Dear GAP Forum, Rudolf Zlabinger wrote:
> I try to extend the number of points acted on by permutation groups to > multiples of the original domain. > For example i tried to find a subgroup of S60 isomorphic to A5, thus > acting on 60 points. > As the samples attached show, I did it by brute force, extending the > generators by acting on all residue classes mod the multiplier. > > This method yields subgroups of the desired target symmetric groups, but > only in one subgroup conjugacy class. The conjugacy class(es) I am > looking for, are those, whose subgroups are transitive, as needed, for > example, to be true rotation groups acting on the vertices of a solid. > > This was possible in the case of A5 extending to S12, but not to S30 or > even S60, as those targets are to big for executing, for example > IsomorphicSubgroups. > > In the case of finite rotation groups there are other methods using > presentations in Linear Algebra too. I used blueprints of solids to get > the right generator permutations for S12 and S60 for the desired rotation > groups. > > 1. Is there a simpler method for extending the domain acted on to a > multiple in general? > > 2. Is there a feasable method for big Symmetric Groups as S60 to find > conjugacy classes of isomorphic subgroups (for example for A5) beeing > transitive? If I understand the request correctly then the aim is to construct transitive permutation representations of a given (permutation) group, of prescribed degree. These representations are parametrized by conjugacy classes of subgroups of the corresponding index: The permutation domain can be identified with the set of right cosets of any subgroup in the class. So all transitive permutation representations of the alternating group $A_5$ of degrees $12$, $30$, and $60$ can be obtained by taking representatives of classes of subgroups of the orders $5$, $2$, and $1$, and then considering the action on the right cosets. (In GAP, one can actually act on the return value of `RightTransversal'.) In particular, there is only one transitive action on $60$ points, up to renumbering of the points. All the best, Thomas Breuer _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
