[GAP Forum] commutator relations in p-group
Siddhartha Sarkar
sidhu at mri.ernet.in
Thu Nov 11 01:21:22 GMT 2004
Dear forum,
I need to know the classification of p-groups satisfy the following
commutator relation:
(1) say the group G is minimally generated by d elements
x_1, x_2,...,x_d.
(2) if d is even, the relation is [x_1, x_2]...[x_{d-1},x_d] = 1
(3) for d odd, the relation is [x_1, x_2]...[x_d,x] = 1
for some element x in G.
Kindly help me in this.
with regards,
Siddhartha
***************************************************************************
Siddhartha Sarkar
School of Mathematics
Harish Chandra Research Institute
Chhatnag Road,Jhusi
Allahabad-211019.
India.
***************************************************************************
On Wed, 10 Nov 2004 forum-request at gap-system.org wrote:
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> Today's Topics:
>
> 1. reza orfi (Reza Orfi)
> 2. Re: reza orfi (Bettina Eick)
> 3. using GAP workspace (Alec Makosi)
> 4. Re: GUAVA 2.0 alpha (Gary Zablackis)
> 5. a problem with matrix algebras and homomorphisms
> (Laurent Bartholdi)
> 6. Re: reza orfi (Marco Costantini)
> 7. Schreier-Sims for matrix gps paper (David Joyner)
> 8. Dangerous bug in Saving Workspaces (was "using GAP
> workspace") (Steve Linton)
> 9. Sophus package for Lie algebras is accepted (Alexander Konovalov)
> 10. p-group of maximal class (Reza Orfi)
> 11. Re: p-group of maximal class (Mowsey)
> 12. Re: a problem with matrix algebras and homomorphisms
> (Thomas Breuer)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Wed, 27 Oct 2004 01:54:26 -0700 (PDT)
> From: Reza Orfi <reza_orfi at yahoo.com>
> Subject: [GAP Forum] reza orfi
> To: Forum at gap-system.org
> Message-ID: <20041027085426.66305.qmail at web53409.mail.yahoo.com>
> Content-Type: text/plain; charset=us-ascii
>
> Dear forum.
> with many thanks.
> I need all groups of order 3^7,3^8,3^9,3^10,5^7,5^8,5^9,5^10.
> but there are not in gap4r4.
> please help me .i need this group .
> Best regard.
>
> __________________________________________________
> Do You Yahoo!?
> Tired of spam? Yahoo! Mail has the best spam protection around
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>
> ------------------------------
>
> Message: 2
> Date: Thu, 28 Oct 2004 10:38:58 +0200 (MESZ)
> From: Bettina Eick <beick at tu-bs.de>
> Subject: Re: [GAP Forum] reza orfi
> To: <forum at gap-system.org>, Reza Orfi <reza_orfi at yahoo.com>
> Message-ID:
> <Pine.HPX.4.33.0410281034300.5018-100000 at rzserv1i.rz.tu-bs.de>
> Content-Type: TEXT/PLAIN; charset=US-ASCII
>
>
> Dear GAP Forum, dear Reza Orfi,
>
> you wrote:
> > I need all groups of order 3^7,3^8,3^9,3^10,5^7,5^8,5^9,5^10.
> > but there are not in gap4r4. please help me .i need this group .
>
> it is true that the groups of your considered orders are not available
> in GAP. For many of these orders an explicit determination of these
> groups has not been done yet and would probably be difficult.
>
> For example, consider the groups of order p^10. For p=2, these groups
> have been enumerated and thus it is known that there are 49487365422
> groups of order 2^10. (See: B. Eick, E. O'Brien. Enumerating p-groups.
> J. Austral. Math. Soc. 67, 191 - 205 (1999).) As this is a very large
> number, we did not attempt to list the groups of order 2^10 explicitly
> or to include them in an electronic database.
>
> This problem is going to be worse for the groups of order 3^10 and 5^10.
> The groups of these orders have not been enumerated or listed yet and it
> seems unlikely that this will happen soon.
>
> For the smallest of your orders, (e.g. 3^7,) it may be possible to list
> all or at least the interesting groups explicitly, if you are willing
> to invest time and effort in it. In principle, the p-groups of order p^n
> can be determined (up to isomorphism) using the p-group generation
> algorithm by E. O'Brien. This is available in the ANUPQ package of GAP.
