where P,Q, and R are the sides of the triangle. To do this, I will construct all 4-colorings of the Tiling T of the plane (the triangle with sides P,Q,R is the fundamental region of T) where all elements of H effect permutations of the 4 colors {1,2,3,4}. For such colorings I defined a homomorphism pi:H->S4, D4 and V (the symmetric, dihedral, and Klein-4 group respectively) which are transitive subgroups of S4. Suppose pi:H->S4, since H=

, pi is completely determined when pi(P), pi(Q) and pi(R) are specified. Since H is to permute the colors in the resulting coloring, each of P,Q,R either fixes or interchanges any two colors; i.e., each of P,Q,R can be mapped to any of the 2-cycle or products of 2-cycle element of s4. ??? If I want only the index 4 subgroups of H when pi(H)=S4, the first step would be to list all the generators of the symmetry group S4 with each generator consisting only of 3 elements from the 9 2-cyles and products of two cycles of S4 (e.g.{(12),(13),(24)},? {(13),(14)(23),(12)},?). I know that it's easier to do this using GAP but I just don't know how. Can you give me a working program or code for this? Thanks. ??? Now, suppose {pi(P),pi(Q),pi(R)} is a permutation assignment to P,Q,R that gives rise to an index 4 subgroup K in H. The entries corresponding to (1234){pi(P),pi(Q),pi(R)}(1234)^-1, (1432){pi(P),pi(Q),pi(R)}(1432)^-1, and (13)(24){pi(P),pi(Q),pi(R)}((12)(34))^-1 will respectively, yield h1Kh1^-1, h2Kh2^-1, h3Kh3^-1 (for some? h1,h2,h3 in H), conjugate subgroups of K in H. My next question is, how do I use GAP to obtain the distinct subgroups of index 4 in H up to conjugacy when pi(H)=S4?? Many thanks. Levi From lee_bkua at yahoo.com Thu Nov 27 10:27:39 2008 From: lee_bkua at yahoo.com (Levie Bicua) Date: Thu Nov 27 10:27:59 2008 Subject: [GAP Forum] subgroups of index 4 up to conjugacy Message-ID: <561814.93237.qm@web33401.mail.mud.yahoo.com> Dear GAP forum members, ??? Thank you for accepting me in this forum. I am working on determining all the index 5 subgroups of the triangle group H=*pqr=

where P,Q, and R are the sides of the triangle. To do this, I will construct all 4-colorings of the Tiling T of the plane (the triangle with sides P,Q,R is the fundamental region of T) where all elements of H effect permutations of the 4 colors {1,2,3,4}. For such colorings I defined a homomorphism pi:H->S4, D4 and V (the symmetric, dihedral, and Klein-4 group respectively) which are transitive subgroups of S4. Suppose pi:H->S4, since H=

, pi is completely determined when pi(P), pi(Q) and pi(R) are specified. Since H is to permute the colors in the resulting coloring, each of P,Q,R either fixes or interchanges any two colors; i.e., each of P,Q,R can be mapped to any of the 2-cycle or products of 2-cycle element of s4. ??? If I want only the index 4 subgroups of H when pi(H)=S4, the first step would be to list all the generators of the symmetry group S4 with each generator consisting only of 3 elements from the 9 2-cyles and products of two cycles of S4 (e.g.