[GAP Forum] RE: Forum Digest, Vol 62, Issue 2
Ndiweni, Odilo
ONdiweni at ufh.ac.za
Thu Jan 29 09:39:05 GMT 2009
Dear all
will love to identify elements of subgroups of dihedral groups using GAP.I am still a novice in the use of the language and will be glad to get all sort of ways.
sincerely
Odilo Ndiweni
Mathematics Dept (Pure and Applied)
UFH
Phone: 040-602-2370
Cell: 0762337318
Email: ondiweni at ufh.ac.za
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Today's Topics:
1. Transfinitely nilpotent groups (Nagy G?bor)
2. bag with IsomorphismPermGroup (Evgeny Vdovin)
3. Workshop on Applied Topology and Algebraic Statistics
(Ellis, Grahamj)
4. Sampling elements in conjugacy classes of a symmetric group
(Don King)
5. Permutation Notation (Don King)
6. Re: Permutation Notation (Bill Allombert)
7. Re: bag with IsomorphismPermGroup (Rene Hartung)
8. Re: Permutation Notation (Alexander Konovalov)
9. Quastion (Elaheh khamseh)
10. Array of ListPerm (Don King)
----------------------------------------------------------------------
Message: 1
Date: Mon, 19 Jan 2009 00:04:04 +0100
From: Nagy G?bor <nagyg at math.u-szeged.hu>
Subject: [GAP Forum] Transfinitely nilpotent groups
To: forum at gap-system.org
Message-ID: <4973B564.8010009 at math.u-szeged.hu>
Content-Type: text/plain; charset=ISO-8859-2; format=flowed
Dear Forum,
Can somebody gime me a construction of a transfinitely upper nilpotent
group of class bigger than omega?
That is, I define the upper central series of G by Z_0(G)=1,
Z_{k+1}(G)/Z_k(G)=Z(G/Z_k(G)) and Z_\kappa(G)=U_{k<\kappa} Z_k(G) for
limit ordinals.
I have Z_\kappa(G)=G for some ordinal \kappa. Are there examples where
\kappa>\omega? In particular, what about \kappa=\omega+1?
Thanks in advance, bye,
Gabor
------------------------------
Message: 2
Date: Thu, 22 Jan 2009 16:54:11 +0600
From: Evgeny Vdovin <vdovin at math.nsc.ru>
Subject: [GAP Forum] bag with IsomorphismPermGroup
To: forum at gap-system.org
Message-ID:
<9451da800901220254n2cf64b75o5b423df31864c951 at mail.gmail.com>
Content-Type: text/plain; charset=ISO-8859-1
Dear all,
This program runs ok on my GAP
F:=FreeGroup(1);
G:=F/[F.1^5]; # Z_5
h:=NqEpimorphismNilpotentQuotient(G,1);;
H:=Image(h);
Size(H); # critical line
IsomorphismPermGroup(H);
but if I remove the "critical line"
F:=FreeGroup(1);
G:=F/[F.1^5]; # Z_5
h:=NqEpimorphismNilpotentQuotient(G,1);;
H:=Image(h);
IsomorphismPermGroup(H);
it runs into error as follows:
gap> F:=FreeGroup(1);
<free group on the generators [ f1 ]>
gap> G:=F/[F.1^5]; # Z_5
<fp group on the generators [ f1 ]>
gap> h:=NqEpimorphismNilpotentQuotient(G,1);
[ f1 ] -> [ g1 ]
gap> H:=Image(h);
Pcp-group with orders [ 5 ]
gap> IsomorphismPermGroup(H);
Error, no method found! For debugging hints type ?Recovery from
NoMethodFound
Error, no 2nd choice method found for `IsomorphismPermGroup' on 1
arguments called from
<function>( <arguments> ) 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>
I have no idea why choosing not to calculate Size(H) makes the program
break.
Can anyone help me out?
