A flat triangular fidget spinner carries three equal weights 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 flat triangular fidget spinner carries three equal weights labelled 1,2,31,2,3. It can be rotated about its central axis by 120∘120^\circ or 240∘240^\circ, and it can be flipped end-for-end about an axis through one weight and the midpoint of the opposite edge. Together with doing nothing, these six operations form the group D3D_3 (isomorphic to C3vC_{3v}, realised here without any mirror). ((a)) Give the sizes of the three conjugacy classes of D3D_3: the identity, the pair of 120∘,240∘120^\circ,240^\circ rotations, and the three end-for-end flips. ((b)) Given that the three weights form a single orbit under D3D_3, use the orbit-stabiliser theorem to find the order of the subgroup that fixes weight 11. ((c)) Write the permutation of the weights produced by the 120∘120^\circ rotation 1→2→3→11\to2\to3\to1 in cycle notation and give its sign.

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