[GAP Forum] Cartesian Group Direct Product

[Sven Reichard]

PAUL HJELMSTAD wrote:
> Can someone tell me what the right function is for the "Cartesian" style 
> group direct product?
> (For example, D4 X S3)? I know they use the double-format for dihedral 
> groups, so this would
> be Dihedral(8) I guess.
> 
> The one I have been playing with isn't right for my purposes, taking 
> sums instead of Cartesian products. I need the form that ends up with 12 
> terms in its cycle-structure G X H
> 
> Thanks
> 

Paul,

the following may be what you were looking for:

DirectProductProductAction := function ( G, H )
     local  sum, domain, product;
     sum := DirectProduct( G, H );
     domain := Cartesian( [ 1 .. LargestMovedPoint( G ) ],
        [ 1 .. LargestMovedPoint( H ) ] + LargestMovedPoint( G ) );
     product := Action( sum, domain, OnTuples );
     return product;
end


gap> D4 := Group((1,2,3,4),(1,3));
Group([ (1,2,3,4), (1,3) ])
gap> S3 := SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> DirectProductProductAction(D4, S3);
Group([ (1,4,7,10)(2,5,8,11)(3,6,9,12), (1,7)(2,8)(3,9),
   (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,2)(4,5)(7,8)(10,11) ])

Sven.
--
Sven Reichard
School of Mathematics and Statistics
The University of Western Australia
35 Stirling Highway
Crawley 6009
Western Australia