PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry
Invariance, Ch Groups: the algebra of symmetry operations
Problem
A regular octahedron has 12 edges. The rotation group of the octahedron acts on the set of edges, and it is transitive: any edge can be rotated onto any other. The rotation that fixes a given edge setwise (mapping it to itself, possibly swapping its two endpoints) consists only of the identity and the 180∘ rotation about the axis through the midpoints of that edge and the opposite edge.
((a)) State the size of the orbit of one edge under the rotation group.
((b)) State the order of the stabiliser of that edge.
((c)) Use the orbit-stabiliser theorem to find the order of the octahedron's rotation group.