[GAP Forum] Cartesian Group Direct Product

[PAUL HJELMSTAD]

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 
Polya Polynomial
has proved difficult, merely because of a problem with variables. Thanks

PGH


>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  
>>non-Cartesian
>>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 
>>issues
>>
>>I need these generators for D4 X S3:
>>
>>(0,3,6,9)(1,4,7,10)(2,5,8,11)
>>(0,4,8)(1,5,9)(2,6,10)(3,7,11)
>>(1,7)(3,9)(5,11)
>>(1,11)(2,10)(3,9)(4,8)(5,7)
>>
>>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:
>
>gap> d4:=DihedralGroup(IsPermGroup,8);
>Group([ (1,2,3,4), (2,4) ])
>gap> s3:=SymmetricGroup(3);
>Sym( [ 1 .. 3 ] )
>gap> d:=DirectProduct(d4,s3);
>Group([ (1,2,3,4), (2,4), (5,6,7), (5,6) ])
>gap> cart:=Cartesian([1..4],[5..7]);
>[ [ 1, 5 ], [ 1, 6 ], [ 1, 7 ], [ 2, 5 ], [ 2, 6 ], [ 2, 7 ], [ 3, 5 ],
>   [ 3, 6 ], [ 3, 7 ], [ 4, 5 ], [ 4, 6 ], [ 4, 7 ] ]
>gap> hom:=ActionHomomorphism(d,cart,OnTuples,"surjective");
><action epimorphism>
>gap> prod:=Image(hom);
>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.
>
>
>Best wishes,
>
>     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
>http://www.math.colostate.edu/~hulpke
>
>