Use Burnside's theorem to find the colourings of an octahedron

Question

The question is to find in how many ways we can colour the edges of an octahedron with $k$ colours by using Burnside's theorem.

I already know that I'm supposed to find the automorphism group to get $|G|$ (I assume that it contains all the rotations we can do) and then the fixed point set for each rotation. After that, I'm pretty lost.

I assume that we could rotate it with an axis through each face, vertex and edge? When I'm trying to do that though I feel pretty lost. Would really appreciate some tips and help with this. Reading material would also be greatly appreciated.

Answer 1

Let's enumerate the different rotations:

• There's the identity operator. This has 8 fixed sets (each face).
• For each vertex, we can turn it by $90^{\circ}$. That's 6 operations, and each has 2 fixed point sets (the top four faces, the bottom 4 faces).
• For each pair of opposing edges, we can turn it by $180^{\circ}$. There are 6 such pairs of edges. Each has 4 fixed point sets.
• For each face, we can turn it by $120^{\circ}$. That's 8 operations, and each has 4 fixed point sets.
• Finally, for each pair of opposing vertices, we can turn them by $180^{\circ}$. There are 3 such pairs, and each has 4 fixed point sets.