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 [EMAIL PROTECTED] if you need more help.)
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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.