{(12),(13),(24)},? {(13),(14)(23),(12)},?). I know that it's easier to do this using GAP but I just don't know how. Can you give me a working program or code for this? Thanks. ??? Now, suppose {pi(P),pi(Q),pi(R)} is a permutation assignment to P,Q,R that gives rise to an index 4 subgroup K in H. The entries corresponding to (1234){pi(P),pi(Q),pi(R)}(1234)^-1, (1432){pi(P),pi(Q),pi(R)}(1432)^-1, and (13)(24){pi(P),pi(Q),pi(R)}((12)(34))^-1 will respectively, yield h1Kh1^-1, h2Kh2^-1, h3Kh3^-1 (for some? h1,h2,h3 in H), conjugate subgroups of K in H. My next question is, how do I use GAP to obtain the distinct subgroups of index 4 in H up to conjugacy when pi(H)=S4?? Many thanks. Levi From lee_bkua at yahoo.com Thu Nov 27 10:35:01 2008 From: lee_bkua at yahoo.com (Levie Bicua) Date: Thu Nov 27 10:35:17 2008 Subject: [GAP Forum] subgroups of index 4 up to conjugacy(corrected message) Message-ID: <395881.29273.qm@web33403.mail.mud.yahoo.com> Dear GAP forum members, ??? Thank you for accepting me in this forum. I am working on determining all the index?4 subgroups of the triangle group H=*pqr=

where P,Q, and R are the sides of the triangle. To do this, I will construct all 4-colorings of the Tiling T of the plane (the triangle with sides P,Q,R is the fundamental region of T) where all elements of H effect permutations of the 4 colors {1,2,3,4}. For such colorings I defined a homomorphism pi:H->S4, D4 and V (the symmetric, dihedral, and Klein-4 group respectively) which are transitive subgroups of S4. Suppose pi:H->S4, since H=

, pi is completely determined when pi(P), pi(Q) and
pi(R) are specified. Since H is to permute the colors in the resulting coloring, each of P,Q,R either fixes or
interchanges any two colors; i.e., each of P,Q,R can be mapped to any of the 2-cycle or products of 2-cycle
element of s4.
??? If I want only the index 4 subgroups of H when pi(H)=S4, the first step would be to list all the generators of
the symmetry group S4 with each generator consisting only of 3 elements from the 9 2-cyles and products of two
cycles of S4 (e.g.{(12),(13),(24)},? {(13),(14)(23),(12)},?). I know that it's easier to do this using GAP
but I just don't know how. Can you give me a working program or code for this? Thanks.
??? Now, suppose {pi(P),pi(Q),pi(R)} is a permutation assignment to P,Q,R that gives rise to an
index 4 subgroup K in H. The entries corresponding to (1234){pi(P),pi(Q),pi(R)}(1234)^-1,
(1432){pi(P),pi(Q),pi(R)}(1432)^-1, and (13)(24){pi(P),pi(Q),pi(R)}((12)(34))^-1 will respectively, yield h1Kh1^-1,
h2Kh2^-1, h3Kh3^-1 (for some? h1,h2,h3 in H), conjugate subgroups of K in H. My next question is, how do I use GAP
to obtain the distinct subgroups of index 4 in H up to conjugacy when pi(H)=S4??
Many thanks.
Levi
From e.obrien at auckland.ac.nz Sun Dec 7 00:21:32 2008
From: e.obrien at auckland.ac.nz (Eamonn O'Brien)
Date: Sun Dec 7 00:21:52 2008
Subject: [GAP Forum] Special Session on Computational Algebra,
BMC/IMS Meeting April 2009
Message-ID: <493B170C.7000306@math.auckland.ac.nz>
Dear Colleagues,
A joint meeting of the British Mathematical Colloquium
and the Irish Mathematical Society will take place at
NUI Galway from 6-9 April, 2009.
As part of the program, we will run a special session on
Computational Algebra, featuring the following presentations:
* Arjeh Cohen (Eindhoven University):
Constructions of curves with given groups of automorphisms
* Bettina Eick (Braunschweig):
Isomorphism testing for algebras (Lie or associative)
* Dane Flannery (NUI Galway):
On deciding finiteness of matrix groups
* Steven Galbraith (Royal Holloway, University of London):
Elliptic curves and public key cryptography
* Gunter Malle (Kaiserslautern):
Computing in Hecke algebras
* Gary McGuire (University College Dublin):
Some computational algebra in cryptography
Abstracts and other details on the Special Session are at
http://www.maths.nuigalway.ie/bmc2009/casession.shtml
which also provides a link to the main BMC/IMS site.
We hope that you can join us for this event.
Best wishes.
