], [ 3 ], [ 4 ] ], [ [ 1, 2 ], [ 2, 3 ], [ 1,
4 ], [ 3, 4 ] ],
[ [ 1, 2, 3 ], [ 2, 3, 4 ], [ 1, 3, 4 ], [ 1, 2, 4 ] ], [ [ 1, 2, 3, 4 ] ],
[ [ 1, 3 ], [ 2, 4 ] ] ]
whence 6 orbits of size 1,4,4,2,4,1.
On 4/13/06, Abhishek wrote:
> Dear GAP forum,
>
> I need to solve the following problem.
>
> Let X be a set and G a group action on X. Let Y={0,1}.
> Consider the collection Y^X of all functions f:X --> Y.
> I need to find the orbits of Y^X induced by G. (The image of any function f
> in Y^X under a \in G is given by (a*f)(x)=f(a*x), for each x in X)
>
> I don't know if this can be solved using GAP since I have little background
> in Group Theory. I'll be glad if someone could point to some specific
> topic(s) in the GAP Manuals or beyond.
>
> Thank you,
> Abhishek
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
--
Laurent Bartholdi \ laurent.bartholdigmailcom
EPFL SB SMA IMB MAD \ T?l?phone: +41 21-6935458
Station 8 \ Secr?taire: +41 21-6935501
CH-1015 Lausanne, Switzerland \ Fax: +41 21-6930339
From reichard at maths.uwa.edu.au Thu Apr 13 15:38:29 2006
From: reichard at maths.uwa.edu.au (reichard@maths.uwa.edu.au)
Date: Thu Apr 13 15:39:03 2006
Subject: [GAP Forum] orbits of a set under induced group action
In-Reply-To: <1ff637850604130543y5393f63dhc00f6a6dd5dfeefe@mail.gmail.com>
References:
<1ff637850604130543y5393f63dhc00f6a6dd5dfeefe@mail.gmail.com>
Message-ID: <58207.203.59.124.92.1144939109.squirrel@203.59.124.92>
Hi Abhishek, hi Laurent,
the approach described by Laurent works fine if X is reasonably small, say
up to 20 elements. For larger X (and hopefully a larger group) you can
construct representatives of the orbits without storing all sets
simultaneously, e.g., by orderly generation[1]. If you are interested I
can make available some code which performs this task.
Regards,
Sven.
[1] READ, R.C. Every-one a winner. Ann. Discr. Math., 1978, 2, 107--120.
--
Sven Reichard
School of Mathematics and Statistics
University of Western Australia
From mckay at encs.concordia.ca Thu Apr 13 16:25:47 2006
From: mckay at encs.concordia.ca (MCKAY john)
Date: Thu Apr 13 16:25:58 2006
Subject: [GAP Forum] orbits of a set under induced group action
In-Reply-To: <58207.203.59.124.92.1144939109.squirrel@203.59.124.92>
References:
<1ff637850604130543y5393f63dhc00f6a6dd5dfeefe@mail.gmail.com>
<58207.203.59.124.92.1144939109.squirrel@203.59.124.92>
Message-ID:
This is I think a standard problem. There is code in
Collected algorithms of ACM in ALGOL going back to the
60's. Try authors Regener, Soicher if my mempory serves me
right.
John
On Thu, 13 Apr 2006 reichard@maths.uwa.edu.au wrote:
> Hi Abhishek, hi Laurent,
>
> the approach described by Laurent works fine if X is reasonably small, say
> up to 20 elements. For larger X (and hopefully a larger group) you can
> construct representatives of the orbits without storing all sets
> simultaneously, e.g., by orderly generation[1]. If you are interested I
> can make available some code which performs this task.
>
> Regards,
> Sven.
>
> [1] READ, R.C. Every-one a winner. Ann. Discr. Math., 1978, 2, 107--120.
>
> --
> Sven Reichard
> School of Mathematics and Statistics
> University of Western Australia
>
>
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
From akshriv at gmail.com Fri Apr 14 15:43:48 2006
From: akshriv at gmail.com (Abhishek)
Date: Fri Apr 14 15:43:58 2006
Subject: [GAP Forum] orbits of a set under induced group action
In-Reply-To:
References:
<1ff637850604130543y5393f63dhc00f6a6dd5dfeefe@mail.gmail.com>
<58207.203.59.124.92.1144939109.squirrel@203.59.124.92>
Message-ID:
Hi John, Sven and Laurent,
Laurent, thanks for pointing out that f's are characteristic functions of X.
I can now solve small problems. But as Sven said I am running into memory
problems when trying to solve large problems.
Sven, thanks for the reference. I am trying to get a copy of it. Since my
library doesn't have a copy of it, it's going to be a few days before I can
get it. I'll probably ask you for your code after I get the paper. Also,
from your comment, I'm guessing that GAP doesn't have any such algorithm
implemented that just finds a representative in each orbit. Did I get this
right?
John, thanks for pointer but unfortunately I have not been able to locate
it. Can you possibly give me a more specific reference?
Regards,
Abhishek
On 4/13/06, MCKAY john wrote:
>
>
> This is I think a standard problem. There is code in
> Collected algorithms of ACM in ALGOL going back to the
> 60's. Try authors Regener, Soicher if my mempory serves me
> right.
>
> John
>
>
> On Thu, 13 Apr 2006 reichard@maths.uwa.edu.au wrote:
>
> > Hi Abhishek, hi Laurent,
> >
> > the approach described by Laurent works fine if X is reasonably small,
> say
> > up to 20 elements. For larger X (and hopefully a larger group) you can
> > construct representatives of the orbits without storing all sets
> > simultaneously, e.g., by orderly generation[1]. If you are interested I
> > can make available some code which performs this task.
> >
> > Regards,
> > Sven.
> >
> > [1] READ, R.C. Every-one a winner. Ann. Discr. Math., 1978, 2, 107--120.
> >
> > --
> > Sven Reichard
> > School of Mathematics and Statistics
> > University of Western Australia
> >
> >
> >
> > _______________________________________________
> > Forum mailing list
> > Forum@mail.gap-system.org
> > http://mail.gap-system.org/mailman/listinfo/forum
> >
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
--
Web: http://people.tamu.edu/~akshriv/
"One has to look out for engineers - they begin with sewing machines and end
up with the atomic bomb." --Marcel Pagnol
From Rudolf.Zlabinger at chello.at Fri Apr 14 19:02:50 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Fri Apr 14 18:55:53 2006
Subject: [GAP Forum] Conway Co2 group
References:
<2B1690C3-7795-4B0E-BC44-63F3AF35F3B9@frii.com>
Message-ID: <000c01c65fed$a51f0800$eec57254@chello.at>
Dear Alexander Hulpke,
Excuse the delay of my answer caused by an accident with my ry right eye, I
was nearly blind for 3 days, so I could use my computer first time again
yesterday.
Thank you for this "crash course" in handling groups of higher complexity.
The information contained is very dense and I will evaluate it as integral
part of my studies of group theory step by step as I proceed, my present
state is introductionary, but my target is to get enough qualification to
handle the full extent of informations contained in ATLAS.
My longterm target is application of group theory, and, more general, the
further application of finite and discrete mathematics in basic physics. A
first approximation will be the automorphisms of graphs. I know, there is a
long way to go for me, but I think the effort will be worthwile. GAP will be
a valuable help to me for following my target.
