[GAP Forum] IdGroup and GroupHomomorphism

[tkohl at math.bu.edu]

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: hulpke at colostate.edu, Phone: ++1-970-4914288
> http://www.math.colostate.edu/~hulpke
> 
> 
> 

--