The rotation group of an equilateral triangular plate that can be flipped over has…
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry
Invariance, Ch Groups: the algebra of symmetry operations
Problem
The rotation group D3 of an equilateral triangular plate that can be flipped over has order 6: the identity, rotations C3 and C32 by 120∘ and 240∘ about the axis through the centre perpendicular to the plate, and three 180∘ flips A1,A2,A3 about in-plane axes each through one vertex and the midpoint of the opposite side. Someone argues: 'A1, A2 and A3 flip the plate about three physically distinct axes through three different vertices, so each must be its own conjugacy class, giving three singleton classes.' Decide whether this is correct, using the fact that conjugating a rotation by g carries its axis to the image of that axis under g.