= Möbius transformations permuting three points
{c}
{title2=$\operatorname{Stab}\{0,1,\infty\}\cong S_3$}
The <Möbius transformations> preserving a three-point set act faithfully on it: one fixing all three is the identity. Every permutation is realized because a <Möbius transformation> can send any ordered distinct triple to any other, by composing the two normalizations to $0,1,\infty$. For that normalized set the six maps are $z$, $1-z$, $1/z$, $1/(1-z)$, $z/(z-1)$ and $(z-1)/z$. Thus its setwise stabilizer is the <symmetric group> $S_3$. Normalizing any other three-point set conjugates its stabilizer to this group.
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