[GAP Forum] Re: [GAP Support] wreath product factorization

[Steve Linton]

Just a brief thought, not a fully-thought out response, but for groups of this
size, you could certainly obtain all the normal subgroups and examine each one
to see if it is a direct product. If it is, and furthermore the quotient acts
faithfully and transitively by conjugation on the factors, and the extension is
split, all of which I think you can check, then the original group must be a
wreath product.

	Steve

> 
> Also, I am more interested in similar decompositions of wreath  
> products. Since my groups are small (at most a few thousand elements)  
> I could maybe write this myself, by looping over the "small groups"  
> database and trying all wreath products until I find one that is  
> isomorphic to the group I want to decompose. Is this reasonable?
> 
> Thanks!
> Roman.
> 
> 
> On Sep 14, 2005, at 13:20, Marco Costantini wrote:
> 
> > Dear Roman,
> > there are two different questions:
> > 1) given a group G which is the output of DirectProduct, find out  
> > its factors,
> > and similar for WreathProduct.
> > 2) given any group G, for instance
> > gap> G := Group( (1,2,3,4,5), (1,2), (6,7,8), (6,7) );
> > find out that G is the direct product of Group( (1,2,3,4,5),  
> > (1,2) ) and
> > Group( (6,7,8), (6,7) ). Similar for wreath product.
> >
> > Question 1 is easy. The information about direct product G is  
> > stored in the
> > record DirectProductInfo(G), and the component DirectProductInfo 
> > ( G ).groups
> > contains the list of factors.
> >
> > gap> G := DirectProduct( SymmetricGroup(5) , SymmetricGroup(3) );;
> > gap> DirectProductInfo( G );
> > rec( groups := [ Sym( [ 1 .. 5 ] ), Sym( [ 1 .. 3 ] ) ], olds :=  
> > [ [ 1 .. 5 ],
> > [ 1 .. 3 ] ],
> >   news := [ [ 1, 2, 3, 4, 5 ], [ 6, 7, 8 ] ], perms := [ (),  
> > (1,6,3,8,5,2,7,4)
> > ], embeddings := [  ],
> >   projections := [  ] )
> > gap> DirectProductInfo( G ).groups;
> > [ Sym( [ 1 .. 5 ] ), Sym( [ 1 .. 3 ] ) ]
> >
> > WreathProductInfo(G) is the analogous for wreath product.
> >
> > gap> G := WreathProduct( SymmetricGroup(5) , SymmetricGroup(3) );;
> > gap> WreathProductInfo(G).groups;
> > [ Sym( [ 1 .. 5 ] ), Sym( [ 1 .. 3 ] ) ]
> >
> > Question 2 is not so easy, but something can be done. A very recent  
> > version of
> > Gap is required; try DirectFactorsOfGroup:
> >
> > gap> G := Group( (1,2,3,4,5), (1,2), (6,7,8), (6,7) );;
> > gap> DirectFactorsOfGroup( G );
> > [ Group([ (6,7,8), (6,8) ]), Group([ (4,5), (1,5,2,4,3) ]) ]
> >
> > You may try also the function StructureDescription:
> >
> > gap> G := Group( (1,2,3,4,5), (1,2), (6,7,8), (6,7) );;
> > gap> StructureDescription( G );
> > "S3 x S5"
> >
> > For further information, see the file lib/grpnames.gd in your Gap
> > installation.
> >
> > For wreath product, I don't know any possibility.
> >
> > (This answer is a draft, more detailed information may come later,  
> > feel free
> > to write to support at gap-system.org if you need more help.)
> >
> > Please note that you have sent your mail to <support at gap- 
> > system.org> , which
> > is not the Gap forum. The address of Gap forum is <forum at gap- 
> > system.org>, and
> > should be used only for topic that are likely to interest many of  
> > the GAP
> > users.
> > For those topics that are more or less local to you, <support at gap- 
> > system.org>
> > is the right address.
> >
> > Best regards,
> > Marco Costantini
> >
> >
> >
> > On Wednesday 14 September 2005 16:58, Roman Schmied wrote:
> >
> >> dear GAP forum member,
> >>
> >> Is there a way to factorize groups with respect to the wreath
> >> product? For instance, given the group
> >>
> >> gap> g := WreathProduct( SymmetricGroup(5) , SymmetricGroup(3) );;
> >>
> >> I would like to decompose g and find that it is indeed the wreath
> >> product of S5 with S3. How can I do this? And what about the direct
> >> product
> >>
> >> gap> g := DirectProduct( SymmetricGroup(5) , SymmetricGroup(3) );;
> >>
> >> can this be factorized into S5 and S3?
> >>
> >> Cheers!
> >> Roman.
> >>
> >
> >
> 
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-- 
Steve Linton	School of Computer Science  &
      Centre for Interdisciplinary Research in Computational Algebra
	     University of St Andrews 	 Tel   +44 (1334) 463269
http://www.dcs.st-and.ac.uk/~sal	 Fax   +44 (1334) 463278