From: Alexander Hulpke <[EMAIL PROTECTED]>
To: PAUL HJELMSTAD <[EMAIL PROTECTED]>
CC: GAP Forum <[EMAIL PROTECTED]>
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: [EMAIL PROTECTED], Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
