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 wooden cube die has symmetry group O, the 24 proper rotations of the cube.
((a)) Find the size of the conjugacy class containing a rotation by 90∘ about an axis through two opposite face centres, and say which rotations it contains.
((b)) Label the die's eight corners 1 to 8. Given that all eight corners lie in a single orbit under O, use the orbit-stabiliser theorem to find the order of the subgroup of O that fixes one corner.
((c)) A 90∘ rotation about a face axis cycles the four corners of the top face among themselves and independently the four corners of the bottom face among themselves. Write this permutation of the eight corners as a product of two cycles and state its sign (+1 if even, −1 if odd).