--
Best Regards
Prof. Vdovin Evgeny
Institute of Mathematics
pr-t Acad. Koptyug, 4
630090, Novosibirsk, Russia
Office +7 383 3634540
Cellular +7 913 9475524
Fax +7 383 3332598
------------------------------
Message: 3
Date: Fri, 23 Jan 2009 17:26:36 -0000
From: "Ellis, Grahamj" <graham.ellis at nuigalway.ie>
Subject: [GAP Forum] Workshop on Applied Topology and Algebraic
Statistics
To: <forum at gap-system.org>
Message-ID:
<47C2E007B3E98F4E8BBC7997F007CE1303AD27DA at EVS1.ac.nuigalway.ie>
Content-Type: text/plain; charset="iso-8859-1"
DE BRUN WORKSHOP ON APPLIED TOPOLOGY AND ALGEBRAIC STATISTICS
(FIRST ANNOUNCEMENT)
>From June 29 to July 10, 2009, the de Brun Centre for Computational Algebra at NUI Galway, Ireland, is running a workshop consisting of the following four 5-lecture courses
* Gunnar Carlsson (Stanford)
Applied Topology
* Marian Mrozek (Krakow)
Computational Homology
* Eva Riccomagno (Genoa)
Algebraic Statistics
*Henry Wynn (LSE)
Algebraic Statistics
plus talks contributed by participants.
The lecture courses are aimed at mathematicians with a general interest in computational aspects of algebra, but who don't necessarily have expertise in the topics of the courses.
The workshop is supported by Science Foundation Ireland. Some funding towards the cost of accommodation is available for a limited
number of participants.
For further information see:
http://hamilton.nuigalway.ie/DeBrunCentre/SecondWorkshop.html
The organizers,
John Burns
Graham Ellis
Emil Skoldberg
------------------------------
Message: 4
Date: Sun, 25 Jan 2009 22:36:38 -0800 (PST)
From: Don King <symmetryholic at yahoo.ca>
Subject: [GAP Forum] Sampling elements in conjugacy classes of a
symmetric group
To: forum at gap-system.org
Message-ID: <681731.60975.qm at web111008.mail.gq1.yahoo.com>
Content-Type: text/plain; charset=us-ascii
Hello,
I`d like to sample some elements in a symmetric group of order n based on the ratio of conjugacy classes.
For instance, if a symmetric group just has five conjugacy classes (this is just for illustration, not an actual symmetric group),
Class 1: 10 memeber
Class2: 20 members
Class3: 30 memebers
Class4: 20 members
Class5: 10 memebers
is it possible to pick random samples like 1,2,3,2,1 elements from each class based on the ratio of each conjugacy classes ?.
Is there any way to pick random samples from each conjugacy class ?.
Thank you.
Sincerely,
Don
__________________________________________________________________
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------------------------------
Message: 5
Date: Mon, 26 Jan 2009 01:31:02 -0800 (PST)
From: Don King <symmetryholic at yahoo.ca>
Subject: [GAP Forum] Permutation Notation
To: forum at gap-system.org
Message-ID: <828310.9109.qm at web111005.mail.gq1.yahoo.com>
Content-Type: text/plain; charset=iso-8859-1
Hello,
I am wondering if I can convert a cycle into a different notation.
gap> ConjugacyClasses(s10);
.....
(1,2,5,10)
......
For instance, above one represents a mapping 1->2, 2->5, 5->10, 10->1.
If I use a permutation notation for above one,
1 2 3 4 5 6 7 8 9 10
2 5 3 4 10 6 7 8 9 1
is it possible to print a permutation notation (or other similar notation) rather than a cycle notation ?
Thanks in advance.