>
> A good strategy in your case might be to try to restrict the groups that
> you need first and then try to list these groups with p-group generation.
> If you find a suitable restriction, then this approach might work.
>
> Hope this helps,
> Bettina
>
>
>
>
> ------------------------------
>
> Message: 3
> Date: Thu, 28 Oct 2004 23:17:32 +0800
> From: Alec Makosi <alec at khakasnet.ru>
> Subject: [GAP Forum] using GAP workspace
> To: GAP forum <forum at gap-system.org>
> Message-ID: <1098976387.1039.8.camel at localhost.localdomain>
> Content-Type: text/plain
>
> Dear forum!
>
> Whether is able GAP to maintain in workspace enumerators?
>
> More precisely, the situation is those:
> I create enumerator for the list in which approximately 200000 elements.
> Each element is a permutation enough a large degree.
> I save it in workspace. Then restart GAP having loaded workspace. Size
> command correctly specifies size of the relevant list.
> But attempt to receive "far enough" elements comes to failure
> (the message, something such as "List Element: <list>[1361] must have
> an assigned value at
> pnt := G.orbit[b * U.lenblock + 1];
> called from ... ").
>
> Certainly it is a pity, if such possibility is not present. And I can
> somewhere am mistaken?
>
> Thank.
>
> Sincerely Alec Makosi.
>
>
>
>
> ------------------------------
>
> Message: 4
> Date: Thu, 28 Oct 2004 09:29:31 -0700 (PDT)
> From: Gary Zablackis <gzabl at yahoo.com>
> Subject: Re: [GAP Forum] GUAVA 2.0 alpha
> To: David Joyner <wdj at usna.edu>, GAP forum <forum at gap-system.org>
> Message-ID: <20041028162931.18338.qmail at web53008.mail.yahoo.com>
> Content-Type: text/plain; charset=us-ascii
>
> David,
>
> I downloaded and installed GUAVA 2.0 on my PC running
> Windows 2K yesterday.
>
> I ran ./configure and make from a CYGWIN bash shell
> and the executables were all correctly created. I
> attempted RequirePackage("guava"); and received
> several undefined variable errors (I forget what the
> exact GAP error message is and am not at my machine).
>
> The following changes fixed the problem:
> In lib\codeman.gi:
> InstallMethod(ConstantWeightSubcode, "method for
> linear code, weight", true,
> [IsLinearCode, IsInt], 0,
> function(C, wt)
> ###Comment out: local S, c, a, CWS,path;
> ##GEZ add the following:
> local S, c, a, CWS,path, F, tmpdir, incode,
> infile, inV, Els, i, D;
> In lib\decoders.gi:
> ###??????? insert Irons' code here ??????
> ###GEZ: I just return 0 for the moment
> return 0;
>
> and all of the tests I have had time to run worked
> except for two that I will report to you later, once I
> have had a chance to make sure that they are not an
> artifact of my
> GAP setup.
>
> I hope this is of some help and thanks for the great
> work,
> Gary Zablackis
>
> __________________________________________________
> Do You Yahoo!?
> Tired of spam? Yahoo! Mail has the best spam protection around
> http://mail.yahoo.com
>
>
>
> ------------------------------
>
> Message: 5
> Date: Thu, 28 Oct 2004 18:48:19 +0200 (CEST)
> From: Laurent Bartholdi <laurent.bartholdi at epfl.ch>
> Subject: [GAP Forum] a problem with matrix algebras and homomorphisms
> To: Gap 4 Forum <forum at gap-system.org>
> Message-ID: <Pine.LNX.4.58.0410281843050.12846 at madpc1.epfl.ch>
> Content-Type: TEXT/PLAIN; charset=iso8859-1
>
> hi!
> i'm trying to set up the following:
>
> A := FreeAssociativeAlgebraWithOne(Integers,"s","f");
> s := A.1; f := A.2; z := Zero(A); o := One(A);
> M := MatrixAlgebra(A,2);
> phi := AlgebraHomomorphismByImages(A,M,[s,f],[[[z,o],[o,z]],[[f,z],[s,z]]]);
>
> i.e. a free associative algebra A, and a map phi from A to 2x2 matrices
> over itself.
>
> unfortunately, gap chokes saying
> Error, <S> and <R> must have same left acting domain called from ...
>
> now on one side it's true that they don't have the same acting domain; all
> that's needed is that <R> _contain_ the left acting domain of <S>.
