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\begin{document}

{\bf Question}

Let $T =\{ 2 + i, 4, -2 + 3i \}$.  There are six M\"obius
transformations $m$ satisfying $m(T) = T$.  Find explicit
expressions for two of them (other than the identity).
\medskip

{\bf Answer}

$m(2+i)=0,\ m(4)=\infty,\ m(-2+3i)=1$:

\begin{eqnarray*} m(z) & = & \ds\frac{z-(2+i)}{z-4} \cdot
\ds\frac{-2+3i-4}{-2+3i-(2+i)}\\ & = & \ds\frac{z-(2+i)}{z-4}
\cdot \ds\frac{-6+3i}{-4+2i}\\ & = &
\ds\frac{(-6+3i)z+15}{(-4+2i)z+(16-8i)}. \end{eqnarray*}

$J(z)=\ds\frac{1}{z}$ permutes $\{0,1,\infty\}$ as does
$p(z)=-z+1$ and so

$mJm^{-1},mpm^{-1}$ permute $\{2+i,4,-2+3i\}$.
\end{document}