Eamonn O'Brien, University of Auckland
Goetz Pfeiffer, NUI Galway
From a.abdollahi at math.ui.ac.ir Tue Dec 9 07:39:03 2008
From: a.abdollahi at math.ui.ac.ir (Alireza Abdollahi)
Date: Tue Dec 9 07:39:22 2008
Subject: [GAP Forum] Two Days Group Theory Seminar in ISFAHAN, IRAN
Message-ID: <165966.71898.qm@web35702.mail.mud.yahoo.com>
Dear Group and GAP Pubbers,
?
A two days seminar on Group Theory will be held at the Depratment of Mathematics, University of Isfahan, Iran, ?during 12-13 March 2009.
?
Th goal of this seminar is to bring together many of the people in Iran who are working on Group Theory, to introduce their works to each other as well as to students and young researches.?We also hope? to? create an atmosphere?to exchange/share/know their ideas and current researchs.?This seminar hopefully??will? serve as a place in which Ph.D./M.Sc. students?get the benifit of ideas/guidance of the others.
?
The tentative plan is to have some talks of 50 and 25 minutes.
?
We sincerely invite?all?interested people in Group Theory???to help us by attending this seminar.
?
The following? URL (in Persian)?can be used?for Iranian?to download the registration form and to get the necessary information.
?
www.sci.ui.ac.ir/~cem/1tdgs.htm
?
The others can? send an e-mail to me?and?ask of???Abstract Talk submission,?visa form, regisrtation?fee, accomodation, Hotel ?Reserving, ...??
?
Best Regards
Alireza Abdollahi
?
?
Department of Mathematics
University of Isfahan,
Isfahan 81746-73441,Iran
abdollahi@member.ams.org
URL: http://sci.ui.ac.ir/~a.abdollahi
From pawel.laskos at gmail.com Tue Dec 9 19:56:56 2008
From: pawel.laskos at gmail.com (=?UTF-8?B?UGF3ZcWCIExhc2tvxZstR3JhYm93c2tp?=)
Date: Thu Dec 11 10:41:47 2008
Subject: [GAP Forum] Obtaining Small Group information
Message-ID: <493ECD88.6080603@gmail.com>
Hello,
I have noticed that GAP Small Groups library provides useful information
on the structure of groups belonging to the layer 1 of the library, but
does not do so for (some) bit more complicated groups. I am rather
dissatisfied by the output
gap> SmallGroupsInformation(1625);
There are 5 groups of order 1625.
They are sorted by normal Sylow subgroups.
1 - 5 are the nilpotent groups.
How can I obtain such a pleasant info like the following?
gap> SmallGroupsInformation(125);
There are 5 groups of order 125.
1 is of type c125.
2 is of type 5x25.
3 is of type 5^2:5.
4 is of type 25:5.
5 is of type 5^3.
And, by the way, what does the colon stand for in the 125,3 and 125,4
type descriptions? I failed to find the explanation in the help pages.
Regards,
Pawe? Lasko?-Grabowski
From h.dietrich at tu-bs.de Thu Dec 11 12:37:14 2008
From: h.dietrich at tu-bs.de (Heiko Dietrich)
Date: Thu Dec 11 12:38:37 2008
Subject: [GAP Forum] Obtaining Small Group information
In-Reply-To: <493ECD88.6080603@gmail.com>
References: <493ECD88.6080603@gmail.com>
Message-ID: <200812111337.14970.h.dietrich@tu-bs.de>
Dear Pawe?,
you can use the command "StructureDescription":
gap> for i in AllSmallGroups(1625) do Display(StructureDescription(i)); od;
C1625
C325 x C5
C13 x ((C5 x C5) : C5)
C13 x (C25 : C5)
C65 x C5 x C5
The output is explained in the manual:
http://www.gap-system.org/Manuals/doc/htm/ref/CHAP037.htm#SECT006
Best,
Heiko
On Tuesday 09 December 2008 20:56, Pawe? Lasko?-Grabowski wrote:
> Hello,
>
> I have noticed that GAP Small Groups library provides useful information
> on the structure of groups belonging to the layer 1 of the library, but
> does not do so for (some) bit more complicated groups. I am rather
> dissatisfied by the output
>
> gap> SmallGroupsInformation(1625);
>
> There are 5 groups of order 1625.
> They are sorted by normal Sylow subgroups.
