Group Actions on Truncated Octahedron

• Obraz35

Homework Statement

Let G be the group of rotational symmetries of the octahedron and consider the action of G on the edges of the truncated octahedron.

Describe the orbits of this action.

Choose one representative element in each orbit. Describe the stabilizers of these representative elements.

The Attempt at a Solution

I am having a lot of trouble visualizing the movements of the edges of the truncated octahedron (which is what I am assuming is what happens when G acts on them). But I think the edges of the square faces make up one orbit and I'm not sure on the others. Any thoughts?

Also, I don't really know what it means to pick a representative element of an orbit. Just pick one edge and see what the stabilizers are?

well if you pick an edge and see what the stabilisers are, this can tell you the size of the orbit, at least that way you know what you're looking for.

when i was looking at this my gut feeling was that the edges of the square faces would be one orbit and those of the hexagonal faces another but after some thought, the edges of the square faces are also edges of the hexagonal faces. I'm going to go for transitive group action for this reason and the fact that using the symmetries of the octahedron i can't find anything other than e to stabilise a particular edge.

Maybe this doesn't make sense, but I don't see how the edges of the square can be in the same orbit as the edges that only border hexagons.