[GAP Forum] Semidirect Product (second variant)

[Marc Keilberg]

The problem is in how you define J.  As is, GAP recognizes it as a finitely
presented group (IsFpGroup(J) returns true) but your subgroup of
automorphisms is not stored as an Fp group. Someone more familiar with the
inner particulars would have to explain exactly why this results in an
error, but it's easy to avoid it.  The easiest way to fix this is to define
J without resorting to quotients of free groups:

J := DirectProduct(CyclicGroup(4),CyclicGroup(4));

Now run the rest of your code and it's error free.

Marc

On Mon, Oct 22, 2018 at 6:42 AM, William Giuliano <
williamgiuliano00 at gmail.com> wrote:

> Dear Forum,
>                      I would like to construct a semidirect product of C4 x
> C4 with S3 in the following way. I feel it's something easy, but I cannot
> understand the error message I get.
>
> gap> free_j:=FreeGroup("a","b");;
>
> gap> AssignGeneratorVariables(free_j);;
>
> #I  Assigned the global variables [ a, b ]
>
> gap> rel_j:=[a^4,b^4,Comm(a,b)];;
>
> gap> J:=free_j/rel_j;;
>
> gap> StructureDescription(J);
>
> "C4 x C4"
>
>
> gap> A:=AutomorphismGroup(J);;
>
> gap> a:=Elements(A);;
>
> gap> x:=a[16];
>
> [ a, b ] -> [ b, (a*b)^3 ]
>
> gap> s:=a[9];
>
> [ a, b ] -> [ b, a ]
>
> gap> H:=Group([x,s]);;
>
> gap> StructureDescription(H);
>
> "S3"
>
>
> gap> IsGroupOfAutomorphisms(H);
>
> true
>
> gap> AutomorphismDomain(H)=J;
>
> true
>
> gap> SemidirectProduct(H,J);
>
> Error, no method found! For debugging hints type ?Recovery from
> NoMethodFound
>
> Error, no 1st choice method found for `IsomorphismFpGroup' on 2 arguments
> at /Users/williamgiuliano/Downloads/gap-4.9.3/lib/methsel2.g:250 called
> from
>
> IsomorphismFpGroup( G, "F" ) at
> /Users/williamgiuliano/Downloads/gap-4.9.3/lib/gpfpiso.gi:21 called from
>
> IsomorphismFpGroup( G ) at /Users/williamgiuliano/Downloads/gap-4.9.3/lib/
> gprd.gi:1127 called from
>
> SemidirectProduct( G, IdentityMapping( G ), N ) at
> /Users/williamgiuliano/Downloads/gap-4.9.3/lib/gprd.gi:1077 called from
>
> <function "unknown">( <arguments> )
>
>  called from read-eval loop at *stdin*:16
>
> you can 'quit;' to quit to outer loop, or
>
> you can 'return;' to continue
>
> brk>
>
>
> Thank you ver much
>
> William
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