> so i tried to double-think gap with "M!.LeftActingDomain := Integers",
> but then i get in to the error that the matrix entries are not in
> Integers.
>
> thanks for your help,
> laurent
>
> --
> Laurent Bartholdi \ laurent.bartholdi<at>epfl<dot>ch
> EPFL, IGAT, Bâtiment BCH \ Téléphone: +41 21-6930380
> CH-1015 Lausanne, Switzerland \ Fax: +41 21-6930385
>
>
>
> ------------------------------
>
> Message: 6
> Date: Thu, 28 Oct 2004 19:18:54 +0200
> From: Marco Costantini <costanti at science.unitn.it>
> Subject: Re: [GAP Forum] reza orfi
> To: Reza Orfi <reza_orfi at yahoo.com>, Forum at gap-system.org
> Message-ID: <200410281918.54797.costanti at science.unitn.it>
> Content-Type: text/plain; charset="iso-8859-1"
>
> Dear all,
>
> On Wednesday 27 October 2004 10:54, Reza Orfi wrote:
> > Dear forum.
> > with many thanks.
> > I need all groups of order 3^7,3^8,3^9,3^10,5^7,5^8,5^9,5^10.
> > but there are not in gap4r4.
>
> As Bettina Eick pointed out, all the groups of these orders are not available,
> and it is suggested to restrict to the groups really needed. In fact, some
> classes of groups can be computed more easily.
>
> I have lists of the groups of orders 2^10, 3^7, 3^8, 3^9, 5^7 and rank (number
> of generators) 1 and 2, and of order 5^6 and rank 1, 2, 3, and 4.
> These lists were calculated with the GrpConst package, and a few weeks of CPU
> time. They include all the groups of the given orders and ranks, but are not
> duplicate free. At the moment, these lists are roughly organized as a gap4.3
> package.
>
> Calculating the groups of higher order in this way would require months of
> machine time, and calculating groups of higher rank would require to solve
> problems related to the memory management.
>
> If you are interested, I can give you these lists.
>
> Best regard,
> Marco Costantini
>
>
>
> ------------------------------
>
> Message: 7
> Date: Fri, 29 Oct 2004 07:06:41 -0400
> From: David Joyner <wdj at usna.edu>
> Subject: [GAP Forum] Schreier-Sims for matrix gps paper
> To: GAP forum <forum at gap-system.org>
> Message-ID: <41822441.2060209 at usna.edu>
> Content-Type: text/plain; charset=us-ascii; format=flowed
>
> FYI: A new paper on implementing Schreier-Sims alg for matri groups in
> GAP by
> H Baarnhielm has appeared on
> the math arcxiv: http://arxiv.org/abs/math/0410593
>
> Here's the abstract:
> This is the report of a project with the aim to make a new
> implementation of the Schreier-Sims algorithm in GAP, specialized for
> matrix groups. The standard Schreier-Sims algorithm is described in some
> detail, followed by descriptions of the probabilistic Schreier-Sims
> algorithm and the Schreier-Todd-Coxeter-Sims algorithm. Then we discuss
> our implementation and some optimisations, and finally we report on the
> performance of our implementation, as compared to the existing
> implementation in GAP, and we give benchmark results. The conclusion is
> that our implementation in some cases is faster and consumes much less
> memory.
>
>
>
> ------------------------------
>
> Message: 8
> Date: Tue, 2 Nov 2004 12:23:47 +0000
> From: Steve Linton <sal at dcs.st-and.ac.uk>
> Subject: [GAP Forum] Dangerous bug in Saving Workspaces (was "using
> GAP workspace")
> To: GAP Forum <forum at gap-system.org>
> Message-ID: <20041102122347.5b2f94df at caolila.dcs.st-and.ac.uk>
> Content-Type: text/plain; charset=US-ASCII
>
>
> Dear GAP Forum,
>
> A few days ago Alex Makosi reported problems with certain objects in a
> saved GAP workspace not loading correctly. It turns out that the cause of this
> problem is the code that saves Boolean lists (blists). Any blist with more than
> 32 entries would almost certainly not be saved correctly, and might load with
> wrong values or even as a completely corrupt object. This bug has been present
> in all releases of GAP 4.
>
> A fix for this problem is available and will be included in release 4.4.4 due
> in a few days. However, all workspaces created by SaveWorkspace up to now must
> be regarded as suspect. Boolean lists are used quite widely in the library (for
> instance in the code for transversals of subgroups of permutation groups), so
> you may be using them even if you do not think you are. We must therefore
> recommend that you recheck any results created using data from saved workspace
> if you have any reason to doubt them.
