> < ^ From:

> < ^ Subject:

Dear Gap-Forum,

Avital Oliver wrote:

If I construct a wreath product using WreathProduct(G, P), when P is a

permutation group of degree k, how can I get a projection onto the k copies

of G in the direct product base of the wreath product? The documentation

makes it seem as if it is only possible to get the projection onto P. Is

this true?

The projections on the components are defined only on a subgroup of the

wreath product, and therefore not provided by GAP.

To get these, you have two possibilities, both requiring to get this

subgroup first:

Option a) constructs the projections from the embeddings in the base group:

gap> a5:=AlternatingGroup(5); Alt( [ 1 .. 5 ] ) gap> s3:=SymmetricGroup(3); Sym( [ 1 .. 3 ] ) gap> w:=WreathProduct(a5,s3); <permutation group of size 1296000 with 8 generators> gap> b:=Kernel(Projection(w)); Group([ (1,2,3,4,5), (3,4,5), (6,7,8,9,10), (8,9,10), (11,12,13,14,15), (13,14,15) ]) gap> Size(b); 216000 gap> agens:=GeneratorsOfGroup(a5); [ (1,2,3,4,5), (3,4,5) ] gap> egens:=List([1..3],i->List(agens,j->Image(Embedding(w,i),j))); [ [ (1,2,3,4,5), (3,4,5) ], [ (6,7,8,9,10), (8,9,10) ], [ (11,12,13,14,15), (13,14,15) ] ] # for example projection on 2nd component, Replace 2 by component number gap> proj2:=GroupHomomorphismByImages(b,a5,Concatenation(egens), > Concatenation(List([1..(2-1)*Length(agens)],i->One(a5)), > agens, > List([1..(3-2)*Length(agens)],i->One(a5))));

Option b) (works only for permutation groups) uses the fact that the base

group is as a permutation group the same as a direct power:

gap> b:=DirectProduct(a5,a5,a5); Group([ (1,2,3,4,5), (3,4,5), (6,7,8,9,10), (8,9,10), (11,12,13,14,15), (13,14,15) ]) gap> Projection(b,2); 2nd projection of Group([ (1,2,3,4,5), (3,4,5), (6,7,8,9,10), (8,9,10), (11,12,13,14,15), (13,14,15) ])

I hope this helps,

Alexander Hulpke

-- Colorado State University, Department of Mathematics,

Weber Building, Fort Collins, CO 80523, USA

email: hulpke@math.colostate.edu, Phone: ++1-970-4914288

http://www.math.colostate.edu/~hulpke

> < [top]