Dear GAP Forum, there is some supplememtal Information to my message concerning "Icosahedron exercises":
Ad 3. (How many labelling sets a specific group can handle, with other words how to describe a set of labelling sets by selecting a specific group. The results are affirmed by following calculation. If you divide the size of S12 by the sizes of the conjugacy classes of the isomorhic subgroups of A5 to S12 you get the number of labelling sets handled by each group conjugacy class also as follows: (variables as given in the procedures in the preceding message) gap> sizes_cc_iisos12:=List(iisos12,c->Size(ConjugacyClassSubgroups(s12,c))); [ 792, 5544, 997920, 1995840, 3991680, 1995840, 1995840 ] gap> ratio_sizes_cc_to_s12:= > List(sizes_cc_iisos12,c->Size(s12)/c); [ 604800, 86400, 480, 240, 120, 240, 240 ] Please compare it to results in the preceding message. The labelling sets thus are evenly distributed to the groups of each class separately. Ad 2. (To get the groups containing a specific permutation) For each group conjugacy class (of the isomorphic....) the normalizer of the group generated by the permutation in question is the only allowed conjugator to the groups for executing the command Orbit. Proof: If two groups in a conjugacy class were connected by another permutation outside the normalizer, they couldnt have the permutation in question in common, as the subgroup containing it wouldnt be fixed. So the only whats to do to get all possible groups containing our permutation is, to repeat the procedure shown in my message for each conjugacy class that moves (in our case) 10 ore more points and forming the union of the groups resulting. These was the classes 3,4,5,6,7 in isos12 in the procedure given. In this way i got (in our case) a total of 55 groups containing the permutation normact. The groups are distributed to classes as follows: gap> Sizes_orbits:=List(listgroups,c->Size(Orbit(norm_normgroup,c))); # variables selfexplaining from context [ 5, 10, 20, 10, 10 ] Kind regards, Rudolf Zlabinger _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
