[GAP Forum] Numberings of Ikosaeders vertices

[Rudolf Zlabinger]

Dear Thomas Breuer,

Thank you at least for the hint to ATLAS, but also for the rich material
given.

To introduce Ikosaeder itself as group only its suffficient to begin with
A5, it can be mirrored, as desired, to a 12 point permutation group by
Isomorphic Subgroups to s12.

My question mainly was, as there is a direct method to find out a group out
of isomophic subgroups from A5 to S12, containing a special permutation. In
our case it was the permutation (2,3,4,5,6)(7,8,9,10,11). I expected
intuitively, that such a group recognizes a special numbering of the
vertices of Ikosaeder.

More general: As the action of all possible numberings, that is S12 in our
case, on a starting set of one representative group for each  conjugacy
class of groups (isomorphic to A5 in our case) is injective, one can expect,
that there is one group for each conjugacy class fitting to a special
numbering.

But in using numberings of vertices only, there is the problem to uniquely
describe this numbering formally up to a selection of a starter set of
groups belonging to a initial numbering, that has to be induced intuitively
from outside the formalism, as to, for example, requesting the inclusion of
special permutations in the starter group(s) (see above).

I didnt explore yet, whether the reverse does hold, in order to have a
unique description of a special numbering itself by the groups selected,
because this relation above seems not to be bijective. That is, one group
out of a conjugacy class may not uniquely describe a special numbering, but
a distinct manyfold of numberings. In this case one had to look, what
manyfold it is, maybe it is determined as a structure quite well, and maybe
the numbering is uniquely determined by the full set of the selected groups,
one group per conjugacy class. I will try to explore this myself as an
exercise, perhaps this problem reveales to be trivial at all.

So the method given by you (out of ATLAS) has the advantage of uniquely
describing the vertices by vectors, independent from a selected group
representation.

Thank you, and best regards, Rudolf Zlabinger