Someone argues that in the rotation group of a cube (order 24)

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

Someone argues that in the rotation group of a cube (order 24), the 120∘120^\circ rotation about a body diagonal and the 240∘240^\circ rotation about the same diagonal cannot be conjugate, and so must fall into different classes, because 'conjugation only moves an axis to another axis of the same kind; it cannot turn a rotation into its own inverse about the same axis.' Take the diagonal along (1,1,1)(1,1,1), so the 120∘120^\circ rotation is R=(001100010)R=\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix} (a cyclic permutation of the axes) and the 240∘240^\circ rotation is R−1=RTR^{-1}=R^{\mathsf T}. Using the cube's 180∘180^\circ rotation AA about the edge-midpoint axis (1,−1,0)/2(1,-1,0)/\sqrt2, given by A=2nnT−1A=2nn^{\mathsf T}-\mathsf1 with n=(1,−1,0)/2n=(1,-1,0)/\sqrt2, decide whether the argument is correct by computing ARA−1ARA^{-1}.

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