= Unit sphere orbit of the defining special unitary action
{title2=$S^{2n-1}\cong SU(n)/SU(n-1)$}
For $n\geq2$, the <special unitary group> acts transitively on the complex unit sphere. Extend any unit vector to an orthonormal basis, then adjust the phase of a remaining column to give determinant one. The <stabilizer subgroup> of the first vector consists of special unitary transformations of its orthogonal complement. Thus the sphere is a homogeneous quotient, not just a set with a norm-preserving action.
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