[GAP Forum] Icosahedron exercises; some supplemental remarks

[Rudolf Zlabinger]

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