A camera-mount head shaped like a regular tetrahedron has four attachment points labelled
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 camera-mount head shaped like a regular tetrahedron has four attachment points labelled 1,2,3,4. Its rotation group T has order 12, and the six edges of the tetrahedron form a single orbit under T.
((a)) Using the orbit-stabiliser theorem, find the order of the subgroup that maps a given edge to itself (as a set).
((b)) The non-identity element of that stabiliser is a 180∘ turn about the axis through the midpoints of that edge and the opposite edge. For the edge joining 1 and 2 (opposite edge 3-4), this turn sends 1↔2 and 3↔4. Write this permutation in cycle notation and give its sign (+1 if even, −1 if odd).