[GAP Forum] Working with restriction of WreathProduct's Projection

[Stefan Kohl]

Dear Jacob, dear Forum,

what you can do is to represent your group as a permutation group
acting on the integers. Then such computations are easy:

gap> LoadPackage("rcwa");
gap> AllTransitiveGroups(DegreeAction,12,Size,60);
[ A_5(12) ]
gap> A5 := last[1]; # your A5 embedded in S12
A_5(12)
gap> F1 := CyclicGroup(IsRcwaGroupOverZ,infinity);
<tame rcwa group over Z with 1 generator, of order infinity>
gap> G := WreathProduct(F1,A5);
<tame rcwa group over Z with 3 generators, of order infinity>
gap> P := RespectedPartition(G);
[ 0(12), 1(12), 2(12), 3(12), 4(12), 5(12), 6(12), 7(12), 8(12), 9(12),
   10(12), 11(12) ]
gap> t0 := G.1*G.2;;   # choose any subgroup generators
gap> t1 := G.2*G.3^2;; #
gap> H := Group(t0,t1);
<rcwa group over Z with 2 generators>
gap> Permutation(t0*t1,P); # compute image under projection
(1,11)(2,3)(4,6)(5,7)(8,9)(10,12)

If you need to form homomorphism objects performing in this way,
then you would need to add suitable methods to the RCWA package
(that being a routine task -- it just hasn't been done so far).

Does this help you?

     Stefan

P.S.: Just in case -- the repository for RCWA is here:
https://github.com/gap-packages/rcwa

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Am 09.02.2018 um 19:44 schrieb Jacob Bond:
> Hi,
>
>
> I have constructed F1 wr A5, where F1 is the free group on 1 generator and A5 is embedded in S12, and have the projection map of the wreath product.  However, I would like to restrict the projection to a subgroup of the wreath product given by two generators t0, t1.  I've tried constructing the restriction in a few different ways:  RestrictedMapping, GroupHomomorphismByImages, and GroupHomomorphismByFunction.  None of these would return, and while GroupHomomorphismByImagesNC returned the homomorphism, it was unable to compute the image even of t0*t1.  Is there a better way to construct the restriction of the projection map?  Or am I probably out of luck?
>
>
> Thank you,
>
> Jacob
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