Don
__________________________________________________________________
Yahoo! Canada Toolbar: Search from anywhere on the web, and bookmark your favourite sites. Download it now at
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Message: 6
Date: Mon, 26 Jan 2009 13:20:17 +0100
From: Bill Allombert <Bill.Allombert at math.u-bordeaux1.fr>
Subject: Re: [GAP Forum] Permutation Notation
To: forum at gap-system.org
Message-ID: <20090126122017.GE25802 at yellowpig>
Content-Type: text/plain; charset=iso-8859-1
On Mon, Jan 26, 2009 at 01:31:02AM -0800, Don King wrote:
> Hello,
> I am wondering if I can convert a cycle into a different notation.
> gap> ConjugacyClasses(s10);
> .....
> (1,2,5,10)
> ......
>
> For instance, above one represents a mapping 1->2, 2->5, 5->10, 10->1.
> If I use a permutation notation for above one,
>
> 1 2 3 4 5 6 7 8 9 10
> 2 5 3 4 10 6 7 8 9 1
>
> is it possible to print a permutation notation (or other similar notation) rather than a cycle notation ?
You can use
Permuted([1..10],(1,2,5,10)^-1);
Cheers,
Bill.
------------------------------
Message: 7
Date: Mon, 26 Jan 2009 13:26:25 +0100
From: Rene Hartung <r.hartung at tu-braunschweig.de>
Subject: Re: [GAP Forum] bag with IsomorphismPermGroup
To: forum at gap-system.org, Evgeny Vdovin <vdovin at math.nsc.ru>
Message-ID: <200901261326.26459.r.hartung at tu-braunschweig.de>
Content-Type: text/plain; charset="iso-8859-1"
Dear Evgeny,
sorry for the delay.
The method NqEpimorphismNilpotentQuotient computes an epimorphism onto a
PcpGroup. Since a PcpGroup might (in general) be infinite, the (current)
method implemented for IsomorphismPermGroup asks for a group lying in the
filter 'IsPcpGroup and IsFinite'.
When computing the size of the group, the filter 'IsFinite' is set and the
current method applies to your (finite) PcpGroup. If, on the other hand, the
PcpGroup does not know whether it is finite or not, then there's no method
implemented. This will be changed in an future update of polycyclic (it is
already contained in the cvs-branch of gap).
Cheers, ren\'e.
On Thursday 22 January 2009 11:54, Evgeny Vdovin wrote:
> Dear all,
> This program runs ok on my GAP
>
> F:=FreeGroup(1);
> G:=F/[F.1^5]; # Z_5
> h:=NqEpimorphismNilpotentQuotient(G,1);;
> H:=Image(h);
> Size(H); # critical line
> IsomorphismPermGroup(H);
>
> but if I remove the "critical line"
>
> F:=FreeGroup(1);
> G:=F/[F.1^5]; # Z_5
> h:=NqEpimorphismNilpotentQuotient(G,1);;
> H:=Image(h);
> IsomorphismPermGroup(H);
>
> it runs into error as follows:
>
>
> gap> F:=FreeGroup(1);
> <free group on the generators [ f1 ]>
> gap> G:=F/[F.1^5]; # Z_5
> <fp group on the generators [ f1 ]>
> gap> h:=NqEpimorphismNilpotentQuotient(G,1);
> [ f1 ] -> [ g1 ]
> gap> H:=Image(h);
> Pcp-group with orders [ 5 ]
> gap> IsomorphismPermGroup(H);
> Error, no method found! For debugging hints type ?Recovery from
> NoMethodFound
> Error, no 2nd choice method found for `IsomorphismPermGroup' on 1
> arguments called from
> <function>( <arguments> ) 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>
>
> I have no idea why choosing not to calculate Size(H) makes the program
> break.
> Can anyone help me out?
------------------------------
Message: 8
Date: Mon, 26 Jan 2009 21:07:26 +0000
From: Alexander Konovalov <alexander.konovalov at gmail.com>
Subject: Re: [GAP Forum] Permutation Notation
To: forum at gap-system.org
Cc: Don King <symmetryholic at yahoo.ca>, Bill Allombert
<Bill.Allombert at math.u-bordeaux1.fr>
Message-ID: <0A46F8F6-752C-4965-A2AF-13D42A5CD87B at gmail.com>
Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes
On 26 Jan 2009, at 12:20, Bill Allombert wrote:
> On Mon, Jan 26, 2009 at 01:31:02AM -0800, Don King wrote:
>> Hello,
>> I am wondering if I can convert a cycle into a different notation.
