[GAP Forum] Semidirect Product (second variant)

[William Giuliano]

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

[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 <

Post by William Giuliano
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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