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 _______________________________________________ Forum mailing list Forum@gap-system.org https://mail.gap-system.org/mailman/listinfo/forum