>
> We apologize for any inconvenience caused by this problem.
>
> Steve Linton
>
> --
> 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
>
>
>
> ------------------------------
>
> Message: 9
> Date: Tue, 2 Nov 2004 22:25:47 +0200
> From: Alexander Konovalov <gap at gap.zssm.zp.ua>
> Subject: [GAP Forum] Sophus package for Lie algebras is accepted
> To: GAP Forum <forum at gap-system.org>
> Message-ID: <1532396636.20041102222547 at gap.zssm.zp.ua>
> Content-Type: text/plain; charset=us-ascii
>
> Dear GAP Forum,
>
> I am very glad to announce that the Sophus package by Csaba Schneider
> has been accepted as a refereed GAP package. The package is now
> available for download from the GAP www-page http://www.gap-system.org
> or FTP sites, and also from the author's homepage at
>
> http://www.sztaki.hu/~schneider/Research/Sophus/
>
> Following author's description of the package, it is written for
> computations with nilpotent Lie algebras over finite prime fields.
> Using this package, you can compute the cover, the list of immediate
> descendants, and the automorphism group of such Lie algebras. You can
> also test if two such Lie algebras are isomorphic.
>
> The immediate descendant function of the package can be used to
> classify small-dimensional nilpotent Lie algebras over a given field.
> For instance, the package author obtained a classification of
> nilpotent Lie algebras with dimension at most 9 over F_2; see
> www.sztaki.hu/~schneider/Research/SmallLie.
>
> The Sophus package works in any operating system where the GAP system
> works. Please note that to use Sophus you will also need to install
> the AutPGrp package (version >= 1.2) by Bettina Eick and Eamonn
> O'Brien, which is available from the GAP site or from
> http://www.tu-bs.de/~beick/so.html.
>
> To use Sophus online help, it is necessary to install the GAP4 package
> GAPDoc by Frank Luebeck and Max Neunhoeffer, which is available from
> the GAP site or http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/.
>
>
> Alexander Konovalov
>
>
>
>
> ------------------------------
>
> Message: 10
> Date: Thu, 4 Nov 2004 13:17:22 -0800 (PST)
> From: Reza Orfi <reza_orfi at yahoo.com>
> Subject: [GAP Forum] p-group of maximal class
> To: Forum at gap-system.org
> Message-ID: <20041104211722.28402.qmail at web53405.mail.yahoo.com>
> Content-Type: text/plain; charset=us-ascii
>
> Dear gap-forum.
> we have two questions about p-groups of maximal class.
> 1)let G be a p-group of maximal class with order p^n (n>=5).
> what do you know about the structure of G/Z(G)?
> for exampel we know that exp(G/Z(G))=p and G/Z(G) is of maximal class
> and has two generators.
> we want to know if anyone found a presentation for G/Z(G) or not.
> 2) let G be p-group of maximal class with order p^n (n>=5).
> we want to know some information about Aut(G).
> with best regards.
> REZA ORFI & SHIRIN FOULADI
>
>
> __________________________________________________
> Do You Yahoo!?
> Tired of spam? Yahoo! Mail has the best spam protection around
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>
> ------------------------------
>
> Message: 11
> Date: Thu, 04 Nov 2004 17:18:45 -0500
> From: Mowsey <gapforum at mowsey.org>
> Subject: [GAP Forum] Re: p-group of maximal class
> To: forum at gap-system.org
> Message-ID: <200411042237.iA4MbUow023404 at gap-system.org>
>
> My apologies in advance for such a partial reply. I still don't
> know enough about groups of maximal class to be very useful.
>
> All p-groups of maximal class are descendents of a group of maximal
> class from a particular family, and the descendents can be described
> fairly explicitly by twistings. This is described in chapter 8
> of Leedham-Green and McKay's "Structure of Groups of Prime Power
> Order". Prof. Eick has made these descriptions concrete in her
> paper "On the determination of the uniserial space groups with a
> given coclass."
>
> I have not myself been able to construct all groups of maximal
> class, but I have written a very simple routine to generate the
> infinite family. Perhaps it may help to at least have one example
> of every order, though this is obviously a far cry from having
> all such groups.