> 1 - 5 are the nilpotent groups.
>
> How can I obtain such a pleasant info like the following?
>
> gap> SmallGroupsInformation(125);
>
> There are 5 groups of order 125.
> 1 is of type c125.
> 2 is of type 5x25.
> 3 is of type 5^2:5.
> 4 is of type 25:5.
> 5 is of type 5^3.
>
> And, by the way, what does the colon stand for in the 125,3 and 125,4
> type descriptions? I failed to find the explanation in the help pages.
>
> Regards,
> Pawe? Lasko?-Grabowski
>
> _______________________________________________
> Forum mailing list
> Forum@mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
From lee_bkua at yahoo.com Thu Dec 11 14:45:38 2008
From: lee_bkua at yahoo.com (Levie Bicua)
Date: Thu Dec 11 14:45:53 2008
Subject: [GAP Forum] Some GAP question
Message-ID: <340284.77373.qm@web33406.mail.mud.yahoo.com>
Dear GAP forum members,
I?m new to this GAP thing and I think this question is trivial to most of you.
Suppose I have a set of 3 elements coming from s4 (e.g. [(1,3),(2,4),(1,2)]) and I want to generate other sets using GAP by the method below:
gap> ()^-1*[(1,3),(2,4),(1,2)]*();
[ (1,3), (2,4), (1,2) ]
gap> (2,3)^-1*[(1,3),(2,4),(1,2)]*(2,3);
[ (1,2), (3,4), (1,3) ]
gap> (2,4)^-1*[(1,3),(2,4),(1,2)]*(2,4);
[ (1,3), (2,4), (1,4) ]
gap> (3,4)^-1*[(1,3),(2,4),(1,2)]*(3,4);
[ (1,4), (2,3), (1,2) ]
gap> (2,3,4)^-1*[(1,3),(2,4),(1,2)]*(2,3,4);
[ (1,4), (2,3), (1,3) ]
gap> (2,4,3)^-1*[(1,3),(2,4),(1,2)]*(2,4,3);
[ (1,2), (3,4), (1,4) ]
The method gave 6 different sets of 3 elements. If I will use another set of 3 elements and repeat the process with again using (),(2,3),(2,4),(3,4),(2,3,4), (2,4,3) as conjugating elements, I will obtain again 6 different sets. But using this process every time I want to obtain a list of different sets as above would be eating much of my time. Is there a more efficient command/method than what I had used? Thanks.
From jjm at mcs.st-andrews.ac.uk Thu Dec 11 15:41:39 2008
From: jjm at mcs.st-andrews.ac.uk (John McDermott)
Date: Thu Dec 11 15:41:45 2008
Subject: [GAP Forum] New update of GAP released
Message-ID: <382E6091-3699-447B-BF18-E1999E711712@mcs.st-andrews.ac.uk>
Dear GAP Forum,
We are delighted to announce the release of GAP 4 release 4 update 11
(GAP
4.4.11 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 the facility for dynamic loading of modules on Cygwin using
a dll
based approach, methods for testing membership in general and special
linear
groups over the integers, better View methods for strings and some other
objects, and more. Full details can be found on the web pages at
http://www.gap-system.org/Download/Updates/gap4r4p11.html.
John McDermott,
for the GAP Group.
--
John McDermott
Scientific Officer
Centre for Interdisciplinary Research in Computational Algebra
School of Computer Science
University of St Andrews
North Haugh, St Andrews, Fife
KY16 9SX
SCOTLAND
(Room 330, Mathematical Institute)
tel +44 1334 463813
mob +44 7941 507531
The University of St Andrews is a charity registered in Scotland : No
SC01353
From jbohanon2 at gmail.com Thu Dec 11 19:51:21 2008
From: jbohanon2 at gmail.com (Joe Bohanon)
Date: Thu Dec 11 19:51:31 2008
Subject: [GAP Forum] Obtaining Small Group information
In-Reply-To: <200812111337.14970.h.dietrich@tu-bs.de>
References: <493ECD88.6080603@gmail.com>
<200812111337.14970.h.dietrich@tu-bs.de>
Message-ID: