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

[Roman Schmied]

Sorry for initial posting to the wrong address.

I am much more interested in question 2. StructureDescription and  
DirectFactorsOfGroup would be what I am looking for in the case of  
direct products, but they seem to crash or give incomplete results  
quite easily (I am running the latest version of GAP):

gap> G := DirectProduct( CyclicGroup(2) , CyclicGroup(3) ,  
SymmetricGroup(4) );;
gap> DirectFactorsOfGroup(G);
     --> crashes

gap> G := DirectProduct( CyclicGroup(2) , CyclicGroup(3) , CyclicGroup 
(4) );;
gap> StructureDescription(G);
"C12 x C2"
     --> incomplete decomposition

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.
>>
>
>