Groups of symmetries questions.
I'm just not sure exactly what these problems are asking me to do.
The first is
"Describe the decomposition of S into G-orbits." for a set S and a group G acting on S.
The second is
"Describe the orbits of poles for the group of rotations of a (insert platonic solid here)."
I think the first is asking for the cosets of the orbits in S.
The second asking for the generator formulas to give all elements of the given orbit.
I'm really not sure though as I can't seem to find reference to them in my text or notes.
Re: Groups of symmetries questions.
"Describe the decomposition of S into G-orbits" is sufficiently vague that I think anything you write down that's close will be acceptable. The question specifically asks about the decomposition of S, not the decomposition of G, so at a minimum you should include something like "The action of G on S decomposes S into equivalence classes, each of which is the G orbit of any one of its members, where the equivalence relation is...".
If you also want to include facts about how that action decomposes G (see how that's not what was asked for?) via cosets of stabilizers, then you're technically adding more than is asked for. But it can't hurt.
"Describe the orbits of poles for the group of rotations of a (insert platonic solid here)" sounds like it wants something trivial (on what seems to be the precise reading of the problem), or fairly demanding. It either really wants, as it says, a description of the orbits of each vertex (which is, under this action, is the set of all the verticies), or it wants some kind of description given for the group action based on how a vertex is mapped under a rotation. The second interpretation would take some thought. The first interpretation requires nothing more than noting that the group of rotations leaving a Platonic solid invariant is transitive on the verticies (i.e. the verticies are all in the same orbit).