Thank you for your valuable contribution to my efforts, best wishes, Rudolf
Zlabinger
----- Original Message -----
From: "Alexander Hulpke"
To: "GAP Forum" Dear Alexander
Sent: Monday, April 10, 2006 4:45 AM
Subject: Re: [GAP Forum] Conway Co2 group
Dear GAP Forum,
On Apr 6, 2006, at 9:29 , Rudolf Zlabinger wrote:
> I try to handle the Conway Co2 group in GAP. I formed a Fpgroup
> with the relators as given in the ATLAS as follows:
As this question concerns two aspects that come up again and again,
allow me a more general answer:
The first aspect is on how to get a particular simple (or close to
simple) group in GAP.
The most ``universal'' source is -- as already mentioned by several
contributors -- probably the ATLAS web pages maintained by Rob Wilson
and collaborators which provides a plethora of representations for
such groups. In particular, there is the AtlasRep package which
provides a very nice interface.
By searching through the forum archive you will find several mails
describing this package in more detail.
A second way to get these groups is by the fact that they often have
a faithful primitive representation of small degree. (For Example Co2
has a representation of degree 2300.)
Thus many of these groups can be found in the primitive groups
library. Even if you don't know the degree, but the order you can
search for it:
AllPrimitiveGroups(IsSimple,true Size,42305421312000);
(This will run a while as it has to read in the whole library to
check. However you only need to do this once.)
The second aspect is working with finitely presented groups.
Often the first example of defining a group in an abstract algebra
course is de facto as a finitely presented group. Unfortunately this
belies a serious problem:
There can not be an algorithm (i.e. there is a proof that no such
algorithm can exist)
which would be able to test for an arbitrary finitely presented
group whether
it is trivial. In particular just testing the equality of words
in a finitely presented
group cannot be solved in general.
There are heuristics which can work in certain cases, but in other
situations these heuristics fail and end up using huge amounts of
memory and runtime. With this in mind, the following rules of thumb
are worth remembering:
- If you have a finite group for which you know the order or even the
isomorphism type don't represent it as a finitely presented group or
-- if you have no other possibility -- convert it into another
representation (say a permutation group as soon as possible.
- If you run any ``generic'' command for finitely presented groups,
GAP internally has to find a way to compare elements. This is either
a faithful permutation representation or a confluent rewriting
system. Both can be potentially very large.
- Even with such an element comparison in place almost all algorithms
used will be very naive, for example enumerating elements. Even just
testing whether the group is finite can be unsurmountable.
- GAP does NOT automatically translate all complicated commands to
permutation groups, because such a permutation representation could
potentially be large and doing so would interfere with some
calculations for huge finitely presented groups. You have to form the
permutation representation by hand.
- If you want to get results as words in generators, still work in a
permutation group, but use the permutation representation to pull the
end result back into a finitely presented (or a free) group.
- It can sometimes be worth to preprocess a finitely presented group
using `SimplifiedFpGroup' or `IsomorphismSimplifiedFpGroup' to
eliminate redundant generators.
- If you know a priori the order of a group (of whatever form) you
can often help GAP by telling it the order:
SetSize(G,known_size);
So how could we go about finding a faithful permutation
representation. The most naive way is `IsomorphismPermGroup' but this
will by default for FP groups use via the representation on the
cosets of a cyclic subgroup. (The reason for this is that in general
such groups are not simple and we need to ensure that we get a
faithful representation.)
This is out of the question for a group as large as Co2.
The next possibility is to use the permutation representation on the
cosets of a subgroup (or a set of subgroups). Here we are in luck, as
CO2 is simple and any nontrivial representation will be faithful. So
we only need to find a subgroup.
To find subgroups one could use quotient algorithms and take
preimages of subgroups of quotients, or one could use the low index
algorithm to find subgroups of small (say 20-50 at most) index.
In our case alas the group is perfect and the smallest subgroup index
is 2300. So neither method will succeed. Thus we are left with a
slightly desperate attempt:
Construct subgroups from generators. (In fact, given the particular
presentation used in the example this is not as desperate as it
sounds. For these Coxeter-type presentations often subgroups
generated by some generators are a good try.
Indeed for the group co2 you defined in your mail, the following
yields a subgroup of reasonable finite index:
gap> s:=Subgroup(co2,GeneratorsOfGroup(co2){[1,2,3,4,5,6]});
Group([ a, b, c, d, e, f ])
gap> Index(co2,s);
47104
Now we can take the permutation action on the cosets to get a
permutation representation:
gap> act:=FactorCosetAction(co2,s);;
gap> p:=Image(act);
gap> IsPrimitive(p);
true
Alas this is primitive, so we don't have an easy way of reducing the
degree of the permutation action via blocks. (In fact it is rather
hard to construct the representation of degree 2300 from this, which
makes the first approach much more promising.)
(As Dima Pasechnik wrote, there are other representations in which
one can easily find generators for the subgroup of index 2300, which
would give us the degree 2300 representation.)
Best wishes,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke@math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
_______________________________________________
Forum mailing list
Forum@mail.gap-system.org
http://mail.gap-system.org/mailman/listinfo/forum
From sh_fouladi at yahoo.com Sat Apr 15 10:41:04 2006
From: sh_fouladi at yahoo.com (shirin fouladi)
Date: Sat Apr 15 10:42:11 2006
Subject: [GAP Forum] a question about central product
Message-ID: <20060415094105.27402.qmail@web53311.mail.yahoo.com>
Dear Gap Forum.
Is there any way, to find the central product of two arbitrary groups in Gap.
Best Regards.
Shirin fouladi
---------------------------------
Love cheap thrills? Enjoy PC-to-Phone calls to 30+ countries for just 2?/min with Yahoo! Messenger with Voice.
From Rudolf.Zlabinger at chello.at Mon Apr 17 13:35:55 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Mon Apr 17 13:28:44 2006
Subject: [GAP Forum] Conway Co2 group (fwd)
References:
Message-ID: <000801c6621b$789f9080$eec57254@chello.at>
Dear Beth,
although this is outside the range of GAP Forum, here my basic ideas, yet in
an very intuitive and initial state, this is only a first possible and
speculative approximation:
Assume, that physical space is finite and consists of elements connected by
some kind of neighbourhood and thus can be modelled by finite graphs. Then
all possible physical structures can be modelled by graphs, whereas flat
space is a special structure only and physical objects, as particles,
emerging from flat space by their special structure, thus physical objects
and space are somewhat the same: graphs.
The symmetries of space and physical objects then would be given by groups,
and, as beeing finite, by finite groups..... Symmetries could be to some
extent "structure stabilizers" over the time and thus they could be the
primary constituent for objects, that could survive long enough to gain
physical "existence", the emergence I mentioned above. In such a way groups
could describe and classify emergent physical objects, for example the
stabile particles known, and not yet known. (A "periodical system" of
particles?)
A thrilling example: Consider a full graph, where all vertices are fully
connected to each other. Lets assume, that lenght is a measure on the
minimal pathlengths between the vertices. Thus the object described by such
a graph would be a region of space of minimal diameter and maximal densitiy
and maximal symmetry and thus maximal stability, properties, that are, for
example, described for a black hole.....
A model like this could describe any structure, that, at the present, cannot
be handled by models used as yet....