>> gap> ConjugacyClasses(s10);
>> .....
>> (1,2,5,10)
>> ......
>>
>> For instance, above one represents a mapping 1->2, 2->5, 5->10, 10-
>> >1.
>> If I use a permutation notation for above one,
>>
>> 1 2 3 4 5 6 7 8 9 10
>> 2 5 3 4 10 6 7 8 9 1
>>
>> is it possible to print a permutation notation (or other similar
>> notation) rather than a cycle notation ?
>
> You can use
> Permuted([1..10],(1,2,5,10)^-1);
Hello,
there is also a function ListPerm(perm) which returns a list that
contains the images of the positive integers under the permutation
perm. That means that list[i] = i^perm, where i lies between 1 and the
largest point moved by perm (see LargestMovedPoint).
For example,
gap> ListPerm((1,2,5,10));
[ 2, 5, 3, 4, 10, 6, 7, 8, 9, 1 ]
gap> ListPerm((1,2,5,9));
[ 2, 5, 3, 4, 9, 6, 7, 8, 1 ]
Note that the length of the list is equal to the
LargestMovedPoint(perm), so dependently on your problem you might be
interested in adding the "tail"
to the resulting list or might be not.
Also, the bigger is input the more visible is that ListPerm is faster,
e.g.:
gap> s:=Random(SymmetricGroup(10000));;
gap> LargestMovedPoint(s);
10000
gap> for i in [1..10000] do ListPerm(s);od;time;
2317
gap> for i in [1..10000] do Permuted([1..10000],s^-1);od;time;
5022
Best wishes,
Alexander
------------------------------
Message: 9
Date: Thu, 29 Jan 2009 10:31:30 +0330
From: Elaheh khamseh <elahehkhamseh at gmail.com>
Subject: [GAP Forum] Quastion
To: forum at gap-system.org
Message-ID:
<db123b0d0901282301v7134d591ifc607717b9b841c2 at mail.gmail.com>
Content-Type: text/plain; charset=ISO-8859-1
Dears
If possible to determin the automorphism of Z_{p^i} in GAP. I want to
know the number of fixed elements under automorphisms.
Yours;
Khamseh.
------------------------------
Message: 10
Date: Thu, 29 Jan 2009 01:10:43 -0800 (PST)
From: Don King <symmetryholic at yahoo.ca>
Subject: [GAP Forum] Array of ListPerm
To: forum at gap-system.org
Message-ID: <31971.97045.qm at web111012.mail.gq1.yahoo.com>
Content-Type: text/plain; charset=iso-8859-1
Hello,
I am having difficulty in converting cycles to the list of permutations.
gap> s := SymmetricGroup(6);
Sym( [ 1 .. 6 ] )
gap> ConjugacyClasses(s);
[ ()^G, (1,2)^G, (1,2)(3,4)^G, (1,2)(3,4)(5,6)^G, (1,2,3)^G, (1,2,3)(4,5)^G,
(1,2,3)(4,5,6)^G, (1,2,3,4)^G, (1,2,3,4)(5,6)^G, (1,2,3,4,5)^G,
(1,2,3,4,5,6)^G ]
gap> c := ConjugacyClass(s,(1,2,3)(4,5));
(1,2,3)(4,5)^G
gap> Size(c);
120
gap> for i in [1.. 120] do ListPerm(c);
> (Syntax Error !)
What I'd like to do is to get 120 list of permutations for above formatted like [1, 3, 4, 5, 2, 6], [ 2,3, 4, 5, 6, 1],etc.
Any help will be highly appreciated.
Don
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