= Smallest axial rotation group forcing second-rank transverse isotropy
{title2=$\langle R_z(2\pi/3),R_x(\pi)\rangle\cong S_3$}
The six-element <dihedral group> generated by the displayed rotations already forces every invariant second-rank <tensor> to have the same form as an <axially invariant second-rank tensor>. The axial threefold rotation eliminates mixed entries and leaves only a planar scalar plus a planar antisymmetric part; the transverse half-turn eliminates the latter. No smaller finite <subgroup> suffices: cyclic rotation <groups> retain an axial antisymmetric <tensor>, while a four-element noncyclic rotation <group> retains arbitrary diagonal <tensors> along its three mutually perpendicular half-turn axes.
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