>
> # pHedralGroupCons(p,n) returns a group of order p^n and nilpotency
> # class p^(n-1). See Leedham-Green&McKay, example 3.1.5.ii, example
> # 7.4.14.i, and proposition 8.2.3.iii
>
> pHedralGroupCons:=function(p,n)
> local F,rels,i,j,gens;
> F:=FreeGroup(IsSyllableWordsFamily,n+1);
> gens:=GeneratorsOfGroup(F);
> gens:=Concatenation(gens,ListWithIdenticalEntries(p+1,One(F)));
> rels:=[];
> Add(rels,gens[1]^p);
> for i in [2..n+1] do Add(rels,gens[i]^p/gens[i+p-1]); od;
> for i in [2..n+1] do
> if(1 = i mod (p-1) or p=2) then Add(rels, Comm(gens[i],gens[1])*
> Product(List([i+2-p..i],j->gens[j+p-1]^(Binomial(p,1+(j-2) mod (p-1))/p))));
> else Add(rels,Comm(gens[i],gens[1])/gens[i+1]);
> fi;
> for j in [2..i-1] do Add(rels,Comm(gens[i],gens[j])); od;
> od;
> rels:=Filtered(rels,x->not IsOne(x));
> return RefinedPcGroup(PcGroupFpGroup(F/rels));
> end;
>
>
>
>
> ------------------------------
>
> Message: 12
> Date: Wed, 10 Nov 2004 18:10:21 +0100
> From: Thomas Breuer <thomas.breuer at math.rwth-aachen.de>
> Subject: Re: [GAP Forum] a problem with matrix algebras and
> homomorphisms
> To: forum at gap-system.org
> Message-ID: <E1CRvzB-0000QC-00 at altair>
>
> Dear GAP Forum,
>
> Laurent Bartholdi wrote
>
> > i'm trying to set up the following:
> >
> > A :=3D FreeAssociativeAlgebraWithOne(Integers,"s","f");
> > s :=3D A.1; f :=3D A.2; z :=3D Zero(A); o :=3D One(A);
> > M :=3D MatrixAlgebra(A,2);
> > phi :=3D AlgebraHomomorphismByImages(A,M,[s,f],[[[z,o],[o,z]],[[f,z],=
> > [s,z]]]);
> >
> > i.e. a free associative algebra A, and a map phi from A to 2x2 matric=
> > es
> > over itself.
> >
> > unfortunately, gap chokes saying
> > Error, <S> and <R> must have same left acting domain called from ...
> >
> > now on one side it's true that they don't have the same acting domain=
> > ; all
> > that's needed is that <R> _contain_ the left acting domain of <S>.
> > so i tried to double-think gap with "M!.LeftActingDomain :=3D Integer=
> > s",
> > but then i get in to the error that the matrix entries are not in
> > Integers.
>
> I am sorry but the GAP machinery is not developed very far in the
> area of algebras and algebra homomorphisms.
> What works is the case of finite dimensional algebras over fields,
> which are treated mainly via linear algebra.
> (As indicated in the question, it is reasonable to extend the
> current behaviour such that also F-linear maps from a space V over F
> into a space W over K are supported, if K is an F-space;
> currently it is necessary to replace W explicitly by the corresponding
> F-space.)
>
> If I understand the underlying wish correctly then it is intended to
> map an element in a free associative algebra to a matrix.
> This can be achieved as follows.
>
> gap> A:= FreeAssociativeAlgebraWithOne( Integers, "s", "f" );;
> gap> z:= Zero(A);; o:= One(A);;
> gap> s:= A.1;; f:= A.2;;
> gap> mats:= [ [[z,o],[o,z]], [[f,z],[s,z]] ];;
> gap> MappedExpressionForElementOfFreeAssociativeAlgebra( s*f, [s,f], mats );
> [ [ (1)*s, <zero> of ... ], [ (1)*f, <zero> of ... ] ]
>
> One final comment:
> Overwriting components in GAP objects with the `!.' operator
> cannot be recommended.
> In fact, I would not trust any result of a computation with such
> an object.
> Note that GAP is able to store a lot of information inside its objects,
> and exchanging part of this by hand will almost certainly break the
> consistency of the object.
>
> All the best,
> Thomas
>
>
>
>
> ------------------------------
>
> _______________________________________________
> Forum mailing list
> Forum at mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>
>
> End of Forum Digest, Vol 12, Issue 1
> ************************************
>
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