I am fascinated by this idea, and, in order to develope it, I am now first
trying to get enough qualification in discrete mathematics. The next step
would be to gain some ideas and informations to "dynamic structures" ("time"
series of discrete structures).
thank you for your interest, best wishes, Rudolf Zlabinger
----- Original Message -----
From: "Beth Holmes"
To:
Cc: "Steve Linton"
Sent: Sunday, April 16, 2006 3:52 PM
Subject: Re: [GAP Forum] Conway Co2 group (fwd)
Rudolf,
Hi. I was interested to see that you want to apply group theory to
physics. My own experience is mostly in computing with ATLAS groups, but
I am currently seeking physics problems that I can apply computational
algebra to. If there is anything that you need any help with that wasn't
in Alexander's email, or any computations that GAP isn't sufficient for,
please get in touch as I am really keen to get involved with that sort of
thing.
Yours,
Beth Holmes
---------- Forwarded message ----------
Date: Fri, 14 Apr 2006 20:02:50 +0200
From: Rudolf Zlabinger
To: Alexander Hulpke
Cc: forum@gap-system.org
Subject: Re: [GAP Forum] Conway Co2 group
My longterm target is application of group theory, and, more general, the
further application of finite and discrete mathematics in basic physics. A
first approximation will be the automorphisms of graphs. I know, there is a
long way to go for me, but I think the effort will be worthwile. GAP will be
a valuable help to me for following my target.
From r_n_tsai at yahoo.com Wed Apr 19 01:11:59 2006
From: r_n_tsai at yahoo.com (R.N. Tsai)
Date: Wed Apr 19 01:15:46 2006
Subject: [GAP Forum] Casimir Invariants of Lie Algebras
Message-ID: <20060419001159.56876.qmail@web33704.mail.mud.yahoo.com>
Dear gap-forum,
Is it possible to calculate Casimir invariants of Lie algebras?
Alternatively can you calculate the center of the the Universal Enveloping Algebra?
Here's an attempt that didn't work :
gap> L:= SimpleLieAlgebra( "B", 2, Rationals );;
gap> UL:= UniversalEnvelopingAlgebra( L );;
gap> Center(UL);
Error, this case will eventually be handled by the Vector Enumerator
which is not available yet called from
OperationAlgebraHomomorphism( A, [ [ Zero( A ) ] ], OnRight ) called from
IsomorphismMatrixFLMLOR( A ) called from
IsFiniteDimensional( A ) called from
( ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
I'm using GAP4 version 4.4.7 under WindowsXp (cygwin)
Thanks,
R.N.
---------------------------------
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From degraaf at science.unitn.it Wed Apr 19 16:29:10 2006
From: degraaf at science.unitn.it (Willem De Graaf)
Date: Wed Apr 19 16:30:01 2006
Subject: [GAP Forum] Casimir Invariants of Lie Algebras
In-Reply-To: <20060419001159.56876.qmail@web33704.mail.mud.yahoo.com>
References: <20060419001159.56876.qmail@web33704.mail.mud.yahoo.com>
Message-ID: <44465746.6020107@science.unitn.it>
Dear R.N. Tsai,
Unfortunately in GAP there is no algorithm implemented
to find the centre of a universal enveloping algebra of a
split semisimple Lie algebra. The only algorithm that
I know for this purpose constructs a set of linear equations
for the generators of the centre. However, for all but a few
small cases that is very inefficient, as the number of
equations quickly becomes huge.
Sorry for not being able to help more.
Best wishes,
Willem de Graaf
From ndroock1 at gmail.com Wed Apr 19 21:26:44 2006
From: ndroock1 at gmail.com (Nilo de Roock)
Date: Wed Apr 19 21:26:56 2006
Subject: [GAP Forum] Request for advice
Message-ID:
Hi GAP forum members,
I appreciate any advice, hints, ideas concerning the following...
A bit of background: I consider myself an abstract algebra self-student,
that is: i have learned everything I know through self-study. That would not
have been possible without Mathematica, and more recently GAP. My career,
work, is in IT consultancy. Mathematics however, is in my
heart, I am sure you'll understand.
Although there is already an Abstract Algebra package for Mathematica
available, it is limited and no longer maintained. (
http://www.central.edu/eaam/index.asp ) It is my goal to develop a Group
Theory package for Mathematica. I'll make it a project of 1, maybe 2 years.
This project is ( nothing more than ) an instrument to bring my
understanding of Group theory to a deeper level. Sofare about my plans.
My ambitions are the following. ( Initially ) I do not want to invent
algorithms, I want to study them and implement them in the Mathematica
programming language. Since GAP is ( or very close to ) the centre of the
state-of-the-art of Computational Group Theory / Algebra, I intend to use
the GAP algorithms, if at all possible. If I understand it correct the
knowledge in GAP is free to use as long as it is referenced. As far as
functionality is concerned I aim for what is available in the nice Open
Source program Group Explorer 2.0 ( http://groupexplorer.sourceforge.net/ )
plus Character Tables, Table of Marks and the Cycle Graph of a group. More
than enough to do, I might say. Group Explorer is rather limited in the
groups it can handle: all groups of order =< 20, three of order 24 and A5. I
intend to be able to handle much more groups than that. The key to solving
that issue is imo using better, faster algorithms. Hence, GAP.
To the point. I am looking for ( the papers which contain ) the algorithms
on which GAP has been built, restricted to Group Theory, Representation
Theory, Vector Spaces and Matrices. What is really an issue in my situation
though, is that, since I don't work at a university, I do not have access to
the paid journals.
Before you may decide this message is outside the context of the GAP forum I
would like to add that it is my ultimate goal to add code to GAP itself and
publish about that, and the sooner the better. In order to achieve that I
must practice first.
Thanks for reading.
Kind regards,
nilo de roock
From niranj at math.ohio-state.edu Wed Apr 19 23:30:51 2006
From: niranj at math.ohio-state.edu (Niranjan Balachandran)
Date: Wed Apr 19 23:31:00 2006
Subject: [GAP Forum] Request for advice
In-Reply-To:
References:
Message-ID:
hi,
i think the book by Akos Seress titled Permutation group Algorithms is a
good source to start looking for things.
bye
niranjan
From joachim.neubueser at math.rwth-aachen.de Thu Apr 20 12:05:29 2006
From: joachim.neubueser at math.rwth-aachen.de (Joachim Neubueser)
Date: Thu Apr 20 12:05:55 2006
Subject: [GAP Forum] Request for advice
In-Reply-To:
References:
Message-ID: <20060420110529.GA8960@math.rwth-aachen.de>
Dear Nilo de Rook,
> To the point. I am looking for ( the papers which contain ) the algorithms
> on which GAP has been built, restricted to Group Theory, Representation
> Theory, Vector Spaces and Matrices.
The page
http://www.gap-system.org/Doc/references.html
of the GAP website gives you an overview of literature on
computational group theory. The book by Derek Holt is to be specially
recommended.
Kind regards Joachim Neubueser
From maasiru at yahoo.com Thu Apr 20 17:33:31 2006
From: maasiru at yahoo.com (muniru asiru)
Date: Thu Apr 20 17:34:04 2006
Subject: [GAP Forum] Counting number of involutions in finite simple groups.
Message-ID: <20060420163331.18621.qmail@web53313.mail.yahoo.com>
I will be grateful if you can offer some help on this
problem.
I have been counting the number of involutions in
finite simple groups and
I like to know whether or not it is possible to use
Gap to find examples of finite simple groups for which
"the number of involutions including the identity
element is a prime number (greater or equal to 5)"
or to relax the condition a little bit, can one use
Gap to find examples of finite simple groups for which
"the number of involutions including the identity
element is odd (greater or equal to 5)".
If G is a group, x in G is called an involution if
x^2=1, where 1 is the identity element in G.
__________________________________________________
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From jdixon at math.carleton.ca Thu Apr 20 17:54:45 2006
From: jdixon at math.carleton.ca (John Dixon)
Date: Thu Apr 20 21:42:49 2006
Subject: [GAP Forum] Counting number of involutions in finite simple
groups.
In-Reply-To: <20060420163331.18621.qmail@web53313.mail.yahoo.com>
References: <20060420163331.18621.qmail@web53313.mail.yahoo.com>
Message-ID: <4447BCD5.7090000@math.carleton.ca>
The condition you ask for will never happen. If G is a finite group of
order g and p^k is a prime power dividing g then a modified version of
the Sylow theorems shows that the number of subgroups of G of order p^k
is congruent to 1 mod p. In your case, the number of subgroups of order
2 is odd, and this implies that the number of involutions (including 1)
is even.
- John Dixon.
muniru asiru wrote:
>
> I will be grateful if you can offer some help on this
> problem.
>
> I have been counting the number of involutions in
> finite simple groups and
> I like to know whether or not it is possible to use
> Gap to find examples of finite simple groups for which
>
>
> "the number of involutions including the identity
> element is a prime number (greater or equal to 5)"
>
> or to relax the condition a little bit, can one use
> Gap to find examples of finite simple groups for which
>
> "the number of involutions including the identity
> element is odd (greater or equal to 5)".
>
> If G is a group, x in G is called an involution if
> x^2=1, where 1 is the identity element in G.
>
> __________________________________________________
> Do You Yahoo!?
> Tired of spam? Yahoo! Mail has the best spam protection around
> http://mail.yahoo.com
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
>
From marta31 at gmail.com Tue Apr 25 04:37:29 2006
From: marta31 at gmail.com (marta asaeda)
Date: Tue Apr 25 04:37:41 2006
Subject: [GAP Forum] galois group
Message-ID: <4dd3929a0604242037o344ed2cdka72d91feca152ee0@mail.gmail.com>
Hello
I am wondering if galois group computation using gap is useable as a
mathematical result. Is there any ambiguity ? Or, if I give a polynomial,
and it computes its galois group, is it mathematically certain that the
polynomial I gave was irreducible ? How does gap know that it is irreducible
?
marta
From mckay at encs.concordia.ca Tue Apr 25 12:57:25 2006
From: mckay at encs.concordia.ca (MCKAY john)
Date: Tue Apr 25 12:57:36 2006
Subject: [GAP Forum] galois group
In-Reply-To: <4dd3929a0604242037o344ed2cdka72d91feca152ee0@mail.gmail.com>
References: <4dd3929a0604242037o344ed2cdka72d91feca152ee0@mail.gmail.com>
Message-ID:
In my view it is unacceptable for a result to be given by computer
unless it is true. The nature of mathematics is that it is about
certainty.
Why may a result not fit these criteria? Galois computations may
be time consuming. It is often faster to compute with numerical
approximations than to work exactly. By themselves, these
approximations are inadequate and may lead to dubious results.
A good example of how to PROVE exactness from numerical approximations
comes from a note by Darmon and Ford
(Communications in Algebra,17, (1989) pp. 2941-2943 Computational
verification of M11 & M12 as Galois groups over Q.)
referred to at the start of the issue of J. Symb. Computing devoted
to computing Galois groups. An illuminating toy example is the cubic
resolvent of a 4-ic monic polynomial in which one needs to decide
for some root labelling if x1x2+x3x4 is a rational integer. How to prove
this from numerical approximations? A priori it is an algebraic integer.
p-adic methods will often provide the means to prove exactness.
Unfortunately some systems do not give the user essential information
needed to establish whether the method is exact - and its truth assured...
as an example, testing primality is often done with many quadratic residue
tests and n tests imply a probability of 1-1/2**n of correctness.
Finally all programs have a possibility of internal error - and there is
always a remote possibility of error elsewhere (do you recall the
"division error" in a chip years ago?)
If in doubt, examine the steps of the "proof" given by GAP.
GAP is one of the very best and, as far as know, GAP provides
provable results for Galois groups... however
CAVEAT EMPTOR!
John McKay
On Mon, 24 Apr 2006, marta asaeda wrote:
> Hello
>
> I am wondering if galois group computation using gap is useable as a
> mathematical result. Is there any ambiguity ? Or, if I give a polynomial,
> and it computes its galois group, is it mathematically certain that the
> polynomial I gave was irreducible ? How does gap know that it is irreducible
> ?
>
> marta
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
From hulpke at math.colostate.edu Tue Apr 25 16:58:06 2006
From: hulpke at math.colostate.edu (Alexander Hulpke)
Date: Tue Apr 25 16:58:02 2006
Subject: [GAP Forum] galois group
In-Reply-To: <4dd3929a0604242037o344ed2cdka72d91feca152ee0@mail.gmail.com>
References: <4dd3929a0604242037o344ed2cdka72d91feca152ee0@mail.gmail.com>
Message-ID:
Dear GAP Forum,
Marta Aseda wrote:
>
> I am wondering if galois group computation using gap is useable as a
> mathematical result.
In general, unless you explicitly turn off verifications or use
functions which deliberately only return proibabilistic results (such
as `ProbabilityShapes' for Galois groups), any result obtained with
(documented functions of) GAP is proven correct. (There is of course
always the philosophical problem of human error in implementation or
use, however the same problems are as well in published papers.) The
aim is that one could use such a result in the same way as a result
from a published paper.
> Is there any ambiguity ? Or, if I give a polynomial,
> and it computes its galois group, is it mathematically certain that
> the
> polynomial I gave was irreducible ? How does gap know that it is
> irreducible
The command `GaloisType' first performs an irreducibility test, using
the standard approach of factorization modulo a prime, Hensel lifting
and testing systematically all combinations of the lifted factors.
(There is a newer algorithm due to van Hoeij, but this is not yet
implemented in GAP.)
You can check this directly by calling `Factors' on the polynomial.
Thus indeed it is certain that -- if no error is issued -- the
polynomial is irreducible. Also the type returned by `GaloisType' is
proven correct -- at the cost that such a test might take very long
for certain polynomials.
Best wishes,
Alexander Hulpke
From l.h.soicher at qmul.ac.uk Wed Apr 26 12:14:14 2006
From: l.h.soicher at qmul.ac.uk (Leonard Soicher)
Date: Wed Apr 26 12:14:41 2006
Subject: [GAP Forum] Future developments for the DESIGN Package
Message-ID: <20060426111414.GA12403@mrcpc02.maths.qmul.ac.uk>
Dear Design Developers, Dear GAP-Forum,
This message is to inform you of developments for a future release of
the DESIGN Package for GAP.
The functionality is being improved to allow for more straightforward
construction of block designs, such as those formed from group orbits
of blocks, and derived and residual designs. Moreover, greater use of
(new and old) theoretical knowledge is being used in the BlockDesigns
function for more efficient discovery and classification of block designs
with given properties. Also, the documentation is being upgraded.
ONE WARNING: I plan to remove the undocumented option to BlockDesigns
for it to compute only the resolvable and simple block designs with the
given properties. This would be a useful option if it worked efficiently,
but at present to compute resolvable designs (using the DESIGN package)
it seems to be usually more efficient to compute all block designs with
the given properties and then to filter out the resolvable ones. I'm
sure it is possible to do better than this, but probably not with the
current setup. If the "resolvable and simple" feature is essential
to your work then I would be interested to hear about it, and might
reconsider my decision to remove this cumbersome "feature". In due
course I may try to write a more efficient implementation of an ability
to produce only resolvable block designs with given properties.
Regards,
Leonard Soicher
From costanti at science.unitn.it Wed Apr 26 18:03:39 2006
From: costanti at science.unitn.it (Marco Costantini)
Date: Wed Apr 26 18:04:15 2006
Subject: [GAP Forum] galois group
In-Reply-To:
References: <4dd3929a0604242037o344ed2cdka72d91feca152ee0@mail.gmail.com>
Message-ID: <200604261903.39695.costanti@science.unitn.it>
Dear all,
Gap is an open source system, so each user can look at the code of each
function, to check both whether it is correct, and whether it does what the
user needs. (Yes, this requires time, but also reading and full understanding
a published paper requires time.)
Of course bugs in Gap happen, however, when bugs are reported, they are
usually fixed within a reasonable time. On the other hand, usually "bugs" in
published papers are not fixed, and it has been reported that one refereed
paper out of four contains at least a "bug".
So I would say that using a result from Gap may be at least as safe as using a
result from a published paper.
Best regards,
Marco Costantini
On Tuesday 25 April 2006 17:58, Alexander Hulpke wrote:
>
> Marta Aseda wrote:
> > I am wondering if galois group computation using gap is useable as a
> > mathematical result.
>
> In general, unless you explicitly turn off verifications or use
> functions which deliberately only return proibabilistic results (such
> as `ProbabilityShapes' for Galois groups), any result obtained with
> (documented functions of) GAP is proven correct. (There is of course
> always the philosophical problem of human error in implementation or
> use, however the same problems are as well in published papers.) The
> aim is that one could use such a result in the same way as a result
> from a published paper.
From dn2447 at yahoo.com Sun Apr 30 22:06:28 2006
From: dn2447 at yahoo.com (D N)
Date: Sun Apr 30 22:06:44 2006
Subject: [GAP Forum] The ring associated to the set of irreducible
characters of a finite group
Message-ID: <20060430210628.73695.qmail@web37406.mail.mud.yahoo.com>
Dear Gap Forum,
Let G be a finite group and let Irr(G) be the set of irreducible
complex characters of G. Then the set Irr(G) gives rise to
a ring.
Given any two finite groups G and G' is there a way to use GAP
to check whether the rings associated to Irr(G) and Irr(G') are
isomorphic?
Best,
D. Naidu
---------------------------------
Get amazing travel prices for air and hotel in one click on Yahoo! FareChase
From akshriv at gmail.com Mon May 1 15:01:17 2006
From: akshriv at gmail.com (Abhishek)
Date: Mon May 1 15:02:39 2006
Subject: [GAP Forum] algorithm used by GAP for OrbitsDomain(...) ?
Message-ID:
Hi,
I want to know the algorithm used by GAP when a call to OrbitsDomain(...)
function is made. The reference manual on page 396 only says that the
OrbitsDomain operation is often faster than orbits. I want to know how/why
is it so.
Any pointers would be greatly appreciated.
Regards
Abhishek
--
Web: http://people.tamu.edu/~akshriv/
From anvita21 at usa.com Tue May 2 02:47:48 2006
From: anvita21 at usa.com (Anvita)
Date: Tue May 2 02:48:27 2006
Subject: [GAP Forum] normalizer of the identity symmetric group
Message-ID: <20060502014748.33FDB83BFA@ws3-1.us4.outblaze.com>
Dear Forum,
This is probably trivial again,
but still I thought I should report it.
All is fine when I do the following:
------------------------
gap> G:=Group(());
Group(())
gap> N:=Normalizer(G,G);
Group(())
gap>
------------------------
However, a similar program makes GAP break:
-----------------------------------------------------------------------------
gap> S:=SymmetricGroup(1);
Sym( [ ] )
gap> N:=Normalizer(S,S);
Error, usage: Group(,...), Group(), Group(,) called from
Group( pg ) called from
NormalizerParentSA( G, U ) called from
oper( super, sub ) called from
( ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk>
-----------------------------------------------------------------------------
Why?
Anvita
--
___________________________________________________
Play 100s of games for FREE! http://games.mail.com/
From craigugoretz at yahoo.com Tue May 2 03:28:27 2006
From: craigugoretz at yahoo.com (Craig Ugoretz)
Date: Tue May 2 03:28:51 2006
Subject: [GAP Forum] GAP feasibility study?
Message-ID: <20060502022827.39524.qmail@web82104.mail.mud.yahoo.com>
Hello,
For the past year I have been doing pre-research into a means of implementing mathematical texts on the subject of neutrosophics, a new form of logic, in a computer program. These texts, per my analysis, depend upon abstract algebra. By browsing the Internet, I discovered the GAP program, and I wonder if it would be suitable to implement the mathematics. However, I have a minimum of knowledge about abstract algebra, so it has been difficult for me to come to a decision. I decided to write this forum to see if anyone with extensive knowledge of the GAP program could help render a decision for/with me regarding its feasibility.
Please refer to this web page: http://www.gallup.unm.edu/~smarandache/philos.htm. I realize that there is a lot of material to take a look at, so any help given this matter would be most appreciated. As motivation, please take a look at some of the books that give examples of neutrosophic philosophy and how neutrosophics could be used to model social political situations. I believe that this new philosophy, as does its founder Dr. Smarandache, has applications to world peace because it is a means of expressing neutralities between conflicting philosophical systems.
Feel free to contact me at any juncture.
Sincerely,
Craig Ugoretz
craigugoretz@yahoo.com
---------------------------------
Yahoo! Messenger with Voice. PC-to-Phone calls for ridiculously low rates.
From hulpke at mac.com Tue May 2 12:02:25 2006
From: hulpke at mac.com (Alexander Hulpke)
Date: Tue May 2 12:02:39 2006
Subject: [GAP Forum] algorithm used by GAP for OrbitsDomain(...) ?
In-Reply-To:
References:
Message-ID:
>
Dear GAP-Forum,
Abishek wrote:
> I want to know the algorithm used by GAP when a call to OrbitsDomain
> (...)
> function is made. The reference manual on page 396 only says that the
> OrbitsDomain operation is often faster than orbits. I want to know
> how/why
> is it so.
OrbitsDomain *must* be given a domain which must be closed under the
group operation.
Orbits takes only seed points for the orbits, the orbits can be much
larger than the set of seed points.
`OrbitsDomain' can be faster as it can for example use an enumeration
of the full domain and use bit lists to remember which points have
been touched. `Orbits' instead must check all seed points whether
they are contained in the orbits found so far.
The reason for having these two operations is that `Orbits' (with
this syntax) was available in GAP3 and therefore has to be available
for compatibility reasons. However now we realize that the syntax of
`OrbitsDomain' in general permits a more efficient operation.
If you write new code I'd recommend you use `OrbitsDomain'.
Best wishes,
Alexander Hulpke
From ken.w.smith at cmich.edu Tue May 2 12:35:27 2006
From: ken.w.smith at cmich.edu (Ken W Smith)
Date: Tue May 2 12:35:46 2006
Subject: [GAP Forum] GAP feasibility study?
In-Reply-To: <20060502022827.39524.qmail@web82104.mail.mud.yahoo.com>
References: <20060502022827.39524.qmail@web82104.mail.mud.yahoo.com>
Message-ID: <1d8c1898a7b0229f9d7fc75f58b85612@cmich.edu>
Hi Craig,
I would be concerned about two issues -- there is a "learning curve"
that has to be climbed in learning abstract algebra; in addition, if
one is not comfortable programming (say in C or C++), there can be also
be a learning curve associated with writing GAP programs....
There is an introductory manual, "Abstract Algebra and GAP", by
Rainbolt and Gallian, see:
http://euler.slu.edu/Dept/Faculty/rainbolt/manual.html
Also -- and this is of interest to other GAP users -- the MAA has a
short online class on "Incorporating the Software GAP into the Teaching
of Abstract Algebra". Check out: http://www.maa.org/prep/2006/
(I'm grateful to a colleague for pointing out this short course. It is
July 10-14, 2006.)
yours,
Ken
On May 1, 2006, at 10:28 PM, Craig Ugoretz wrote:
> Hello,
>
> For the past year I have been doing pre-research into a means
> of implementing mathematical texts on the subject of neutrosophics, a
> new form of logic, in a computer program. These texts, per my
> analysis, depend upon abstract algebra. By browsing the Internet, I
> discovered the GAP program, and I wonder if it would be suitable to
> implement the mathematics. However, I have a minimum of knowledge
> about abstract algebra, so it has been difficult for me to come to a
> decision. I decided to write this forum to see if anyone with
> extensive knowledge of the GAP program could help render a decision
> for/with me regarding its feasibility.
>
> Please refer to this web page:
> http://www.gallup.unm.edu/~smarandache/philos.htm. I realize that
> there is a lot of material to take a look at, so any help given this
> matter would be most appreciated. As motivation, please take a look
> at some of the books that give examples of neutrosophic philosophy and
> how neutrosophics could be used to model social political situations.
> I believe that this new philosophy, as does its founder Dr.
> Smarandache, has applications to world peace because it is a means of
> expressing neutralities between conflicting philosophical systems.
>
> Feel free to contact me at any juncture.
>
> Sincerely,
> Craig Ugoretz
>
> craigugoretz@yahoo.com
>
>
> ---------------------------------
> Yahoo! Messenger with Voice. PC-to-Phone calls for ridiculously low
> rates.
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From maasiru at yahoo.com Wed May 3 15:55:04 2006
From: maasiru at yahoo.com (muniru asiru)
Date: Wed May 3 15:56:09 2006
Subject: [GAP Forum] I need a help
Message-ID: <20060503145504.91361.qmail@web53306.mail.yahoo.com>
I should be grateful if you can offer to help.
I have always found it easy to use the library of
small groups in GAP whenever I need
to make specific reference to some small group in my
calculations; for example
k1:=OneSmallGroup(Size,8,IsAbelian,false);
My problem is to determine all subgroups of k1 as
defined using GAP functions.
Note: I know that k1 is "D8", but I do not want to
identify k1 in this manner.
Sincerely,
M.A. Asiru
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From Rudolf.Zlabinger at chello.at Sat May 6 09:50:51 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Sat May 6 09:42:49 2006
Subject: [GAP Forum] Ikosaeder group
Message-ID: <004801c670ea$2dab5860$eec57254@chello.at>
I dealt with the finite rotation group for the ikosaeder.
My task was to find a permutation group acting on the 12 vertices of
ikosaeder recognizing a special numbering of the vertices as such: look at
the ikosaeder from a top vertex, numbered as 1, then on the 2 x 5 vertices
forming 2 somehow parallel cycles numbered by 2,3,4,5,6 and 7,8,9,10,11,
followed by the bottom vertex numbered by 12. Thus the permutation group
looked for should contain the rotation around the axis 1,12 executed by the
permutation
(2,3,4,5,6)(7,8,9,10,11).
Unfortunately the representative isomorphism Groups of symmetric group 12
itself did not contain the desired rotation, so I had to look for a
conjugate group of them containing it. I for the first glance found no
direct method for doing that, so I used the following sequence:
s12:=SymmetricGroup(12);
normact:=(2,3,4,5,6)(7,8,9,10,11);;
# the desired rotation
ccnormact:=ConjugacyClass(s12,normact);
# the conjugacyclass containing the rotation
Read(Ikosaeder);
# ikosaeder is introduced as group of perm matrices on 12 points
# representing the vertices of ikosaeder numbered as desired
isos12:=IsomorphicSubgroups(s12,ikosaeder);
# to mirror the matrices to permgroups acting on 12 points
iisos12:=List(isos12,c->Image(c));
# to get groups; none of the representative groups contains normact
<=======
enumiisos12:=List(iisos12,c->Enumerator(c));
# to prepare for efficiency
intersects:=List(enumiisos12,c->Intersection(c,ccnormact));
# to get intersects of groups to conjugacy class containing the desired
rotation
grpact:=(2,3,6,7,4)(5,9,12,10,8);
# one permutation of the intersect of the 7th group
repop70:=RepresentativeAction(s12,grpact,normact);
# to get the desired conjugator
cgroup70:=iisos12[7]^repop70;
# to get the desired conjugated group
# normact in cgroup70; the desired rotation is finally in the resulting
group
# true
My question to you is: Is there a more direct method for doing the above?
Thank you, friendly greetings from Vienna, Rudolf Zlabinger
From lars at public.noschinski.de Sun May 7 19:20:14 2006
From: lars at public.noschinski.de (Lars Noschinski)
Date: Sun May 7 19:21:52 2006
Subject: [GAP Forum] Creating a collection?
Message-ID: <20060507182014.GA12254@lars.home.noschinski.de>
Hello!
I'm trying to build a set of all mappings between to arbitrary sets. I
tried to create the individual mappings with GeneralMappingByElements,
made a list of tuples, but I don't know how to create a collection from
this list, so I can feed it to GeneralMappingsByElements.
Maybe there is also a more direct way?
From thomas.breuer at math.rwth-aachen.de Mon May 8 08:42:09 2006
From: thomas.breuer at math.rwth-aachen.de (Thomas Breuer)
Date: Mon May 8 08:42:23 2006
Subject: [GAP Forum] Ikosaeder group
In-Reply-To: <004801c670ea$2dab5860$eec57254@chello.at>
References: <004801c670ea$2dab5860$eec57254@chello.at>
Message-ID: <20060508074209.GA2795@math.rwth-aachen.de>
Dear GAP Forum,
Rudolf Zlabinger wrote
> I dealt with the finite rotation group for the ikosaeder.
>
> My task was to find a permutation group acting on the 12 vertices of
> ikosaeder recognizing a special numbering of the vertices as such: look at
> the ikosaeder from a top vertex, numbered as 1, then on the 2 x 5 vertices
> forming 2 somehow parallel cycles numbered by 2,3,4,5,6 and 7,8,9,10,11,
> followed by the bottom vertex numbered by 12. Thus the permutation group
> looked for should contain the rotation around the axis 1,12 executed by the
> permutation
> (2,3,4,5,6)(7,8,9,10,11).
>
> Unfortunately the representative isomorphism Groups of symmetric group 12
> itself did not contain the desired rotation, so I had to look for a
> conjugate group of them containing it. I for the first glance found no
> direct method for doing that, so I used the following sequence:
> [...]
Perhaps the most explicit way to deal with the symmetries of a regular
icosahedron is to start from a three dimensional model for it.
Following the description on p. 2 of the Atlas Of Finite groups,
the vertices have the coordinates $(0, \pm b, \pm 1)^C$
where $b = ( 1 + \sqrt{5} ) / 2$ and the superscript $C$ denotes cyclic
permutation of the coordinates.
In GAP,
we can write down symmetries of the isosahedron by the following matrices.
b:= -E(5)^2-E(5)^3;
mat1:= [ [ 0, 1, 0 ], # permutation of coordinates
[ 0, 0, 1 ],
[ 1, 0, 0 ] ];;
mat2:= 1/2 * [ [ b-1, 1, -b ], # order 5 rotation
[ -1, b, b-1 ],
[ b, b-1, 1 ] ];;
The permutation action on the twelve vertices can be obtained as follows.
p1:= [ 0, b, 1 ]; # one vertex
g:= Group( mat1, mat2 );
vertices:= Set( Orbit( g, p1 ) );
act:= Action( g, vertices );
And now one can play with the groups.
IsPrimitive( g, vertices ); # no
bl:= Blocks( g, vertices );;
bl[1]; # opposite vertices form a block
act2:= Action( g, bl, OnSets ); # an action on six points
All the best,
Thomas
P.S.:
Here is a LaTeX picture of the icosahedron.
\documentclass{article}
\usepackage{epic}
\begin{document}
\setlength\unitlength{0.4mm}
\begin{picture}(200,220)(-50,-100)
\thicklines
% foreground
\begin{drawjoin}
% outer circle
\jput(57,-72){} % P_1
\jput(91,13){} % P_7
\jput(34.5,85){} % P_9
\jput(-57,72){} % P_3
\jput(-91,-13){} % P_5
\jput(-34.5,-85){} % P_{11}
\jput(57,-72){} % P_1
% inner circles
\jput(57,7.5){} % P_2
\jput(34.5,85){} % P_9
\jput(-34.5,45){} % P_{10}
\jput(-91,-13){} % P_5
\jput(-21,-52){} % P_6
\jput(57,-72){} % P_1
\end{drawjoin}
\begin{drawjoin}
\jput(-21,-52){} % P_6
\jput(57,7.5){} % P_2
\jput(-34.5,45){} % P_{10}
\jput(-21,-52){} % P_6
\end{drawjoin}
% remaining lines
\drawline(-21,-52)(-34.5,-85) % P_6 to P_{11}
\drawline(57,7.5)(91,13) % P_2 to P_7
\drawline(-34.5,45)(-57,72) % P_{10} to P_3
% background
\begin{dottedjoin}{3}
% inner circles
\jput( 34.5, -45){} % P_{12}
\jput( 91, 13){} % P_7
\jput( 21, 52){} % P_8
\jput( -57, 72){} % P_3
\jput( -57,-7.5){} % P_4
\jput(-34.5, -85){} % P_{11}
\jput( 34.5, -45){} % P_{12}
\jput( 21, 52){} % P_8
\jput( -57,-7.5){} % P_4
\jput( 34.5, -45){} % P_{12}
\jput( 57, -72){} % P_1
\end{dottedjoin}
% remaining lines
\dottedline{3}(-91,-13)(-57,-7.5) % P_5 to P_4
\dottedline{3}(21,52)(34.5,85) % P_8 to P_9
\end{picture}\ \begin{picture}(200,220)(-50,-100)
\thicklines
% axes
\drawline(-87,-50)(87,50)
\put(-92,-55){\makebox(0,0){$x$}}
\put(-87,-50){\vector(-2,-1){0}}
\drawline(-87,50)(87,-50)
\put(92,-55){\makebox(0,0){$y$}}
\put(87,-50){\vector(2,-1){0}}
\put(0,-100){\vector(0,1){200}}
\put(0,105){\makebox(0,0){$z$}}
% circles at points
\put( 57, -72){\circle{3}}
\put( 57, 7.5){\circle{3}}
\put( -57, 72){\circle{3}}
\put( -57,-7.5){\circle{3}}
\put( -91, -13){\circle{3}}
\put( -21, -52){\circle{3}}
\put( 91, 13){\circle{3}}
\put( 21, 52){\circle{3}}
\put( 34.5, 85){\circle{3}}
\put(-34.5, 45){\circle{3}}
\put(-34.5, -85){\circle{3}}
\put( 34.5, -45){\circle{3}}
% point names
\put( 64, -79){\makebox(0,0){$1$}}
\put( 64, 12.5){\makebox(0,0){$2$}}
\put( -64, 79){\makebox(0,0){$3$}}
\put( -64,-12.5){\makebox(0,0){$4$}}
\put( -98, -14){\makebox(0,0){$5$}}
\put( -20, -60){\makebox(0,0){$6$}}
\put( 98, 14){\makebox(0,0){$7$}}
\put( 20, 60){\makebox(0,0){$8$}}
\put( 41.5, 91){\makebox(0,0){$9$}}
\put(-43.5, 47){\makebox(0,0){$10$}}
\put(-41.5, -91){\makebox(0,0){$11$}}
\put( 43.5, -47){\makebox(0,0){$12$}}
% join points in a plane
\begin{dottedjoin}{3}
\jput(-91,-13){} % P_5
\jput(-21,-52){} % P_6
\jput(91,13){} % P_7
\jput(21,52){} % P_8
\jput(-91,-13){} % P_5
\end{dottedjoin}
\begin{dottedjoin}{3}
\jput(57,-72){} % P_1
\jput(57,7.5){} % P_2
\jput(-57,72){} % P_3
\jput(-57,-7.5){} % P_4
\jput(57,-72){} % P_1
\end{dottedjoin}
\begin{dottedjoin}{3}
\jput(34.5,85){} % P_9
\jput(-34.5,45){} % P_{10}
\jput(-34.5,-85){} % P_{11}
\jput(34.5,-45){} % P_{12}
\jput(34.5,85){} % P_9
\end{dottedjoin}
\end{picture}
\end{document}
From joachim.s at web.de Mon May 8 09:18:33 2006
From: joachim.s at web.de (Joachim Schittenhelm)
Date: Mon May 8 09:19:26 2006
Subject: [GAP Forum] Fields & Polynomials
Message-ID: <445EFED9.7080302@web.de>
Dear Forum,
with given F=GF(q) and an irreducible polynomial f over F of degree n,
how can i create the field of polynomials over F of degree at most n-1 that
is isomorphic to GF(q^n).
Greetings
Joachim Schittenhelm
From dfh at maths.warwick.ac.uk Mon May 8 15:31:25 2006
From: dfh at maths.warwick.ac.uk (Derek Holt)
Date: Mon May 8 15:31:43 2006
Subject: [GAP Forum] Ac ceptance of the HAP package
Message-ID: <20060508143125.GA3799@maths.warwick.ac.uk>
Dear GAP Forum,
I announce, with great pleasure, that the HAP package, by Graham Ellis,
has been accepted as a refereed GAP package and is available for download
from the GAP Web site, or from the authors Web page at
http://hamilton.nuigalway.ie/Hap/www/
The following description and example of the functionality of the package is
taken form that site.
HAP is a homological algebra library for use with the GAP computer algebra
system, and is still under development. Its initial focus is on computations
related to the cohomology of groups. Both finite and infinite groups are
handled, with main emphasis on integer coefficients.
HAP can be used to make basic calculations in the cohomology of finite and
infinite groups. For example, to calculate the integral homology Hn(D201,Z)
of the dihedral group of order 402 in dimension n=99 we could perform the
following commands.
gap> F:=FreeGroup(2);; x:=F.1;; y:=F.2;;
gap> G:=F/[x^2,y^201,(x*y)^2];; G:=Image(IsomorphismPermGroup(G));;
gap> GroupHomology(G,99);
[ 2, 3, 67 ]
gap> time;
11918
Derek Holt.
From Rudolf.Zlabinger at chello.at Tue May 9 13:31:00 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Tue May 9 13:22:47 2006
Subject: [GAP Forum] Numberings of Ikosaeders vertices
Message-ID: <001001c67364$6defe540$eec57254@chello.at>
Dear Thomas Breuer,
Thank you at least for the hint to ATLAS, but also for the rich material
given.
To introduce Ikosaeder itself as group only its suffficient to begin with
A5, it can be mirrored, as desired, to a 12 point permutation group by
Isomorphic Subgroups to s12.
My question mainly was, as there is a direct method to find out a group out
of isomophic subgroups from A5 to S12, containing a special permutation. In
our case it was the permutation (2,3,4,5,6)(7,8,9,10,11). I expected
intuitively, that such a group recognizes a special numbering of the
vertices of Ikosaeder.
More general: As the action of all possible numberings, that is S12 in our
case, on a starting set of one representative group for each conjugacy
class of groups (isomorphic to A5 in our case) is injective, one can expect,
that there is one group for each conjugacy class fitting to a special
numbering.
But in using numberings of vertices only, there is the problem to uniquely
describe this numbering formally up to a selection of a starter set of
groups belonging to a initial numbering, that has to be induced intuitively
from outside the formalism, as to, for example, requesting the inclusion of
special permutations in the starter group(s) (see above).
I didnt explore yet, whether the reverse does hold, in order to have a
unique description of a special numbering itself by the groups selected,
because this relation above seems not to be bijective. That is, one group
out of a conjugacy class may not uniquely describe a special numbering, but
a distinct manyfold of numberings. In this case one had to look, what
manyfold it is, maybe it is determined as a structure quite well, and maybe
the numbering is uniquely determined by the full set of the selected groups,
one group per conjugacy class. I will try to explore this myself as an
exercise, perhaps this problem reveales to be trivial at all.
So the method given by you (out of ATLAS) has the advantage of uniquely
describing the vertices by vectors, independent from a selected group
representation.
Thank you, and best regards, Rudolf Zlabinger
From mckay at encs.concordia.ca Tue May 9 14:38:44 2006
From: mckay at encs.concordia.ca (MCKAY john)
Date: Tue May 9 14:38:57 2006
Subject: [GAP Forum] Numberings of Ikosaeders vertices
In-Reply-To: <001001c67364$6defe540$eec57254@chello.at>
References: <001001c67364$6defe540$eec57254@chello.at>
Message-ID:
May I put in a word? I have worked with the groups:
[a,b,c] = 1/a+1/b+1/c > 1.
These are finite subgps of SO3.
[a,b,c] is the icosahedral group = dodecahedral group.
use the flags. One has a vertex incident with an edge incident
with a face. The operations x,y,z fix each of these.
For your icosahedron we have [2,3,5]. You can make a
dodecahedron by constructing a pentagon (fold a strip of paper)
and five others adjacent to it. Repeat for the
other half. Put them on top of each other with corners
alternating and an elastic band round them. You can easily
number vertices, edges, and faces. V-E+F=2 30-20+12=2.
Unlike the symmetric group, as you say, here we have two
choices for the element of order 5. Take either z or z^2
of order 5 and they are in distinct conjugacy classes in
[2,3,5].
Good luck!
By the way, the figures in stone were known to late neolithic
man in Scotland (Skara Brae) thousands of years before Plato.
John McKay
On Tue, 9 May 2006, Rudolf Zlabinger wrote:
> Dear Thomas Breuer,
>
> Thank you at least for the hint to ATLAS, but also for the rich material
> given.
>
> To introduce Ikosaeder itself as group only its suffficient to begin with
> A5, it can be mirrored, as desired, to a 12 point permutation group by
> Isomorphic Subgroups to s12.
>
> My question mainly was, as there is a direct method to find out a group out
> of isomophic subgroups from A5 to S12, containing a special permutation. In
> our case it was the permutation (2,3,4,5,6)(7,8,9,10,11). I expected
> intuitively, that such a group recognizes a special numbering of the
> vertices of Ikosaeder.
>
> More general: As the action of all possible numberings, that is S12 in our
> case, on a starting set of one representative group for each conjugacy
> class of groups (isomorphic to A5 in our case) is injective, one can expect,
> that there is one group for each conjugacy class fitting to a special
> numbering.
>
> But in using numberings of vertices only, there is the problem to uniquely
> describe this numbering formally up to a selection of a starter set of
> groups belonging to a initial numbering, that has to be induced intuitively
> from outside the formalism, as to, for example, requesting the inclusion of
> special permutations in the starter group(s) (see above).
>
> I didnt explore yet, whether the reverse does hold, in order to have a
> unique description of a special numbering itself by the groups selected,
> because this relation above seems not to be bijective. That is, one group
> out of a conjugacy class may not uniquely describe a special numbering, but
> a distinct manyfold of numberings. In this case one had to look, what
> manyfold it is, maybe it is determined as a structure quite well, and maybe
> the numbering is uniquely determined by the full set of the selected groups,
> one group per conjugacy class. I will try to explore this myself as an
> exercise, perhaps this problem reveales to be trivial at all.
>
> So the method given by you (out of ATLAS) has the advantage of uniquely
> describing the vertices by vectors, independent from a selected group
> representation.
>
> Thank you, and best regards, Rudolf Zlabinger
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
From Rudolf.Zlabinger at chello.at Tue May 9 16:31:37 2006
From: Rudolf.Zlabinger at chello.at (Rudolf Zlabinger)
Date: Tue May 9 16:23:25 2006
Subject: [GAP Forum] Indexing of Ikosaedrons vertices
References: <001001c67364$6defe540$eec57254@chello.at>
Message-ID: <003201c6737d$a9910e80$eec57254@chello.at>
Dear MCKAY john,
Thank you for your hint to the finite rotation groups, you are right, the
icosahedron (dodecahedron) has 3 pole classes, with 2,3,5 cycle rotations,
resulting in 30,20,12 cosets.
My theoretical problem was, to translate this structure to mere permutation
representations. Distributing numbers 1 to 12 to the vertices has to some
degree influence to the selection of distinct permutation groups
representing the icosahedron indexed by these numbers.
As in our example indexing the icosahedrons vertices from top to down as 1,
(2,3,4,5,6),(7,8,9,10,11), 12 lead to the request, that a representing group
has to contain the permutation (2,3,4,5,6)(7,8,9,10,11) as the rotation
around the axis 1,12.
The problem is, that there is no formal way to describe a special indexing
of the vertices, as to do it intuitively as above. So i am looking for a
method to derive the indexing(s) from the selected permutation group(s) in
reverse order if this is possible.
One way to do it is to derive the vertex indices from one of the 5 cycles,
taking the 2 missing points (vertices) as axis. This results in 4 different
possible kinds of indexing the vertices, these are the 5 cycle permutation ^
1,2,3,4. In this way, the indexing is (partially) determined and described
by a selected special permutation group.
Thank you so far, best wishes, Rudolf Zlabinger
----- Original Message -----
From: "MCKAY john"
To: "Rudolf Zlabinger"
Cc: "Thomas Breuer" ;
Sent: Tuesday, May 09, 2006 3:38 PM
Subject: Re: [GAP Forum] Numberings of Ikosaeders vertices
May I put in a word? I have worked with the groups:
[a,b,c] = ~~