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Dear GAP-Forum,

Enrique Artal asked:

I have a question about Presentation of subgroups

of finitely presented subgroups. It is maybe trivial

but I do not find an answer in the manuals.

If we give a subgroup by a list of generators,

is it possible to describe these generators

in term of the new generators obtained by

either PresentationSubgroup or PresentationNormalClosure?

The `PresentationSubgroup' functions are a bit technical and make it

slightly complicated to get the necessary information to connect the

presentation to the subgroup. (The generators of the subgroup are held in

the attribute `PrimaryGeneratorWords' of the presentation, but that might

not be the only information you want.)

It is therefore usually much more convenient to use `IsomorphismFpGroup(u)'

or `IsomorphismFpGroupByGenerators(u,gens)' (in case a presentation in a

given generating system is desired) to obtain a finitely presented group

isomorphic to the subgroup u. You can take preimages of the generators

of the range to get the corresponding group elements.

For example:

gap> f:=FreeGroup(2);; gap> g:=f/[f.1^2,f.2^3,(f.1*f.2)^5]; <fp group on the generators [ f1, f2 ]> gap> u:=Subgroup(g,[g.1*g.2]); Group([ f1*f2 ]) gap> hom:=IsomorphismFpGroup(u); [ f2^-1*f1^-1 ] -> [ _x1 ] gap> new:=Range(hom); <fp group on the generators [ _x1 ]> gap> List(GeneratorsOfGroup(new),i->PreImagesRepresentative(hom,i)); [ f2^-1*f1^-1 ]

or

gap> hom:=IsomorphismFpGroupByGenerators(u,[u.1]);

[ f1*f2 ] -> [ _x1 ]

gap> new:=Range(hom);

<fp group on the generators [ _x1 ]>

gap> List(GeneratorsOfGroup(new),i->PreImagesRepresentative(hom,i));

[ f1*f2 ]

(Note that at the moment it is not yet possible to compute arbitrary *images*

under such a homomorphism in general. This feature exists already in our

development version and will be added in the next release.)

I hope this is of help,

Alexander Hulpke

PS: I also just amended the manual for the next version so that it will

point to `IsomorphismFpGroup' more prominently as a solution for this type

of problem. AH

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