Dear Alexander,
I cannot thank you enough,
> gap> RepresentativeAction(SymmetricGroup(12),A,B);
> (2,3)(4,8)(6,7)(9,12)(10,11)
is exactly what I need.
-Tim
On Mon, 23 Apr 2018, Hulpke,Alexander wrote:
> Dear Forum, Dear Tim Kohl,
>
> > gap> A:=Group([ (1,2)(3,5)(4,6)(7,9)(8,10)(11,12),
> > (1,3)(2,5)(4,7)(6,9)(8,11)(10,12), (1,4,8)(2,10,6)(3,7,11)(5,12,9) ]);
> > Group([ (1,2)(3,5)(4,6)(7,9)(8,10)(11,12),
> > (1,3)(2,5)(4,7)(6,9)(8,11)(10,12), (1,4,8)(2,10,6)(3,7,11)(5,12,9) ])
> > gap> B:=Group([ (1,5)(2,3)(4,9)(6,7)(8,12)(10,11),
> > (1,3)(2,5)(4,11)(6,12)(7,8)(9,10), (1,8,4)(2,6,10)(3,11,7)(5,9,12) ]);
> > Group([ (1,5)(2,3)(4,9)(6,7)(8,12)(10,11),
> > (1,3)(2,5)(4,11)(6,12)(7,8)(9,10), (1,8,4)(2,6,10)(3,11,7)(5,9,12) ])
> > gap> IdGroup(A);
> > [ 12, 4 ]
> > gap> IdGroup(B);
> > [ 12, 4 ]
> > gap> GroupHomomorphismByImages(A,B);
> > fail
> > gap> GroupHomomorphismByImages(A,AllSmallGroups(12)[4]);
> > [ (1,2)(3,5)(4,6)(7,9)(8,10)(11,12), (1,3)(2,5)(4,7)(6,9)(8,11)(10,12),
> > (1,4,8)(2,10,6)(3,7,11)(5,12,9) ] -> [ f1, f2, f3 ]
> > gap> GroupHomomorphismByImages(B,AllSmallGroups(12)[4]);
> > fail
> >
> > I guess my question is, how does IdGroup determine
> > that a given group is in an isomorphism class of one
> > of the groups in the SmallGroups library.
>
> Briefly, it first determines a number of isomorphism-invariant properties. If
> this does not leave a unique candidate it tries to find elements in a pcgs
> that correspond to the presentation in the library group.
>
> > I realize that the generators of A satisfy the relations of the generators
> > of AllSmallGroups(12)[4]
> > which is why GroupHomomorphismByImages(A,AllSmallGroups(12)[4]) succeeds
> > whereas those of B
> > do not match which is why
> > GroupHomomorphismByImages(B,AllSmallGroups(12)[4]) fails.
> >
> > Is there a way to 'correct' the generating set of B so that I *can*
> > construct a homomorphism from A to B
> > (or from B to AllSmallGroups(12)[2]) which I could also use.
>
> Generically, you could call
> iso:=IsomorphismGroups(A,B);
> to find such an isomorphism.
>
> > I am trying to take advantage of the ismomorphic = conjugate property of
> > regular permutation
> > groups of the same degree to construct an element of S_12 which conjugates
> > A to B, but this
> > homomorphism failure is getting in the way.
>
> gap> RepresentativeAction(SymmetricGroup(12),A,B);
> (2,3)(4,8)(6,7)(9,12)(10,11)
>
> will find such a permutation.
>
> Best,
>
> Alexander Hulpke
>
> -- Colorado State University, Department of Mathematics,
> Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
> email: hul...@colostate.edu, Phone: ++1-970-4914288
> http://www.math.colostate.edu/~hulpke
>
>
>
--
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