= Sharp three-transitivity of Möbius transformations
{title2=$\forall(z_1,z_2,z_3),(w_1,w_2,w_3)\ \exists!M:\ M(z_i)=w_i$}
For ordered triples of distinct points of the <Riemann sphere>, a unique <Möbius transformation> sends the first triple to the second. Normalize each triple to $0,1,\infty$ using $T_z(z)=(z-z_1)(z_2-z_3)/((z-z_3)(z_2-z_1))$, with the usual limits at infinity, and compose $T_w^{-1}\circ T_z$. Any two choices differ by a Möbius transformation fixing those three standard points. Fixing infinity makes it affine; fixing zero and one then makes it the identity. Thus the <Möbius group> action is sharply three-transitive, a stronger property than mere transitivity on triples.
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