[GAP Forum] Cartesian Group Direct Product
phjelmstad at msn.com
Sat Jan 5 18:05:33 GMT 2008
Dear Alexander Hulpke, Dear GAP-Forum,
Thanks! I realized after I sent my last message, that I was not
understanding the meaning of "Cartesian" correctly, but I see you understood
what I meant. Someone else called what I am looking for the "cross-product"
representation, or what you call the transitive product representation.
Finding the CycleIndex should be relatively easy here, but expanding to the
has proved difficult, merely because of a problem with variables. Thanks
>From: Alexander Hulpke <hulpke at math.colostate.edu>
>To: PAUL HJELMSTAD <phjelmstad at msn.com>
>CC: GAP Forum <forum at gap-system.org>
>Subject: Re: [GAP Forum] Cartesian Group Direct Product
>Date: Fri, 4 Jan 2008 12:35:13 -0700
>Dear Paul Hjelmsted, Dear GAP-Forum,
>>The problem with this is that it merely seems to shuffle between the
>>form of the permutations, that is, for example, sending (123) to (567)
>>(Second Embedding) or merely leaving at at (123) (First Embedding) but I
>>may be doing something wrong.
>>I am not getting anything Cartesian-wise. Perhaps I must leave D4
>>(actually called Dihedral(8)),
>>as a pc-group and not a perm group?
>>After I get this right, I need to generate the CycleIndex, and then
>>expand it in a manner you indicated, to get the full Polya Polynomial,
>>whose coefficients will be useful to me (especially to find how many
>>octads there are under D8 X S3 (Dihedral(16) X Symmetric(3)) and other
>>I need these generators for D4 X S3:
>>Or something with the same meaning
>OK. You want a different representation for the direct product. GAP gives
>you by default the intransitive action (which has smaller degree), you
>would prefer the transitive product action.
>The easiest way to construct this group is to let the intransitive direct
>product act on the cartesian product of the domains:
>Group([ (1,2,3,4), (2,4) ])
>Sym( [ 1 .. 3 ] )
>Group([ (1,2,3,4), (2,4), (5,6,7), (5,6) ])
>[ [ 1, 5 ], [ 1, 6 ], [ 1, 7 ], [ 2, 5 ], [ 2, 6 ], [ 2, 7 ], [ 3, 5 ],
> [ 3, 6 ], [ 3, 7 ], [ 4, 5 ], [ 4, 6 ], [ 4, 7 ] ]
>Group([ (1,4,7,10)(2,5,8,11)(3,6,9,12), (4,10)(5,11)(6,12),
> (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,2)(4,5)(7,8)(10,11) ])
>Up to labelling (which is due to the arrangement of the pairs in `cart')
>these are the generators you listed. You can use the map `hom' to go back
>to the intransitive direct product and use its decomposition functions.
> Alexander Hulpke
>-- Colorado State University, Department of Mathematics,
>Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
>email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
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